Herbert Amann Linear and Quasilinear Parabolic Problems - Herbert... · 2020. 1. 17. · Herbert...

Monographs in Mathematics

106

Herbert Amann

Linear and Quasilinear Parabolic ProblemsVolume II: Function Spaces

Series Editors:Herbert AmannUniversität Zürich, Zürich, SwitzerlandJean-Pierre BourguignonIHES, Bures-sur-Yvette, FranceWilliam Y. C. ChenNankai University, Tianjin, China

Associate Editors:Huzihiro Araki, Kyoto University, Kyoto, JapanJohn Ball, Heriot-Watt University, Edinburgh, UKFranco Brezzi, Università degli Studi di Pavia, Pavia, ItalyKung Ching Chang, Peking University, Beijing, ChinaNigel Hitchin, University of Oxford, Oxford, UKHelmut Hofer, Courant Institute of Mathematical Sciences, New York, USAHorst Knörrer, ETH Zürich, Zürich, SwitzerlandDon Zagier, Max-Planck-Institute, Bonn, Germany

The foundations of this outstanding book series were laid in 1944. Until the endof the 1970s, a total of 77 volumes appeared, including works of such distinguishedmathematicians as Carathéodory, Nevanlinna and Shafarevich, to name a few. Theseries came to its name and present appearance in the 1980s. In keeping itswell-established tradition, only monographs of excellent quality are published inthis collection. Comprehensive, in-depth treatments of areas of current interest arepresented to a readership ranging from graduate students to professionalmathematicians. Concrete examples and applications both within and beyond theimmediate domain of mathematics illustrate the import and consequences of thetheory under discussion.

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Monographs in MathematicsVol. 106

http://www.springer.com/series/4843

Herbert Amann

Linear and QuasilinearParabolic ProblemsVolume II: Function Spaces

Herbert AmannInstitut für MathematikUniversität ZürichZürich, Switzerland

ISSN 1017-0480 ISSN 2296-4886 (electronic)Monographs in MathematicsISBN 978-3-030-11762-7 ISBN 978-3-030-11763-4 (eBook)https://doi.org/10.1007/978-3-030-11763-4

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Preface

Nil humani quin corrigi possit1

L. NaumannSystematik der Kochkunst

Dresden 1887

In this volume we present a systematic and detailed exposition of the theory offunction spaces in an Euclidean setting. Particular emphasis is put on Besov andBessel potential spaces which form the frame for the study of parabolic differentialequations in the next volume.

The presentation includes several new features which lets it stand out fromother accounts. First, it consistently develops anisotropic spaces. Second, it ex-pounds the whole theory for functions and distributions taking their values inBanach spaces on which we impose only the necessary restrictions. Thus none inthe case of Besov spaces, except for reflexivity assumptions in duality theorems.Third, the theory is set forth for spaces whose elements are defined on rectangu-lar corners of Euclidean spaces. By this we pave the way for the investigation offunction spaces on Riemannian manifolds, possibly possessing corners and othersingularities. This is also put on hold for the third volume.

Our approach builds basically on two cornerstones: on Fourier analysis andmultiplier theorems, and on extension-restriction techniques. By this we can give aunified presentation incorporating, in particular, Sobolev–Slobodeckii and Holderspace scales. The rather detailed study of these spaces, which are of great impor-tance for the investigation of differential equations, is a further characteristic traitof our treatise.

This volume consists of three chapters and an appendix. The first chapter,which is of rather technical nature, collects preparatory material. It supplies a firmbasis for the main text which covers Chapters VII and VIII. The first one thereofcontains a systematic treatment of anisotropic vector-valued function spaces oncorners. In the second one we give a detailed and unified account of trace andboundary operators.

For the reader’s convenience, in the appendix we include a downscaled versionof L. Schwartz’ theory of vector-valued distributions by admitting only Banachspaces as targets. Particular weight is given on tensor products and convolutionssince, in the main text, we make use of such results. It should be mentioned that,already in 2003, I had put a preliminary, slightly more comprehensive version ofthis appendix on my homepage.

1There is nothing on earth that could not be improved.

v

vi Preface

In essence, this volume forms a profound expansion and amelioration of myearlier lecture notes ‘Anisotropic Function Spaces and Maximal Regularity forParabolic Problems. Part 1: Function Spaces’ [Ama09]. Besides of adding muchmore material, I have corrected numerous flaws and imprecisions which observingreaders have brought to my attention.

Once more, I could rely on the help of Pavol Quittner, Gieri Simonett, andChristoph Walker. They read critically and carefully large parts of the first draft,pointed out plenty of mistakes and misprints, and suggested very advantageouschanges and improvements. Sincere thanks are given to all of them for their gen-erous support.

Last but not least, I could again experience the immensely valuable supportof my wife Gisela who transformed countless barely readable preliminary versionsand revisions into TEX files and provided this perfect layout on hand. I am morethan deeply grateful to her.

Zurich, January 2019 Herbert Amann

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Notations and Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Chapter VI Auxiliary Material

1 Restriction-Extension Pairs

1.1 Smooth Functions on Corners . . . . . . . . . . . . . . . . . . . . . . 4Corners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Restriction-Extension Operators . . . . . . . . . . . . . . . . . . . 6Approximation by Test Functions . . . . . . . . . . . . . . . . . . 9

1.2 Tempered Distributions on Corners . . . . . . . . . . . . . . . . . . 10Duality Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . 12The Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2 Sequence Spaces

2.1 Duality of Sequence Spaces . . . . . . . . . . . . . . . . . . . . . . . 19Definitions and Embeddings . . . . . . . . . . . . . . . . . . . . . 19Duality Pairings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2 Weighted Sequence Spaces . . . . . . . . . . . . . . . . . . . . . . . 25Image Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Embeddings and Duality . . . . . . . . . . . . . . . . . . . . . . . 26

2.3 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Unweighted Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 28Weighted Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3 Anisotropy

3.1 Anisotropic Dilations . . . . . . . . . . . . . . . . . . . . . . . . . . 32Weight Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Dilations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

vii

viii Contents

3.2 Quasinorms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.3 Parametric Augmentations . . . . . . . . . . . . . . . . . . . . . . . 42

Augmented Quasinorms . . . . . . . . . . . . . . . . . . . . . . . . 43Positive Homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . 45Differentiating Inverses . . . . . . . . . . . . . . . . . . . . . . . . 47Slowly Increasing Functions . . . . . . . . . . . . . . . . . . . . . 49

3.4 Fourier Multipliers and Multiplier Spaces . . . . . . . . . . . . . . . 52Elementary Fourier Multiplier Theorems . . . . . . . . . . . . . . 52Fourier Multiplier Spaces . . . . . . . . . . . . . . . . . . . . . . . 54Resolvent Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.5 Multiplier Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Resolvent Estimates for Homogeneous Symbols . . . . . . . . . . . 58Functions of Homogeneous Symbols . . . . . . . . . . . . . . . . . 59Dunford Integral Representations . . . . . . . . . . . . . . . . . . 61Powers and Exponentials . . . . . . . . . . . . . . . . . . . . . . . 65

3.6 Dyadic Partitions of Unity . . . . . . . . . . . . . . . . . . . . . . . 67Preliminary Fourier Multiplier Theorems . . . . . . . . . . . . . . 68

3.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Chapter VII Function Spaces

1 Classical Spaces

1.1 Bounded Continuous Functions . . . . . . . . . . . . . . . . . . . . . 78Banach Spaces of Bounded Continuous Functions . . . . . . . . . 78Vector Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

1.2 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84Regular Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 84Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

1.3 Restrictions and Extensions . . . . . . . . . . . . . . . . . . . . . . . 861.4 Distributional Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 901.5 Reflexivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 931.6 Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 961.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

2 Besov Spaces

2.1 The Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99Preliminary Estimates . . . . . . . . . . . . . . . . . . . . . . . . . 100A Retraction-Coretraction Pair . . . . . . . . . . . . . . . . . . . 102The Final Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 104

2.2 Embedding Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 104Little Besov Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 107Embeddings With Varying Target Spaces . . . . . . . . . . . . . . 109

2.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1092.4 Fourier Multiplier Theorems . . . . . . . . . . . . . . . . . . . . . . 112

Contents ix

2.5 Operators of Positive Type . . . . . . . . . . . . . . . . . . . . . . . 115Resolvent Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 115A Representation Theorem . . . . . . . . . . . . . . . . . . . . . . 117Bounded Imaginary Powers . . . . . . . . . . . . . . . . . . . . . . 118Interpolation-Extrapolation Scales . . . . . . . . . . . . . . . . . . 119

2.6 Renorming by Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 120Equivalent Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . 121Sandwich Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 123Sobolev Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . 124

2.7 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125Real and Complex Interpolation . . . . . . . . . . . . . . . . . . . 125Interpolation with Different Target Spaces . . . . . . . . . . . . . 126Embeddings of Intersection Spaces . . . . . . . . . . . . . . . . . . 127Interpolation of Classical Spaces . . . . . . . . . . . . . . . . . . . 128

2.8 Besov Spaces on Corners . . . . . . . . . . . . . . . . . . . . . . . . 1282.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

3 Intrinsic Norms, Slobodeckii and Holder Spaces

3.1 Commuting Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . 1333.2 Semigroups and Interpolation . . . . . . . . . . . . . . . . . . . . . . 137

Preliminary Estimates . . . . . . . . . . . . . . . . . . . . . . . . . 138Renorming Intersections of Interpolation Spaces . . . . . . . . . . 141

3.3 Translation Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . 1463.4 Renorming Besov Spaces . . . . . . . . . . . . . . . . . . . . . . . . 1493.5 Intersection-Space Characterizations . . . . . . . . . . . . . . . . . . 151

Intersection Space Representations . . . . . . . . . . . . . . . . . . 152Equivalent Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . 153Nikol′skiı Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

3.6 Besov–Slobodeckii and Besov–Holder Spaces . . . . . . . . . . . . . 155Mixed Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . 155Slobodeckii, Holder, and Little Holder Spaces . . . . . . . . . . . 157

3.7 Little Holder Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 159Very Little Holder Spaces . . . . . . . . . . . . . . . . . . . . . . . 164

3.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

4 Bessel Potential Spaces

4.1 Basic Facts, Embeddings, and Real Interpolation . . . . . . . . . . . 1684.2 A Marcinkiewicz Multiplier Theorem . . . . . . . . . . . . . . . . . . 1714.3 Renorming by Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 1734.4 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1744.5 Complex Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . 176

A Holomorphic Semigroup . . . . . . . . . . . . . . . . . . . . . . 176Interpolation with Different Target Spaces . . . . . . . . . . . . . 178

x Contents

4.6 Intersection-Space Characterizations . . . . . . . . . . . . . . . . . . 1824.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

5 Triebel–Lizorkin Spaces

5.1 Maximal Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 188Preliminary Estimates for Sequences . . . . . . . . . . . . . . . . 189Estimates for a Single Function . . . . . . . . . . . . . . . . . . . 193

5.2 Definition and Basic Embeddings . . . . . . . . . . . . . . . . . . . . 196Equivalent Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . 197Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

5.3 Fourier Multiplier Theorems . . . . . . . . . . . . . . . . . . . . . . 2015.4 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2025.5 Renorming by Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 203

Sandwich Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 2035.6 Sobolev Embeddings and Related Results . . . . . . . . . . . . . . . 204

Multiplicative Inequalities . . . . . . . . . . . . . . . . . . . . . . 204Optimal Sobolev-Type Embeddings . . . . . . . . . . . . . . . . . 206Sharp Embeddings of Intersection Spaces . . . . . . . . . . . . . . 207

5.7 Gagliardo–Nirenberg Type Estimates . . . . . . . . . . . . . . . . . 208Nonhomogeneous Inequalities . . . . . . . . . . . . . . . . . . . . 208Homogeneous Estimates . . . . . . . . . . . . . . . . . . . . . . . 212Isotropic Multiplicative Inequalities . . . . . . . . . . . . . . . . . 215Sobolev Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . 216Parabolic Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 218

5.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

6 Point-Wise Multiplications

6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222Continuity of Derivatives . . . . . . . . . . . . . . . . . . . . . . . 223Point-Wise Products . . . . . . . . . . . . . . . . . . . . . . . . . . 224

6.2 Multiplications in Classical Spaces . . . . . . . . . . . . . . . . . . . 225Spaces of Bounded Continuous Functions . . . . . . . . . . . . . . 225Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226Spaces of Negative Order . . . . . . . . . . . . . . . . . . . . . . . 228

6.3 Multiplications in Besov Spaces of Positive Order . . . . . . . . . . . 2296.4 Multiplications in Besov Spaces of Negative Order . . . . . . . . . . 235

The Reflexive Case . . . . . . . . . . . . . . . . . . . . . . . . . . 235The Non-Reflexive Case . . . . . . . . . . . . . . . . . . . . . . . . 237

6.5 Multiplications in Bessel Potential Spaces . . . . . . . . . . . . . . . 2446.6 Space-Dependent Bilinear Maps . . . . . . . . . . . . . . . . . . . . 2456.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

Contents xi

7 Compactness

7.1 Equicontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249Compact Sets in BUC . . . . . . . . . . . . . . . . . . . . . . . . 249Compact Sets in Lq . . . . . . . . . . . . . . . . . . . . . . . . . . 250

7.2 Compact Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . 253Compact Embeddings of Besov Spaces . . . . . . . . . . . . . . . 253Compact Embeddings of Holder, Sobolev–Slobodeckii, and Bessel

Potential Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2557.3 Function Spaces on Intervals . . . . . . . . . . . . . . . . . . . . . . 256

Classical Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256Besov Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257A Retraction-Coretraction Theorem . . . . . . . . . . . . . . . . . 260Interpolations and Embeddings . . . . . . . . . . . . . . . . . . . 264The Rellich–Kondrachov Theorem . . . . . . . . . . . . . . . . . . 265

7.4 Aubin–Lions Type Theorems . . . . . . . . . . . . . . . . . . . . . . 266The General Result . . . . . . . . . . . . . . . . . . . . . . . . . . 266Limit Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

7.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

8 Parameter-Dependent Spaces

8.1 Sobolev Spaces and Bounded Continuous Functions . . . . . . . . . 2718.2 Besov and Bessel Potential Spaces . . . . . . . . . . . . . . . . . . . 2748.3 Intersection-Space Characterizations . . . . . . . . . . . . . . . . . . 2778.4 Fourier Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2788.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

Chapter VIII Traces and Boundary Operators

1 Traces

1.1 Trace Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2821.2 The Retraction Theorem . . . . . . . . . . . . . . . . . . . . . . . . 2861.3 Traces on Half-Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 299

Parameter-Dependence . . . . . . . . . . . . . . . . . . . . . . . . 302General Besov spaces . . . . . . . . . . . . . . . . . . . . . . . . . 303

1.4 Spaces of Vanishing Traces . . . . . . . . . . . . . . . . . . . . . . . 304Sobolev–Slobodeckii Spaces . . . . . . . . . . . . . . . . . . . . . . 304

1.5 Weighted Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310Weighted Lebesgue Spaces . . . . . . . . . . . . . . . . . . . . . . 310Hardy Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 312Weighted Space Characterizations of Sobolev–Slobodeckii Spaces . 314

1.6 Further Characterizations of Spaces with Vanishing Traces . . . . . 317General Besov Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 320

xii Contents

Bessel Potential Spaces . . . . . . . . . . . . . . . . . . . . . . . . 322Holder Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

1.7 Representation Theorems for Spaces of Negative Order . . . . . . . 324Spaces of Mildly Negative Order . . . . . . . . . . . . . . . . . . . 324Spaces of Strongly Negative Order . . . . . . . . . . . . . . . . . . 326Duality of Sums and Intersections . . . . . . . . . . . . . . . . . . 327Weighted Space Representations . . . . . . . . . . . . . . . . . . . 330

1.8 Traces for Corners . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330Traces on a Single Face . . . . . . . . . . . . . . . . . . . . . . . . 331Vanishing Traces on Corners . . . . . . . . . . . . . . . . . . . . . 332Faces of Higher Codimensions . . . . . . . . . . . . . . . . . . . . 332Compatibility Conditions . . . . . . . . . . . . . . . . . . . . . . . 334The Retraction Theorem for Corners . . . . . . . . . . . . . . . . 335

1.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

2 Boundary Operators

2.1 Boundary Operators on Half-Spaces . . . . . . . . . . . . . . . . . . 343Normal Boundary Operators . . . . . . . . . . . . . . . . . . . . . 344

2.2 Systems of Boundary Operators . . . . . . . . . . . . . . . . . . . . 347The Boundary Operator Retraction Theorem . . . . . . . . . . . . 348Embeddings with Boundary Conditions . . . . . . . . . . . . . . . 350

2.3 Transmission Operators . . . . . . . . . . . . . . . . . . . . . . . . . 353Patching Together Half-Spaces . . . . . . . . . . . . . . . . . . . . 357

2.4 Interpolation With Boundary Conditions . . . . . . . . . . . . . . . 358Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358The Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 360Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365Complex Interpolation of Bessel Potential Spaces . . . . . . . . . 367

2.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

Appendix Vector-Valued Distributions

1 Tensor Products and Convolutions

1.1 Locally Convex Topologies . . . . . . . . . . . . . . . . . . . . . . . 370The Uniform Boundedness Principle . . . . . . . . . . . . . . . . . 370Hypocontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371Montel Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372Strict Inductive Limits . . . . . . . . . . . . . . . . . . . . . . . . 372Smooth Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 373Test Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373Rapidly Decreasing Smooth Functions . . . . . . . . . . . . . . . . 374Slowly Increasing Smooth Functions . . . . . . . . . . . . . . . . . 374Spaces of Vector-Valued Distributions . . . . . . . . . . . . . . . . 374

Contents xiii

1.2 Convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375Convolutions of Distributions and Test Functions . . . . . . . . . 375Translation-Invariant Operators . . . . . . . . . . . . . . . . . . . 378Convolutions of Two Distributions . . . . . . . . . . . . . . . . . . 378Elementary Properties of Convolutions . . . . . . . . . . . . . . . 380Convolutions of Temperate Distributions . . . . . . . . . . . . . . 381

1.3 Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383Multiplications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383Leibniz’ Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384Approximation by Test Functions . . . . . . . . . . . . . . . . . . 385Density by Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . 387Approximation by Tensor Products . . . . . . . . . . . . . . . . . 388Approximation by Polynomials . . . . . . . . . . . . . . . . . . . . 389Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390

1.4 Topological Tensor Products and the Kernel Theorem . . . . . . . . 390Algebraic Tensor Products . . . . . . . . . . . . . . . . . . . . . . 390Basic Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392Projective Tensor Products . . . . . . . . . . . . . . . . . . . . . . 393Nuclear Maps and Spaces . . . . . . . . . . . . . . . . . . . . . . . 393Projective Tensor Products and Maps of Finite Rank . . . . . . . 394Approximation by Maps of Finite Rank . . . . . . . . . . . . . . . 395Completeness of Spaces of Linear Operators . . . . . . . . . . . . 397The Abstract Kernel Theorem . . . . . . . . . . . . . . . . . . . . 398Tensor Product Characterizations of Some Distribution Spaces . . 398

1.5 Extending Bilinear Maps . . . . . . . . . . . . . . . . . . . . . . . . 400General Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . 405

1.6 Point-Wise Multiplication . . . . . . . . . . . . . . . . . . . . . . . . 405A Characterization of OM . . . . . . . . . . . . . . . . . . . . . . 406The General Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 407Basic Properties of Multiplications . . . . . . . . . . . . . . . . . . 409

1.7 Scalar Products and Duality Pairings . . . . . . . . . . . . . . . . . 413Parseval’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 414Duality Pairings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415

1.8 Tensor Products of Distributions and Kernel Theorems . . . . . . . 416Approximation by Tensor Products . . . . . . . . . . . . . . . . . 416Tensor Products of Distributions . . . . . . . . . . . . . . . . . . . 416Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421Topological Tensor Products of Distributions . . . . . . . . . . . . 425Kernel Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 427

1.9 Convolutions of Vector-Valued Distributions . . . . . . . . . . . . . . 428The Basic Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 428Lp Functions with Compact Supports . . . . . . . . . . . . . . . . 430Convolutions of Regular Distributions . . . . . . . . . . . . . . . . 430

xiv Contents

Tensor Products and Convolutions . . . . . . . . . . . . . . . . . . 431Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433Convolution Algebras . . . . . . . . . . . . . . . . . . . . . . . . . 436Convolutions of Bounded and Integrable Functions . . . . . . . . 437The Convolution Theorem . . . . . . . . . . . . . . . . . . . . . . 439

2 Vector Measures and the Riesz Representation Theorem

Measures of Bounded Variation . . . . . . . . . . . . . . . . . . . 441Integrals with Respect to Vector Measures . . . . . . . . . . . . . 442Vector Measures as Distributions . . . . . . . . . . . . . . . . . . 444Convolutions Involving Vector Measures . . . . . . . . . . . . . . 444The Riesz Representation Theorem . . . . . . . . . . . . . . . . . 446

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461

Notations and Conventions

We use the notations and conventions introduced on pages 1–7 of Volume I, withthe following exceptions, applied in the main body of this volume (but not in theappendix):

• All abstract vector spaces are over the complex field.

• K does not stand for either the real or the complex number field.Instead, it symbolizes corners which are introduced in Subsection VI.1.1.

For the reader’s convenience, we reproduce here some of the very basic notationestablished in Volume I.

Let X and Y be nonempty sets. Then Y X is the set of all maps u : X → Y .If A is a subset of X, the characteristic function of A (in X) is denoted by χA.We also put 1 := 1X := χX .

The set of all x ∈ X in which definitions and relations hold is often de-noted by [ . . . ], where . . . stand for the definitions and relations. For example, ifu ∈ RX then

[u ≥ 0] :=

x ∈ X ; u(x) ≥ 0

etc.Suppose X and Y are Hausdorff topological spaces. Then C(X,Y ) is the set

of all continuous maps in Y X , endowed with the compact-open topology. We write

X → Y or i : X → Y ,

if X is continuously injected in Y , that is, X ⊂ Y and the natural injection

i : X → Y , x 7→ x

is continuous. If X is a dense subset of Y , we write Xd⊂ Y . Thus X

d→ Y means

that X is densely and continuously injected in Y .We often write 1X , or simply 1, for the identity mapping, idX : X → X,

x 7→ x, if no confusion seems likely.Given a subset M of a vector space, we put

qM := M \0 .

Assume X and Y are topological vector spaces. Then L(X, Y ) is the vectorspace of all continuous linear maps from X in Y , and

L(X) := L(X, X) .

In this case X → Y means also that X is a vector subspace of Y , that is, i belongsto L(X, Y ). Moreover,

Lis(X,Y ) :=

T ∈ L(X,Y ) ; T is bijective and T−1 ∈ L(Y,X)

1

2 Notations and Conventions

is the set of all topological linear (toplinear) isomorphisms from X onto Y , and

Laut(X) := Lis(X,X)

is the group of all toplinear automorphisms of X, the general linear group, alsodenoted by GL(X).

By a locally convex space (LCS) we mean a Hausdorff locally convex topo-logical vector space. If X and Y are LCSs, then L(X, Y ) is equipped with thebounded convergence topology.

Throughout this volume, unless specified otherwise, c denotes constants ≥ 1which may be different from occurrence to occurrence, but are always independentof the free variables appearing at a given place.

Chapter VI

Auxiliary Material

Our in-depth study of vector-valued function spaces is based on a considerableamount of technical tools. Among them stand out extension and restriction theo-rems and Fourier multiplier estimates.

In the first section of this chapter we study smooth Banach-space-valuedfunctions and tempered distributions on rectangular Euclidean corners. We estab-lish a restriction-extension formalism which plays a fundamental role throughoutthe whole volume. On its basis we can –– in the next chapter –– introduce func-tion spaces on corners and transfer their properties, derived by Fourier analyticmethods on the ‘full’ Euclidean space, to the corner setting.

Section 2 essentially summarizes the more or less well-known theory of se-quence spaces. We emphasize duality and interpolation properties. That sectionprovides a firm basis for many later considerations, not only in the following chap-ters, but in the next volume too.

Section 3 is the most technical one. First, we introduce weight systems and re-lated quasinorms, as well as parameter-dependent versions thereof. Next we launchFourier multiplier spaces and derive multiplier estimates, for homogeneous func-tions in particular. These results are of use not only at various later places, but inthe next volume also.

Preparing for the Fourier analytic approach to Besov spaces, we investigatein the last subsection to some extent dyadic properties of unity.

The reader is strongly advised to browse with some care –– even at a first read-ing –– at least through the beginnings of Sections 1 and 3 to grasp the definitionsof r-e pairs and weight systems.

© Springer Nature Switzerland AG 2019H. Amann, Linear and Quasilinear Parabolic Problems, Monographsin Mathematics 106, https://doi.org/10.1007/978-3-030-11763-4_6

3

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4 VI Auxiliary Material

1 Restriction-Extension Pairs

The theory of function spaces on Rd is most easily established by means of Fourieranalysis. Afterwards it is important to be able to transfer the so derived results tohalf-spaces or, more generally, to corners. In this section we develop the technicaltools for this approach.

In the first subsection we introduce the concepts of (model) corners and thebasic operators which allow an efficient handling of restrictions and extensions ofsmooth functions to and from corners, respectively. These operators are, in thesecond subsection, extended to spaces of tempered distributions. This leads to themain result of this section, namely Theorem 1.2.3. In the last subsection it is shownthat spaces of tempered distributions on corners can be characterized by duality,similarly as for scalar-valued distributions on Euclidean spaces. However, the re-sults of this section apply to functions and distributions with values in arbitraryBanach spaces.

The simple but important concept underlying this approach, and being ofgreat value in later sections also, is that of retractions and corresponding coretrac-tions. These notions have already been introduced in Subsection I.2.3. There someuseful general facts have been established as well. To facilitate the exposition andto provide the necessary precision we now introduce the following more precisemanner of speaking:

Let X and Y be LCSs, r ∈ L(X ,Y) a retraction, and rc ∈ L(Y,X )a coretraction. Then we express this by saying:(r, rc) is a retraction-coretraction pair (an r-c pair, for short) for (X ,Y).

Recall that this means that rc is a right inverse for r.

1.1 Smooth Functions on Corners

In this subsection we introduce the concept of corners and establish the basicrestriction-extension pair for rapidly decaying smooth functions.

Corners

Suppose 1 ≤ k ≤ d. A standard k-corner in Rd is a Cartesian product subset1

K = Kdk = I1 × · · · × Ik × Rd−k

of Rd such that Ij ∈R+, (0,∞)

for 1 ≤ j ≤ k. We denote by J = JK the subset

of 1, . . . , k such that Ij = R+ iff j ∈ J , and J∗ := 1, . . . , k\J . We also setI∗j := (0,∞) if j ∈ J , and I∗j := R+ if j ∈ J∗. Then K∗ = I∗1 × · · · × I∗k × Rd−k isthe adjoint corner of K. Note K∗∗ = K. Moreover, K = K iff J = 1, . . . , k, and

1K should not be confused with the same notation used in volume I to denote either the realor the complex field. In this volume K always denotes a corner.

VI.1 Restriction-Extension Pairs 5

then K∗ = K. We call J type of K and write more precisely K = K(J) to expressthe fact that K is of type J .

Let K be a standard k-corner of type J and 1 ≤ j ≤ k. Then

∂jK := I1 × · · · × Ij−1 × 0 × Ij+1 × · · · × Ik × Rd−k

is the j-face of K. Observe that ∂jK ⊂ K iff j ∈ J , and ∂jK ∩K = ∅ if j ∈ J∗.

x1

x2

x3

On the left there is the stan-dard 3-corner K of type 1, 3in R3. The lightly shaded face∂2K does not belong to K.

For I ∈ R,±R+,±(0,∞)

we put

I ×j

∂jK := I1 × · · · × Ij−1 × I × Ij+1 × · · · × Ik × Rd−k .

Deviating from definition (III.4.1.4), it is now more convenient to put

qk,m(u) := max|α|≤m

supx∈Rd

〈x〉k |∂αu(x)|E , k,m ∈ N , (1.1.1)

where 〈x〉 := (1 + |x|2)1/2. It is a family of seminorms generating the topology ofthe Frechet space S(Rd, E), the space of smooth and rapidly decreasing E-valuedfunctions on Rd. We denote by S(K, E) the Frechet space of smooth2 and rapidlydecreasing E-valued functions on K, whose topology is generated by the familyof seminorms

qKk,m ; k, m

, where qKk,m is obtained from (1.1.1) by replacing Rd

by K. Then, by definition,

S(K, E) is the closed linear subspace of S(K, E) consisting ofall u satisfying ∂αu|xj=0 = 0 , α ∈ Nd , j ∈ J∗ .

(1.1.2)

Let Sd be the group of permutations of 1, . . . , d. We define a linear op-eration of Sd on Rd by setting sx := (xs(1), . . . , xs(d)) for s ∈ Sd and x ∈ Rd. Itinduces a linear representation Us ; s ∈ Sd of Sd on L1,loc(Rd, E) by defining

Usu(x) := u(s−1x) , a.a. x ∈ Rd , u ∈ L1,loc(Rd, E) .

2Differentiable at x ∈ ∂jK means ‘differentiable from the right’.


It is obvious that it restricts to a continuous linear representation on S(Rd, E). Infact, it follows from (1.1.1) that

UsS(Rd, E) = S(Rd, E) , s ∈ Sd . (1.1.3)

A subset K of Rd is a (general) k-corner in Rd if there exist a standardk-corner K in Rd and a permutation s ∈ Sd such that K = sK := sx ; x ∈ K .Then K∗ := sK∗, and

S(K, E) := UsS(K, E) . (1.1.4)

Restriction-Extension Operators

The subsequent constructions are based on the following technical lemma.

1.1.1 Lemma There exists h ∈ C∞((0,∞),R

)satisfying

∫ ∞

0

ts |h(t)| dt < ∞ , (−1)`

∫ ∞

0

t`h(t) dt = 1 , h(1/t) = −th(t) (1.1.5)

for s ∈ R, ` ∈ Z, and t > 0.

Proof We denote by C\R+ → C, z 7→ z1/4 the branch of z1/4 which satisfies(x + i0)1/4 = x1/4 for x ≥ 0. Then (x − i0)1/4 = ix1/4, x ≥ 0. Put

f(z) := (1 + z)−1 exp(−(1 − i)z1/4 − (1 + i)z−1/4

), z ∈ C\R+ .

Then

f(x + i0) = (1 + x)−1e−(x1/4+x−1/4)(cos(x1/4 − x−1/4) + i sin(x1/4 − x−1/4)

)

and f(x − i0) = f(x + i0) for x ∈ R+.Let Γ be a piece-wise smooth path in C\R+ running from ∞− i0 to ∞ + i0

such that −1 is to its left. Then, by Cauchy’s theorem,∫

Γ

z`f(z) dz = 2i

∫ ∞

0

(1 + x)−1xè−(x1/4+x−1/4) sin(x1/4 − x−1/4) dx

for ` ∈ Z. Since zkf(z) → 0 for k ∈ N as |z| → ∞, we can apply the residue theoremto deduce ∫

Γ

z`f(z) dz = 2πi Res(z`f,−1) = 2πi(−1)è−2√

2 ,

thanks to (−1)1/4 = (1 + i)/√

2. Thus, putting

h(t) := π−1e2√

2(1 + t)−1e−(t1/4+t−1/4) sin(t1/4 − t−1/4)

for t > 0, we see that the second claim is true. The other assertions are now clear. ¥


Now we suppose that K = I1 × · · · × Ik × Rd−k is a standard k-corner in Rd

of type J . We set

xi := (x1, . . . , xi−1, xi+1, . . . , xd) and(xi; xi

):= x = (x1, . . . , xd) .

Then, given 1 ≤ i ≤ k, ` ∈ Z, and x ∈ (−∞, 0) ×i

∂iK,

εìu(x) := (−1)`

∫ ∞

0

t`h(t)u(−txi; xi

)dt , u ∈ S(K, E) , (1.1.6)

and εi := ε0i , where h is as in Lemma 1.1.1. Note

∂i(εìu)(x) = ε`+1

i ∂iu(x) , ∂j(εìu)(x) = ε`

i∂ju(x) , j 6= i , (1.1.7)

for x ∈ (−∞, 0) ×i

∂iK.

For Kdk let

Kk,k−1 = Kdk,k−1 := I1 × · · · × Ik−1 × Rd−k+1 , K1,0 := Rd .

Then

K ∪ ((R\Ik) ×

k∂kK

)= Kk,k−1 (1.1.8)

and(K∗)k,k−1 = I∗1 × · · · × I∗k−1 × Rd−k−1 = (Kk,k−1)∗ =: K∗

k,k−1 .

Suppose k ∈ J . Given ` ∈ N and u ∈ S(K, E),

E`ku(x) :=

u(x) , x ∈ K ,

ε`ku(x) , x ∈ (−∞, 0) ×

k∂kK ,

(1.1.9)

and Ek := E0k. It follows from Lemma 1.1.1 that

E`k ∈ L(S(K, E),S(Kk,k−1, E)

). (1.1.10)

The operator of point-wise restriction Rk is defined by

Rku(x) := u(x) , x ∈ K , u ∈ S(Kk,k−1, E) ,

and satisfies

Rk ∈ L(S(Kk,k−1, E),S(K, E))

. (1.1.11)

It is clear that (Rk, E`k) is an r-c pair for

(S(Kk,k−1, E),S(K, E))

if k ∈ J .


Let k ∈ J∗. Then the trivial extension operator Ek is given for u ∈ S(K, E) by

Eku(x) :=

u(x) , x ∈ K ,

0 , x ∈ (−R+) ×k

∂kK .(1.1.12)

It follows from (1.1.2) that

Ek ∈ L(S(K, E),S(Kk,k−1, E))

. (1.1.13)

Note that εk is also well-defined if u ∈ S(Kk,k−1, E) and x ∈ K. Hence

Rku(x) := u(x) − εku(x) , x ∈ K , (1.1.14)

is meaningful for u ∈ S(Kk,k−1, E). By means of Lemma 1.1.1 it is verified that

∂`kRku|xk=0 = 0 , ` ∈ N . (1.1.15)

This implies

Rk ∈ L(S(Kk,k−1, E),S(K, E))

. (1.1.16)

It is obvious that (Rk, Ek) is an r-c pair for(S(Kk,k−1, E),S(K, E)

)if k ∈ J∗.

We set(Rk, Ek

):=

(Rk, Ek) , if k ∈ J ,

(Rk, Ek) , if k ∈ J∗ .

Then (1.1.8), (1.1.11), and (1.1.16) imply

RK := Rk Rk−1 · · · R1 ∈ L(S(Rd, E),S(K, E))

. (1.1.17)

Similarly, we get from (1.1.8), (1.1.10), and (1.1.13)

EK := E1 E2 · · · Ek ∈ L(S(K, E),S(Rd, E))

. (1.1.18)

Clearly, (RK, EK) is an r-c pair for(S(Rd, E)),S(K, E)

).

Now we drop the assumption that K be a standard corner, that is, we assume

• K is a k-corner in Rd.

Thus there exist a standard k-corner K in Rd and s ∈ Sd such that K = sK. Weput

RK := Us RK U−1s , EK := Us EK U−1

s . (1.1.19)

It follows from, (1.1.3), (1.1.4), (1.1.17), and (1.1.18) that

RK ∈ L(S(Rd, E),S(K, E))

, EK ∈ L(S(K, E),S(Rd, E))

. (1.1.20)


Moreover, (RK, EK) is an r-c pair. We express (1.1.20) and the latter fact by saying:

(RK, EK) is a restriction-extension pair, an r-e pair, for(S(Rd, E),S(K, E)

).

Suppose s′ ∈ Sd and K′ is a standard k-corner in Rd such that K = s′K′. Sets1x := s−1(s′x) so that s1K′ = K. Then

Us′ RK′ U−1s′ = Us Us1 RK′ U−1

s1 U−1s = Us RK U−1

s = RK .

Similarly, Us′ EK′ U−1s′ = Us EK U−1

s = EK. Thus (RK, EK) is well-defined by(1.1.19) independently of the particular representation of K.

Approximation by Test Functions

The following simple observation will repeatedly be of use.

1.1.2 Lemma Let X and Y be LCSs and (r, rc) an r-c pair for (X ,Y). SupposeX0 is a dense subset of X . Then r(X0) is dense in Y.

Proof Suppose y ∈ Y and U is a neighborhood of y in Y. Then, since r is surjec-tive, r−1(U) is a neighborhood of r−1(y) in X . Hence there exists x ∈ X0 ∩ r−1(y)by the density of X0 in X. Then r(x) ∈ U ∩ r(X0). This proves the claim. ¥

It is the purpose of the following considerations to extend (RK, EK) to an r-epair for

(S ′(Rd, E),S ′(K, E)). For this we need some preparation.

We denote by D(K, E) the subspace of S(K, E) consisting of all functionswith compact support, endowed with the usual LF topology (analogous to thedefinition of D(Rd, E)). It is the space of E-valued test functions on K.

1.1.3 Lemma

(i) Let K be a standard k-corner. Set Ij := R for k + 1 ≤ j ≤ d. Then

d−1⊗

i=1

S(Ii) ⊗ S(Id, E)d⊂ S(K, E) .

(ii) If K is a corner in Rd, then D(K, E)d

→ S(K, E).

Proof (1) It is known (e.g., Corollary 1.8.2 in the Appendix or Theorem 3.9.2in [Tre67]) that

S(R)⊗(d−1) ⊗ S(R, E)d⊂ S(Rd, E) . (1.1.21)

Suppose u ∈ S(K, E). Then EKu ∈ S(Rd, E). Hence there exists a sequence (uj) inS(R)⊗(d−1) ⊗ S(R, E) with uj → EKu in S(Rd, E). It follows RKuj → RKEKu = u


in S(K, E). Given v := v1 ⊗ · · · ⊗ vd in S(Rd)⊗(d−1) ⊗ S(R, E), it is clear thatRKv belongs to

⊗d−1i=1 S(Ii) ⊗ S(Id, E). Hence RKuj ∈

⊗d−1i=1 S(Ii) ⊗ S(Id, E) for

j ∈ N. This proves (i).(2) We fix ϕ ∈ D(R) with ϕ(x) = 1 for |x| ≤ 1 and put ϕr(x) := ϕ(x/r) for

x ∈ R and r > 0. Suppose u ∈ S(R+, E). Then ϕru ∈ D(R+, E) for r > 0. Notethat 〈x〉k ≤ 〈x〉k+1/r for |x| ≥ r. Moreover, ∂α(ϕr − 1) vanishes for |x| < r andα ∈ N. Hence, by Leibniz’ rule,

supx>0

〈x〉k |∂m(ϕru− u)|E (x) ≤ c(m)m∑

j=0

r−j supx>r

〈x〉k |∂m−ju|E (x)

≤ c(m)r−1qk+1,m(u)

for k,m ∈ N and r > 1. Thus ϕru → u in S(R+, E) as r →∞. This shows thatD(R+, E) is dense in S(R+, E).

(3) Let τy be the right translation, that is,

(τyu)(x) = u(x − y) , u : R→ E , x ∈ R , y ∈ R+ .

Then τy ; y ≥ 0 is a strongly continuous semigroup on S(R, E) (cf. the proofof Theorem VII.3.3.1 below). We set I := (0,∞) and ρy := RI τy EI . Since(RI , EI) is an r-e pair for

(S(R, E),S(I, E)), it follows that ρy ; y ≥ 0 is a

strongly continuous semigroup on S(I, E). Note that supp(ρyu) ⊂ [y,∞) for y > 0and u ∈ S(I, E). Hence ϕr(ρyu) ∈ D(I, E).

It follows from

u− ϕr(ρyu) = u − ρyu − (ϕr − 1)ρyu ,

by choosing first y sufficiently small and then r sufficiently large, that each ele-ment u of S(I, E) can be arbitrarily closely approximated in S(I, E) by elementsof D(I, E). Hence D(I, E) is dense in S(I, E).

(4) Clearly, D(K, E) → S(K, E). We deduce from steps (2) and (3) and thedensity of D(R, E) in S(R, E) that each u ∈ ⊗d−1

i=1 S(Ii)⊗ S(Id, E) can be arbi-trarily closely approximated in S(Rd, E) by elements of

⊗d−1i=1 D(Ii) ×D(Id, E).

Since each such element belongs to D(K, E), we see from (1.1.21) that D(K, E) isdense in S(K, E). This proves (ii). ¥

1.2 Tempered Distributions on Corners

Throughout this subsection

• K is a corner in Rd.

As already mentioned, we now extend the r-e pair (RK, EK) to distributions. Inanalogy to the full-space case, the space of tempered distributions on K is defined


by

S ′(K, E) := L(S(K∗), E)

. (1.2.1)

Note that3

(u, ϕ) 7→ 〈u, ϕ〉K :=∫

Kuϕdx (1.2.2)

is a continuous bilinear map S(K, E)× S(K∗) → E. It follows from (1.1.2), partialintegration, and Fubini’s theorem that

〈∂αu, ϕ〉K = (−1)|α|〈u, ∂αϕ〉K , α ∈ Nd . (1.2.3)

Similarly as in the classical case, we define Tu ∈ S ′(K, E) for u ∈ S(K, E) by

Tuϕ := 〈u, ϕ〉K , ϕ ∈ S(K∗) .

Since u 7→ Tu is injective we can identify u ∈ S(K, E) with Tu ∈ S ′(K, E) so that

S(K, E) → S ′(K, E) . (1.2.4)

We use the notation 〈u, ϕ〉K also for the value of u ∈ S ′(K, E) on ϕ ∈ S(K∗)and call 〈·, ·〉K E-valued

(S ′(K, E),S(K∗))

duality pairing induced by (1.2.2). The(K)-distributional derivative ∂αu of u ∈ S ′(K, E) is then defined for α ∈ Nd by

〈∂αu, ϕ〉K := (−1)|α|〈u, ∂αϕ〉K , ϕ ∈ S(K∗) . (1.2.5)

By (1.2.3) and (1.2.4) it extends the classical derivative on K. Furthermore,

∂α ∈ L(S(K, E)) ∩ L(S ′(K, E)

). (1.2.6)

It should be observed that ∂αu is not the distributional derivative of u in the senseof Schwartz distributions if K 6= K, since S(K∗) is then substantially larger thanthe space of test functions D(K).

In the following, we shall often use pull-back operators. For this reason wecollect in the next remarks some of their basic properties.

1.2.1 Remarks Let X and Y be LCSs.

(a) Suppose f ∈ L(X ,Y). Then the pull-back f∗u of u ∈ L(Y, E) is defined byf∗u := u f . Thus

f ∈ L(X ,Y) =⇒ f∗ ∈ L(L(Y, E),L(X , E))

.

If Z is a further LCS and g ∈ L(Y,Z), then

(g f)∗ = f∗ g∗ , (idX )∗ = idL(X ,E) .

3uϕ := ϕu.


Consequently,

f ∈ Lis(X ,Y) =⇒ f∗ ∈ Lis(L(Y, E),L(X , E)

),

and (f∗)−1 = (f−1)∗.

(b) Let i : X → Y and u ∈ L(Y, E). Then i∗u = u i = u |X , considered as a

continuous linear map from X into E. If X d→ Y , then i∗ is injective. Hence it

is justified to identify L(Y, E) with a linear subspace of L(X , E) by identifyingu ∈ L(Y, E) with u |X ∈ L(X , E). It follows

X d→ Y =⇒ L(Y, E) → L(X , E) . (1.2.7)

(c) Let (r, rc) be an r-c pair for (X ,Y). Then (a) implies that((rc)∗, r∗

)is an r-c

pair for(L(X , E),L(Y, E)

).

(d) Since L(X ,C) = X ′ we get f∗ = f ′ ∈ L(Y ′,X ′) for f ∈ L(X ,Y). Thus (1.2.7)generalizes

X d→ Y =⇒ Y ′ → X ′

(cf. Proposition V.1.4.8). ¥

Duality Formulas

It follows from Lemma 1.1.3(ii) and (1.2.7) that

S ′(K, E) → D′(K, E) := L(D(K∗), E)

.

In particular, S ′(K, E) → D′(K, E) which shows that

each u ∈ S ′(K, E) is an E-valued Schwartz distribution on K. (1.2.8)

1.2.2 Lemma It holds

〈EKu, ϕ〉Rd = 〈u, RK∗ϕ〉K , u ∈ S(K, E) , ϕ ∈ S(Rd) ,

and〈RKu, ϕ〉K = 〈u, EK∗ϕ〉Rd , u ∈ S(Rd, E) , ϕ ∈ S(K∗) .

Proof (1) Let K be a standard k-corner of type J . Suppose k ∈ J , u ∈ S(K, E),and ϕ ∈ S(K∗

k,k−1). Then, by (1.1.9),

〈EKu, ϕ〉Kk,k−1 =∫

Kϕudx +

∫

(−∞,0)×k

∂kKϕεku dx . (1.2.9)


We write u(y; ·) :=(xk 7→ u

(y; xk

))etc. Then, by Lemma 1.1.1 and Fubini’s the-

orem,∫ 0

−∞ϕ(y; ·)εku(y; ·) dy =

∫ 0

−∞ϕ(y; ·)

∫ ∞

0

h(t)u(−ty; ·) dt dy

=y 7→−y

∫ ∞

0

∫ ∞

0

h(t)u(ty; ·)ϕ(−y; ·) dt dy

=y 7→z:=ty

∫ ∞

0

∫ ∞

0

h(t)u(z; ·)ϕ(−z/t; ·) dz dt/t

=t 7→s:=1/t

∫ ∞

0

∫ ∞

0

h(1/s)u(z; ·)ϕ(−sz; ·) dz ds/s

=h(1/s)=−sh(s)

−∫ ∞

0

∫ ∞

0

u(z; ·)h(s)ϕ(−sz; ·) dz ds

= −∫ ∞

0

u(z; ·)(εkϕ)(z; ·) dz .

By integrating this relation over ∂kK we get from (1.2.9) and (1.1.14)

〈Eku, ϕ〉Kk,k−1 =∫

Kuϕdx −

∫

Kuεkϕ dx = 〈u, Rkϕ〉K .

(2) Let K be as in (1) and assume k ∈ J∗. Then we obtain by (1.1.12)

〈Eku, ϕ〉Kk,k−1 =∫

Kuϕ dx = 〈u, Rkϕ〉K

for u ∈ S(K, E) and ϕ ∈ S(K∗k,k−1) since Rkϕ is the point-wise restriction.

(3) From (1) and (2) and the definition of (RK, EK) we find

〈EKu, ϕ〉Rd = 〈u, RK∗ϕ〉K , u ∈ S(K, E) , ϕ ∈ S(Rd) ,

if K is a standard corner.(4) Now suppose K = sK where K is a standard corner and s ∈ Sd. Then,

x 7→ sx being an isometry,

〈Usu, ϕ〉K =∫

Ku(s−1x)ϕ(x) dx =

∫

Ku(y)ϕ(sy) dy = 〈u, U−1

s ϕ〉K

and, since sRd = Rd, we infer from (1.1.3), (1.1.4), (1.1.19), and step (3)

〈EKu, ϕ〉Rd = 〈Us EK U−1s u, ϕ〉Rd = 〈EK U−1

s u, U−1s ϕ〉Rd

= 〈U−1s u, RK∗ U−1

s ϕ〉K = 〈u, Us RK∗ U−1s ϕ〉K = 〈u, RK∗ϕ〉K

for u ∈ S(K, E) and ϕ ∈ S(Rd). This proves the first assertion. It is obvious thatthe above arguments give the second claim as well. ¥


By (1.1.20) we know that (RK∗ , EK∗) is an r-c pair for(S(Rd),S(K∗)

). Hence,

by (1.2.1) and Remark 1.2.1(c),

(E∗K∗ , R

∗K∗) :=

((EK∗)∗, (RK∗)∗

)is an r-c pair for

(S ′(Rd, E),S ′(K, E)). (1.2.10)

Lemma 1.2.2 shows that

E∗K∗ ∈ L(S ′(Rd, E),S ′(K, E)

)

is an extension of RK ∈ L(S(Rd, E),S(K, E)) (1.2.11)

and

R∗K∗ ∈ L(S ′(K, E),S ′(Rd, E)

)

is an extension of EK ∈ L(S(K, E),S(Rd, E)).

(1.2.12)

It is known (cf. Theorem 1.3.6 in the Appendix) that S(Rd, E)d

→ S ′(Rd, E). Henceit follows from (1.2.4), (1.2.11), and (1.2.12) that the following diagrams4

S(Rd, E) S(K, E)

S ′(Rd, E) S ′(K, E)

RK

E∗K∗

d

-

-?

¤¡

?

¤¡ S(K, E) S(Rd, E)

S ′(K, E) S ′(Rd, E)

EK

R∗K∗

d

-

-?

¤¡

?

¤¡

are commuting. Moreover, (1.2.10) and Lemma 1.1.2 imply that S(K, E) is alsodense in S ′(K, E). From this we infer that (E∗

K∗ , R∗K∗) is uniquely determined

by (RK, EK). For this reason we write again (RK, EK) for (E∗K∗ , R

∗K∗) without fear-

ing confusion and say that (RK, EK) is a restriction-extension pair (an r-e pair) for(S ′(Rd, E),S ′(K, E)).

The Main Theorem

The above facts are summarized in part (i) of the following basic restriction-extension theorem for smooth functions and tempered distributions.

1.2.3 Theorem Let K be a corner in Rd.(i) (RK, EK) is an r-e pair for

(S ′(Rd, E),S ′(K, E))

and the diagrams

S(Rd, E) S(K, E)

S ′(Rd, E) S ′(K, E)

RK

RK

d d

-

-?

¤¡

?

¤¡ S(K, E) S(Rd, E)

S ′(K, E) S ′(Rd, E)

EK

EK

d d

-

-?

¤¡

?

¤¡

are commuting.4In all diagrams occurring in this volume, arrows represent continuous linear maps.


(ii) RKu is the restriction of u ∈ S ′(Rd, E) to S ′(K, E) in the sense of Schwartzdistributions.

(iii) ∂α RK = RK ∂α, α ∈ Nd.

Proof (1) Claim (i) has been proved by the above considerations.

(2) Since RKu = (EK)∗u for u ∈ S ′(Rd, E) and EK = E1 · · · Ek if K is astandard k-corner, we get

〈RKu, ϕ〉K = 〈u, ϕ〉K , u ∈ S ′(Rd, E) , ϕ ∈ S(K) , (1.2.13)

from (1.1.12). It follows from the continuity of Us on S(K) and the density asser-tions of part (i) that

〈Usu, ϕ〉K = 〈u, U−1s ϕ〉K , u ∈ S ′(K, E) , ϕ ∈ S(K) .

From this and (1.2.13) we infer that the latter holds for arbitrary corners, whichproves (ii).

(3) Using (1.2.3), (1.2.5), (ii), and the continuity of ∂α on S(K, E) andS ′(K, E) we get

〈∂αRKu, ϕ〉K = (−1)|α|〈RKu, ∂αϕ〉K = (−1)|α|〈u, ∂αϕ〉K = 〈∂αu, ϕ〉K= 〈RK∂αu, ϕ〉K

for u ∈ S ′(Rd, E) and ϕ ∈ S(K). This implies (iii). ¥

1.2.4 Remarks (a) The r-e pair (RK, EK) is universal in the following sense: sup-pose E1 is a Banach space with E1 → E and let (R1

K, E1K) be the r-e pair for(S ′(Rd, E1),S ′(K, E1)

)constructed above. Then the diagrams

S ′(Rd, E1) S ′(Rd, E)

S ′(K, E1) S ′(K, E)

R1K RK

-¤£

-¤£? ?

S ′(K, E1) S ′(K, E)

S ′(Rd, E1) S ′(Rd, E)

E1K EK

-¤£

-¤£? ?

are commuting. Furthermore, the particular form of (RK, EK) is independent ofthe choice of E.

(b) Let X ∈ S,S ′. Then RKu = RKu for u ∈ EKX (K, E).

Proof (1) Suppose X = S and k ∈ J∗. Let u = Ekv for some v ∈ S(K, E). Thenεku(x) = 0 for x ∈ K, since u

(xk; xk

)= 0) for xk ≤ 0. Hence definition (1.1.14)

implies Rku(x) = u(x) = Rku(x) for u(x) ∈ K. From this we get the claim in thepresent case, due to RK = Rk Rk−1 · · · R1 if K is a standard k-corner.


(2) If X = S ′, then the assertion follows from (1) by continuous extensionbased on Theorem 1.2.3. ¥

The following completeness assertion will be of use in later sections.

1.2.5 Proposition Let K be a corner in Rd and X ∈ Rd,K. Then S ′(X, E) is acomplete LCS.

Proof (1) Suppose X = Rd. It is well-known that S(Rd) and S ′(Rd) are completeand reflexive (e.g., Theorem 1.1.2 in the Appendix). Hence Lemma 1.4.7 ibidemguarantees that S ′(Rd, E) = L(S(Rd), E

)= L((S ′(Rd)

)′, E

)is complete.

(2) Assume X = K. It follows from Theorem 1.2.3(i) and Lemma I.2.3.1 thatS ′(K, E) is isomorphic to a closed linear subspace of S ′(Rd, E). Since the latterspace is complete by (1), S ′(K, E) is also complete. ¥

1.3 Duality

The theorem proved in this subsection shows that S ′(K, E) can be characterizedby vector-valued duality. It is the basis for the duality theory of vector-valuedBesov and Bessel potential spaces.

For the sake of a uniform presentation we put (Rd)∗ := Rd in what follows.

1.3.1 Theorem Let K be a corner in Rd and X ∈ Rd,K. Then

S ′(X, E′) = S(X∗, E)′ (1.3.1)

with respect to the unique duality pairing satisfying

〈u′, u〉S(X∗,E) =∫

X∗

⟨u′(x), u(x)

⟩E

dx (1.3.2)

for u′ ∈ S(X, E′) and u ∈ S(X∗, E).

Proof (1) Suppose X = Rd. Then the assertion follows from L. Schwartz’ theoryof vector-valued distributions [Schw57b]. In fact, 〈·, ·〉S(Rd,E) is uniquely deter-mined by its values on (ϕ′ ⊗ e′, ϕ ⊗ e) ∈ (D(Rd)⊗ E′)× (D(Rd) ⊗ E

)and it is a

hypocontinuous bilinear form on S ′(Rd, E) × S(Rd, E). More precisely, the claimis implied by Corollary 1.4.10 and Theorem 1.7.5 of the Appendix and the densityof D in S.

(2) Assume X = K. It is a consequence of Theorem 1.2.3 and Lemma I.2.3.1that EK∗S(K∗, E) is a closed linear subspace of S(Rd, E) and

EK∗ ∈ Lis(S(K∗, E), EK∗S(K∗, E)

). (1.3.3)


The same is true if EK∗ and S(K∗, E) are replaced by EK and S ′(K, E′), respec-tively. Hence

〈u′, u〉S(K∗,E) := 〈EKu′, EK∗u〉S(Rd,E) , u′ ∈ S ′(K, E′) , u ∈ S(K∗, E) ,

defines a hypocontinuous bilinear form 〈·, ·〉S(K∗,E) on S ′(K, E′) ⊗ S(K∗, E).

(3) Suppose u′ ∈ S(K, E′) and u = ϕ⊗ e ∈ D(K∗)⊗ E ⊂ S(K∗)⊗ E. Then

EK∗u = (EK∗ϕ)⊗ e ∈ S(Rd) ⊗ E ,

Lemma 1.2.2, and (1.2.2) imply

〈u′, u〉S(K∗,E) = 〈EKu′, EK∗u〉S(Rd,E) =⟨〈EKu′, EK∗ϕ〉Rd , e

⟩E

=⟨〈u′, RK∗EK∗ϕ〉K∗ , e

⟩E

=⟨〈u′, ϕ〉K∗ , e

⟩E

=∫

K∗

⟨u′(x), ϕ(x) ⊗ e

⟩E

dx =∫

K∗

⟨u′(x), u(x)

⟩E

dx .

(1.3.4)

By Theorem 1.3.6 of the Appendix we know that S(Rd)⊗ E is dense in S(Rd, E).Suppose u ∈ EK∗S(K∗, E) and (uj) is a sequence in S(Rd) ⊗ E converging to u.Since PK∗ := EK∗RK∗ is a continuous projection in S(Rd, E) onto EK∗S(K∗, E),it follows that (PK∗uj) is a sequence in EK∗S(K∗, E) converging to u. If

v =n∑

k=0

ϕk ⊗ ek ∈ S(Rd) ⊗ E ,

then ψk := RK∗ϕk ∈ S(K∗) implies

PK∗v =n∑

k=0

EK∗ψk ⊗ ek ∈ EK∗S(K∗)⊗ E .

This shows that EK∗S(K∗) ⊗ E is dense in EK∗S(K∗, E). Since D(K∗) is densein S(K∗) by Lemma 1.1.3, it follows that EK∗D(K∗)⊗ E is dense in EK∗S(K∗, E).By Theorem 1.2.3, S(K, E′) is dense in S ′(K, E′). These density properties andthe hypocontinuity of 〈·, ·〉S(K∗,E) imply that the latter bilinear form is uniquelydetermined by its restriction to S(K, E′)× (D(K∗)⊗ E

). From this and (1.3.4) we

see that (1.3.2) applies.

(4) Suppose u′ ∈ S ′(K, E′) and 〈u′, u〉S(K∗,E) = 0 for all u ∈ S(K∗, E). Then〈u′, ϕ ⊗ e〉S(K∗,E) = 0 for all (ϕ, e) ∈ S(K∗) × E. It follows, similarly as in step (3),that 〈u′, ϕ ⊗ e〉S(K∗,E) =

⟨〈u′, ϕ〉K∗ , e⟩

E. Hence we derive from

⟨〈u′, ϕ〉K∗ , e⟩

E= 0

for all e ∈ E that 〈u′, ϕ〉K∗ = 0 for all ϕ ∈ S(K∗). Thus u′ = 0. We deduce similarlythat, given u ∈ S(K∗, E), 〈u′, u〉S(K∗,E) = 0 for all u ∈ S ′(K, E′) implies u = 0.Hence 〈·, ·〉S(K∗,E) is separating. This proves the theorem. ¥


1.3.2 Corollary The dual of

RK∗ ∈ L(S(Rd, E),S(K∗, E))

equals EK ∈ L(S ′(K, E′),S ′(Rd, E′))

and the dual of

EK∗ ∈ L(S(K∗, E),S(Rd, E))

equals RK ∈ L(S ′(Rd, E′),S ′(K, E′)).

Proof Let u ∈ S(K, E′) and v = ϕ⊗ e ∈ S(Rd) ⊗ E. Then, by Lemma 1.2.2 andK∗∗ = K,

〈EKu, ϕ〉S(Rd,E) =⟨〈EKu, ϕ〉Rd , e

⟩E

=⟨〈u, RK∗ϕ〉K∗ , e

⟩E

= 〈u, RK∗v〉S(K∗,E) .

From this, S(K, E′)d

→ S ′(K, E′), and EK ∈ L(S ′(K, E′),S ′(Rd, E′))

it follows

〈EKu, v〉S(Rd,E) = 〈u, RK∗v〉S(K∗,E) (1.3.5)

for u ∈ S ′(K, E′) and v ∈ S(Rd) ⊗ E. By employing the density of S(Rd)⊗ E inS(Rd, E), we thus find that (1.3.5) holds for v ∈ S(Rd, E). This implies the firstassertion. The second one follows by similar arguments. ¥

1.4 Notes

As mentioned in the introduction to this section, the basic result is Theorem 1.2.3,the restriction-extension theorem for smooth functions and tempered distributionson corners. Of course, its nucleus is the corresponding result for half-spaces, thatis, 1-corners. This, in turn, is based on the crucial Lemma 1.1.1 which is takenfrom [Ham75].

In the case of a closed half-space H := R+ × Rd−1, the extension operator EHis kind of a continuous version of Seeley’s extension operator [See64]. It has, how-ever, the advantage that it gives rise to the important duality formulas of Corol-lary 1.3.2.

The section at hand is a reworking and amplification of Sections 4.1–4.3of [Ama09], which was inspired by [Ham75].

VI.2 Sequence Spaces 19

2 Sequence Spaces

By means of localization techniques we will naturally be led to consider spaces ofBanach-space-valued sequences. Such localizations occur in connection with Besovspaces, the solvability theory of elliptic and parabolic boundary vale problems, andin the study of function spaces and differential equations on Riemannian manifolds.For this reason we present in this section some of the basic results on sequencespaces for the purpose of easy reference. In particular, we prove a general dualitytheorem for Banach-space-valued `p-spaces, introduce weighted sequence spaces,and collect their fundamental interpolation properties.

2.1 Duality of Sequence Spaces

With regard to later applications, where concrete enumerations are not convenient,we consider ‘sequences’ on arbitrary countable index sets.

Definitions and Embeddings

We denote by A a nonempty countable set (an index set) which we give the discretetopology. Thus A is a σ-compact Hausdorff space.

Let Xα be for each α ∈ A an LCS. We set X := (Xα) = (Xα)α∈A and endow

c(X ) :=∏α

Xα

with the product topology, that is, the coarsest locally convex topology for whichall projections

prβ : c(X ) → Xβ , x = (xα) 7→ xβ

are continuous. Bycc(X ) :=

⊕α

Xα

we mean the locally convex direct sum. Thus cc(X ) is the linear subspace of c(X )consisting of all compactly supported sequences equipped with the finest locallyconvex topology for which all natural injections

jβ : Xβ → cc(X ) , xβ 7→ (xβδαβ)α (2.1.1)

are continuous, where δαβ := 0 if α 6= β and δαα := 1, the Kronecker symbol.If Xα = X for α ∈ A, then we write c(X ) and cc(X ) for c(X ) and cc(X ),

respectively. Thusc(X ) = XA

is the LCS of all X -valued functions on A with the topology of point-wise conver-gence. Since every map from a discrete space into a topological space is continuous


and A is σ-compact, XA = C(A,X ). Moreover,

cc(X ) = Cc(A,X )

is the space of compactly supported continuous X -valued functions on A with theobvious LF topology.

Let 〈·, ·〉α be the (X ′α,Xα)-duality pairing. Then, letting X ′ := (X ′

α),

〈〈·, ·〉〉 : c(X ′) × cc(X ) → C , (x′, x) 7→∑α

〈x′α, xα〉α

is a separating continuous bilinear form, and

cc(X )′ = c(X ′) (2.1.2)

with respect to 〈〈·, ·〉〉, that is, 〈〈·, ·〉〉 is the(cc(X )′, cc(X )

)-duality pairing (cf. Corol-

lary 1 in Section IV.4.3 of H.H. Schaefer [Sch71]).Suppose E = (Eα), where each Eα = (Eα, |·|α) = (Eα, |·|Eα

) is a Banachspace and 1 ≤ r ≤ ∞. Then `r(E) = `r(A, E) is the linear subspace of c(E) con-sisting of all x = (xα) such that5

‖x‖`r(E) :=

(∑α|xα|rα

)1/r, r < ∞ ,

supα|xα|α , r = ∞ ,

is finite. It is a Banach space with the norm ‖·‖`r(E). It is easily seen that

cc(E)d

→ `q(E) → `r(E) → c(E) , 1 ≤ q < r ≤ ∞ . (2.1.3)

We denote byc0(E) the closure of cc(E) in `∞(E).


`q(E)d

→ c0(E) , 1 ≤ q < ∞ . (2.1.4)

If Eα = E for α ∈ A, then

`r(E) = `r(A, E) := Lr(A, κ, E) , 1 ≤ r ≤ ∞ ,

where κ is the counting measure on A, and

c0(E) = C0(A, E) .

Suppose Yα is an LCS with Yα → Xα for α ∈ A. In this case we write Y → X .

Furthermore, Y d→ X means Yα

d→ Xα for α ∈ A. It is easy to see that

Y → X =⇒ c(Y) → c(X ) , cc(Y) → cc(X ) (2.1.5)

with dense embeddings if Y d→ X .

5α aα = supKbA β∈K aβ if aα ∈ R+ for α ∈ A.


Duality Pairings

Let F := (Fα) with each Fα being a Banach space. If F → E, then

`r(F ) → `r(E) , 1 ≤ r ≤ ∞ , c0(F ) → c0(E) . (2.1.6)

If Fd

→ E, then these embeddings are dense as well, provided r < ∞.

By Holder’s inequality6,∣∣∣∑α

〈x′α, xα〉α∣∣∣ ≤

∑α

|〈x′α, xα〉α| ≤∑α

|x′α|E′α|xα|Eα ≤ ‖x′‖`r′ (E′) ‖x‖`r(E)

for x′ ∈ `r′(E′), x ∈ `r(E), and 1 ≤ r ≤ ∞, where r′ := r/(r − 1) is the dual ex-

ponent of r. Note that∑α

〈x′α, xα〉α = 〈〈x′, x〉〉 , x′ ∈ `r′(E′) , x ∈ cc(E) .

Hence we can put

〈〈x′, x〉〉 :=∑α

〈x′α, xα〉α , x′ ∈ `r′(E′) , x ∈ `r(E) , 1 ≤ r ≤ ∞ ,

without creating confusion. In this connection we sometimes refer to 〈〈·, ·〉〉 as(`r′ , `r)-duality pairing. The next theorem justifies this usage. Here E is reflexive ifeach Eα is reflexive.

2.1.1 Theorem

(i) With respect to 〈〈·, ·〉〉,(`r(E)

)′ = `r′(E′) , 1 ≤ r < ∞ , (2.1.7)

and(c0(E)

)′ = `1(E′) . (2.1.8)

(ii) If E is reflexive and 1 < r < ∞, then `r(E) is reflexive as well.

Proof (1) Let F = (F, ‖·‖) be a nontrivial Banach space. We denote the dualnorm by ‖·‖′. Thus

‖f ′‖′ = supf∈F\0

|〈f ′, f〉|‖f‖ , f ′ ∈ F ′ .

6Let limK denote the limit with respect to the family of compact subsets of A directed byinclusion. Then α〈yα, xα〉α = limK β∈K〈yβ , xβ〉.


Suppose f ′ 6= 0 and ε > 0. Then there exists gε ∈ F with ‖g‖ε = 1 and

|〈f ′, gε〉| ≥ (1 − ε) ‖f ′‖′ .

Define ϕ ∈ [0, 2π) by e−iϕ := 〈f ′, gε〉/|〈f ′, gε〉|. Then fε := eiϕgε belongs to F ,

satisfies ‖fε‖ = 1, and

〈f ′, fε〉 = eiϕ〈f ′, gε〉 = |〈f ′, gε〉| ≥ (1 − ε) ‖f ′‖′ .

(2) For K b A and x ∈ c(E) we set (PKx)α := xα if α ∈ K, and (PKx)α := 0otherwise. Then

PK ∈ L(`q(E)

), 1 ≤ q ≤ ∞ , P 2

K = PK , im(PK) ⊂ cc(E) .

By replacing E by E′ in this definition we obtain the finite rank projectionP ′

K ∈ L(`q(E′)

).

Let x ∈ `q(E), y ∈ `q′(E′), and K b A. Then

〈〈y, PKx〉〉 =∑

α∈K

〈yα, xα〉α = 〈P ′Ky, x〉 .

Moreover, if x ∈ `q(E) with q < ∞, then7 limK PKx = x in `q(E).

(3) Assume 1 ≤ q ≤ ∞. For y ∈ `q′(E′) we set

Tyx := 〈〈y, x〉〉 , x ∈ `q(E) .

Then Holder’s inequality shows that Ty ∈ (`q(E)

)′ and

‖Ty‖(`q(E))′ ≤ ‖y‖`q′ (E′) , y ∈ `q′(E′) . (2.1.9)

Thus

(y 7→ Ty) ∈ L(`q′(E′), (`q(E))′

). (2.1.10)

(4) Let 0 < ε < 1, 1 < r < ∞, and y ∈ cc(E′). By step (1) we can choose foreach yα 6= 0 an xα ∈ Eα satisfying |xα|α = 1 and 〈yα, xα〉α ≥ (1 − ε) |yα|′α.

We set zα := (|yα|′α)r′−1xα, where zα := 0 if yα = 0. Then z = (zα) ∈ cc(E)and

〈〈y, z〉〉 =∑α

(|yα|′α)r′−1〈yα, xα〉α ≥ (1− ε)∑α

(|yα|′α)r′ = (1 − ε) ‖y‖r′`r′ (E′) .

7Clearly, we could fix an enumeration of A and consider ‘standard’ sequences indexed by N.However, in later applications it is more convenient to avoid explicit enumerations.


Moreover,

‖z‖`r(E) =(∑

α

(|yα|′α)r(r′−1) |xα|rα)1/r

=(∑

α

(|yα|′α)r′)1/r

= ‖y‖r′/r`r′ (E′) = ‖y‖r′−1

`r′ (E′) .

Hence

〈〈y, z〉〉 ≥ (1− ε) ‖y‖`r′ (E′) ‖z‖`r(E) . (2.1.11)

(5) Let 0 < ε < 1, y ∈ `1(E′), and K b A. Then setting r := ∞, so thatr′ = 1, and defining zα as above,

〈〈y, PKz〉〉 =∑

α∈K

〈yα, zα〉α ≥ (1 − ε)∑

α∈K

|yα|′α

= (1 − ε) ‖P ′Ky‖`1(E′) ‖PKz‖`∞(E) .

(2.1.12)

(6) Lastly, assume 0 < ε < 1, r = 1, and y ∈ `∞(E′). We fix β ∈ A with|yβ |′β ≥ (1 − ε) ‖y‖`∞(E′). Then we choose xβ ∈ Eβ with |xβ |β = 1 and

〈yβ , xβ〉β ≥ (1− ε) |yβ |′β .

It follows that z := (δαβxβ)α belongs to `1(E), has norm 1, and satisfies

〈〈y, z〉〉 = 〈yβ , xβ〉β ≥ (1− ε)2 ‖y‖`∞(E′) ‖z‖`1(E) . (2.1.13)

(7) Suppose 0 < ε < 1, 1 ≤ r < ∞, and y ∈ cc(E′) if r > 1, and y ∈ `∞(E′)if r = 1. Then we get from (2.1.11)

‖Ty‖(`r(E))′ = supx∈`r(E)

|〈〈y, x〉〉|‖x‖`r(E)

≥ |〈〈y, z〉〉|‖z‖`r(E)

≥ (1 − ε) ‖y‖`r′ (E′)

if r > 1, and from (2.1.13), analogously,

‖Ty‖(`1(E))′ ≥ (1 − ε)2 ‖y‖`∞(E′) .

This being true for every ε ∈ (0, 1),

‖Ty‖(`r(E))′ ≥ ‖y‖`r′ (E′) . (2.1.14)

If r > 1, then cc(E′) is dense in `r′(E′). Thus we infer from (2.1.10) that (2.1.14)holds for each y in `r′(E′). Consequently, by (2.1.9),

‖Ty‖(`r(E))′ = ‖y‖`r′ (E′) , y ∈ `r′(E′) . (2.1.15)

Hence y 7→ Ty is a linear isometry from `r′(E′) into(`r(E)

)′ if 1 ≤ r < ∞.


(8) Assume y ∈ `1(E′) and 0 < ε < 1. Then, given K b A, it follows from(2.1.12) that

‖Ty‖(c0(E))′ = supx∈cc(E)

|〈〈y, x〉〉|‖x‖`∞(E)

≥ |〈〈y, PKz〉〉|‖PKz‖`∞(E)

≥ (1 − ε) ‖P ′Ky‖`1(E′) .

Consequently,‖Ty‖(c0(E))′ ≥ ‖P ′

Ky‖`1(E′) .

Since this is true for every K b A, and since limK P ′Ky = y in `1(E′), we get

‖Ty‖(c0(E))′ ≥ ‖y‖`1(E′) .

By Holder’s inequality

|Tyx| = |〈〈y, x〉〉| ≤ ‖y‖`1(E′) ‖x‖`∞(E) = ‖y‖`1(E′) ‖x‖c0(E)

for x ∈ c0(E). Thus Ty |c0(E) belongs to(c0(E)

)′ and

‖Ty‖(c0(E))′ ≤ ‖y‖`1(E′) , y ∈ `1(E′) . (2.1.16)

Hence y 7→ Ty is a linear isometry from `1(E′) into(c0(E)

)′.(9) Given K b A, we set

jK := cc(E) → cc(E) , x 7→ ∑α∈Kjαxα ,

where jα : Eα → cc(E) is the canonical injection (2.1.1).

Suppose 1 ≤ r < ∞. Let f belong to(`r(E)

)′. We define y = (yα) ∈ c(E′)by yα := f jα and set yK := P ′

Ky. Then, by step (2),

〈〈yK, x〉〉 = 〈〈P ′Ky, x〉〉 = 〈〈y, PKx〉〉

=∑

α∈K

〈yα, xα〉α =∑

α∈K

f(jαxα) = f(jKx) (2.1.17)

for x ∈ `r(E). Hence

|TyK(x)| ≤ ‖f‖(`r(E))′ ‖jKx‖`r(E) = ‖f‖(`r(E))′

(∑

α∈K

|xα|rα)1/r

≤ ‖f‖(`r(E))′ ‖x‖`r(E) .

From this, yK ∈ cc(E′) ⊂ `r′(E′), and (2.1.15) we deduce

‖yK‖`r′ (E′) = ‖TyK‖(`r(E))′ ≤ ‖f‖(`r(E))′ , K b A .

Thusy ∈ `r′(E′) , ‖y‖`r′ (E′) ≤ ‖f‖(`r(E))′ .



Ty(jKx) = f(jKx) , x ∈ `r(E) .

Since Ty ∈ (`r(E)

)′ by step (3), and limK jKx = x in `r(E), we find Ty(x) = f(x)for x ∈ `r(E), that is, Ty = f . This shows that (y 7→ Ty) is a surjection from`r′(E′) onto

(`r(E)

)′. Thus it follows from step (7) that assertion (2.1.7) is true.

(10) Assume f ∈ (c0(E)

)′. We define y ∈ c(E′) as in the preceding step.Then (2.1.17) applies for x ∈ c0(E). From this and step (8) we get, similarly asabove,

‖yK‖`1(E′) = ‖TyK‖(c0(E))′ ≤ ‖f‖(c0(E))′ , K b A .

Hence y ∈ `1(E′). Furthermore, since limK jKx = x in c0(E), we deduce fromTy(jKx) = f(jKx) that Ty(x) = f(x) for all x in c0(E). Thus Ty |c0(E) = f . Dueto the results of step (8), this implies the validity of (2.1.8).

(11) If E is reflexive, then (E′)′ = E. Hence the last assertion is obvious. ¥

2.2 Weighted Sequence Spaces

In the study of Besov spaces, weighted sequences play a fundamental role. In thisconnection the weights are powers of 2, naturally related to dyadic decompositionsstudied in the next section. In the section at hand, we transpose the precedingresults to weighted sequence spaces.

Image Spaces

Let X and Y be LCSs and ϕ ∈ L(X ,Y). The image space ϕX is the image of X in Yunder ϕ endowed with the unique locally convex topology for which ϕ, defined bythe commutativity of the diagram

X ϕX ⊂ Y

X/ ker(ϕ)

ϕ

ϕ-

@@@R ¡

¡¡µ (2.2.1)

is a toplinear isomorphism. Here the unlabeled arrow represents the canonicalprojection.

2.2.1 Remarks (a) ϕX is an LCS with ϕX → Y , and ϕ is a continuous surjectiononto ϕX . Let P be a generating family of seminorms for X and set

p(y) := inf

p(x) ; x ∈ ϕ−1(y)

, y ∈ ϕX , p ∈ P .


Then the family p ; p ∈ P generates the topology of ϕX . If X is a Banachspace, then ϕX is a Banach space as well with the quotient norm

y 7→ ‖y‖ϕX := inf‖x‖X ; x ∈ ϕ−1(y)

.

In particular, if ϕ is injective, then

‖y‖ϕX = ‖ϕ−1y‖X , y ∈ ϕX .

Proof This follows from the closedness of ker(ϕ) and standard properties of quo-tient spaces. ¥

(b) Let X0 and Z be LCSs such that i : X0 → X and Y → Z. We write ϕX0 for(ϕ i)X0. Then

ϕX0 → ϕX → Z .

If X0d

→ X , then ϕX0d

→ ϕX .

Proof The first assertion is obvious. The second one follows from (2.2.1) and thecontinuity of the canonical projection. ¥

(c) If X is a reflexive or separable Banach space, then ϕX is also reflexive orseparable, respectively.

Proof Quotients of Banach spaces modulo closed linear subspaces possess theseproperties. ¥

Let a ∈ C\0. We identify it with the multiplication operator x 7→ ax forx ∈ E. Then aE = (E, |·|aE) and aE

.= E. In fact, |·|aE = |·|/|a|. Furthermore,given x′ ∈ E′,

|x′|(aE)′ = supx∈aE

|〈x′, x〉E ||x|aE

= supy∈E

|〈ax′, y〉E ||y|E = |ax′|E′ = |x′|a−1E′ .

Hence

a−1E′ = (aE)′ with respect to 〈·, ·〉E . (2.2.2)

Embeddings and Duality

We define weighted sequence spaces as follows:

for s ∈ R we set 2−sE :=((2−skE)k∈N

).

Then

`sr(E) := `r(2−sE) , 1 ≤ r ≤ ∞ , cs

0(E) := c0(2−sE) . (2.2.3)


Thus, for x ∈ EN it holds:

x ∈ `sr(E) ⇐⇒ (2ksxk)k ∈ `r(E)

andx ∈ cs

0(E) ⇐⇒ 2ks |xk|E → 0 as k →∞ .

Hence, denoting by κs the weighted counting measure which assigns the value 2ks tok ⊂ N,

`sr(E) = Lr(N, κs, E) .

Note`0r(E) = `r(E) , c0

0(E) = C0(N, E) .

In the following theorem, which is for most of its parts a direct consequenceof the results of Subsection 2.1, we collect the basic properties of these spaces.

2.2.2 Theorem Suppose s, t ∈ R and 1 ≤ q, r ≤ ∞.(i) It holds:

cc(E)d

→ `sq(E)

d→ `s

r(E)d

→ cs0(E) → `s

∞(E) , 1 ≤ q < r < ∞ , (2.2.4)

and

`sq(E) → `t

r(E) , s > t . (2.2.5)

(ii) Let E0 and E1 be Banach spaces with E1 → E0. Then

`sr(E1) → `s

r(E0) , cs0(E1) → cs

0(E0) .

If E1d

→ E0, then

`sr(E1)

d→ `s

r(E0) , r < ∞ , cs0(E1)

d→ cs

0(E0) .

(iii) With respect to8 〈〈·, ·〉〉,(`sr(E)

)′ = `−sr′ (E′) , r < ∞ ,

(cs0(E)

)′ = `−s1 (E′) . (2.2.6)

If E is reflexive and 1 < r < ∞, then `sr(E) is also reflexive.

Proof (1) Assertion (2.2.4) follows immediately from (2.2.3), (2.1.3), and (2.1.4).(2) If s > t, then

‖x‖`tr(E) = ‖(2ktxk)‖`r(E) ≤ ‖(2k(t−s))‖`r ‖(2ksxk)‖`∞(E)

≤ c ‖x‖`s∞ ≤ c ‖x‖`sq(E) ,

where the last estimate follows from (2.2.4). This proves (2.2.5).(3) Note that E1 → E0 implies 2−sE1 → 2−sE0, using obvious notation,

and E1d

→ E0 gives 2−sE1d

→ 2−sE0. Hence (ii) follows from (2.1.6).(4) Claim (iii) is a direct consequence of (2.2.2), (2.2.3), and Theorem 2.1.1. ¥

8Here we set A := N.


2.3 Interpolation

For the reader’s convenience and for easy reference we collect some of the interpo-lation properties of sequence spaces.

Unweighted Spaces

We suppose that

(Eα,0, Eα,1) is for each α ∈ A an interpolation couple

so that Eα,j → Xα for j = 0, 1 and some locally convex space Xα. Then `rj(Ej)

with Ej := (Eα,j)α satisfies `rj(Ej) → c(X ) for j = 0, 1 and 1 ≤ rj ≤ ∞. Hence(

`r0(E0), `r1(E1))

is an interpolation couple as well.Throughout the rest of this chapter

• 0 < θ < 1 .

• sθ := (1 − θ)s0 + θs1 , s0, s1 ∈ R .

• 1/r(θ) := (1 − θ)/r0 + θ/r1 , 1 ≤ r0, r1 ≤ ∞ .

(2.3.1)

We use the notations (·, ·)θ,r, [·, ·]θ, and (·, ·)0θ,∞ for the real, the complex, andthe continuous interpolation functor, respectively, introduced in Subsection I.2.4.Then we set

Eα,θ,r := (Eα,0, Eα,1)θ,r , Eα,[θ] := [Eα,0, Eα,1]θ ,

andEθ,r := (Eα,θ,r) , E[θ] := (Eα,[θ])

for 1 ≤ r ≤ ∞.

2.3.1 Theorem

(i) Suppose 1 ≤ r0, r1 < ∞. Then(`r0(E0), `r1(E1)

)θ,r(θ)

.= `r(θ)(Eθ,r(θ))

and [`r0(E0), `r1(E1)

]θ

.= `r(θ)(E[θ]) .

(ii) If 1 ≤ r0 < r1 = ∞, then[`r0(E0), `∞(E1)

]θ

.=[`r0(E0), c0(E1)

]θ

.= `r(θ)(E[θ]) .

(iii)[c0(E0), c0(E1)

]θ

.= c0(E[θ]).

Proof See [Tri95, Theorem 1.18.1 and Remarks 2 and 3]. ¥


Weighted Spaces

For an easy way to derive interpolation theorems for weighted sequence spaces weprove a simple technical result.

2.3.2 Lemma Suppose (·, ·)θ is either one of the interpolation functors (·, ·)θ,r,1 ≤ r ≤ ∞, (·, ·)0θ,∞, or [·, ·]θ. Let a0, a1 > 0. Then

(a0E0, a1E1)θ = a1−θ0 aθ

1(E0, E1)θ .

Proof (1) For the K-functional (cf. Example I.2.4.1) we get

K(t, x, a0E0, a1E1) = inf

a−10 ‖x0‖E0 + ta−1

1 ‖x1‖E1 ; x = x0 + x1

= a−10 K(a0a

−11 t, x, E0, E1)

for t > 0. Hence

‖x‖(a0E0,a1E1)θ,r=

∥∥t−θK(t, x, a0E0, a1E1)∥∥

Lr((0,∞),dt/t)

= aθ−10 a−θ

1

∥∥t−θK(t, x, E0, E1)∥∥

Lr((0,∞),dt/t)

= aθ−10 a−θ

1 ‖x‖(E0,E1)θ,r= ‖x‖a1−θ

0 aθ1(E0,E1)θ,r

.

This proves the assertion if (·, ·)θ = (·, ·)θ,r. From this and the definition of thecontinuous interpolation functor it is clear that it applies for (·, ·)θ = (·, ·)0θ,∞ also.

(2) We use the notations of Example I.2.4.2. For f ∈ F(E0, E1) we set

fa0,a1(z) := a1−z0 az

1f(z) , z ∈ S .

Then fa0,a1 ∈ F(a0E0, a1E1), and A := (f 7→ fa0,a1) is an isometric isomorphismfrom F(E0, E1) onto F(a0E0, a1E1). Thus

‖x‖[a0E0,a1E1]θ = inff∈F(a0E0,a1E1)

‖f‖F(a0E0,a1E1) ; f(θ) = x

= infg∈F(E0,E1)

‖Ag‖F(a0E0,a1E1) ; Ag(θ) = x

= infg∈F(E0,E1)

‖g‖F(E0,E1) ; g(θ) = aθ−10 a−θ

1 x

= ‖aθ−10 a−θ

1 x‖[E0,E1]θ = ‖x‖a1−θ0 aθ

1[E0,E1]θ.

This gives the assertion for (·, ·)θ = [·, ·]θ. ¥

Let F = (Fα) be a sequence of Banach spaces (indexed by A) and a = (aα)and b = (bα) belong to (0,∞)A. Set aF := (aαFα) and ab := (aαbα).


2.3.3 Corollary Suppose aj = (aj,α) ∈ (0,∞)A for j = 0, 1.(i) If 1 ≤ r0, r1 < ∞, then

(`r0(a0E0), `r1(a1E1)

)θ,r(θ)

.= `r(θ)(a1−θ0 aθ

1Eθ,r(θ))

and [`r0(a0E0), `r1(a1E1)

]θ

.= `r(θ)(a1−θ0 aθ

1E[θ]) .

(ii) If 1 ≤ r0 < r1 = ∞, then[`r0(a0E0), `∞(a1E1)

]θ

.=[`r0(a0E0), c0(a1E1)

]θ

.= `r(θ)(a1−θ0 aθ

1E[θ]) .

(iii)[c0(a0E0), c0(a1E1)

]θ

.= c0(a1−θ0 aθ

1E[θ]).

Proof Theorem 2.3.1 and Lemma 2.3.2. ¥

Due to definition (2.2.3), most of the following interpolation theorem is simplya reformulation of this corollary.

2.3.4 Theorem Suppose s0, s1 ∈ R.(i) If 1 ≤ r0, r1 < ∞, then

(`s0r0

(E0), `s1r1

(E1))θ,r(θ)

.= `sθ

r(θ)(Eθ,r(θ)) .

(ii) For 1 ≤ r0, r1 < ∞,[`s0r0

(E0), `s1r1

(E1)]θ

.= `sθ

r(θ)(E[θ]) .

(iii) If 1 ≤ r0 < ∞, then[`s0r0

(E0), cs10 (E1)

]θ

.= `sθ

r(θ)(E[θ]) .

(iv)[cs00 (E0), cs1

0 (E1)]θ

.= csθ0 (E[θ]).

Proof (i), (iii), and (iv) are immediate consequences of this corollary, due to(2.2.3) and Lemma 2.3.2. For (ii) we refer to [Tri95, Theorem 1.18.1]. ¥

In the case of the real interpolation functor the assumption r0, r1 < ∞ can bedropped if we restrict ourselves to the case E0 = E1. In fact, the following strongerresult prevails.

2.3.5 Theorem Suppose s0, s1 ∈ R with s0 6= s1. If 1 ≤ r, r0, r1 ≤ ∞, then(`s0r0

(E), `s1r1

(E))θ,r

.= `sθr (E) .

Proof [BeL76, Theorem 5.6.1] or [Tri95, Theorem 1.18.2]. ¥


2.4 Notes

The results of this section are more or less common knowledge. They are includedfor easy reference.

The proof of Theorem 2.1.1 is an elaboration of the one by J.-L. Lions andJ. Peetre [LP64] who considered the case Eα = E for α ∈ N. The demonstrationof Lemma 2.3.2 uses arguments of [Tri95, Proof of Lemma 3.4.1].


3 Anisotropy

The study of parabolic evolution equations leads to differential operators in whichthe time and space derivatives possess different strength. This fact is most easilytaken care of by means of anisotropic function spaces. Such spaces are characterizedby the fact that weights of various order are assigned to the underlying variables.This situation can be conveniently handled by introducing appropriate actions ofthe multiplicative group

((0,∞), q ) on the underlying Euclidean spaces, which are

determined by the weights.In this section we introduce the basic framework for an efficient handling of

anisotropic function spaces. More precisely, in the first subsection we define theconcept of a weight system and study, in some detail, the action of the associateddilation on the basic elementary function spaces occurring in analysis.

For quantitative considerations the concept of a quasinorm is very useful.A quasinorm is, basically, a function which is positively homogeneous of degree 1with respect to the anisotropic dilation. Thus it is a substitute for the Euclideannorm which is, in general, not well-adapted to the anisotropic structure. Quasi-norms are introduced in the second subsection and basic examples are given.

In connection with resolvent estimates for differential operators, studied inlater chapters, it is very useful to consider parameter-dependent function spacesand operators. For this reason we introduce, in the third subsection, parameter-dependent quasinorms and present some technical estimates for later use.

The most important concept on which our approach to linear parabolic equa-tions is based, is that of a Fourier multiplier. Thus, in the next subsection, thegeneral definition of operator-valued Fourier multipliers is given and some of theirelementary properties are expounded. Subsection 3.5 is devoted to multiplier esti-mates for some important symbol classes.

The last subsection contains the definition of dyadic partitions of unity onEuclidean spaces, adapted to the anisotropic structures. Using this concept, weprove a rather technical Fourier multiplier lemma which is the fundament for thegeneral Fourier multiplier theorem for operator-valued symbols on Besov spaces,presented in Subsection VII.2.4 below.

3.1 Anisotropic Dilations

The notations and definitions introduced in this subsection will be applied through-out the whole book. The reader is advised to become familiar with the ratherconcise setting.

Weight Systems

A weight system for Rd is a triple [`, d, ν], where

• ` ∈qN , d = (d1, . . . , d`), ν = (ν1, . . . , ν`) ∈ (

qN)` , d1 + · · · + d` = d .

VI.3 Anisotropy 33

Then

Rd = Rd1 × · · · × Rd` (3.1.1)

is the d-clustering of Rd. We write

ξ = (ξ1, . . . , ξd) = (ξ1, . . . , ξ`) , ξi = (ξ1i , . . . , ξdi

i ) , 1 ≤ i ≤ ` ,

according to the interpretation of ξ as an element of Rd or of Rd1 × · · · × Rd` .We call [`, d, ν] reduced weight system if ` < d, and it is non-reduced otherwise.

If ` = d, then d = 1 = 1d = (1, . . . , 1). The weight system is ν-homogeneous if ` = 1.Then d = (d) and ν = (ν). In this case we write [1, d, ν] for [1, d, ν]. In general,

ν := LCM(ν) = LCM(ν1, . . . , ν`) ,

the least common multiple of ν1, . . . , ν`.With [`, d, ν] we associate its non-reduced version [d,1, ω], where

ω = ω(ν) = (ω1, . . . , ωd) := (ν1, . . . , ν1, ν2, . . . , ν2, . . . , ν`, . . . ν`) (3.1.2)

with di copies of νi. Thus the non-reduced version of [1, d, ν] equals [d,1, ν1]. NoteLCM(ω1, . . . , ωd) = ν and

|ω| := ω1 + · · · + ωd = d q ν := d1ν1 + · · · + d`ν` .

3.1.1 Example (Parabolic weight systems) We assume m, ν ∈qN with ν ≥ 2 and

put d := m + 1. Then[2, (m, 1), (1, ν)

]is a parabolic weight system on Rd, which

we henceforth denote by [m, ν]. Observe that we exhibit the ‘spatial dimension’ mand not d. The non-reduced version of [m, ν] is

[m + 1, 1m+1, (1m, ν)

].

Parabolic weight systems play an important role in connection with parabolicdifferential operators studied in detail in later chapters. This explains the name‘spatial dimension’. Correspondingly, the last coordinate variable is often called‘time variable’. ¥

Dilations

Let [`, d, ν] be a weight system for Rd. The (anisotropic) dilation associated with it

is defined by

t q ξ = t qν

ξ := (tν1ξ1, . . . , tν`ξ`) , t ≥ 0 , ξ ∈ Rd . (3.1.3)

Note that

s q (t q ξ) = (st) q ξ , s, t ≥ 0 , ξ ∈ Rd . (3.1.4)


Thus (0,∞) × (Rd)q → (Rd)

q, (t, ξ) 7→ t q ξ is a continuous action of the multi-

plicative group((0,∞), q ) on (Rd)

q. If [d,1, ω] is the non-reduced weight system

associated with [`, d, ν], then

t qν

ξ = t qω

ξ , t ≥ 0 , ξ ∈ Rd , (3.1.5)

where, on the left side, ξ is d-clustered, of course.

For u : (Rd)q → E we set

(σtu)(ξ) = (σνt u)(ξ) := u(t q ξ) , t > 0 , ξ ∈ (Rd)

q. (3.1.6)

Note that (3.1.4) implies

σs(σtu) = σstu , s, t > 0 , u : (Rd)q → E . (3.1.7)

We use the universal notation σt = σνt in which we do not indicate the image space E.

It will always be clear from the context which E is being considered.

Let G be the multiplicative group9((0,∞), q ) and X an LCS. A linear rep-

resentation of G on X is a map G → L(X ), t 7→ Tt satisfying TsTt = Tst andT1 = 1X . Note that Tt ; t ∈ G is a commutative subgroup of the automorphismgroup Laut(X ), and

(Tt)−1 = Tt−1 = T1/t , t ∈ G .

The representation is continuous, resp. strongly continuous, if t 7→ Tt is continuousfrom G into L(X ), resp. into Ls(X ). This is the case iff the respective continuityholds at t = 1.

3.1.2 Lemma t 7→ σt is a continuous linear representation of G on D(Rd, E) whichis strongly continuous on L1,loc(Rd, E).

Proof We simply write D for D(Rd, E) etc.

(1) Recall that

D = lim−−→KbRd

DK , where DK :=

u ∈ C∞ ; supp(u) ⊂ K

,

endowed with the topology induced by the seminorms

u 7→ pm,K(u) := max|α|≤m

supK

|∂αu|E , m ∈ N ,

(cf. Subsection 1.1 of the Appendix).

9It can be replaced by any multiplicatively written continuous commutative group.

VI.3 Anisotropy 35

(2) The chain rule implies10, for k ∈ N,

∂α(σtu) = tαppppωσt(∂αu) , u ∈ Ck , α q ω ≤ k . (3.1.8)

Suppose u ∈ Dt ppppK . Then we get from (3.1.8)

pm,K(σtu) = max|α|≤m

supK

|∂α(σtu)|E= max

|α|≤m

(tα

ppppω supK

|σt(∂αu)|E) ≤ c(t,m)pm,t ppppK(u) .

(3.1.9)

Hence(u 7→ σtu) ∈ L(Dt ppppK ,DK) .

Thus, by DK → D, it follows (u 7→ σtu) ∈ L(Dt ppppK ,D) for K b Rd. Consequently,(u 7→ σtu) ∈ L(D), by the properties of LF spaces. This and (3.1.7) show thatt 7→ σt is a linear representation of G on D.

(3) Note that|t q ξ − ξ| ≤ max

1≤i≤`|tνi − 1| |ξ| .

Hence, given11 u ∈ C1,

u(t q ξ) − u(ξ) =∫ 1

0

∂u(ξ + τ(t q ξ − ξ)

)dτ(t q ξ − ξ)

implies, for ξ ∈ M ⊂ Rd,

|σtu − u|E (ξ) ≤ max1≤i≤`

|tνi − 1| |ξ| supMt

|∂u|L(Rd,E) , (3.1.10)

whereMt :=

(1− τ)ξ + τ(t q ξ) ; ξ ∈ M, 0 ≤ τ ≤ 1

.

By (3.1.8),

∂α(σtu− u) = tαppppω(

σt(∂αu) − ∂αu)

+ (tα ppppω − 1)∂αu .

From this and (3.1.10) we deduce, for ξ ∈ M and |α| ≤ m,

|∂α(σtu− u)(ξ)|E ≤ tαppppω max

1≤i≤`|tνi − 1| |ξ| max

|β|≤m+1supMt

|∂βu|E

+ |tα ppppω − 1| supM

|∂αu| .(3.1.11)

(4) Now suppose M b K b Rd and u ∈ DM . Then u ∈ DK and there existsε ∈ (0, 1) such that |t− 1| ≤ ε implies Mt ⊂ K. Moreover, 0 < t < 2. Let m ∈ N.

10We use standard multi-index notation. In particular, |α| = α1 + · · · + αd andα qω = α1ω1 + · · · + αdωd for α ∈ Nd. (This can cause no confusion since (3.1.3) is not definedon Nd.) Moreover, somewhat inconsistently, the coordinates of multi-indices carry lower indices.

11∂ is the Frechet derivative on Rd.


We infer from (3.1.11) that

pm,K(σtu − u) ≤ (c(m,K) max

1≤i≤`|tνi − 1| + |tα ppppω − 1|)pm+1,K(u)

for u ∈ DK and |t− 1| ≤ ε. This implies σtu → u in D for t → 1, uniformly withrespect to u in bounded sets. Hence σt → 1D in L(D) as t → 1.

(5) Let u ∈ L1,loc. By using the transformation rule

d(t q ξ) = t|ω| dξ (3.1.12)

for the Lebesgue measure dξ on Rd, a change of variables gives∫

t ppppK|σtu(ξ)|E dξ =

∫

t ppppK|u(t pppp ξ)|E dξ = t−|ω|

∫

K

|u(η)|E dη , K b Rd .

This implies σt ∈ L(L1,loc), uniformly for t in compact subsets of G.

(6) Recall that D d→ L1,loc. Hence it follows from step (4) that

(t 7→ σt) ∈ C(G,L(D, L1,loc)

). (3.1.13)

Suppose K b Rd, ε > 0, and u ∈ L1,loc. By (3.1.13), the density of D in L1,loc,and (5) we can find v ∈ D such that

∫

K

|u − v|E dξ +∫

K

|σt(u− v)|E dξ ≤ ε/2, , |t− 1| ≤ 1/2 .

Hence we infer from (3.1.13) that there exists δ ∈ (0, 1/2) such that∫

K

|σtu − u|E dξ ≤∫

K

|σt(u − v)|E dξ +∫

K

|u − v|E dξ +∫

K

|σtv − v|E dξ ≤ ε

for |t− 1| ≤ δ. This proves that t 7→ σt is strongly continuous on L1,loc. ¥

Since σt ∈ L(D(Rd)), Remark 1.2.1(a) implies

(σt)∗ ∈ L(D′(Rd, E))

, t > 0 , (3.1.14)

where⟨(σt)∗u, ϕ

⟩Rd = 〈u, σtϕ〉Rd , u ∈ D′(Rd, E) , ϕ ∈ D(Rd) . (3.1.15)

Clearly,

(σs)∗(σt)∗ = (σst)∗ , s, t > 0 . (3.1.16)

Let u ∈ L1,loc(Rd, E) be a regular distribution. Then σtu ∈ L1,loc(Rd, E) by Lem-ma 3.1.2. By (3.1.12) and a substitution of variables,

〈σtu, ϕ〉Rd =∫

Rd

⟨u(t q ξ), ϕ(ξ)

⟩E

dξ = t−|ω|〈u, σ1/tϕ〉Rd (3.1.17)

for ϕ ∈ D(Rd).

VI.3 Anisotropy 37

Now we define an action of G on D′(Rd, E), associated with [`, d, ν], by setting

σtu := t−|ω|(σ1/t)∗u , u ∈ D′(Rd, E) , t > 0 ,

that is,

〈σtu, ϕ〉Rd = t−|ω|〈u, σ1/tϕ〉Rd , ϕ ∈ D(Rd) . (3.1.18)

It follows from (3.1.14) and (3.1.16) that t 7→ σt is a linear representation of Gon D′(Rd, E). Furthermore, (3.1.17) and (3.1.18) show that it is an extensionover D′(Rd, E) of the point-wise defined representation on L1,loc(Rd, E).

3.1.3 Proposition The map t 7→ σt, associated with [`, d, ν], is a strongly contin-uous linear representation of G on D′(Rd, E). If

X ∈ D, S, C, C0, BC, BUC, L1,loc, Lq with 1 ≤ q ≤ ∞, S ′ ,

then X (Rd, E) is an invariant linear subspace of D′(Rd, E) and t 7→ σt is contin-uous on X (Rd, E) if X ∈ D,S and strongly continuous otherwise, unless X be-longs to BC, BUC, L∞. Moreover12,(i) ∂α σt = tα

ppppωσt ∂α on D′(Rd, E).(ii) ‖σtu‖q = t−|ω|/q ‖u‖q, u ∈ Lq(Rd, E), 1 ≤ q ≤ ∞.(iii) F σt = t−|ω|σ1/t F and F−1 σt = t−|ω|σ1/t F−1 on S ′(Rd, E).

Proof (1) As for the first assertion, it remains to show that t 7→ σt is stronglycontinuous on D′(Rd, E). By step (4) of the preceding proof we know that σtϕ → ϕin D(Rd) as t → 1, uniformly for ϕ in bounded sets. This and (3.1.18) showthat, given u ∈ D′(Rd, E), limt→1〈σtu, ϕ〉E = 〈u, ϕ〉 uniformly with respect to ϕ inbounded sets of D(Rd). Hence limt→1 σtu = u in D′(Rd, E), that is, with respectto the topology of uniform convergence on bounded subsets of D(Rd). This provesthis claim.

(2) Suppose u ∈ S(Rd, E). As in the proof of (3.1.9) we find

qk,m(σtu) = max|α|≤m

supRd

〈·〉k |∂α(σtu)|E

= max|α|≤m

supξ∈Rd

tαppppω 〈ξ〉k〈t q ξ〉k 〈t q ξ〉k |∂αu(t q ξ)|E ≤ c(t, k, m)qk,m(u)

for k,m ∈ N. Hence σt ∈ L(S(Rd, E)). From (3.1.11) we derive

qk,m(σtu − u) ≤ c(t, k, m)(max1≤i≤`

|tνi − 1| + |tα ppppω − 1|)qk+1,m+1(u) ,

where t 7→ c(t, k, m) is continuous on G. This shows that σtu → u in S(Rd, E)as t → 1, uniformly for u in bounded sets, that is, (t 7→ σt) ∈ C

(G,L(S(Rd, E))

).

This proves the assertion for X = S.12F := (u 7→ u) is the Fourier transformation on S′(Rd, E) (cf. Section III.4.2).


(3) Let u ∈ S ′(Rd, E). It follows from S ′ → D′ that (3.1.18) applies for u inS ′(Rd, E) and ϕ in S(Rd). Thus, due to (2), the arguments which we applied toX = D′ yield the claim for X = S ′ also.

(4) Let X := C. For u ∈ X and K b Rd we set pK(u) := p0,K(u). Then

pK(σtu) = pt ppppK(u) , t > 0 .

Hence σt ∈ L(X ). Since u is uniformly continuous on compact sets, it follows thatσtu → u in X as t → 1.

(5) Let X ∈ BC,BUC, L∞. Then, clearly, σtu ∈ X for u ∈ X . Moreover,

‖σtu‖∞ = ‖u‖∞ , u ∈ X . (3.1.19)

This implies σt ∈ L(X ) and also proves (ii) if q = ∞.

(6) Since D d→ C0 → BC, we infer from Lemma 3.1.2 and (3.1.19) that σt be-

longs to L(C0). Since t 7→ σt is strongly continuous on D, we get from

σtu − u = σt(u− v) + (σtv − v) + (v − u) , v ∈ D ,

and (3.1.19) that t 7→ σt is strongly continuous on C0 (cf. step (6) of the precedingproof).

(7) Suppose 1 ≤ q < ∞. Then (ii) follows by a substitution of variables. Hence

σt ∈ L(Lq) and ‖σt‖Lq is bounded on compact subsets of G. From this and D d→ Lq

we obtain the strong continuity of t 7→ σt on Lq by an approximation argument asin (6).

(8) By step (2) of the proof of Lemma 3.1.2 we know that (i) holds on D.Now we get its validity on D′ by the definition of the distributional derivatives.

(9) Suppose u ∈ S(Rd). Then, using once more a substitution of variables,

F(σtu)(ξ) =∫

Rd

e−i 〈ξ,x〉u(t q x) dx = t−|ω|σ1/t(Fu)(ξ)

for ξ ∈ Rd. This proves that the first part of (iii) holds on S(Rd). Now we get itsvalidity on S ′(Rd, E) from the definitions of F and σt for tempered distributions.The second part of (iii) is then obtained by applying F−1 from the left and fromthe right.

Since the cases X ∈ D, L1,loc are covered by Lemma 3.1.2, the propositionis proved. ¥

VI.3 Anisotropy 39

Throughout the rest of this volume

• [`, d, ν] is a weight system for Rd

and [d,1, ω] is its non-reduced version.• ν = LCM(ν) = ω.

• t 7→ σt is the linear representation of((0,∞), q )

on D′(Rd, E) induced by the anisotropic dilation (3.1.3).

(3.1.20)

3.2 Quasinorms

Let X be a nonempty set. For f, g : X → R+ we write

f ∼ g ⇐⇒ g/c ≤ f ≤ cg (3.2.1)

for some constant c ≥ 1. Then ∼ is an equivalence relation on (R+)X .

Let z ∈ C. A distribution u ∈ D′(Rd, E) is positively z-homogeneous (positivelyhomogeneous of degree z) with respect to [`, d, ν] if

σtu = tzu , t > 0 .

If u ∈ C(Rd, E) is positively r-homogeneous for some r > 0, then u(0) = 0.

A function Q is a ν-quasinorm on Rd if

• Q ∈ C(Rd,R+) ∩ C∞((Rd)

q)with Q(ξ) > 0 for ξ 6= 0.

• Q is even and positively 1-homogeneous with respect to [`, d, ν].• Q satisfies the triangle inequality

Q(ξ + ξ′) ≤ Q(ξ) + Q(ξ′) , ξ, ξ′ ∈ Rd .

The following examples and observations will be readopted at various later occa-sions.

3.2.1 Examples (a) Set

N(ξ) := Nν(ξ) :=(∑

i=1

|ξi|2ν/νi

)1/2ν

, ξ ∈ Rd .

Then N is a ν-quasinorm, the natural ν-quasinorm.


Proof We set d(0) := 0 and d(i) := d1 + · · · + di for 1 ≤ i ≤ `. Then

∂kN(ξ) =|ξi|2(ν/νi−1)

νiN2ν−1(ξ)ξk , d(i − 1) < k ≤ d(i) , 1 ≤ i ≤ ` . (3.2.2)

From this and ν/νi ∈qN we obtain N ∈ C∞(

(Rd)q)

by induction.

By means of the elementary inequality,

(x + y)1/k ≤ x1/k + y1/k , x, y ≥ 0 , k ∈ N ,

we get from Minkowski’s inequality

N(ξ + η) =(∑

i=1

|ξi + ηi|2ν/νi

)1/2ν

≤(∑

i=1

(|ξi|1/νi + |ηi|1/νi)2ν

)1/2ν

≤ N(ξ) + N(η)

for ξ, η ∈ Rd. The remaining assertions are clear. ¥

(b) We denote by E(ξ), for ξ 6= 0, the unique t > 0 satisfying |t−1 q ξ| = 1 and setE(0) := 0. Then E is a ν-quasinorm on Rd, the Euclidean ν-quasinorm.

Proof (1) It is obvious that E is well-defined and even. From

1 =∣∣(E(t q ξ))−1 q (t q ξ)∣∣ =

∣∣(E(t q ξ))−1t q ξ∣∣ = |E(ξ)−1 q ξ|

we get E(ξ) = E(t q ξ)/t for t > 0 and ξ 6= 0. Hence E is positively 1-homogeneouson (Rd)

q.

(2) Suppose |ξ − η| ≤ 1 and set ν := maxν1, . . . , ν`. Then E(ξ − η) ≤ 1 and,consequently,

1 =∑

i=1

|ξi − ηi|2E(ξ − η)2νi

≤ |ξ − η|2E2ν(ξ − η)

.

Hence E(ξ − η) ≤ |ξ − η|1/ν for |ξ − η| ≤ 1. Thus the triangle inequality impliesE ∈ C(Rd).

(3) Now we use (3.1.5). Then we get, for k ∈ 1, . . . , d and h ∈qR,

0 = |E(ξ + hek) q (ξ + hek)|2 − |E(ξ) q ξ|2 =(a(ξ, h)

∣∣b(ξ, h))Rd , (3.2.3)

where

a(ξ, h) := E(ξ + hek) q (ξ + hek) + E(ξ) q ξ (3.2.4)

VI.3 Anisotropy 41

and

b(ξ, h) := E(ξ + hek) q (ξ + hek) − E(ξ) q ξ= E(ξ + hek) q ξ − E(ξ) q ξ + E(ξ + hek) q ekh .

(3.2.5)

Note that

E(ξ + hek) q ξ − E(ξ) q ξ =((

E(ξ + hek)−ωj − E(ξ)−ωj)ξj

)1≤j≤d

andE(ξ + hek)−ωj − E(ξ)−ωj = cj(ξ, h)

(E(ξ + hek) − E(ξ)

),

where

cj(ξ, h) :=−1

E(ξ + hek)E(ξ)

ωj−1∑

i=0

E(ξ + hek)−iE(ξ)−ωj+1+i .

From this, (3.2.4), (3.2.5), and step (2) we deduce

(a(ξ, h)

∣∣E(ξ + hek) q ξ − E(ξ) q ξ)Rd

(E(ξ + hek) − E(ξ)

)−1

→ −2E(ξ)−1d∑

j=1

ωj

(ξjE(ξ)ωj

)2

as h → 0. Similarly,(a(ξ, h)

∣∣E(ξ + hek) q ek

)Rd → 2E−2ωk(ξ)ξk .

By combining this with (3.2.3) and (3.2.5), we see that

E(ξ + hek) − E(ξ)h

→ ξkE(ξ)1−2ωk

( d∑

j=1

ωj

(ξj

/E(ξ)ωj

)2)−1

(3.2.6)

as h → 0. This shows that E is differentiable for ξ 6= 0, and that ∂kE(ξ) is givenby the right side of (3.2.6). From this result and step (2) we deduce, by induction,that E belongs to C∞(

(Rd)q).

(4) To prove the triangle inequality, we set

a(ξ, η) :=∑

i=1

|ξi + ηi|2(E(ξ) + E(η)

)2νi, ξ, η ∈ (Rd)

q.

Then the claim follows, provided we show

a(ξ, η) ≤ 1 =∑

i=1

|ξj + ηi|2E(ξ + η)2νi

.


Since a(t q ξ, t q η) = a(ξ, η), we can assume E(ξ) + E(η) = 1. Thus 0 < E(ξ) ≤ 1implies

1 =∑

i=1

|ξi|2E(ξ)2νi

≥ |ξ|2E(ξ)2

.

Hence |ξ| ≤ E(ξ). Similarly, |η| ≤ E(η). Thus

a(ξ, η) = |ξ + η|2 ≤ (|ξ| + |η|)2 ≤ (E(ξ) + E(η)

)2 = 1

for ξ, η ∈ (Rd). ¥

3.2.2 Remarks (a) Let Q be a ν-quasinorm on Rd and set

dQ(ξ, ξ′) := Q(ξ − ξ′) , ξ, ξ′ ∈ Rd .

Then dQ is a translation-invariant metric on Rd.

(b) If [`, d, ν] is the trivial weight system [d,1,1], then N is the Euclidean normon Rd.

(c) Suppose [m, ν] is a parabolic weight system on Rm. Write (ξ, τ) for the generalpoint of Rm × R and set

P(ξ, τ) :=(|ξ|2ν + |τ |2)1/2ν

, (ξ, τ) ∈ Rm × R .

Then P is the natural (1, ν)-quasinorm on Rm+1, the ν-parabolic quasinorm. More-over,

δP

((ξ, τ), (ξ′, τ ′)

):=

(|ξ − ξ′|2 + |τ − τ ′|2/ν)1/2

defines the ν-parabolic metric on Rm+1. It is equivalent to dP. ¥

3.3 Parametric Augmentations

For technical reasons it is often convenient to consider parameter-dependent set-tings. To introduce them, we recall that a cone in a vector space X is a nonemptysubset C of X satisfying R+C ⊂ C. Thus 0 ⊂ C, and C is trivial if it equals 0.

In analogy to (3.1.3), we define on C` an anisotropic dilation by

t qν

z := (tν1z1, . . . , tν`z`) , t ≥ 0 , z ∈ C` .

Then we fix for each i = 1, . . . , ` a closed cone Hi in C containing the positivehalf-line R+ + i0, if it is nontrivial. We set H := H1 × · · · × H` and Z := Rd × H.The general point of Z is denoted by ζ or (ξ, η) with η = (η1, . . . , η`) ∈ H. Notethat Z is a closed cone in Rd × C` which is invariant under the action

t qν

ζ := (t qν

ξ, t qν

η) , t ≥ 0 , ζ ∈ Z . (3.3.1)

We embed R+ naturally in C` by identifying t ∈ R+ with (t + i0, 0, . . . , 0) ∈ C`.Then R+ is the positive half-line in C`.

VI.3 Anisotropy 43

Throughout the rest of this section we assume

• H = H1 × · · · × H` is a closed cone in C`

containing the positive half-line.• Z = Rd × H.

(3.3.2)

To simplify the notation, we put

t q ζ := t qν

ζ , σt := σνt , t ≥ 0 , ζ ∈ Z ,

so that σt is now defined on functions whose domain is Z. We say that Z is aparametric augmentation of Rd, associated with the weight system [`, d, ν]. It is trivial

if H = R+, that is, H1 = R+ and H2 = · · · = H` = 0.

Augmented Quasinorms

For a : Z → E we put

aη : Rd → E , ξ 7→ a(ξ, η) , η ∈ H .

We also set

[η] = [η]ν :=(∑

i=1

|ηi|2ν/νi

)1/2ν

.

Identifying C` with R2` by identifying z = x + iy with (x, y) ∈ R2, we see that[·]ν is the natural ν-quasinorm on R2` with respect to the reduced weight system[`, 2 · 1`, ν].

By a ν-quasinorm on Z we mean the restriction to Z of a ν-quasinorm on13

Rd × C` = Rd × R2` with respect to the reduced weight system[d + `, (d, 2 · 1`), (ν, ν)

]

for Rd × R2d. Thus, if Q is a ν-quasinorm on Z, then

Q ∈ C(Z) , Qη ∈ C∞(Rd) , η ∈q

H .

The natural ν-quasinorm on Z is defined by

Λ(ζ) :=(N2ν(ξ) + [η]2ν

)1/2ν, ζ = (ξ, η) ∈ Z . (3.3.3)

3.3.1 Remarks (a) Observe that the conventions for H mean that we can consideras many parameters as there are factors in the d-clustering (3.1.1) of Rd. This

13a = b means: a is identified with b.


choice of H and the definition of [η]ν are made for notational simplicity, since theyallow to stick throughout to the weight vector ν. It is obvious that C` could be re-placed by Cn for any n ∈

qN, and [η]ν by [η]µ for any weight vector µ = (µ1, . . . , µn)

satisfying LCM(µ) = ν. This latter restriction can also be dropped if we work withthe natural (ν, µ)-quasinorm, defined by

Λ(ν,µ)(ζ) :=(N2ρ

ν (ξ) + [η]2ρµ

)1/2ρ,

where µ := LCM(µ) and ρ := LCM(ν, µ). Such generalizations are left to thereader.

(b) For most applications either the trivial parametric augmentation Z = Rd × R+

or the complex parametric augmentation Z = Rd × C suffices, where C is naturallyidentified with C× 0 × · · · × 0 ⊂ C`. Thus, in these cases,

Λ(ζ) =(N2ν(ξ) + |η|2ν/ν1

)1/2ν. (3.3.4)

Clearly, instead of using the natural identifications we could use, for any i belongingto 1, . . . , `, the embedding η 7→ (0, . . . , 0, η, 0, . . . , 0), where η is at position i. Inother words, everything remains valid if ν1 in (3.3.4) is replaced by νi for anyi ∈ 1, . . . , `. This means that the parameter η ∈ R+, resp. η ∈ C, can be givenany of the weights ν1, . . . , ν`. This fact will sometimes be used to simplify thepresentation. ¥

Let Q be a ν-quasinorm on Z. Then

SQ := [Q = 1] =ζ ∈ Z ; Q(ζ) = 1

is the Q-quasisphere in Z, and

rQ :qZ → SQ , ζ 7→ Q(ζ)−1 q ζ (3.3.5)

is the Q-retraction. Since, by the 1-homogeneity of Q,

Q(rQ(ζ)

)= Q(ζ)−1Q(ζ) = 1 , ζ ∈

qZ ,

it follows that rQ is indeed a continuous (that is, topological) retraction fromqZ

onto SQ.

3.3.2 Lemma

(i) Suppose M ∈ C( qZ, (0,∞)

)is positively 1-homogeneous. Then [M = 1] is com-

pact and M ∼ Λ.

VI.3 Anisotropy 45

(ii) Assume z ∈ C and a ∈ C(qZ, E) is positively z-homogeneous. If α ∈ Nd and

∂αξ a ∈ C(

qZ, E), then ∂α

ξ a is positively (z − α q ω)-homogeneous and, givenany ν-quasinorm Q,

∂αξ a(ζ) = Qz−α ppppω(ζ)(∂α

ξ a) rQ(ζ) (3.3.6)

and

|∂αξ a(ζ)|E ≤ QRez−α ppppω(ζ) ‖(∂α

ξ a) rQ‖∞ < ∞ , ζ ∈qZ . (3.3.7)

Proof (1) It is obvious that SΛ is bounded and closed in Z, hence in Rd × C`,since H is closed in C`. Thus SΛ is compact in Rd × C`, therefore in Z. Now itfollows from the continuity of M that 1/c ≤ M(ζ) ≤ c for ζ ∈ SΛ. Consequently,

M(ζ) = M(Λ(ζ) q rΛ(ζ)

)= Λ(ζ)M

(rΛ(ζ)

), ζ ∈

qZ ,

implies M ∼ Λ. Thus Λ ≤ cM which gives [M = 1] ⊂ [Λ ≤ c]. Hence [M = 1] isbounded. Now, similarly as above, we see that it is compact. This proves (i).

(2) We differentiate σta = tza with respect to ξ and use Proposition 3.1.3(i)to obtain

∂αξ (σta) = tα

ppppωσt(∂αξ a) = tz∂α

ξ a .

This shows that ∂αξ a is positively (z − α q ω)-homogeneous. By replacing in the

above relation t by 1/t we find

∂αξ a(ζ) = tz−α ppppωσ1/t(∂α

ξ a)(ζ) , ζ ∈qZ .

We substitute Q(ζ) for t and note that

σ1/Q(ζ)(∂αξ a)(ζ) = ∂α

ξ a(rQ(ζ)

), ζ ∈

qZ .

Then we arrive at (3.3.6).The last estimate is obvious by the compactness of SQ and the continuity

of ∂αξ a. ¥

3.3.3 Remark Let Q be a ν-quasinorm on Rd. We define the Q-quasisphere in Rd

and the Q-retraction by replacing Z by Rd in the above definitions. Then the as-sertions of the lemma apply in this situation as well, where now Λ is to be replacedby N. In particular, it follows that all ν-quasinorms on Rd are equivalent. ¥

Positive Homogeneity

For z ∈ C and k ∈ N we denote by Hkz (Z, E) the vector space of all positively

z-homogeneous a ∈ C(qZ, E) satisfying ∂α

ξ a ∈ C(qZ, E) for α ∈ Nd with α q ω ≤ k.


We endow it with the norm ‖·‖Hkz, defined by

‖a‖Hkz

:= maxα ppppω≤k

‖(∂αξ a) rΛ‖∞ . (3.3.8)

3.3.4 Lemma Hkz (Z, E) is a Banach space. If Q is a ν-quasinorm and ‖·‖Q

Hkz

is

defined by replacing Λ in (3.3.8) by Q, then ‖·‖QHk

z∼ ‖·‖Hk

z.

Proof (1) It is obvious that ‖·‖Hkz

is a norm. Let (aj) be a Cauchy sequence.

It follows from (3.3.6) that (∂αξ aj) is a Cauchy sequence in C(

qZ, E) with respect

to the topology of uniform convergence on compact subsets ofqZ. By the com-

pleteness of C(qZ, E) there exist aα ∈ C(

qZ, E) with ∂α

ξ aj → aα in C(qZ, E). Since

C(Rd, E) → D′(Rd, E) and ∂α belongs to L(D′(Rd, E)), we obtain ∂αaj,η → ∂αa0

η

in D′(Rd, E) for η ∈q

H. Consequently, ∂αξ a0 = aα ∈ C(

qZ, E) and ∂α

ξ aj → ∂αξ a0 in

C(qZ, E) for α q ω ≤ k. In particular, (∂α

ξ aj) converges, uniformly on the compact

set SΛ, towards ∂αξ a0, and a0 is z-homogeneous. Since rΛ maps

qZ onto SΛ, it follows

from (3.3.8) that Hkz (Z, E) is complete.

(2) Note rQ(ζ) = Q(ζ)−1 q ζ = (Λ/Q)(ζ) q rΛ(ζ) for ζ ∈qZ. Thus we deduce

from the (z − α q ω)-homogeneity of ∂αξ a that

(∂αξ a) rQ = (Λ/Q)z−α ppppω(∂α

ξ a) rΛ , a ∈ Hkz (Z, E) , α q ω ≤ k .

Observe that Λ/Q is positively 0-homogeneous onqZ and continuous on the compact

set SΛ. Hence there exists c = c(Rez) ≥ 1 such that

1/c ≤ |Λ/Q|Rez−α ppppω ≤ c , α q ω ≤ k .

This implies ‖·‖QHk

z∼ ‖·‖Hk

z. ¥

3.3.5 Remark Due to Remark 3.3.3, this lemma and its proof remain true ifH = 0, that is, Z = Rd. ¥

Clearly, Hmz (Z, E) → Hk

z (Z, E) for m > k. Hence we can set

H∞z (Z, E) :=

⋂

k

Hkz (Z, E) ,

endowed with the obvious projective limit topology which makes it a Frechet space.

3.3.6 Lemma Let k ∈ N ∪ ∞ and z1, z2 ∈ C. If

E1 × E2 → E3 , (e1, e2) 7→ e1e2

VI.3 Anisotropy 47

is a multiplication14, then its point-wise extension is a multiplication

Hkz1

(Z, E1) ×Hkz2

(Z, E2) → Hkz1+z2

(Z, E3) .

Proof Note

σt(ab) = σt(a)σt(b) , (a, b) ∈ C(qZ, E1)× C(

qZ, E2) .

Using this, we deduce the assertion from Leibniz’ rule. ¥

Differentiating Inverses

Next we consider elements of Hkz

(Z,Lis(E1, E0)

). For this we first establish a

semi-explicit formula for derivatives of inverses which will be useful repeatedly.

3.3.7 Lemma Suppose X is open in Rd, k ∈qN, and a ∈ Ck

(X,Lis(E1, E0)

). Set

a−1(x) :=(a(x)

)−1 for x ∈ X. Then

∂αa−1 =|α|∑

j=1

∑

β∈Bj,α

εβa−1(∂β1a)a−1 · · · · · (∂βj a)a−1 (3.3.9)

for 0 < α q ω ≤ k, where

Bj,α :=

β := β1, . . . , βj ; βi ∈ Nd, |βi| > 0,∑j

i=1 βi = α

and εβ ∈ Z. In particular, ∂αa−1 ∈ C(X,L(E0, E1)

)for α q ω ≤ k.

Proof Recall that Lis(E1, E0) is open in L(E1, E0) and

inv : Lis(E1, E0) → Lis(E0, E1) , b 7→ b−1 (3.3.10)

is smooth with(∂ inv(b)

)c = −b−1c b−1 , b ∈ Lis(E1, E0) , c ∈ L(E1, E0) , (3.3.11)

(e.g., [AmE06, Proposition VII.7.2]). From this we infer a−1 ∈ C(X,Lis(E1, E0)

)and, by the chain rule,

∂ja−1 = −a−1(∂ja)a−1 , 1 ≤ j ≤ d .

Now (3.3.9) follows by induction. The last assertion is then read off (3.3.9). ¥

14A multiplication X1 ×X2 → X3 of LCSs is a continuous bilinear map. Deviating from theusage in Volume I, we do not require that it has norm at most one if the Xj are normed spaces.However, the present definition is not more general –– but more practical –– than the one ofVolume I (cf. Example 1.3.1(g) in the Appendix). Furthermore, it is now convenient to writee1e2 for e1

q e2.


As a first application of this formula we derive an estimate for the inverse ofa positively homogeneous operator-valued map.

3.3.8 Lemma Suppose z ∈ C and k ∈ N. If a ∈ Hkz

(Z,Lis(E1, E0)

), then a−1 be-

longs to Hk−z

(Z,Lis(E0, E1)

)and

‖a−1‖Hk−z

≤ c(‖a‖Hk

z, ‖a−1 rΛ‖∞

) ‖a−1 rΛ‖∞ .

Proof (1) Since a ∈ C( qZ,Lis(E1, E0)

), (3.3.10) implies a−1 ∈ C

( qZ,Lis(E0, E1)

).

It is a consequence of a(ζ)a−1(ζ) = 1E0 that

1E0 = σt(aa−1) = σt(a)σt(a−1) = tzaσt(a−1)

and, similarly, 1E1 = σt(a−1a) = tzσt(a−1)a. Hence

σt(a−1) = t−za−1 , t > 0 . (3.3.12)

Thus a−1 is positively (−z)-homogeneous. Moreover, it follows from Lemma 3.3.7that ∂α

ξ a−1 ∈ C( qZ,L(E0, E1)

)for α q ω ≤ k.

(2) If b ∈ C( qZ,L(E1, E2)

)and c ∈ C

( qZ,L(E0, E1)

), then

(bc) rΛ = (b rΛ)(c rΛ) . (3.3.13)

Suppose 0 < α q ω ≤ k, 1 ≤ j ≤ |α|, and β1, . . . , βj ∈ Bj,α. Then we get from(3.3.13) ∥∥(

a−1(∂β1ξ a)a−1 · · · · · (∂βj

ξ a)a−1) rΛ

∥∥∞

≤ ‖a−1 rΛ‖j+1∞

j∏

i=1

‖(∂βi

ξ a) rΛ‖∞ .

Now the assertion follows from (3.3.9). ¥

3.3.9 Example Suppose z ∈ C. Then Λz ∈ H∞z (Z) and

‖Λz‖Hkz≤ c (1 + |z|)k ‖Λ‖Hk

1, k ∈ N . (3.3.14)

Proof From Example 3.2.1(a) we infer that ∂αξ Λ ∈ C(

qZ) for α ∈ N. Hence Λ be-

longs to H∞1 (Z). We get by induction from

∂jΛz = zΛz−1∂jΛ , 1 ≤ j ≤ d , (3.3.15)

that ∂αξ Λz ∈ C(

qZ) for α ∈ Nd. Thus Λz ∈ H∞

z (Z) since Λz is z-homogeneous. From(3.3.15) we also derive the second part of the claim. ¥

VI.3 Anisotropy 49

Slowly Increasing Functions

Recall that OM (Rd, E) is the LCS of smooth slowly increasing E-valued functionson Rd (see Subsection III.4.1 or Subsection 1.1 of the Appendix) and that

S(Rd, E) → OM (Rd, E) → S ′(Rd, E) . (3.3.16)

3.3.10 Lemma Let z ∈ C. Then

(a 7→ aη) ∈ L(H∞z (Z, E),OM (Rd, E)

), η 6= 0 ,

uniformly with respect to z in bounded sets.

Proof Using |ξi| ≤ |ξ| ≤ max1, |ξ|, we find

[η] ≤ Λ(ζ) ≤ c([η]

)(1 + |ξ|2ν)1/2ν , ζ = (ξ, η) ∈ Z . (3.3.17)

Define Qi : Z → R by

Q1(ζ) :=([η]2ν + |ξ|2ν

)1/2ν, Q2(ξ, η) := (|η|2 + |ξ|2)1/2 .

Then Q1 and Q2 are quasinorms with respect to the trivial parametric augmenta-tion of the reduced trivial weight system [1,1,1]. Thus Q1 ∼ Q2 by Lemma 3.3.2(i).In particular,

(1 + |·|2ν)1/2ν = Q1(·, 1) ∼ Q2(·, 1) = 〈·〉 .

From this and (3.3.17) we deduce

[η] ≤ Λη ≤ c(|η|)〈·〉 . (3.3.18)

Let B be a bounded subset of C, α ∈ Nd, and ϕ ∈ S(Rd). We fix k ∈ N withRez − α q ω ≤ k for z ∈ B. Then we infer from (3.3.7) and (3.3.18) that aη isslowly increasing and, recalling (1.1.1),

|ϕ∂αξ aη|E =

∣∣Λz−α ppppωη ϕ

((∂α

ξ a) rΛ

)η

∣∣E

≤ ΛRez−α ppppωη 〈·〉−kqk,0(ϕ) ‖a‖Hα ppppω

z≤ c(|η|) qk,0(ϕ) ‖a‖Hα ppppω

z

for η 6= 0 and z ∈ B. This proves the assertion. ¥

Now we can improve Example 3.3.9.

3.3.11 Lemma Suppose η 6= 0. Then15 z 7→ Λzη1OM is a continuous representation

of (C, +) in Laut(OM (Rd)

).

15Recall that, given an LCS X , we write 1X for idX ∈ L(X ).


Proof (1) Example 3.3.9 and Lemma 3.3.10 imply Λzη1OM

∈ OM (Rd). Conse-quently, see Theorem 1.6.4 of the Appendix, Λz

η1OM= (u 7→ Λz

ηu) ∈ L(OM (Rd)).

Thus, since Λz1+z2 = Λz1Λz2 and Λ0 = 1Rd , it suffices to prove the asserted conti-nuity. It follows from

Λh − 1 = h

∫ 1

0

Λth log Λ dt , h ∈ C ,

that

|∂αξ (Λh − 1)| ≤ |h|

∫ 1

0

|∂αξ (Λth log Λ)| dt , h ∈ C . (3.3.19)

(2) We evaluate ∂αξ (Λz log Λ) for α ∈ Nd and z ∈ C. Note that

∂αj (Λz log Λ) = (Λz−1∂jΛ)(1 + z log Λ) , 1 ≤ j ≤ d . (3.3.20)

Suppose |α| ≥ 2. Fix j ∈ 1, . . . , d with α − ej ≥ 1. Then, by (3.3.20) and Leibniz’rule,

∂αξ (Λz log Λ) = ∂

α−ej

ξ ∂j(Λz log Λ)

=∑

β≤α−ej

(α − ej

β

)∂β

ξ (Λz−1∂jΛ)∂α−ej−βξ (1 + z log Λ)

= ∂α−ej

ξ (Λz−1∂jΛ)(1 + z log Λ)

+ z∑

β<α−ej

(α − ej

β

)∂β

ξ (Λz−1∂jΛ)∂α−ej−βξ log Λ .

(3.3.21)

Let α − ej − β > 0. Then we can fix ` ∈ 1, . . . , d with α − ej − β − e` ≥ 0. Thus

∂α−ej−βξ log Λ = ∂

α−ej−β−e`

ξ (Λ−1∂`Λ) . (3.3.22)

(3) From Example 3.3.9 we know that Λz ∈ H∞z (Z). Hence it follows from

Lemmas 3.3.2(ii) and 3.3.6 that

Λz−1∂jΛ ∈ H∞z−ωj

(Z) , 1 ≤ j ≤ d .

Using Lemma 3.3.2(ii) once more, we thus obtain

∂γξ (Λz−1∂jΛ) = Λz−ωj−γ ppppω(

∂γξ (Λz−1∂jΛ)

) rΛ (3.3.23)

for γ ∈ Nd and 1 ≤ j ≤ d. By also employing Lemma 3.3.6,

|∂γξ (Λz−1

η ∂jΛη)| ≤ ΛRez−ωj−γ ppppωη ‖Λz−1∂jΛ‖Hk

z−ωj

≤ c(k)ΛRez−ωj−γ ppppωη ‖Λz−1‖Hk

z−1‖Λ‖Hk+1

1−ωj

for γ ∈ Nd with γ q ω ≤ k.

VI.3 Anisotropy 51

(4) It follows from Example 3.3.9 that

|∂γξ (Λz−1

η ∂jΛη)| ≤ c(k) |z|k ΛRez−ωj−γ ppppωη , 1 ≤ j ≤ d , γ q ω ≤ k . (3.3.24)

From this we get

|∂α−ej

ξ (Λz−1η ∂jΛη)| ≤ c(k) |z|k ΛRez−α ppppω

η

and, due to (3.3.22),

|∂βξ (Λz−1

η ∂jΛη)∂α−ej−βξ log Λ| ≤ c(k) |z|k ΛRez−α ppppω

η

for α q ω ≤ k and β < α − ej . Using these estimates, we obtain from (3.3.20) and(3.3.21) that

|∂αξ (Λz−1

η log Λη)| ≤ c(k) |z|k ΛRez−α ppppωη (1 + |z| | log Λη|) (3.3.25)

for 0 < α q ω ≤ k and z ∈ C.(5) Suppose h ∈ C with |h| ≤ 1, and 0 ≤ t ≤ 1. Then

−1 − k ≤ −1 − α q ω ≤ Re(th− α q ω) ≤ 1− α q ω ≤ 1 , α q ω ≤ k ,

and (3.3.25) imply, due to Λη ≥ [η] and log Λη ≤ c(η)Λη,

|∂αξ (Λth

η log Λη)| ≤ c(k, η)Λ2η , α q ω ≤ k .

Thus it follows from (3.3.19) that

|∂α(Λhη − 1)| ≤ c(k, η) |h|Λ2

η , α q ω ≤ k , |h| ≤ 1 . (3.3.26)

We fix z ∈ C. By Leibniz’ rule,

∂αξ (Λz+h − Λz) = ∂α

ξ

(Λz(Λh − 1)

)=

∑

β≤α

(αβ

)∂α−β

ξ Λz∂βξ (Λh − 1)

for h ∈ C. From this, (3.3.26), Lemma 3.3.2(ii), and Example 3.3.9 we get

|∂α(Λz+hη − Λz

η)| ≤ c(k, η) |h|Λ|z|+2η , α q ω ≤ k , |h| ≤ 1 .

We choose k ∈ N with k ≥ |z| + 2. Then this estimate and (3.3.18) imply

|∂αξ (Λz+h

η − Λzη)| ≤ c(k, η) |h| 〈·〉k , α q ω ≤ k , |h| ≤ 1 .

Consequently, given z ∈ C, α ∈ Nd, and η 6= 0, there exist c ≥ 1 and k ∈ N suchthat

‖ϕ∂α(Λz+hη − Λz

η)‖∞ ≤ c |h| qk,0(ϕ) , α ∈ Nd , |h| ≤ 1 , ϕ ∈ S(Rd) .

This proves Λz+hη → Λz

η in OM (Rd) as h → 0. Now the assertion follows by ap-pealing once more to Theorem 1.6.4 of the Appendix. ¥


3.4 Fourier Multipliers and Multiplier Spaces

Large parts of what is discussed in this volume depend on Fourier analytic tech-niques. For this reason we now embark on investigations of classes of functionswhose members can serve as symbols of Fourier multiplier operators.

Elementary Fourier Multiplier Theorems

Let

E1 × E2 → E3 , (e1, e2) 7→ e1e2 (3.4.1)

be a multiplication. For m ∈ S ′(Rd, E1) we define a linear map m(D) with domain

dom(m(D)

):=

u ∈ S ′(Rd, E2) ; mu ∈ S ′(Rd, E3)

by16

m(D)u := F−1mF = F−1(mu) , u ∈ dom(m(D)

),

provided the point-wise extension of (3.4.1), (m, u) → mu, is well-defined. Thenm(D) maps dom

(m(D)

) ⊂ S ′(Rd, E2) into S ′(Rd, E3). It is the Fourier multiplier

with symbol m.

Our first ‘Fourier multiplier theorem’ concerns smooth symbols.

3.4.1 Theorem Let X ∈ S,S ′. Then(m 7→ m(D)

) ∈ L(OM (Rd, E1),L(X (Rd, E2),X (Rd, E3)

)).

Proof Theorem 1.6.4 of the Appendix guarantees that there exists a unique hypo-continuous bilinear map

OM (Rd, E1)×X (Rd, E2) → X (Rd, E3) , (3.4.2)

the point-wise multiplication induced by (3.4.1), which coincides on S with thenatural point-wise multiplication. Since F ∈ Laut

(X (Rd, E)), we see that

OM (Rd, E1)×X (Rd, E2) → X (Rd, E3) , (m,u) 7→ mu

is a hypocontinuous bilinear map as well. Consequently, the bilinear map

OM (Rd, E1) ×X (Rd, E2) → X (Rd, E3) , (m,u) 7→ F−1(mu)

is also hypocontinuous. This implies the claim. ¥

16We do not indicate its dependence on (3.4.1) which will always be clear from the context.

VI.3 Anisotropy 53

Recalling (3.3.3), we set

Jzη := Λz

η(D) , z ∈ C , η ∈ H , (3.4.3)

andJz := Jz

1 , J := J1 .

Since, in the classical isotropic case (where the reduced weight system is the trivialone), J−s is a Bessel kernel for s > 0, in the present general setting J−s might becalled ν-anisotropic Bessel kernel.

3.4.2 Proposition Let X ∈ S,S ′. Then z 7→ Jz is a strongly continuous linearrepresentation of (C, +) in Laut

(X (Rd, E)).

Proof This is an easy consequence of Lemma 3.3.11 and Theorem 3.4.1. ¥

3.4.3 Example As usual, ∆ = ∆d = −|D|2 = ∂21 + · · · + ∂2

d denotes the Laplace op-erator on Rd. We write ∆xi

if only the variables xi := (x1i , . . . , x

dii ) are considered.

The following differential, respectively pseudodifferential, operators

A1 := 1 − ∆ , A2 := 1 − ∆x1 + ∆2x2

,

A3 := 1 − ∂2t + ∆2

m , A4 :=√

1− ∂2t + ∆3

m

are toplinear automorphisms of S(Rd, E) and S ′(Rd, E).

Proof Observe that Ak = J2ν for 1 ≤ k ≤ 3, where the weight system [`, d, ν]equals the trivial one [1, d, 1] if k = 1 (the isotropic case),

[2, (d1, d2), (2, 1)

]if

k = 2, and the parabolic weight system [m, 2] if k = 3. Furthermore, A4 = Jν withrespect to the parabolic weight system [m, 3]. Thus the assertions follow from thepreceding proposition. ¥

Our principal interest, in what follows, are Fourier multiplier theorems foroperator-valued symbols. Here we content ourselves with a very simple thoughuseful result.

By17 L1(Rd, E) → S ′(Rd, E) and F ∈ L(S ′(Rd, E)), the image FL1(Rd, E)

of L1(Rd, E) under F is a well-defined Banach space. The Riemann-Lebesguelemma, the density of S(Rd, E) in L1(Rd, E) and in C0(Rd, E), and the invarianceof S(Rd, E) under F imply

S(Rd, E)d

→ FL1(Rd, E)d

→ C0(Rd, E) . (3.4.4)

3.4.4 Theorem Suppose E1 × E2 → E3, (e1, e2) 7→ e1e2 is a multiplication. If

X ∈ BUC, C0, Lq, 1 ≤ q < ∞ ,

17See (VII.1.2.1) below.


then(a 7→ a(D)

) ∈ L(FL1(Rd, E1),L(X (Rd, E2),X (Rd, E3)

)). (3.4.5)

Moreover,(a 7→ a(D)

) ∈ L(FL1(Rd, E1),L(L∞(Rd, E2), BUC(Rd, E3)

)). (3.4.6)

The norms of these linear maps are bounded by 1.

Proof (1) It follows from Theorem 1.9.9 of the Appendix that convolution is amultiplication of Banach spaces

L1(Rd, E1) ×X2(Rd, E2) → X3(Rd, E3) , (a, u) 7→ a ∗ u

of norm at most 1 if

(X2,X3) ∈

(BUC, BUC), (C0, C0), (Lq, Lq), (L∞, BUC)

,

where 1 ≤ q < ∞. From this we infer that the map(a 7→ (u 7→ F−1a ∗ u)

)pos-

sesses properties (3.4.5) and (3.4.6).(2) Suppose a ∈ S(Rd, E1) and u ∈ S ′(Rd, E2). Then we deduce from (3.3.16)

and (3.4.2) that au ∈ S ′(Rd, E3) and, by the convolution theorem (Theorem 1.9.10of the Appendix), a(D)u = F−1(au) = F−1a ∗ u. Thus, by step (1) and since17

X2(Rd, E2) → S ′(Rd, E2),

‖a(D)u‖X3(Rd,E3) ≤ ‖a‖FL1(Rd,E1) ‖u‖X2(Rd,E2) , u ∈ X2(Rd, E2) ,

provided a ∈ S(Rd, E1). Now the assertion follows from the density of the latterspace in FL1(Rd, E1). ¥

Fourier Multiplier Spaces

Now we introduce a space of E-valued functions on Rd, a Fourier multiplier space,as follows:

We set k(ν) := |ω| (2 + max ν). Then

M(Rd, E) = Mν(Rd, E)

is the linear subspace of C(Rd, E) consisting of all u satisfying∂αu ∈ C(Rd, E) for α q ω ≤ k(ν) and

‖u‖M := maxα ppppω≤k(ν)

‖Λα ppppω1 ∂αu‖∞ < ∞ ,

endowed with the norm ‖·‖M.

(3.4.7)

In Subsections 3.6, VII.2.4, and VII.5.3 it will be shown that the elements ofM(Rd, E) are Fourier multipliers for large classes of function spaces. We take k(ν)as regularity index to allow for easy proofs. Lesser regularity requirements wouldsuffice, but this is not important for what follows and we do not give details (see,however, the notes to Section 2).

VI.3 Anisotropy 55

3.4.5 Lemma

(i) M(Rd, E) is a Banach space.(ii) If E1 × E2 → E3 is a multiplication, then its point-wise extension is a mul-

tiplicationM(Rd, E1)×M(Rd, E2) →M(Rd, E3) .

Proof (1) We write M for M(Rd, E) etc. Let (aj) be a Cauchy sequence in M.Then (Λα ppppω

1 ∂αaj) is a Cauchy sequence in BC, for α q ω ≤ k(ν). Hence there existbα ∈ BC with Λα ppppω

1 ∂αaj → bα in BC for α q ω ≤ k(ν). Set aα := Λ−α ppppω1 bα ∈ C.

Then ∂αaj → aα in C, hence in D′. By the continuity of the distributional deriva-tive on D′, it follows ∂αaj → ∂αa0 in D′. Hence ∂αa0 = aα ∈ C for α q ω ≤ k(ν).Consequently, Λα ppppω

1 ∂αaj → Λα ppppω1 ∂αa0 in C. Since ‖Λα ppppω

1 ∂αaj‖∞ ≤ c for j ∈ N,it follows ‖Λα ppppω

1 ∂αa0‖∞ < ∞ for α q ω ≤ k(ν). Thus a0 ∈M and, letting k →∞in ‖aj − ak‖M, we deduce from the definition of Cauchy sequences that aj → a0

in M. Thus M is complete.(2) Claim (ii) follows from Leibniz’ rule. ¥

Resolvent Estimates

For later use, we present here and in the next subsection a series of results whichwill help to determine Fourier multipliers in concrete situations. The reader maywant to skip the rest of this section at the first reading and return when theseresults are actually employed.

We recall that

Sϑ := [ | arg z| ≤ ϑ] ∪ 0 ⊂ C , 0 ≤ ϑ ≤ π ,

where arg z ∈ (−π, π] is the principal value of the argument of z.In the following, we assume

• E1 → E0 , κ ≥ 1 , 0 ≤ ϑ < π .

Then

A ∈ P(L(E1, E0); κ, ϑ)

(3.4.8)

iff18 A ∈ L(E1, E0), Sϑ ⊂ ρ(−A), and

(1 + |λ|)1−j |(λ + A)−1|L(E0,Ej) ≤ κ , λ ∈ Sϑ , j = 0, 1 . (3.4.9)

Thus A ∈ Lis(E1, E0) and, consequently, A ∈ C(E0) by Lemma I.1.1.2. This def-inition renders more precise the class P(E; κ, ϑ) introduced in Subsection III.4.7

18Recall that ρ(−A) is the resolvent set of −A, where A is considered as a linear operatorin E0 with domain E1.


since we now specify the domain of A. On the other hand, here we do not require

that E1d

→ E0, that is, A be densely defined.It should be observed that all the subsequent estimates, involving linear op-

erators belonging to classes characterized by suitable constants, depend on theindicated quantities only and not on the individual linear maps.

3.4.6 Lemma The following estimates apply.(i) If a ∈M(

Rd,Lis(E1, E0))

and ‖a−1‖∞ ≤ κ, then a−1 ∈M(Rd,Lis(E0, E1)

)and ‖a−1‖M ≤ c(κ, ‖a‖M),

(ii) Assume a ∈M(Rd,L(E1, E0)

)and a(Rd) ⊂ P(L(E1, E0); κ, ϑ

). Then

(1+ |λ|)1−j ‖(λ+a)−1‖M(Rd,L(E0,Ej)) ≤ c(κ, ‖a‖M) , λ ∈ Sϑ , j = 0, 1 .

Proof We prove (ii). Then (i) follows by setting λ = 0 in the subsequent compu-tations.

We use the notation of Lemma 3.3.7. Then, if 0 < α q ω ≤ k(ν),

Λα ppppω1 ∂α(λ + a)−1

=|α|∑

j=1

∑

β∈Bj,α

εβ(λ + a)−1Λβ1 ppppω1 ∂β1a(λ + a)−1 · · ·Λβj

ppppω1 ∂βj a(λ + a)−1 .

(3.4.10)

From (3.4.9), applied to a, and since a ∈M(Rd,L(E1, E0)

), we get

|Λβi ppppω1 ∂βia(λ + a)−1|L(E0) ≤ κ ‖a‖M , 1 ≤ i ≤ j .

Hence we obtain from (3.4.10)

|Λα ppppω1 ∂α(λ + a)−1|L(E0,Ej) ≤ c(κ, ‖a‖M) |(λ + a)−1|L(E0,Ej)

for α q ω ≤ k(ν) and λ ∈ Sϑ. The assertion follows by applying (3.4.9) once more. ¥

Now we turn to parameter-dependent symbols and define

Hz(Z, E) := Hk(ν)z (Z, E) , z ∈ C .

These spaces are closely related to the multiplier space M(Rd, E) as is confirmedby the following observations.

3.4.7 Remarks (a) Suppose Rez ≤ 0 and [η] ≥ 1. Then

(a 7→ aη) ∈ L(Hz(Z, E),M(Rd, E))

VI.3 Anisotropy 57

and‖aη‖M ≤ [η]Rez ‖a‖Hz .

Proof Note Λ1/Λη ≤ max1, 1/[η] for η ∈q

H. Hence we get from (3.3.7)

Λα ppppω1 |∂α

ξ aη| ≤ (Λ1/Λη)α ppppωΛRezη ‖(∂α

ξ a) rΛ‖∞ ≤ [η]Rez ‖a‖Hz

for α q ω ≤ k(ν) and [η] ≥ 1. ¥

(b) In connection with parameter-dependent homogeneous symbols, (3.3.6) sug-gests to endow M(Rd, E) with the equivalent η-dependent norm

‖u‖Mη := maxα ppppω≤k(ν)

‖Λα ppppωη ∂αu‖∞ .

Thus Mη(Rd, E) :=(M(Rd, E), ‖·‖Mη

)for η ∈

qH and M1 = M.

Although ‖·‖M and ‖·‖Mηare equivalent, they are not η-uniformly equiva-

lent. However, if a ∈ C(qZ, E), then

‖aη‖Mη= ‖σ[η]aη‖M , η ∈

qH .

This fact will be important when we consider η-dependent function spaces.

Proof First we note

σ1/[η]Λz1 = [η]−zΛz

η , z ∈ C , η ∈q

H . (3.4.11)

From this and Proposition 3.1.3(i) we get

Λα ppppω1 ∂α(σ[η]aη) = [η]α ppppωΛα ppppω

1 σ[η](∂αaη)

= [η]α ppppωσ[η]

((σ1/[η]Λα ppppω

1 )∂αaη

)= σ[η](Λα ppppω

η ∂αaη) .(3.4.12)

Thus, by Proposition 3.1.3(ii),

‖Λα ppppω1 ∂α(σ[η]aη)‖∞ = ‖Λα ppppω

η ∂αaη‖∞ (3.4.13)

for α ∈ Nd and η ∈q

H. This implies the claim. ¥

3.5 Multiplier Estimates

We introduce a class of parameter-dependent symbols which will play a funda-mental role in connection with elliptic differential operators. Namely, we assumethat

E1 → E0 , κ ≥ 1 , 0 ≤ ϑ < π , s ≥ 0 . (3.5.1)


Then, by definition,

a ∈ Ps

(Z,L(E1, E0); κ, ϑ

)(3.5.2)

iff

(i) a ∈ Hs

(Z,L(E1, E0)

), ‖a‖Hs ≤ κ .

(ii) (1 + |λ|)1−j∣∣(λ + (a rΛ)η

)−1(ξ)∣∣L(E0,Ej)

≤ κ

for λ ∈ Sϑ, ξ ∈ Rd, and j = 0, 1, η-uniformly, that is,uniformly with respect to η ∈

qH.

(3.5.3)

Henceforth, it is understood that (3.5.1) applies if Ps

(Z,L(E1, E0); κ, ϑ

)is referred

to.It is no restriction of generality for what follows to use the same bound κ ≥ 1

in (i) and (ii), as follows from

P(L(E1, E0); κ0, ϑ0

) ⊂ P(L(E1, E0); κ1, ϑ1

), κ0 ≤ κ1 , ϑ0 ≥ ϑ1 . (3.5.4)

This convention simplifies the estimates below.

Resolvent Estimates for Homogeneous Symbols

The following lemma is an analogue of Lemma 3.4.6 for homogeneous symbols.

3.5.1 Lemma If a ∈ Ps

(Z,L(E1, E0); κ, ϑ

), then

Λjsη

([η]s + |λ|)1−j ‖(λ + aη)−1‖Mη(Rd,L(E0,Ej)) ≤ c(κ)

for λ ∈ Sϑ, η ∈q

H, and j = 0, 1.

Proof From a = Λsa rΛ we get

(λ + a)−1 = Λ−s(Λ−sλ + a rΛ)−1 . (3.5.5)

Since Λs(ζ)Sϑ ⊂ Sϑ, we infer from (3.5.3)

|(λ + aη)−1|L(E0,Ej) = Λ−sη

∣∣(Λ−sη λ + (a rΛ)η

)−1∣∣L(E0,Ej)

≤ κΛ−sη (1 + Λ−s

η |λ|)j−1 = κΛ−jsη (Λs

η + |λ|)j−1

≤ κΛ−jsη ([η]s + |λ|)j−1

(3.5.6)

for λ ∈ Sϑ and η ∈q

H. Hence, using (3.3.7), it follows

Λβ ppppωη |∂βaη(λ + aη)−1|L(E0) ≤ ‖(∂β

ξ a) rΛ‖∞ κ ≤ c(κ)

for β q ω ≤ k(ν), λ ∈ Sϑ, and η ∈q

H. From this and (3.5.6) we get the assertionby replacing Λ1 by Λη in the proof of Lemma 3.4.6. ¥

VI.3 Anisotropy 59

Functions of Homogeneous Symbols

For easy reference we include the following simple well-known perturbation theo-rem which will be of repeated use below.

3.5.2 Lemma Suppose E1 → E2 → E0. If A ∈ Lis(E1, E0) and B ∈ L(E2, E0) sat-isfy |BA−1|L(E0) ≤ 1/2, then A + B ∈ Lis(E1, E0) and

|(A + B)−1|L(E0,Ej) ≤ 2 |A−1|L(E0,Ej) , j = 0, 1 .

Proof This follows from A + B = (1E0 + BA−1)A and a Neumann series esti-mate. ¥

Assume A ∈ P(L(E); κ, ϑ)

and set |·| := |·|L(E). Then |A−1| ≤ κ. Hence, bythe preceding lemma, |(λ− A)−1| ≤ 2κ for |λ| ≤ 1/2κ. More generally, if λ0 ∈ Sϑ,then (cf. (III.4.7.11))

(1 + |λ|) |(λ − A)−1| ≤ 2κ + 1 for |λ − λ0| ≤ (1 + |λ0|)/2κ .

In particular, [ |λ − λ0| ≤ |λ0|/2κ] ⊂ ρ(−A) for λ0 ∈ Sϑ. We set

ϑ(κ) := ϑ + arcsin(1/2κ) .

Then it follows that [ |z| ≤ 1/2κ] ∪ Sϑ(κ) ⊂ ρ(−A) and

(1 + |λ|) |(λ− A)−1| ≤ 3κ for λ ∈ [ |z| ≤ 1/2κ] ∪ Sϑ(κ) .

Next, we put

ϕ = ϕ(κ, ϑ) := π − ϑ(κ) , W = Wκ,ϑ := Sϕ ∩ [ |z| ≥ 1/2κ] . (3.5.7)

Then W is closed,

σ(A) ⊂ W ⊂ W ⊂ Sπ−ϑ , (3.5.8)

and

(1 + |λ|) |(λ− A)−1| ≤ 3κ (3.5.9)

for λ ∈ C\W .Suppose

a ∈ Ps

(Z,L(E); κ, ϑ

).

Sπ−ϑ\W

W

Γ

W for ϕ > π/2

Then (3.5.8) and (3.5.9) apply to a rΛ(ζ) for each ζ ∈qZ. Let Σ be any posi-

tively oriented contour (that is, a piece-wise smooth simple closed curve) in Sπ−ϑ


containing σ(a rΛ(ζ)

)in its interior. For τ > 0 we denote by τΣ the contour ob-

tained from Σ by applying the dilation z 7→ τz. Then we infer from a = Λsa rΛ

that Λs(ζ)Σ is a positively oriented contour containing

σ(a(ζ)

)= Λs(ζ)σ

(a rΛ(ζ)

)

in its interior.Let h : Sπ−ϑ → C be holomorphic. Since Λs(ζ)Σ ⊂ Sπ−ϑ, the Cauchy inte-

gral

h(a)(ζ) :=1

2πi

∫

Λs(ζ)Σ

h(λ)(λ − a(ζ)

)−1dλ (3.5.10)

is well-defined in L(E) for ζ ∈qZ. It follows from Cauchy’s theorem that h(a)(ζ)

is independent of the particular contour Σ. In fact, the well-known Dunford cal-culus (cf. [DuS57, Section VII.1]) shows that h(a)(ζ) depends on the values of hon σ

(a(ζ)

)only.

It is the purpose of the following considerations to show that Σ in (3.5.10)can be replaced by Γ := ∂W , the positively oriented boundary of W , and thatthen the integral converges in Mη

(Rd,L(E)

)for each η ∈

qH. For this we prepare

some technical results.

3.5.3 Lemma Suppose U is open in C and h : U → C is holomorphic, V is openin Rd and g ∈ Cm(V,U) for some m ∈

qN, and α ∈ Nd satisfies |α| = m. Then

∂α(h g) =m∑

j=1

h(j) g∑

(k,β)∈B(j,α)

ω(k, β)(∂β1g)k1 · · · (∂βmg)km ,

where

B(j, α) :=

(k, β) ; k = (k1, . . . , km) ∈ Nm, β = (β1, . . . , βm) ∈ (Nd\0)m

,

k1 + · · · + km = j,∑m

i=1 kiβi = α

and ω(k, β) ∈ N.

Proof Let m = 1 and α = ei. Then, by the chain rule,

∂α(h g) = ∂i(h g) = (h′ g)∂ig = (h′ g)∂αg .

Hence the assertion holds in the present case. From this, ∂α+ei = ∂i∂α, and the

product rule we obtain the claim by induction. ¥

3.5.4 Lemma Suppose h : Sψ → C is holomorphic for some ψ ∈ (ϕ, π − ϑ). Letthere exist δ > 0 and R0 ≥ 1 such that |z|δ |h(z)| ≤ 1 for z ∈ Sψ satisfying |z| ≥ R0.

VI.3 Anisotropy 61

Then, given η ∈q

H, there is an Rη ≥ R0 such that

Λα ppppωη

∣∣∂αξ

(h(Λs

ηz))∣∣ ≤ c(|α|)[η]−δs |z|−δ

for α ∈ (Nd)qand z ∈ W with |z| ≥ Rη.

Proof We infer from Λs ∈ Hs(Z) and Lemma 3.3.2(ii) that

∣∣(∂βξ (Λs

ηz))k∣∣ ≤ c(k, β)Λ(s−β ppppω)k

η |z|k

for k ∈qN and β ∈ Nd\0. Thus Lemma 3.5.3 implies

|∂αξ h(Λs

ηz)| ≤ c(α)|α|∑

j=1

|h(j)(Λsηz)|Λjs−α ppppω

η |z|j (3.5.11)

for z ∈ Sψ.

We set r := sin(ψ − ϕ)/2. Then the disc [ |λ− z| ≤ r |z| ] is for z ∈ W con-tained in Sψ. Hence this is also true for the disc

[ |λ− Λsη(ζ)z| ≤ rΛs

η(ζ) |z| ] forζ ∈

qZ and z ∈ W . Consequently, by Cauchy’s formula,

h(j)(Λs(ζ)z

)=

j!2πi

∫

|λ−Λs(ζ)z|=rΛs(ζ) |z|

h(λ) dλ

(λ − Λs(ζ)z)j+1, j ∈ N .

We put Rη := max1, 1/(1 − r)[η]s

R0. If z ∈ W satisfies

|z| ≥ Rη and |λ− Λsη(ζ)z| = rΛs

η(ζ) |z| ,

then|λ| ≥ (1− r)Λs

η(ζ) |z| ≥ (1− r)[η]s |z| ≥ R0 .

Thus, invoking the assumption,

∣∣h(j)(Λs

η(ξ)z)∣∣ ≤ j! r−j(1− r)−δ[η]−δs

(Λs

η(ξ) |z|)−j |z|−δ

for ξ ∈ Rd, z ∈ W , and j ∈ N. By inserting this estimate into the right side of(3.5.11), the assertion follows. ¥

Dunford Integral Representations

Now we are ready for the proof of the representation theorem for holomorphicfunctions of multipliers.

3.5.5 Proposition Suppose h : Sψ → C is holomorphic for some ψ ∈ (ϕ, π − ϑ).Let there exist δ > 0 such that |z|δ h(z) → 0 as |z| → ∞. Denote by Γ the positively


oriented boundary of W . Then h(aη) ∈Mη

(Rd,L(E)

)and

‖h(aη)‖Mη≤ c(κ, ϑ) sup

ξ∈Rd

maxz∈Γ

∣∣h(Λs

η(ξ)z)∣∣ (3.5.12)

for a ∈ Ps

(Z,L(E); κ, ϑ

)and η ∈

qH. Furthermore,

h(aη) =1

2πi

∫

ΛsηΓ

h(λ)(λ − aη)−1 dλ (3.5.13)

in Mη

(Rd,L(E)

).

Proof (1) Throughout this proof ζ ∈qZ, α, β ∈ Nd with α q ω, β q ω ≤ k(ν), and

a ∈ Ps

(Z,L(E); κ, ϑ

).

Since |a rΛ(ζ)| ≤ ‖a‖Hs ≤ κ and the norm of a linear operator is an upperbound for its spectral radius, it follows that

σ(a rΛ(ζ)

) ⊂ W ∩ [ |z| ≤ κ] . (3.5.14)

(2) We denote by ΣR the positively oriented boundary of W ∩ [ |z| ≤ R] forR ≥ 2κ. Then we claim

∣∣(z − a rΛ(ζ))−1∣∣ ≤ 3κ/|z| , z ∈ ΣR . (3.5.15)

Indeed, if |z| ≥ 2κ ≥ 2 |a rΛ(ζ)|, then

∣∣(z − a rΛ(ζ))−1∣∣

= |z|−1∣∣(1− z−1a rΛ(ζ)

)−1∣∣ ≤ 2/|z|

by Lemma 3.5.2. If |z| = 1/2κ, then

|z| ∣∣(a rΛ(ζ))−1∣∣ ≤ 1/2 ,

since (3.5.3) guarantees that∣∣(a rΛ(ζ)

)−1∣∣ ≤ κ.Hence, invoking Lemma 3.5.2 once more, we ar-rive at

ΓR

Σ′R

ΣR = Σ′R + ΓR

∣∣(z − a rΛ(ζ))−1∣∣ ≤

∣∣(a rΛ(ζ))−1∣∣ ∣∣(z(a rΛ(ζ))−1 − 1

)−1∣∣ ≤ 2κ = 1/|z| .

Lastly, if | arg z| = ϕ, then (3.5.9) implies∣∣(z − a rΛ(ζ)

)−1∣∣ ≤ 3κ/(1 + |z|) ≤ 3κ/|z| .

Thus (3.5.15) follows from κ ≥ 1. Consequently,∣∣(Λs(ζ)z − a(ζ)

)−1∣∣ = Λ−s(ζ)∣∣(z − a rΛ(ζ)

)−1∣∣ ≤ 3κ/Λs(ζ) |z| (3.5.16)

for z ∈ ΣR.

VI.3 Anisotropy 63

(3) Fix ζ0 ∈qZ. By the upper semicontinuity of the spectrum and its com-

pactness, there exists a neighborhood U of ζ0 inqZ such that σ

(a(U)

)is contained

in the interior of Λs(ζ0)ΣR. Thus, by Cauchy’s theorem,

h(a)(ζ) =1

2πi

∫

Λs(ζ0)ΣR

h(λ)(λ − a(ζ)

)−1dλ , ζ ∈ U .

Consequently,

∂αξ h(a)(ζ) =

12πi

∫

Λs(ζ0)ΣR

h(λ)∂αξ (λ− a)−1(ζ) dλ , ζ ∈ U .

This holds, in particular, for ζ = ζ0. Hence, ζ0 being arbitrary inqZ, we find

∂αξ h(a)(ζ) =

12πi

∫

Λs(ζ)ΣR

h(λ)∂αξ (λ− a)−1(ζ) dλ (3.5.17)

for ζ ∈qZ.

(4) Suppose β 6= 0. From (3.3.7) and (3.5.16) we derive

Λβ ppppω ∣∣(∂βξ (λ − a)(λ− a)−1

)|λ=Λsz

∣∣ = Λβ ppppω |∂βξ a| |(Λsz − a)−1|

≤ 3κΛs ‖a‖Hs

/Λs |z| ≤ c(κ)

for z ∈ ΣR. Using this, Lemma 3.3.7 and (3.5.16) once more, it follows

Λα ppppω ∣∣(∂αξ (λ− a)−1

)|λ=Λsz

∣∣ ≤ c(κ) |(Λsz − a)−1| ≤ c(κ)Λ−s |z|−1 (3.5.18)

for z ∈ ΣR. Now we deduce from (3.5.17)

Λα ppppωη |∂αh(aη)| ≤ 1

2π

∣∣∣∫

ΣR

h(Λsηz)Λα ppppω

η

(∂α(λ− aη)−1

)|λ=Λs

ηzΛs

η dz∣∣∣

≤ c(κ)∫

ΣR

|h(Λsηz) dz/z|

≤ c(κ, ϑ) supξ∈Rd

maxz∈ΣR

|h(Λsηz)| .

This shows that h(aη) belongs to Mη

(Rd,L(E)

)and

‖h(aη)‖Mη≤ c(κ, ϑ) sup

ξ∈Rd

maxz∈ΣR

|h(Λsηz)| (3.5.19)

for any R ≥ 2κ. Since |Λsη(ξ)z| ≥ [η]s |z|, the assumption implies h

(Λs

η(ξ)z) → 0

as |z| → ∞, uniformly for ξ ∈ Rd. Hence, by choosing R sufficiently large, we inferfrom the maximum principle for holomorphic functions that

maxz∈ΣR

∣∣h(Λs

η(ξ)z)∣∣ = max

z∈Γ

∣∣h(Λs

η(ξ)z)∣∣ .

From this and (3.5.19) it follows that (3.5.12) applies.


(5) We fix Rη ≥ 2κ such that([η]s |z|)δ ∣∣h(

[η]sz)∣∣ ≤ 1 , |z| ≥ Rη , z ∈ W .

Then Λsη(ξ) |z| = [η]s

∣∣(Λsη(ξ)

/[η]s

)z∣∣ and

∣∣(Λsη(ξ)

/[η]s

)z∣∣ ≥ |z| give

(Λs

η(ξ) |z|)δ ∣∣h(Λs

η(ξ)z)∣∣ ≤ 1 , ξ ∈ Rd , z ∈ W , |z| ≥ Rη . (3.5.20)

We setΣ′

R := ΣR ∩ [ |z| = R] = Rei t ; −ϕ ≤ t ≤ ϕ .

Then we get from (3.5.20) and (3.5.16)∣∣∣∫

Λsη(ξ)Σ′

R

h(λ)(λ− aη(ξ)

)−1dλ

∣∣∣

≤∫

Σ′R

∣∣h(Λs

η(ξ)z)∣∣ ∣∣(Λs

η(ξ)z − aη(ξ))−1∣∣ Λs

η(ξ) d|z|

≤ c(κ, ϑ)[η]−δsR−δ

(3.5.21)

for ξ ∈ Rd and R ≥ Rη.Put ΓR := ΣR ∩ Γ and

IR(aη) :=1

2πi

∫

ΛsηΓR

h(λ)(λ − aη)−1 dλ

for R ≥ 2κ. By Cauchy’s theorem,

h(aη) =1

2πi

(∫

ΛsηΓR

+∫

ΛsηΣ′

R

)h(λ)(λ − aη)−1 dλ .

From this and (3.5.21) we deduce

h(aη) = limR→∞

IR(aη)

=1

2πi

∫

ΛsηΓ

h(λ)(λ− aη)−1 dλ

in BC(Rd,L(E)

).

(6) Let ∆R′,R be the positively oriented bound-ary of W ∩ [R ≤ |z| ≤ R′] for 2κ ≤ R < R′ < ∞.Since λ 7→ h(λ)(λ − a)−1 is holomorphic in aneighborhood of W ∩ [ |z| ≥ 2κ], Cauchy’s the-orem implies

ΓR′ \ΓR

ΓR Σ′R

Σ′R′

∆R′R

0 =1

2πi

∫

Λs∆R′,R

h(λ)(λ− a)−1 dλ

= IR′(a)− IR(a) +1

2πi

(∫

ΛsΣ′R′

−∫

ΛsΣ′R

)h(λ)(λ− a)−1 dλ .

VI.3 Anisotropy 65

Thus

Λα ppppωη ∂α

(IR′(aη) − IR(aη)

)=

Λα ppppωη

2πi∂α

(∫

Σ′R

−∫

Σ′R′

)h(Λs

ηz)(Λsηz − aη)−1Λs

η dz .

From this, Lemma 3.5.4, (3.5.18), and Leibniz’ rule we deduce, for η ∈q

H,

∣∣Λα ppppωη ∂α

(IR′(aη)− IR(aη)

)∣∣ ≤ c[η]−δs(∫

Σ′R

+∫

Σ′R′

)|z|−1−δ d|z|

≤ c[η]−δsR−δ → 0

as R →∞. Thus(IR(aη)

)R≥2κ

is a Cauchy net in Mη

(Rd,L(E)

)for R →∞. Since

Mη is a Banach space, there exists I(η) ∈Mη

(Rd,L(E)

)such that IR(aη) con-

verges in M(Rd,L(E)

)towards I(η), hence in BC

(Rd,L(E)

)by M → BC. Now

step (5) implies I(η) = h(aη). This proves the last part of the claim. ¥

Powers and Exponentials

Now we consider two most important choices for h, namely, power functions andexponentials. Recall that λ 7→ log λ = log |λ| + i arg λ and

λ 7→ hz(λ) := λz = ez log λ , λ ∈ C , (3.5.22)

are the principal value of the logarithm and the power function, respectively. Bythe Dunford calculus,

az(ζ) = hz(a)(ζ) =1

2πi

∫

Λs(ζ)Γ

λz(λ − a(ζ)

)−1dλ ∈ L(E) , ζ ∈

qZ ,

for a ∈ Ps

(Z,L(E); κ, ϑ

)and z ∈ C.

3.5.6 Proposition Suppose Rez < 0. Then

azη =

12πi

∫

ΛsηΓ

λz(λ − aη)−1 dλ (3.5.23)

in Mη

(Rd,L(E)

)and

‖azη‖Mη ≤ c(κ, ϑ)[η]sReze|Imz|ϕ (3.5.24)

for η ∈q

H and a ∈ Ps

(Z,L(E); κ, ϑ

).

Proof Since Re(z log λ) = Rez log |λ| − (Imz) arg λ and |λz| = eRe(z log λ), it fol-lows

|λz| ≤ |λ|Rez e|Imz| (π−ϑ) , λ ∈ Sπ−ϑ .


Hence, given 0 < δ < −Rez, we see that |λδ|λz → 0 as λ →∞ in Sπ−ϑ. Further-more,

|λz| ≤ ([η]s/2κ

)Reze|Imz|ϕ , λ ∈ Λs

ηΓ .

Thus the assertion follows from Proposition 3.5.5. ¥

Of outstanding importance is the symbol class

Ps

(Z,L(E); κ

):= Ps

(Z,L(E); κ, π/2

). (3.5.25)

In this case ϕ = ϕ(κ) ∈ (0, π/2). Consequently,

W ⊂ [Rez ≥ ω(κ)

], ω(κ) := cos

(ϕ(κ)

)/2κ . (3.5.26)

Now we consider the entire analytic function

λ 7→ et(λ) := e−tλ , t > 0 .

Then

e−ta(ζ) := et(a)(ζ)

=1

2πi

∫

Λs(ζ)Σ

e−tλ(λ − a(ζ)

)−1dλ

with ζ ∈qZ is, for a ∈ Ps

(Z,L(E); κ

), the

Dunford representation of the analytic semi-group e−ta(ζ) ; t ≥ 0 on E, generated bya(ζ) ∈ L(E).

0 ωW

[Rez ≥ ω]

W ⊂ [Rez ≥ ω]

3.5.7 Proposition If t > 0, then

e−taη =1

2πi

∫

ΛsηΓ

e−tλ(λ − a(ζ)

)−1dλ

in Mη

(Rd,L(E)

)and

‖e−taη‖Mη≤ c(κ)e−t[η]sω(κ)

for η ∈q

H and a ∈ Ps

(Z,L(E); κ

).

Proof It is obvious that |z|δ et(z) → 0 as |z| → ∞ in Sψ for any ψ ∈ (ϕ, π/2). By(3.5.26),

|e−tλ| = e−tReλ ≤ e−t[η]sω(κ) , λ ∈ ΛsηW .

Hence Proposition 3.5.5 implies the claim. ¥

VI.3 Anisotropy 67

Lastly, we exhibit a simple example of a symbol in Ps

(Z,L(E); κ

).

3.5.8 Example Put a := Λs1E . Then a ∈ Ps

(Z,L(E); κ

), where

κ := sup

(1 + |λ|)/|1 + λ| ; λ ∈ Sπ/2

.

Proof Since a rΛ = 1E , it holds σ(a rΛ) = 1. Thus Sπ/2 is contained inthe resolvent set of −a rΛ and

∣∣(λ + a rΛ(ζ))−1∣∣ = |1 + λ|−1 ≤ κ(1 + |λ|)−1 for

λ ∈ Sπ/2. Now the assertion follows from Example 3.3.9. ¥

3.6 Dyadic Partitions of Unity

In this subsection we introduce partitions of unity of Rd which are compatiblewith the dilation (3.1.3). They are fundamental for the Fourier-analytic theory ofBesov spaces, discussed in detail in Section 2.

A nonempty subset M of Rd is symmetric (with respect to the origin), ifx ∈ M implies −x ∈ M . It is ν-starshaped, if t q M ⊂ M for 0 ≤ t ≤ 1.

SupposeΩ is a bounded open 0-neighborhood in Rd

which is symmetric and ν-starshaped,

a ν-admissible 0-neighborhood of Rd. We set

Ω0 := 2 q Ω , Σ := 2 q Ω0\2−1 q Ω0 , Ωj := 2j−1 q Σ , j ≥ 1 . (3.6.1)

Then Ωk is open and symmetric for k ∈ N, Ωj ∩ Ωk = ∅ for |j − k| ≥ 2, and⋃k≥0 Ωk = Rd. We say that (Ωk) is the ν-dyadic open covering of Rd induced by Ω.

Assume that ψ is Ω-adapted, that is,

ψ ∈ D(Ω0) , ψ = ψ

, ψ |(3/2) q Ω = 1 = 1Ω . (3.6.2)

We putψ := ψ − σ2ψ , ψ0 = ψ , ψj := σ2−j ψ , j ≥ 1 .

Then

ψk ∈ D(Ωk) , ψk = ψ

k ,

n∑

j=0

ψj = σ2−nψ , k, n ∈ N . (3.6.3)

Since

limn→∞

σ2−nψ(ξ) = limn→∞

ψ(2−n q ξ) = ψ(0) = 1 , ξ ∈ Rd , (3.6.4)


we see that

∞∑

j=0

ψj = 1 on Rd . (3.6.5)

Thus (ψk) is a smooth partition of unity on Rd, subordinate to the open cover-ing (Ωk). We call

((Ωk), (ψk)

)ν-dyadic partition of unity on Rd induced by (Ω, ψ).

3.6.1 Examples (a) Let Q be a ν-quasinorm on Rd. Then Ω := [Q < 1] is aν-admissible 0-neighborhood. Note that

Ω0 = [Q < 2] , Ωj = [2j−1 < Q < 2j+1] , j ≥ 1 .

(b) Let d1, d2 ∈qN satisfy d1 + d2 = d. Assume [ì, di, νi] is a weight system for Rdi

such that[`, d, ν] =

[`1 + `2, (d1, d2), (ν1, ν2)

].

Let Ωi be a νi-admissible 0-neighborhood in Rdi and suppose ψi is Ωi-adapted.Then Ω := Ω1 × Ω2 is a νi-admissible 0-neighborhood of Rd and ψ := ψ1 ⊗ ψ2 isΩ-adapted. ¥

Preliminary Fourier Multiplier Theorems

Now we fix a ν-admissible 0-neighborhood Ω of Rd and an Ω-adapted ψ, and let((Ωk), (ψk)

)be the ν-dyadic partition of unity on Rd induced by (Ω, ψ).

3.6.2 Lemma Suppose X ∈ S,S ′. Then∑∞

k=0 ψk(D)u = u in X (Rd, E) if u ∈ X .

Proof It is an easy consequence of Proposition 3.1.3(i) and (3.6.4) that

limn→∞

∥∥ϕ(∂α(σ2−nψ − 1)

)∥∥∞ = 0 , ϕ ∈ S(Rd) .

Hence we see from (3.6.3) that (3.6.5) converges in OM (Rd). Now the assertionfollows from Theorem 3.4.1. ¥

In the next lemma we collect some Fourier multiplier results for the se-quence (ψk) which will lead, among other things, to a Fourier multiplier theoremfor Besov spaces.

3.6.3 Lemma

(i) Let E0 × E1 → E2 be a multiplication of Banach spaces and

X ∈ BUC, C0, Lq, 1 ≤ q < ∞ .

VI.3 Anisotropy 69

Then(a 7→ (ψka)(D)

) ∈ L(M(Rd, E0), `∞

(L(X (Rd, E1),X (Rd, E2))))

and(a 7→ (ψka)(D)

) ∈ L(M(Rd, E0), `∞

(L(L∞(Rd, E1), BUC(Rd, E2))))

.

(ii) ψk(D) ∈ L(S(Rd, E)) ∩ L(S ′(Rd, E)

), k ∈ N .

(iii) Let 1 ≤ q ≤ ∞ and s ∈ R. Then there exists ` ∈ N such that

supk∈N

‖2ksψk(D)u‖q ≤ c q`,`(u) , u ∈ S , (3.6.6)

and (ψk(D)

) ∈ `s∞

(L(S(Rd, E), C0(Rd, E)))

.

Proof (1) Let a ∈M(Rd, E0). If we show

‖ψka‖FL1 ≤ c ‖a‖M , k ∈ N , (3.6.7)

then assertion (i) follows from Theorem 3.4.4.

(2) If k ∈qN, then ψka = σ2−k(ψσ2ka) and Proposition 3.1.3(ii) and (iii) imply

F−1(ψka) = 2|ω| kσ2kF−1(ψσ2ka)

and, consequently,

‖ψka‖FL1 = ‖F−1(ψσ2ka)‖1 = ‖ψσ2ka‖FL1 , k ∈qN .

Hence (3.6.7) is satisfied if

‖F−1(ψa)‖1 + ‖F−1(ψσ2ka)‖1 ≤ c ‖a‖M , k ∈qN . (3.6.8)

(3) By Leibniz’ rule and Proposition 3.1.3(i),

Dα(ψσ2ka) =∑

β≤α

(αβ

)2kβ ppppωDα−βψσ2k(Dβa) .

We denote by χY the characteristic function of a subset Y of Rd. Then, sincesupp(Dγψ) ⊂ Σ for γ ∈ Nd, we get

|Dα(ψσ2ka)(ξ)|E0 ≤ c∑

β≤α

2kβ ppppω |∂βa(2k q ξ)|E0χΣ(ξ)

for ξ ∈ Rd and k ∈qN.


We fix r > 1 with 1/r ≤ N(ξ) ≤ r for ξ ∈ Σ. Then

2kβ ppppω ≤ (2krN(ξ)

)β ppppω =(rN(2k q ξ))β ppppω

, ξ ∈ Σ , k ≥ 1 . (3.6.9)

Since N ≤ Λ1, it follows

|Dα(ψσ2ka)(ξ)|E0 ≤ c∑

β≤α

Λβ ppppω1 (2k q ξ) |∂βa(2k q ξ)|E0 χΣ(ξ)

≤ c ‖a‖MχΣ(ξ)(3.6.10)

for ξ ∈ Rd, α q ω ≤ k(ν), and k ∈qN.

Similarly, |∂βa|E0 ≤ Λβ ppppω1 |∂βa|E0 ≤ ‖a‖M implies

|Dα(ψa)|E0 ≤ c ‖a‖MχΩ0 , α q ω ≤ k(ν) . (3.6.11)

From (3.6.10) and (3.6.11) we see that these functions are integrable and

‖Dα(ψa)‖1 + supk≥1

‖Dα(ψσ2ka)‖1 ≤ c ‖a‖M , α q ω ≤ k(ν) .

Hence, by the Riemann-Lebesgue lemma (cf. Section III.4.2),

(−1)|α|xαF−1(ψa) = F−1(Dα(ψa)

)

and(−1)|α|xαF−1(ψσ2ka) = F−1

(Dα(ψσ2ka)

)

belong to C0(Rd, E0) and

‖xαF−1(ψa)‖∞ + supk≥1

∥∥xαF−1(ψσ2ka)‖∞ ≤ c ‖a‖M (3.6.12)

for α q ω ≤ k(ν). Thus, choosing α = 0 and α := 2 · 1d,

|F−1(ψa)(x)|E0 + supk≥1

|F−1(ψσ2ka)(x)|E0

≤ c ‖a‖M min1,

∏dj=1 |xj |−2

(3.6.13)

for x ∈ Rd. Consequently, we obtain (3.6.8) by integrating over Rd. This provesassertion (i).

(4) Since ψk ∈ D(Rd) → OM (Rd), claim (ii) follows from Theorem 3.4.1.(5) Let u ∈ S = S(Rd, E). Then, using F−1

((1− ∆)|ω|v

)= 〈·〉2 |ω|F−1v for

v ∈ S ′, we estimate

‖ψk(D)u‖q = ‖〈·〉−2 |ω|〈·〉2 |ω|ψk(D)u‖q

≤ c ‖〈·〉2 |ω|ψk(D)u‖∞ = c∥∥F−1F(〈·〉2 |ω|F−1(ψku)

)∥∥∞

= c∥∥F−1

((1 − ∆)|ω|(ψku)

)∥∥∞ ≤ c ‖(1− ∆)|ω|(ψku)‖1 ,

(3.6.14)

the last inequality being once more a consequence of the Riemann-Lebesgue lemma.

VI.3 Anisotropy 71

From

∂αψk = ∂ασ2−k ψ = 2−kα ppppωσ2−k∂αψ (3.6.15)

we get‖∂αψk‖∞ ≤ ‖∂αψ‖∞ , α ∈ Nd , k ∈ N .

Recall that ψk is supported in Ωk and write ξ = 2k q ξ with ξ ∈ Σ for k ≥ 1. Thenwe deduce from N(ξ) ≥ 1/r that 2k ≤ r2kN(ξ) = rN(ξ). Hence

2ks |∂αψk(ξ)| ≤ (rN(ξ)

)|s| |∂αψk(ξ)| ≤ c(α, s)〈ξ〉|s| , ξ ∈ Rd , (3.6.16)

for α ∈ Nd, where the last inequality follows from (3.3.18) and (3.6.15).We fix m ∈ N with m ≥ |s|. Then (3.6.16),

(1 − ∆)|ω| = (1 + D21 + · · · + D2

d)|ω| ,

and Leibniz’ rule imply

2ks ‖〈·〉2 |ω|(1 − ∆)|ω|(ψku)‖∞ ≤ c∑

|α|≤2 |ω|‖〈·〉m+2 |ω|∂αu‖∞ .

Hence it follows from (3.6.14) that

2ks ‖ψk(D)u‖q ≤ c 2ks ‖(1 − ∆)|ω|(ψku)‖1

≤ c 2ks‖〈·〉2 |ω|(1 − ∆)2 |ω|(ψku)‖∞ ≤ c qm+2 |ω|,2 |ω|(u)(3.6.17)

for u ∈ S and k ∈ N. Due to F ∈ L(S), there exists ` ∈ N such that

qm+2 |ω|,2 |ω|(u) ≤ c q`,`(u) , u ∈ S .

Thus, by (3.6.17), we get (3.6.6). Since ψk(D)u ∈ S ⊂ C0 by (ii), the second partof (iii) is clear in this case as well. ¥

For later use we record the following observations on the preceding proof.

3.6.4 Remarks (a) It holds

supk≥0

∥∥Λ2 |ω|1 F−1

(σ2k(ψk+ja)

)∥∥1≤ c(j) ‖a‖M

for a ∈M(Rd, E0) and j ∈ N.

Proof (1) Assume j = 0. Let ` := |ω|. From (3.3.18) and the multinomial theoremwe deduce

Λ2`1 (x) ≤ c 〈x〉2` = c

(1 + (x1)2 + · · · + (xd)2

)` = c∑

|β|≤`

(`β

)x2β . (3.6.18)


Moreover, β q ω + 2 |ω| ≤ max ω |β| + 2 |ω|. Thus, setting α := β + 21, it followsthat α q ω ≤ k(ν) if |β| ≤ `. Hence (3.6.12) and (3.6.18) imply, letting α = 0, resp.α = β + 21 with |β| ≤ `,

∣∣(Λ2`1 F−1(ψa)

)(x)

∣∣E0

+ supk≥1

∣∣(Λ2`1 F−1(ψσ2ka)

)(x)

∣∣E0

≤ c ‖a‖M min1,

∏dj=1|xj |−2

for x ∈ Rd. Now σ2k(ψka) = ψσ2ka for k ≥ 1 implies the claim.

(2) Suppose j > 0. Then supp(ψj) ⊂ Ωj = 2j q Σ. Now the assertion followsby replacing ψ and Σ by ψj and Ωj , respectively, in step (2) of the proof of Lem-ma 3.6.3. ¥

(b) In the proof of part (i) of this lemma we need only derivatives of order atmost 2 |ω|. Hence this part remains valid if M is replaced by M2 |ω|. ¥

3.7 Notes

This section consists of revisions and augmentations of Chapter 2 of [Ama09].

The anisotropic dilations which we consider are tailored to fit our purposes,the investigation of parabolic differential operators in particular. Reduced weightsystems are introduced so that we can work with the minimal number of para-meters. This will become clear when we consider anisotropic function spaces anddifferential operators in later sections (also see Example 3.4.3).

Weight systems introduced here are a convenient tool for collecting relevantdata. The anisotropic dilations associated with a weight system form a subclassof the general one-parameter group of dilations on the Euclidean d-space. A sys-tematic study of the latter is given in [SteW78]. In step (4) of the proof of Exam-ple 3.2.1(b) we follow J. Johnsen and W. Sickel [JoS07] (also see [CaT75]).

All papers in which anisotropic dilations play a role, and which we knowof (e.g., [CaT75], [Yam86]), employ the natural quasinorm E, which is alreadyintroduced in [FaR66]. In contrast, we base our investigations on the equivalentnatural quasinorm N which is easier to work with and particularly well-adaptedfor our purposes.

It should be noted that our use of ‘quasinorm’ is somewhat nonstandardand has a different meaning than the same notion employed by other authors(e.g., [RS96], [Tri83]).

The multiplier estimates in Subsection 3.5 are of a preparatory nature. Par-ticular care is taken to exhibit their dependence on the constants characterizingthe symbol classes. In order to reduce the number of those constants, we do notgive optimal estimates but content ourselves with qualitative bounds. In particu-lar, if A ∈ Ps

(Z;L(E); κ

), then the estimates depend on κ only, (and on [`, d, ν]

VI.3 Anisotropy 73

and E, which are fixed throughout). This will considerably facilitate the proof ofa priori estimates for concrete parabolic differential equations in later chapters.

The dyadic partition of unity introduced in Subsection 3.6 is an anisotropicvariant of the well-known Littlewood–Paley decomposition (e.g., [Ste93]). Start-ing with J. Peetre’s paper [Pee67], it has been the basis for the Fourier-analyticapproach to general isotropic Besov, Bessel potential, and Triebel–Lizorkin spaces(see one of the many books by H. Triebel, e.g; [Tri83], for detailed expositions andhistoric comments).

The anisotropic case is usually modeled straightforwardly after the isotropicone by means of Example 3.6.1(a) with Q = E (e.g., [Yam86] or [Joh95]). In con-trast to those works, we use the slightly more general and flexible definition (3.6.1).This does pay off in the proof of the trace theorem in Subsection VIII.1.2.

The preliminary Fourier multiplier results of Lemma 3.6.3 constitute ananisotropic extension of the results of Section 4 in [Ama97].

For simplicity, we do not consider dyadic decompositions of (Rd)q, although

simple modifications would suffice (cf. [Ama97]). Such extensions would be neededif we wanted to study homogeneous function spaces on Rd, which is not our inten-tion.

Chapter VII

Function Spaces

This chapter forms the core of this volume. Besides of the classical function spaces,the basic scales of Besov and Bessel potential spaces are introduced and studiedin great detail. Of course, Banach-space-valued anisotropic spaces are consideredthroughout.

The first section is devoted to the spaces of bounded continuous functionsand the Sobolev spaces. Since we do not make any assumption on the targetBanach spaces, the vector-valued Sobolev spaces do not fit into the framework ofthe general Besov and Bessel potential spaces. For this reason a separate study isappropriate.

The following two sections explore Besov spaces. Particular attention is paidto the subclass of little Holder spaces. The latter play a seminal role in the studyof nonlinear parabolic boundary value problems, realized in the third volume ofthis treatise.

Up to this point, no restrictions on the target Banach spaces are necessary,except for reflexivity assumptions in duality theorems. This is different in the caseof Bessel potential spaces whose investigation is executed in Section 4. For mostof their profound properties, restrictions on the geometry of the target Banachspaces have to be imposed.

In Section 5, Triebel–Lizorkin spaces are introduced and their main prop-erties are explored. In our setup they play only a technical role. The principalreason for their inclusion is the fact that they provide sharpenings of some embed-ding theorems without restrictions on the target Banach spaces. This is used toprove anisotropic vector-valued replications of the classical Gagliardo—Nirenberginequalities.

There are three more sections in this chapter. In the first one we establishoptimal and nearly optimal point-wise multiplier theorems. They are essential forthe study of boundary value problems under minimal regularity assumptions.


75


76 Function Spaces

Section 7 explores compact embedding theorems. After having related Besovand Bessel potential spaces on compact intervals to our general setup, we providefar-reaching extensions of the well-known Aubin–Lions theorem.

In the last section we introduce the technical tool of parameter-dependentfunction spaces and establish a number of seminorm estimates. This material isuseful for proving resolvent estimates for elliptic and parabolic differential opera-tors in the next volume.

VII.1 Classical Spaces 77

1 Classical Spaces

Classical Sobolev spaces and spaces of bounded continuous functions are, of course,well-known. However, in this section we are interested in vector-valued anisotropicversions of those spaces, defined on corners. Unfortunately, in the vector-valuedcase some of the most important function spaces, namely spaces of bounded con-tinuous functions and Sobolev spaces, do not fit into any of the general scales ofBanach spaces discussed in the subsequent sections. For this reason they have tobe treated separately.

After having introduced appropriate definitions in the first two subsections,we prove in the third subsection the basic restriction-extension theorems. Thisallows to carry over results proved on the full Euclidean space to spaces definedon corners. As a first application we show in Subsection 1.4 that, even on corners,anisotropic Sobolev spaces, which are defined by a completion procedure, can becharacterized by integrability conditions of distributional derivatives. This is thenused, in the following subsection, to establish their reflexivity, given appropriaterestrictions, and to give an explicit representation of spaces of negative order. Thelast subsection contains some elementary embedding properties.

It should be observed that, in this whole section, there are no restrictionson the Banach space, except for occasional reflexivity assumptions. Nevertheless,we can establish most of the properties, known in the scalar-valued setting, forvector-valued anisotropic Sobolev spaces.

It is assumed throughout that

• K is a corner in Rd.

• X ∈ Rd,K.

Moreover,1

∂ := (∂1, . . . , ∂d) = (∂x1 , . . . , ∂xd) , ∂xi := (∂x1i, . . . , ∂

xdii

) , 1 ≤ i ≤ ` .

We indiscriminately use

∇ := ∂ , ∇xj := ∂xj , ∇xi := ∂xi , 1 ≤ j ≤ d , 1 ≤ i ≤ ` .

If convenient, we identify ∇ = ∂ with the Frechet derivative on X. Then

∇m := ∇ ∇m−1 , ∇mxi

:= ∇xi ∇m−1xi

, ∇0 := ∇0xi

:= id

for m ∈qN. Finally, we recall that assumption (VI.3.1.20) holds throughout.

1The reader is cautioned to distinguish carefully between xi ∈ R, a coordinate of x ∈ Rm, andxi ∈ Rdi .

78 VII Function Spaces

1.1 Bounded Continuous Functions

First we show that uniform continuity ‘from the right’ already implies uniformcontinuity.

1.1.1 Lemma Suppose K is closed and u : X→ E. Then u is uniformly continuousiff

limy→0y∈X

u(x + y) = u(x), uniformly for x ∈ X . (1.1.1)

Proof It is clear that the condition is necessary.Let x, z ∈ X and denote by (e1, . . . , ed) the standard basis of Rd. Define

hi := zi − xi ∈ R and hj := hjej for 1 ≤ j ≤ d. Set x0 := x and xj := x +∑j

i=1 hi

for 1 ≤ j ≤ d. Then xd = xd−1 + hd = z and

|u(z) − u(x)|E ≤d∑

j=1

|u(xj−1 + hj) − u(xj)|E .

Put yj := |hj | ej . Then yj ∈ X and

|u(xj−1 + hj) − u(xj−1)|E =

|u(xj−1 + yj−1) − u(xj−1)|E , if hj ≥ 0 ,

|u(xj−1 − yj + yj) − u(xj−1 − yj)|E , if hj < 0 .


|u(xj−1 + hj) − u(xj−1)|E ≤ supx∈X

|u(x + yj)− u(x)|E → 0

as yj → 0. This proves that (1.1.1) is sufficient. ¥

For the reader’s convenience, we recall the definition of C0(X, E). Let X be aσ-compact metric space. Then u ∈ C(X, E) vanishes at infinity, if for each ε > 0there exists a compact set K such that |u(x)|E < ε for x /∈ K.

Suppose X is a σ-compact metric space which is a dense subset of some topo-logical space Y . If u ∈ C0(X,E) and y ∈ Y \X, then u(x) → 0 as x → y in Y . Henceu has a unique extension u ∈ C(Y, E), and u |Y \X = 0. Thus we can (and do)identify u with u. This applies, in particular, to X := K and Y := K if K is notclosed.

Banach Spaces of Bounded Continuous Functions

We simply write ‖·‖r for ‖·‖Lr(X,E), 1 ≤ r ≤ ∞, if (X, E) is clear from the context.Furthermore, ‖·‖∞ is also used for the supremum norm of B(X, E), the Banach

space of bounded E-valued functions on X.


Recall that j ∈ J∗ iff ∂jK ∩K = ∅. We define

BC(K, E) :=

u ∈ BC(K, E) ; u |∂jK = 0, j ∈ J∗ (1.1.2)

andBUC(K, E) :=

u ∈ BC(K, E) ; u is uniformly continuous

.

ThenC0(X, E) → BUC(X, E) → BC(X, E) → B(X, E)

and each space on the left of one of the symbols → is a closed linear subspace ofthe one on the right. Furthermore,

S(X, E)d

→ C0(X, E) , S(X, E)d

→ Lq(X, E) , (1.1.3)

as is well-known and easily seen.Let m ∈ νN. We put

|||·|||m/ν,r :=∑

α ppppω≤m

‖∂α · ‖r , 1 ≤ r ≤ ∞ . (1.1.4)

We denote by

BCm/ν(X, E) the linear subspace of BC(X, E) such that∂αu exists for α ∈ Nd with α q ω ≤ m (in the classical sense)and belongs to BC(X, E).

(1.1.5)

It is endowed with the norm |||·|||m/ν,∞. If X = K, then u ∈ BCm/ν(K, E) implies,due to (1.1.2),

∂kj u|∂jK = 0 for 0 ≤ k ≤ m/νj and j ∈ J∗. (1.1.6)

1.1.2 Lemma BCm/ν(X, E) is a Banach space.

Proof (1) Let (uj) be a Cauchy sequence. Then (∂αuj) is a Cauchy sequence inBC(X, E) for α q ω ≤ m. Hence there exist uα ∈ BC(X, E) with ‖∂αuj − uα‖∞converging to 0 as j →∞, and u0 = u.

Assume α = ek. Then

uj(x + hek) − u(x) = h

∫ 1

0

∂kuj(x + thek) dt , x, x + hek ∈ X .

Letting j →∞, we get

h−1(u(x + hek) − u(x)

)=

∫ 1

0

uek(x + thek) dt .

From this we infer that ∂ku exists and equals uek . Now we find inductively that∂αu exists and equals uα ∈ BC(X, E) for α q ω ≤ m. This proves the completenessof BCm(X, E). ¥


1.1.3 Remark BC0/ν(K, E) = BC(K, E). ¥

Spaces of bounded and uniformly continuously differentiable functions are intro-duced by

BUCm/ν(X, E) := u ∈ BCm/ν(X, E) ;∂αu is uniformly continuous for α q ω ≤ m .

(1.1.7)

This is a closed linear subspace of BCm/ν(X, E), hence a Banach space.The mean-value theorem implies

BCm/ν(X, E) → BUCk/ν(X, E) , k < m . (1.1.8)

Hence the Frechet space

BC∞(X, E) :=⋂

k∈NBCk(X, E)

satisfies

BC∞(X, E) =⋂

m∈νNBCm/ν(X, E) =

⋂

m∈νNBUCm/ν(X, E) . (1.1.9)

It follows from (VI.1.1.2) and (1.1.6) that S(X, E) → BUCm/ν(X, E). Hence

Cm/ν0 (X, E), the closure of S(X, E) in BUCm/ν(X, E) , (1.1.10)

is a well-defined Banach space.

1.1.4 Theorem

(i) C0/ν0 (X, E) = C0(X, E).

(ii) u ∈ Cm/ν0 (X, E) iff u ∈ BUCm/ν(X, E) and ∂αu ∈ C0(X, E) for α q ω ≤ m.

Proof We set

Xm(X, E) :=

u ∈ BUCm/ν(X, E) ; ∂αu ∈ C0(X, E) for α q ω ≤ m

.

(1) Clearly, S(X, E) ⊂ Xm(X, E). Hence Cm/ν0 (X, E) ⊂ Xm(X, E). Thus it

remains to show that S(X, E) is dense in Xm(X, E).(2) Let ε > 0 and u ∈ Xm(X, E). Then there exists ρ > 0 such that

|∂αu(x)|E < ε/3 , |x|∞ ≥ ρ , α q ω ≤ m . (1.1.11)

We fix χ ∈ D(Rd, [0, 1]

)satisfying χ(x) = 1 for |x|∞ ≤ 1 and χ(x) = 0 for |x|∞ ≥ 2,

and set χr(x) := χ(x/r) for r > 0. Leibniz’ rule implies (cf. step (2) of the proof


of Lemma VI.1.1.3)

‖∂α(χru − u)‖∞ ≤ ‖(χr − 1)∂αu‖∞ + c(α)r−1 |||u|||m/ν,∞ (1.1.12)

for α q ω ≤ m, where c(0) := 0. Note

‖(χr − 1)∂αu‖∞ ≤ sup|x|∞≥r

|∂αu(x)|E .

From this, (1.1.11), and (1.1.12) we see that we can fix r0 ≥ ρ such that

u0 := χr0u ∈ BUCm/ν(X, E)

satisfies

|||u0 − u|||m/ν,∞ ≤ 2ε/3 . (1.1.13)

Suppose X = K. Theorem 1.3.1, which we prove below, implies that (RK, EK)is an r-e pair for

(BUCn/ν(Rd, E), BUCn/ν(K, E)

), n ∈ νN. If X = Rd, then we

set (RX, EX) := (id, id). We put v0 := EXu0. Note that

v0(x) = 0 , x ∈ X ∩ [ |·|∞ ≥ 2r0] . (1.1.14)

Let ϕξ ; ξ > 0 be a mollifier.2 Recall that

ϕξ ∗ v0 ∈ BUC∞(Rd, E) , supp(ϕξ ∗ v0) ⊂ supp(v0) + [ |·|∞ ≤ ξ] , (1.1.15)

and ϕξ ∗ v0 → v0 in BUCm/ν(Rd, E) as ξ → 0. Hence we can fix η > 0 such that

|||ϕη ∗ v0 − v0|||m/ν,∞ < ε/3 . (1.1.16)

Assume K is closed. We set v := RX(ϕη ∗ v0). Then v ∈ BUC∞(X, E). More-over, (1.1.14) and (1.1.15) imply supp(v) ⊂ [ |·|∞ ≤ 2r0 + 2η]. Thus v belongs toD(X, E) ⊂ S(X, E). Since RX is the operator of point-wise restriction and RXv0

equals u0, we get from (1.1.13) and (1.1.16)

|||v − u|||m/ν,∞ ≤ |||v − RXv0|||m/ν,∞ + |||RXv0 − u|||m/ν,∞≤ |||ϕη ∗ v0 − v0|||m/ν,∞ + |||u0 − u|||m/ν,∞ < ε .

This proves that S(X, E) is dense in Xm(X, E) if K is closed.(3) Suppose K is not closed. We can assume K = L×M with L := (0,∞)` for

some ` ∈ 1, . . . , d whereM is a closed corner in Rd−`. We denote by Tt ; t ∈ R the group of right translations on Rd defined by

Ttw(x) := w(x − th) , x, h := (1`, 0) ∈ Rd ,

for w : Rd → E. It is easy to see (cf. Theorem X.7.6 in [AmE08]) that it is stronglycontinuous on BUCm/ν(Rd, E).

2Subsection III.4.


It follows from (VI.1.1.18) that EK = EL EM, where EL is the extension byzero. We set vt := TtEKu0 for t > 0. Then vt ∈ BUCm/ν(Rd, E), vt → EKu0 inBUCm/ν , and

vt(y) = 0 ,

y ∈ x ∈ Rd ; xi ≤ t for 1 ≤ i ≤ ` ∪ x ∈ K ; |x|∞ ≥ 2r0 .(1.1.17)

We fix τ > 0 such that

|||vτ − EKu0|||m/ν,∞ ≤ ε/6 ‖RK‖ , (1.1.18)

where ‖RK‖ is the norm of RK ∈ L(BUCm/ν(Rd, E), BUCm/ν(K, E)

). Next we

choose ζ ∈ (0, τ/2) such that w0 := ϕζ ∗ vτ satisfies

‖w0 − vτ‖m/ν,∞ ≤ ε/6 ‖RK‖ . (1.1.19)

Then w0 ∈ BC∞(Rd, E) and supp(w0) b K, as follows from (1.1.15) and (1.1.17).Hence w := RKw0 ∈ D(K, E) ⊂ S(K, E) and RKw0(x) = w0(x) for x ∈ K by Re-mark VI.1.2.4(b). Thus we find

‖w − u0‖BCm/ν(K,E) = ‖RK(w0 − EKu0)‖BCm/ν(K,E)

≤ ‖RK‖ ‖w0 − EKu0‖BCm/ν(Rd,E)

≤ ‖RK‖(|||w0 − vτ |||m/ν,∞ + |||vτ − EKu0|||m/ν,∞

) ≤ ε/3 ,

due to (1.1.18) and (1.1.19). Using (1.1.13), we find |||w − u|||m/ν,∞ < ε. HenceS(K, E) is also dense in Xm(K, E) if K is not closed. The theorem is proved. ¥

Vector Measures

Let X and Y be LCSs. We refer to Proposition V.1.4.8 for a proof and preciseexplanation of

X d→ Y =⇒ Y ′ → X ′ (1.1.20)

with respect to 〈·, ·〉Y .We denote by MBV (X∗, E) the Banach space of E-valued vector measures

on X∗ of bounded variation. Then (the generalized Riesz representation) Theo-rem 2.0.4 of the Appendix guarantees

C0(X∗, E)′ = MBV (X∗, E′) (1.1.21)

with respect to the duality pairing

〈µ, u〉C0 :=∫

X∗u dµ , µ ∈MBV (X∗, E′) , u ∈ C0(X∗, E) . (1.1.22)


HenceTµ :=

(u 7→ 〈µ, u〉C0

) ∈ S(X∗, E)′ = S ′(X, E′)

by Theorem VI.1.3.1. Furthermore, the map µ 7→ Tµ is linear, continuous, andinjective from MBV (X∗, E′) into S ′(X, E′). Thus we identify MBV (X∗, E′) witha linear subspace of S ′(X, E′) by identifying µ with Tµ. Then

MBV (X∗, E′) → S ′(X, E′) (1.1.23)

This shows that (1.1.21) holds with respect to 〈·, ·〉C0.

Suppose E is reflexive. Set

C−m/ν0 (X, E) :=

(C

m/ν0 (X∗, E′)

)′ (1.1.24)

with respect to 〈·, ·〉C0. Then

S(X∗, E′)d

→ Cm/ν0 (X∗, E′)

d→ C0(X∗, E′) , (1.1.25)

(1.1.20), and (1.1.21) imply

MBV (X∗, E) =(C0(X∗, E′)

)′→ C

−m/ν0 (X, E) → S ′(X, E) . (1.1.26)

Using Remark 2.0.1 of the Appendix, we identify L1(X∗, E′) = L1(X, E′)(Theorem 1.2.1 below) with a closed linear subspace of MBV (X∗, E′)

by identifying f ∈ L1(X∗, E′) with f dx ∈MBV (X∗, E′).

Then〈f dx, ϕ〉C0(X∗,E) =

∫

X

⟨f(x), ϕ(x)

⟩E

dx = 〈f, ϕ〉S(X∗,E)

for f ∈ L1(X, E′) and ϕ ∈ S(X∗, E). Since S(X∗, E)d

→ C0(X∗, E), it follows thatthe duality pairing

〈·, ·〉C0: MBV (X∗, E′) × C0(X∗, E) → C

is a (natural) restriction-extension (restriction in the first variable and extensionin the second one) of the duality pairing

〈·, ·〉S(X∗,E) : S ′(X, E′) × S(X∗, E) → C .

Thus it it feasible to simply write

〈·, ·〉 (1.1.27)

for either of them if no confusion seems likely.


1.2 Sobolev Spaces

In this subsection we introduce anisotropic E-valued Sobolev spaces. For this weneed the concept of regular E-valued distributions.

Regular Distributions

SupposeX ∈ BC, BUC, C0, Lr , 1 ≤ r ≤ ∞ ,

and set r := ∞ if X ∈ BC, BUC,C0. Then, given u ∈ X (X, E), Holder’s inequal-ity implies

|〈u, ϕ〉X|E =∣∣∣∫

Xuϕdx

∣∣∣E≤ ‖u‖r ‖ϕ‖r′ ≤ c ‖u‖r q2d,0(ϕ)

for ϕ ∈ S(X∗), where c := ‖〈·〉−2d‖r′ . Thus

Tu :=(ϕ 7→ 〈u, ϕ〉X

) ∈ L(S(X∗), E)

= S ′(X, E)

and(u 7→ Tu) ∈ L(X (X, E),S ′(X, E)

).

It is clear that u 7→ Tu is injective. Hence we can (and do)

identify X (X, E) with a linear subspace of S ′(X, E)by identifying u with the regular tempered X-distribution Tu.

Thus

X (X, E)d

→ S ′(X, E) , (1.2.1)

the density being a consequence of S(X, E) ⊂ X (X, E) and S(X, E)d

→ S ′(X, E),guaranteed by Theorem VI.1.2.3(i) (cf. (1.3.16) of the Appendix). It follows that,given α ∈ Nd and u ∈ X (X, E), the X-distributional derivative ∂αu is definedin S ′(X, E).

Basic Definitions

Let 1 ≤ q < ∞. We define (E-valued anisotropic Lq) Sobolev spaces of order m/νon X by3

Wm/νq (X, E) is the completion of S(X, E) in Lq(X, E)

with respect to the norm |||·|||m/ν,q.(1.2.2)

3In the standard isotropic case ` = 1 = ν, that is, ω = 1, we write s for s/1, whenever s ∈ R.


Since |||·|||m/ν,q ≥ |||·|||0/ν,q = ‖·‖q, this definition is meaningful and the second

embedding in (1.1.3) implies W0/ν

q (X, E) = Lq(X, E). Hence, by (1.2.1),

S(X, E)d

→ Wm/νq (X, E)

d→ Lq(X, E)

d→ S ′(X, E) . (1.2.3)

By Holder’s inequality,

〈u′, u〉Lr :=∫

X

⟨u′(x), u(x)

⟩E

dx

is well-defined for (u′, u) ∈ Lr′(X, E′) × Lr(X, E) and is called Lr duality pairing.Since X and X∗ differ by a d-dimensional Lebesgue null set only,

Lr(X, E) = Lr(X∗, E) , (1.2.4)

using obvious identifications.4 Thus (VI.1.3.2) and (1.2.1) imply

〈u′, u〉S(X∗,E) = 〈u′, u〉Lr

ifeither (u′, u) ∈ S(X, E′)× Lr(X, E) and r < ∞ ,

or (u′, u) ∈ Lr′(X, E′) × S(X∗, E) and r > 1 .

From this and the density of S in S ′ and in Lq it follows that 〈·, ·〉S(X∗,E) isuniquely determined by 〈·, ·〉Lr

, and vice versa. For this reason

we simply write 〈·, ·〉 for either of them and call it duality pairing,

respectively Lr or X distributional duality pairing, according to the context, if clar-ity requires it. This is consistent with (1.1.27).

For the reader’s convenience we recall the following fundamental duality re-sult.

1.2.1 Theorem Suppose E is reflexive or E′ is separable. Then

Lq(X, E)′ = Lq′(X, E′) , 1 ≤ q < ∞ ,

with respect to 〈·, ·〉. If q > 1, then Lq(X, E) is reflexive iff E is reflexive.

Proof (1) The first assertion follows from [DU77, Theorem 1 on p. 98, Theorem 1on p. 79, Corollary 4 on p. 82] and by the usual procedure for extending results fromfinite measure spaces to the σ-finite case (e.g., [HeS65, Proof of Thorem 20.19].Also see [KuJF77, Section 2.2] and [GaGZ74, Satz IV.1.14] for special cases.

4Recall that the elements of Lr are equivalence classes of measurable functions differing bynull sets only.


(2) Let q > 1. If E is reflexive, then the reflexivity of Lq(X, E) is a consequenceof (1). The converse assertion is implied by the isometry of E to the closed linearsubspace ϕe ; e ∈ E of Lq(X, E), where ϕ ∈ Lq(X) satisfies ‖ϕ‖q = 1, and thefact that a closed linear subspace of a reflexive space is reflexive. ¥

1.2.2 Remark It should be noted that this theorem remains true if X is replaced byan arbitrary σ-compact metrizable space and the Lebesgue measure by a positiveRadon measure. ¥

Sobolev spaces of negative order are defined ‘by duality’:

Suppose m ∈ νN and E is reflexive. Then

W−m/νp (X, E) :=

(W

m/νp′ (X∗, E′)

)′, 1 < p < ∞ ,

with respect to 〈·, ·〉.(1.2.5)

This definition is consistent with Theorem 1.2.1, and (1.1.20), (1.2.3), and (1.2.4)imply

Lp(X, E) → W−m/νp (X, E) → S ′(X, E) . (1.2.6)

1.2.3 Remark We put

‖u‖m/ν,r :=∑

i=1

m/νi∑

j=0

‖∇jxi

u‖r , ‖u‖0m/ν,r := ‖u‖r +

∑

i=1

‖∇m/νixi

u‖r

for m ∈ νN and 1 ≤ r ≤ ∞. Then

‖·‖0m/ν,r ≤ ‖·‖m/ν,r ≤ |||·|||m/ν,r .

In Subsection 4.3 below it is shown that

‖·‖0m/ν,q ∼ ‖·‖m/ν,q ∼ |||·|||m/ν,q , (1.2.7)

provided E belongs to suitable restricted classes of Banach spaces and K is closed.Equivalences (1.2.7) apply, in particular, in the most important case where E isfinite-dimensional. ¥

1.3 Restrictions and Extensions

It is the purpose of this subsection to prove that (RK, EK) restricts to an r-e pairfor the spaces introduced in the preceding subsections. This result is basic for thestudy of more general spaces of distributions on corners.


1.3.1 Theorem Let m ∈ νN and 1 ≤ q < ∞. Suppose either

X ∈ BCm/ν , BUCm/ν , Cm/ν0 ,Wm/ν

q (1.3.1)

or E is reflexive, q > 1, and

X ∈ C−m/ν0 ,W−m/ν

q . (1.3.2)

Then (RK, EK) is (that is, restricts to) an r-e pair for(X (Rd, E),X (K, E)

).

Proof It is obvious that we can assume that K is a standard k-corner.(1) We denote by ~ convolution in the multiplicative group

((0,∞), q) with

respect to the Haar measure dt/t and set L∗r := Lr

((0,∞), dt/t

). Thus

v ~ w(s) =∫ ∞

0

v(s/t)w(t) dt/t =∫ ∞

0

v(st)w(1/t) dt/t .

Hence v ~ w = w ~ v.Young’s inequality says

‖v ~ w‖L∗r≤ ‖v‖L∗

1‖w‖L∗

r.

From this we infer∫ ∞

0

|v ~ w|q ds =∫ ∞

0

∣∣∣∫ ∞

0

t1/qv(t)(s/t)1/qw(s/t) dt/t∣∣∣q

ds/s

≤ cq(v)q

∫ ∞

0

|s1/qw(s)|q ds/s = cq(v)q

∫ ∞

0

|w(s)|q ds ,

(1.3.3)

where cq(v) :=∫ ∞0

t−1+1/q |v(t)| dt.(2) Assume k ∈ J = JK. Set hn(t) := (−1)ntnh(t) for n ∈ N. We deduce from

(VI.1.1.6) and (VI.1.1.5)

‖εku‖qLq((−∞,0)×

k∂kK,E) =

∫ 0

−∞

∫

∂kK

∣∣∣∫ ∞

0

h(t)u(− ty; xk

)dt

∣∣∣q

dxk dy

=∫

∂kK

∫ ∞

0

∣∣∣∫ ∞

0

h(1/t)u(ty; xk

)dt/t

∣∣∣q

dy dxk

=∫

∂kK

∫ ∞

0

∣∣h ~ u(y; xk

)∣∣q dy dxk

for u ∈ S(K, E). Hence, by (1.3.3),

‖εku‖Lq((−∞,0)×k

∂kK,E) ≤ cq(h)(∫

∂kK

∫ ∞

0

∣∣u(y; xk

)∣∣q dy dxk)1/q

= cq(h) ‖u‖Lq(K,E) ,

where cq(h) < ∞ by (VI.1.1.5).


Using the first part of (VI.1.1.7), we find analogously

‖∂nk εku‖Lq((−∞,0)×

k∂kK,E) = ‖εn+1

k ∂nk u‖Lq((−∞,0)×

k∂kK,E)

≤ c(hn+1) ‖∂nk u‖Lq(K,E) .

(1.3.4)

Thus, by employing the second part of (VI.1.1.7) and writing α = (αk; αk) forα ∈ Nd, we find

‖∂αεku‖Lq((−∞,0)×k

∂kK,E) = ‖∂αk

k εk∂αku‖Lq((−∞,0)×k

∂kK,E)

≤ c(αk) ‖∂αu‖Lq(K,E) .(1.3.5)

From this and (VI.1.1.9) it follows

‖∂αEku‖Lq(Kk,k−1,E) ≤ c(α) ‖∂αu‖Lq(K,E) , α ∈ Nd , (1.3.6)

for u ∈ S(K, E).

Since Rk is the operator of point-wise restriction from Kk,k−1 to K, it isobvious that

‖∂αRku‖Lq(K,E) ≤ c ‖∂αu‖Lq(Kk,k−1,E) , α ∈ Nd , (1.3.7)

for u ∈ S(Kk,k−1, E).

(3) Suppose k ∈ J∗. Since Ek is the operator of trivial extension, it is clearthat

‖∂αEku‖Lq(Kk,k−1,E) ≤ ‖∂αu‖Lq(K,E) , α ∈ Nd ,

for u ∈ S(K, E). Recalling (VI.1.1.14), we infer from (1.3.4) and (1.3.5)

‖∂αRku‖Lq(K,E) ≤ c(α) ‖∂αu‖Lq(Kk,k−1,E) (1.3.8)

for u ∈ S(Kk,k−1, E). Now (1.3.6)–(1.3.8) and (VI.1.1.17), (VI.1.1.18) give

‖EKu‖W

m/νq (Rd,E)

≤ c ‖u‖W

m/νq (K,E)

, u ∈ S(K, E) , (1.3.9)

and‖RKu‖

Wm/ν

q (K,E)≤ c ‖u‖

Wm/ν

q (Rd,E), u ∈ S(Rd, E) .

Since S is dense in Wm/ν

q , it follows from (1.3.9) that there exists a unique exten-sion EK ∈ L(

Wm/ν

q (K, E), Wm/νq (Rd, E)

)of EK |S(K, E). From this, W

m/νq → S ′,

and the continuity of EK on S ′(K, E) we deduce that EK = EK |Wm/νq (K, E). Sim-

ilarly, we see that the unique extension RK ∈ L(W

m/νq (Rd, E),Wm/ν

q (K, E))

ofRK |S(Rd, E) equals RK |Wm/ν

q (Rd, E). This proves the assertion if X = Wm/ν

q .


(4) Assume k ∈ J . We deduce from (VI.1.1.5)–(VI.1.1.7) that

‖∂αEku‖B(Kk,k−1,E) ≤ c(m) ‖∂αu‖B(K,E) , α q ω ≤ m ,

if u ∈ BCm/ν(K, E), due to (1.1.4) and (1.1.6). Hence

‖Eku‖BCm/ν(Kk,k−1,E) ≤ c ‖u‖BCm/ν(K,E) , u ∈ BCm/ν(K, E) .

Trivially,‖∂αRku‖B(K,E) ≤ ‖∂αu‖B(Kk,k−1,E) , α q ω ≤ m ,

for u ∈ BCm/ν(Kk,k−1, E).

(5) Let k ∈ J∗. If u ∈ BCm/ν(K, E), then we infer from (1.1.2) that Eku be-longs to BCm/ν(Kk,k−1, E) and

‖Eku‖BCm/ν(Kk,k−1,E) ≤ ‖u‖BCm/ν(K,E).

Using (VI.1.1.14) we find, as in step (4),

‖Rku‖BCm/ν(K,E) ≤ c ‖u‖BCm/ν(Kk,k−1,E)

for u ∈ BCm/ν(Kk,k−1, E). From this and step (4) it follows that (RK, EK) is anr-e pair for

(BCm/ν(Rd, E), BCm/ν(K, E)

), where RKu and EKv are defined by

(VI.1.1.17) and (VI.1.1.18), resp., with u ∈ BCm/ν(Rd, E) and v ∈ BCm/ν(K, E).It is clear that Lemma VI.1.2.2 remains valid if S(K, E) and S(Rd, E) are re-

placed by BCm/ν(K, E) and BCm/ν(Rd, E), resp. Using this fact, BCm/ν → BC,and (1.2.1), we infer that RK on BCm/ν(Rd, E) and EK on BCm/ν(K, E) coincidewith the restriction of RK and EK resp., defined on tempered distributions. Thisproves the claim if X = BCm/ν .

(6) Suppose u ∈ BUC(K, E). Let ` ∈ N and ε > 0 be given. We fix T ≥ 1such that

2 ‖u‖∞∫ ∞

T

|h`(t)| dt < ε/2 .

Then we choose δ > 0 such that∫ T

0

|h`(t)|∣∣u(−txk, xk

)− u(−tyk, yk

)∣∣E

dt < ε/2

for

(x, y) ∈ (−∞, 0) ×k

∂kK with |x − y| < δ . (1.3.10)

Consequently, |ε`ku(x) − ε`

ku(y)|E < ε if (1.3.10) is satisfied. From this and thearguments used in steps (2) and (3) it follows that EKu and RKu are uniformlycontinuous if u is so. Now we easily derive that the assertion holds if X = BUCm/ν .


(7) By Theorem VI.1.2.3, (RK, EK) is an r-e pair for(S(Rd, E),S(K, E)

).

From this and step (6) we get the claim for X = Cm/ν0 .

(8) Suppose E is reflexive and q > 1. By the above, we know that (RK∗ , EK∗)is an r-e pair for

(W

m/νq′ (Rd, E′),Wm/ν

q′ (K∗, E′)). Thus Remark VI.1.2.1(c) guar-

antees that((EK∗)′, (RR∗)′

)is an r-e pair for

(W

−m/νq (Rd, E),W−m/ν

q (K, E)),

due to (1.2.5) and E′′ = E. Now the claim for X = W−m/ν

q follows from Corol-lary VI.1.3.2.

(9) Lastly, by replacing Wm/ν

q′ in step (8) by Cm/ν0 , we see that the assertion

applies to X = C−m/ν0 . ¥

1.3.2 Corollary

(i) RKu = RKu for u ∈ EKX (K, E).

(ii) (RK, EK) is an r-e pair for(BC∞(Rd, E), BC∞(K, E)

).

Proof The first assertion is immediate from Remark VI.1.2.4(b) and the secondone is obvious. ¥

1.4 Distributional Derivatives

Let m ∈ νqN. For clarity, we temporarily denote the X-distributional derivative of

order α of u ∈ S ′(X, E) by ∂αwu.

Suppose u ∈ BCm/ν(X, E) and α q ω ≤ m. It follows from (VI.1.1.2) and(1.1.6) that

〈∂αu, ϕ〉X = (−1)|α|〈u, ∂αϕ〉X , ϕ ∈ S(X∗, E) .

Hence

∂αu = ∂αwu , u ∈ BCm/ν(X, E) , α q ω ≤ m . (1.4.1)

Assume u ∈ Wm/ν

q (X, E). Definition (1.2.2) guarantees the existence of asequence (uj) in S(X, E) such that (∂αuj) converges for α q ω ≤ m in Lq(X, E)towards some uα, where u0 = u. We set ∂α

s u := uα and call it strong (X-)derivative

of u of order α. The construction of the completion implies that ∂αs u is uniquely

determined by u, independently of the particular sequence (uj).Since ∂α

w is an extension of the classical derivative on S(X, E), it holds

〈∂αuj , ϕ〉X = (−1)|α|〈uj , ∂αϕ〉X , ϕ ∈ S(X∗, E) .

Thus, letting j →∞, we get

〈∂αs u, ϕ〉X = (−1)|α|〈u, ∂αϕ〉X , ϕ ∈ S(X∗, E) .


Hence

∂αs u = ∂α

wu , u ∈ Wm/νq (X, E) , α q ω ≤ m . (1.4.2)

The following theorem shows that, conversely, the elements of BUCm/ν(X, E),respectively W

m/νq (X, E), can be characterized by their distributional derivatives.

1.4.1 Theorem Let K be closed in Rd, X ∈ Rd,K , and m ∈ νN.(i) Suppose X ∈ BC, BUC. Then

u ∈ Xm/ν(X, E) iff ∂αwu ∈ X (X, E) for α q ω ≤ m .

Moreover, ∂αu = ∂αwu.

(ii) Let 1 ≤ q < ∞. Then u ∈ Wm/ν

q (X, E) iff ∂αwu ∈ Lq(X, E) for α q ω ≤ m.

Furthermore, ∂αs u = ∂α

wu.

Proof (1) It follows from (1.4.1) that u ∈ Xm/ν(X, E) implies ∂αwu ∈ X (X, E)

and ∂αwu = ∂αu for α q ω ≤ m.

Similarly, (1.4.2) implies that it is a consequence of u ∈ Wm/ν

q (X, E) that∂α

wu ∈ Lq(X, E) and ∂αwu = ∂α

s u for α q ω ≤ m.(2) Assume X = Rd. Let u ∈ X (Rd, E) be such that v := ∂d,wu, the distribu-

tional derivative of u with respect to the last coordinate, also belongs to X (Rd, E).Fix t0 ∈ R and set

w(y, t) :=∫ t

t0

v(y, τ) dτ , (y, t) ∈ Rd−1 × R = Rd .

Thenw ∈ C(Rd, E) → L1,loc(Rd, E) → D′(Rd, E)

and

〈w, ∂dϕ〉Rd =∫

Rd−1

∫

R

∫ t

t0

v(y, τ) dτ ∂dϕ(y, t) dt dy

= −∫

Rd−1

∫

Rv(y, t)ϕ(y, t) dt dy = −〈v, ϕ〉Rd

for ϕ ∈ D(Rd). This shows that the distributional derivative of w with respect tothe last coordinate equals v. Hence ∂d(u− w) = v − v = 0. Thus it follows fromExample 1.8.6(b) of the Appendix that u− w = u1 ⊗ 1 for some u1 ∈ D′(Rd−1, E).Since u and w are continuous, u1 ∈ C(Rd−1, E). Observe that u = w + u1 ⊗ 1 ispoint-wise differentiable with respect to xd and ∂du = ∂dw = v ∈ X (Rd, E).

From this we obtain, by permutation of coordinates and induction, that∂α

wu ∈ X (Rd, E) for α ∈ Nd with α q ω ≤ m implies u ∈ Xm/ν(Rd, E).(3) Suppose X = Rd. Let u ∈ Lq(Rd, E) satisfy ∂α

wu ∈ Lq(Rd, E) for α ∈ Nd

with α q ω ≤ m. We choose ψ ∈ C∞(Rd, [0, 1]

)such that ψ(x) = 1 for |x| ≤ 1,


and ψ(x) = 0 for |x| ≥ 2. For r ∈ (0, 1] and x ∈ Rd we put ψr(x) := ψ(rx). Thenψr(x) = 1 for |x| ≤ 1/r, and ψr(x) = 0 for |x| ≥ 2/r. Moreover, ∂γψr = r|γ|(∂γψ)r

implies the estimates ‖∂γψr‖∞ ≤ ‖∂γψ‖∞ for 0 < r ≤ 1 and γ ∈ Nd. By Leibniz’rule

∂αw(ψru) =

∑

β≤α

(αβ

)∂α−βψr∂

βwu . (1.4.3)

Consequently,

‖∂αw(ψru)‖q ≤ c max

β≤α‖∂β

wu‖q , α q ω ≤ m , 0 < r ≤ 1 . (1.4.4)

Furthermore, (ψru− u)(x) = 0 for |x| ≤ 1/r, and (1.4.3) and (1.4.4) imply

‖∂αw(ψru− u)‖q ≤ c

(∫

[ |x|≥1/r]

|∂αwu|q dx

)1/q

→ 0 as r → 0

for α q ω ≤ m. This shows that ∂αw(ψru) ∈ Lq(Rd, E) for α q ω ≤ m and

|||ψru − u|||wm/ν,q → 0 as r → 0 , (1.4.5)

where |||·|||wm/ν,q means that ∂α in (1.1.4) is replaced by ∂αw.

Let ε > 0 be given and fix r ∈ (0, 1) with

|||ψru − u|||wm/ν,q < ε/2 . (1.4.6)

Assume ϕξ ; ξ > 0 is a mollifier. Since ψru has compact support, ϕξ ∗ (ψru)too has compact support. Hence

ϕξ ∗ (ψru) ∈ D(Rd, E) → S(Rd, E) , ξ > 0 ,

by (III.4.2.13). Furthermore, by (III.4.2.10) and (III.4.2.23), (III.4.2.24),

∂α(ϕξ ∗ (ψru)

)= ϕξ ∗ ∂α

w(ψru) −−→ξ→0

∂αw(ψru) , α q ω ≤ m ,

in Lq(Rd, E). This shows that there exists η > 0 such that

|||ϕη ∗ (ψru) − ψru|||wm/ν,q < ε/2 .

From this and (1.4.6) it follows that there is a sequence (uj) in S(Rd, E) suchthat |||uj − u|||wm/ν,q → 0 as j →∞. Thus (uj) is a Cauchy sequence in the norm

|||·|||m/ν,q, that is, in Wm/ν

q (Rd, E), and ∂αuj → ∂αwu in Lq(Rd, E) for α q ω ≤ m.

This proves ∂αwu = ∂α

s u for α q ω ≤ m.


(4) Suppose X = K and X ∈ BC, BUC, Lq. Set

Xm(K, E) :=

u ∈ X (K, E) ; ∂αwu ∈ X (K, E), α q ω ≤ m

.

Since K∗ = K, it is clear that EK∗∂αϕ = ∂αϕ for ϕ ∈ S(K∗) and α ∈ Nd, providedwe identify ϕ with its trivial extension over Rd.

Let u ∈ Xm(K, E) and ϕ ∈ S(K∗). Then, if α q ω ≤ m,

〈∂αwu, ϕ〉K = (−1)|α|〈u, ∂αϕ〉K = (−1)|α|〈RKEKu, ∂αϕ〉K .

Due to Lemma VI.1.2.2 and the above observation, the last term equals

(−1)|α|〈EKu, EK∗∂αϕ〉K = (−1)|α|〈EKu, ∂αϕ〉Rd = 〈∂α

wEKu, ϕ〉Rd .

By steps (2) and (3), we know that ∂αwEKu equals ∂αEKu if X ∈ BC, BUC,

and ∂αwEKu = ∂α

s EKu for X = Lq. Writing ∂ = ∂ if X ∈ BC,BUC, and ∂ = ∂s

if X = Lq, we see that

〈∂αwu, ϕ〉K =

⟨∂αEKu, ϕ

⟩Rd =

⟨∂αEKu, EK∗ϕ

⟩Rd =

⟨RK∂αEKu, ϕ

⟩K .

Since K is closed, (1.4.1), (1.4.2), and Theorem VI.1.2.3(iii) guarantee that

RK∂αEKu = ∂αRKEKu = ∂αu .

Consequently,

〈∂αwu, ϕ〉 =

⟨∂αu, ϕ

⟩, ϕ ∈ S(K∗) , α q ω ≤ m .

Thus ∂αwu = ∂αu for α q ω ≤ m and u ∈ X (K, E). This proves the theorem. ¥

Henceforth, we write again ∂α for the X-distributional derivatives.

1.5 Reflexivity

In this subsection we extend some well-known facts from the theory of classicalSobolev spaces to the present setting.

1.5.1 Theorem Let E be reflexive, 1 < p < ∞, and m ∈ νZ. Then Wm/ν

p (X, E) isreflexive.

Proof Since the dual of a reflexive space is reflexive, we can assume m ∈ νN.We endow α ∈ Nd ; α q ω ≤ m with the lexicographical ordering, denote

its cardinality by N , and equip Z :=(Lp(X, E)

)N with the `1 norm. It followsfrom Theorem 1.2.1 and the fact that products of reflexive spaces are reflexivethat Z is reflexive. The map u 7→ (∂αu)α ppppω≤m is an isometry from the dense


linear subspace S(X, E) of Wm/ν

p (X, E) into Z. Hence it has a unique isometricextension S over W

m/νp (X, E). Since W

m/νp (X, E) is complete, im(S) is a closed

linear subspace of Z. Consequently, Wm/ν

p (X, E) is reflexive, being isomorphic toa closed linear subspace of a reflexive Banach space. ¥

For the reader’s convenience we recall some basic facts from linear functionalanalysis. Let Y be a subset of some Banach space X. Then

Y ⊥ :=

x′ ∈ X ′ ; 〈x′, y〉X = 0, y ∈ Y

,

the annihilator, or polar set, of Y (in X ′), is a closed linear subspace of X ′.

1.5.2 Lemma Suppose Y is a linear subspace of a Banach space X. Then

f : X ′/Y ⊥ → Y ′ , x′ + Y ⊥ 7→ x′ |Yis an isometric isomorphism.

Proof Let z′ ∈ x′ + Y ⊥. Then z′ − x′ ∈ Y ⊥. Hence (z′ − x′) |Y = 0. Thus f is awell-defined linear map. Given y′ ∈ Y ′, by the Hahn-Banach theorem there is anx′ ∈ X ′ with x′ ⊃ y′. Consequently, f(x′ + Y ⊥) = x′ |Y = y′. This shows that f issurjective. For any z′ ∈ x′ + Y ⊥ it holds

‖f(x′ + Y ⊥)‖Y ′ = ‖z′ |Y ‖Y ′ ≤ ‖z′‖X′ .

Thus‖f(x′ + Y ⊥)‖Y ′ ≤ inf

z′∈x′+Y ⊥‖z′‖X′ = ‖x′ + Y ⊥‖X′/Y ⊥ .

On the other hand, the Hahn-Banach theorem guarantees the existence of v′ ∈ X ′

with v′ ⊃ x′ |Y and ‖v′‖X′ = ‖x′ |Y ‖Y ′ . Hence

‖f(x′ + Y ⊥)‖Y ′ = ‖x′ |Y ‖Y ′ = ‖v′‖X′ ≥ ‖x′ + Y ⊥‖X′/Y ⊥ .

Therefore f is an isometry. In particular, it is injective. ¥

Now we can give an explicit description of Sobolev spaces of negative order.

1.5.3 Theorem Let E be reflexive, m ∈ νqN, and 1 < p < ∞. Then u belongs to

W−m/ν

p (X, E) iff there exist uα ∈ Lp(X, E) such that

u =∑

α ppppω≤m

(−1)|α|∂αuα . (1.5.1)

Then

‖u‖W

−m/νp (X,E)

= inf maxα ppppω≤m

‖uα‖p ∼ inf∑

α ppppω≤m

‖uα‖p , (1.5.2)


the infimum being taken over all representations of u of the form (1.5.1). Further-more,

〈v, u〉 =∑

α ppppω≤m

〈∂αv, uα〉 , v ∈ Wm/ν

p′ (X∗, E′) , (1.5.3)

for any such representation.

Proof We use the notation of the proof of Theorem 1.5.1 and write X for theBanach space

(Lp′(X∗, E′)

)N . By Y we mean the closed linear subspace thereofwhich is the image of the isometry

T : Wm/ν

p′ (X∗, E′) → X , u 7→ (∂αu)α ppppω≤m .

It follows that

T ′ : Y ′ → (W

m/νp′ (X∗, E′)

)′ = W−m/νp (X, E)

is an isometric isomorphism.By Lemma 1.5.2,

f : X ′/Y⊥ → Y ′ , v + Y⊥ 7→ v |Y

is an isometric isomorphism, where

v = (vα) ∈ X ′ =((

Lp(X, E))N

, ‖·‖`∞(Lp)

).=

(Lp(X, E)

)N,

with `∞(Lp) := `∞(1, . . . , N, Lp(X, E)

)(cf. Theorem VI.2.1.1). Hence u belongs

to W−m/ν

p (X, E) iff there exists u + Y⊥ ∈ X ′/Y⊥ such that

u = T ′f(u + Y⊥) = T ′(u |Y) . (1.5.4)

If v ∈ S(X∗, E′), then

〈u, v〉 = 〈v, u〉W

−m/νp (X,E)

=⟨v, T ′(u |Y)

⟩W

−m/νp (X,E)

= 〈Tv, u〉X ′ = 〈u, T v〉X =∑

α ppppω≤m

〈uα, ∂αv〉 .

Assume v = ϕ⊗ e′ ∈ S(X∗) ⊗ E′. Then

〈uα, ∂αv〉 =⟨〈uα, ∂αϕ〉X∗ , e′

⟩E′

= (−1)|α|⟨〈∂αuα, ϕ〉X∗ , e′

⟩E′ = (−1)|α|〈∂αuα, v〉 (1.5.5)

by (VI.1.2.3). Theorem 1.3.6 of the Appendix guarantees that S(Rd) ⊗ E′ is densein S(Rd, E′). This, the fact that (RK, EK) is an r-e pair for

(S(Rd, E′),S(K∗, E′)),


and Lemma VI.1.1.2 imply that S(K∗) ⊗ E′ is dense in S(K∗, E′). Hence we getfrom (1.5.5), Lp(X, E) → S ′(X, E), and (VI.1.2.6) that

〈u, v〉 =∑

α ppppω≤m

〈uα, ∂αv〉 =⟨ ∑

α ppppω≤m

(−1)|α|∂αuα, v⟩

, v ∈ S(X∗, E′) . (1.5.6)

Now the density of S(X∗, E′) in Wm/ν

p′ (X∗, E′) implies that (1.5.1) and (1.5.3)

hold for this particular choice of u ∈ (Lp(X, E)

)N .We deduce from (1.5.4) and the isometry of T ′ and f that

‖u‖W

−m/νp (X,E)

= ‖f(u + Y⊥)‖Y′ = ‖u + Y⊥‖X ′/Y⊥ . (1.5.7)

Note that v ∈ Y⊥ iff v ∈ X ′ =(Lp(X, E)

)N and

0 = 〈v, T v〉X ′ =∑

α ppppω≤m

〈∂αv, vα〉 =⟨v,

∑

α ppppω≤m

(−1)|α|∂αvα

⟩(1.5.8)

for v ∈ S(X∗, E′). Now the assertions follow from (1.5.5)–(1.5.8). ¥

1.6 Embeddings

In this short subsection we present some easy embedding theorems for Sobolevspaces. More sophisticated ones will be given in later sections (cf. Theorems 2.6.5,2.6.6, 4.1.4, 4.3.2, and 5.6.5).

Let E0 and E1 be Banach spaces with E1 → E0. Suppose

X ∈ BC, BUC,C0,Wq . (1.6.1)

Then

Xm1/ν(X, E1) → Xm0/ν(X, E0) , m0,m1 ∈ νN , m1 ≥ m0 . (1.6.2)

Indeed, this is clear if X = Rd. Otherwise, it follows from this fact and Theo-rem 1.3.1.

Next we suppose that (E0, E1) is a densely injected Banach couple, that is,

E1d

→ E0. Then we prove the density of embedding (1.6.2) if X 6= BC. For this weneed the following observation.

1.6.1 Lemma If E1d

→ E0, then S(X, E1)d

→ S(X, E0).

Proof (1) It is clear that S(X, E1) → S(X, E0). By Corollary V.2.4.2 and embed-dings (III.4.1.5) we know

D(Rd, E1)d

→ D(Rd, E0)d

→ S(Rd, E0) .

Now the assertion is clear if X = Rd (cf. (1.3.15) and (1.3.16) of the Appendix).


(2) Let X = K. Then the claim follows from step (1), Theorem VI.1.2.3, andLemma VI.1.1.2. ¥

The following density statements are now easily obtained from this lemma.

1.6.2 Theorem Let (E0, E1) be a densely injected Banach couple and m0,m1 ∈ νZwith m1 ≥ m0, and 1 ≤ q < ∞. Then5

Wm1/νq (X, E1)

d→ Wm0/ν

q (X, E0) (1.6.3)

and

Cm1/ν0 (X, E1)

d→ C

m0/ν0 (X, E0) ,

provided m0 ≥ 0. If E0 and E1 are reflexive, then (1.6.3) holds for m1 ≤ 0 also.

Proof (1) If m0 ≥ 0, then the assertions follow from (1.6.2), Lemma 1.6.1, anddefinitions (1.1.10) and (1.2.2).

(2) Suppose E0 and E1 are reflexive and m1 ≤ 0. Recall

E1d

→ E0 =⇒ E′0

d→ E′

1 (1.6.4)

(cf. Proposition V.1.4.8). From this and step (1) we get

W−m0/ν

q′ (X∗, E′0)

d→ W

−m1/νq′ (X∗, E′

1) .

Since these spaces are reflexive, we obtain the second statement, due to definition(1.2.5), by invoking (1.6.4) once more. ¥

1.6.3 Remark If E1d

→ E0 and E0 and E1 are reflexive, then

C−m1/ν0 (X, E1) → C

−m0/ν0 (X, E0)

for m0,m1 ∈ νN with m0 ≥ m1.

Proof This follows from definition (1.1.10) and the arguments of step (2) of thepreceding proof. However, since the C0 spaces are not reflexive, we get the desiredembedding from (1.1.20). ¥

Our next theorem guarantees that the elements of BUCm/ν(X, E) can beapproximated by smooth functions.

5See Theorem 5.6.5 below for an important improvement of this embedding.


1.6.4 Theorem If m ∈ νN, then BC∞(X, E)d

→ BUCm/ν(X, E).

Proof The continuity of this embedding follows from (1.1.9).(1) If X = Rd, then the assertion is obtained by a standard mollification

argument (e.g., [AmE08, Theorem X.7.11] and Subsections 1.2 and 1.3 of theAppendix).

(2) Let X = K. Then we get the claim by invoking step (1), Theorem 1.3.1and its corollary, and Lemma VI.1.1.2. ¥

1.7 Notes

Theorem 1.3.1 is an amplification and sharpening of Theorem 4.4.3(i) of [Ama09].Part (2) of the proof of Theorem 1.4.1 follows L. Schwartz [Schw66, Theoreme VII](also see L. Hormander [Hor83, Theorem 3.1.7]). In the isotropic scalar-valuedsetting, the fact that strong and weak derivatives of Sobolev functions coincide,has first been proved by N.G. Meyers and J. Serrin [MeS64]. The remaining resultsof this section are straightforward extensions of classical results for scalar-valuedisotropic spaces.

VII.2 Besov Spaces 99

2 Besov Spaces

This and the next section contain a comprehensive exposition of the theory ofanisotropic Banach-space-valued Besov spaces on corners. It extends the Fourieranalytic approach from the isotropic scalar-valued case on Euclidean spaces to thepresent setting.

After having introduced the somewhat technical definition of Besov spaces,we present, in the second subsection, the basic embedding theorems for these scalesof Banach spaces. In addition, we put some emphasis on little Besov spaces, whichwill play an important role in connection with differential equations.

In Subsection 2.3 we prove a duality theorem for anisotropic Besov spaces,which, under the assumption that the target space E is reflexive or has a separabledual, is the perfect analogue of the duality theorem for isotropic scalar-valuedBesov spaces. The next subsection contains the fundamental lifting and Fouriermultiplier theorems. On the basis of these results we derive, in Subsection 2.6,useful renorming theorems and clarify the relations between Besov spaces and theclassical spaces studied in Section 1.

Subsection 2.7 contains the interpolation properties enjoyed by Besov spaces.Everything mentioned above is derived for Besov spaces on the ‘full’ Euclideanspace. In the last subsection we provide extensions to ‘corner spaces’ by buildingon the restriction-extension results of Section VI.1 and Subsection 1.3.

Throughout this section

• s ∈ R , 1 ≤ q, r ≤ ∞ . (2.0.1)

We simply write X for X (Rd, E) if the latter is a vector subspace of S ′ = S ′(Rd, E),provided no confusion seems likely. We also remind the reader that assumption(VI.3.1.20) holds throughout.

2.1 The Definition

We supposeX ∈ BUC, C0, Lq .

We fix a ν-admissible 0-neighborhood Ω of Rd and an Ω-adapted ψ ∈ D(Rd).Then

((Ωk), (ψk)

)is the ν-dyadic partition of unity on Rd induced by (Ω, ψ). By

applying Lemma VI.3.6.3 with E1 = E2 = E3 := C and a = 1 = 1Rd , we get

ψk(D) ∈ L(X ) , ‖ψk(D)‖L(X ) ≤ c , (2.1.1)

and

ψk(D)(L∞) ⊂ BUC , ‖ψk(D)‖L(L∞,BUC) ≤ c (2.1.2)

for k ∈ N.


Preliminary Estimates

We introduce a vector subspace of S ′ by

`s/νr X :=

u ∈ S ′ ; (ψk(D)u) ∈ `s

r(X )

(2.1.3)

and endow it with the norm

u 7→ ‖u‖`

s/νr X :=

∥∥(ψk(D)u

)∥∥`s

r(X ).

Moreover,

cs/ν0 X :=

(u ∈ S ′ ; (ψk(D)u) ∈ cs

0(X ), ‖·‖

`s/ν∞ X

). (2.1.4)

Of course, these definitions depend on the choice of (Ω, ψ). The next lemma shows,however, that another choice of these quantities leads to the same spaces, exceptfor equivalent norms.

2.1.1 Lemma Let Ωi be a ν-admissible 0-neighborhood of Rd and assume ψi isΩi-adapted for i = 1, 2. Denote by

((Ωi

k), (ψik)

)the ν-dyadic partition of unity

on Rd induced by (Ωi, ψi). Write `s/ν,ir X , resp. c

s/ν,i0 X , if (ψk) in (2.1.3), resp. in

(2.1.4), is replaced by (ψik). Then

`s/ν,1r X .= `s/ν,2

r X and cs/ν,10 X .= c

s/ν,20 X .

Proof We can fix m ∈ N with 2−m q Ω1 ⊂ Ω2 ⊂ 2m q Ω1. Then (recall (VI.3.6.1))

Σ2 = 2 q Ω2\2−1 q Ω2 ⊂ 2 q (2m q Ω1)∖

2−1 q (2−m q Ω1)

= 2m q (2 q Ω1)∖

2−m q (2−1 q Ω1) .(2.1.5)

We put, for k ∈ N,

χ1k,m :=

m+1∑

i=−m−1

ψ1k+i , ψ1

j := 0 for j < 0 .

From (VI.3.6.3) we infer that

χ1k,m =

k+m+1∑

j=0

ψ1j −

k−m−2∑

j=0

ψ1j = σ2−(k+m+1)ψ1 − σ2−(k−m−2)ψ1 .

Note that

ψ1(2−(k+m+1) q ξ) = 1 if 2−(k+m+1) q ξ ∈ (3/2) q Ω1 ,


that is, if ξ ∈ 2k+m q (3 q Ω1). Thus, in particular,

σ2−(k+m+1)ψ1(ξ) = 1 if ξ ∈ 2k q (2m q (2 q Ω1))

.

Furthermore, ψ1(2−(k−m−2) q ξ) = 0 if 2−(k−m−2) q ξ /∈ 2 q Ω1, that is, if

ξ /∈ 2k−m−1 q Ω1 = 2k q (2−m q (2−1 q Ω1))

.

This shows that

χ1k,m(ξ) = 1 if ξ ∈ 2k q ((2m q (2 q Ω1))

∖2−m q (2−1 q Ω1)

)

for k ≥ 1, and χ10,m(ξ) = 1 if ξ ∈ 2m q Ω1. Thus, by Ω2 ⊂ 2m q Ω1, and since (2.1.5)

impliesΩ2

k = 2k q Σ2 ⊂ 2k q ((2m q (2 q Ω1))∖

2−m q (2−1 q Ω1))

,

we obtain χ1k,m |Ω2

k = 1 for k ∈ N. From this and supp(ψ2k) ⊂ Ω2

k we get

ψ2k = ψ2

kχ1k,m , k ∈ N . (2.1.6)

Now it follows from (2.1.1) that

‖ψ2k(D)u‖X ≤ c ‖χ1

k,m(D)u‖X ≤ c

m+1∑

i=−m−1

‖ψ1k+i(D)u‖X

for u ∈ X and k ∈ N. This implies

2ks ‖ψ2k(D)u‖X ≤ c 2(m+1) |s|

m+1∑

i=−m−1

2(k+i)s ‖ψ1k+i(D)u‖X .

Consequently,∥∥(

ψ2k(D)u

)∥∥`s

r(X )≤ c(m, s)

∥∥(ψ1

k(D)u)∥∥

`sr(X )

,

that is, `s/ν,1r X → `

s/ν,2r X . Now we obtain the assertion by interchanging the roles

of((Ω1

k), (ψ1k)

)and

((Ω2

k), (ψ2k)

). ¥

2.1.2 Corollary Set

χk := ψk−1 + ψk + ψk+1 , ψ−1 := 0 , k ∈ N .

Then ψkχk = χkψk = ψk for k ∈ N.

Proof This follows from (2.1.6) if Ωi := Ω and ψi := ψ for i = 1, 2, since thenm = 0. ¥


A Retraction-Coretraction Pair

We introduce linear maps

R : `sr(X ) → S ′ , (vk) 7→ ∑

kχk(D)vk

andRc : `s/ν

r X → `sr(X ) , u 7→ (

ψk(D)u)

.

The following lemma shows, in particular, that they are well-defined.

2.1.3 Lemma

(i) `s/νr X is a Banach space and (R, Rc) is a retraction-coretraction pair for(`sr(X ), `s/ν

r X )and for

(cs0(X ), cs/ν

0 X ).

(ii) `s/νr L∞ = `

s/νr BUC.

(iii) `s/νr C0 is a closed linear subspace of `

s/νr BUC.

(iv) cs/ν0 X is a closed linear subspace of `

s/ν∞ X .

Proof (1) Suppose v = (vk) ∈ `sr(X ) and 0 ≤ m < n. By the evenness of ψk (cf.

(III.4.2.2)),

⟨ n∑

k=m

χk(D)vk, ϕ⟩Rd

=n∑

k=m

〈vk, χk(D)ϕ〉Rd , ϕ ∈ S(Rd) . (2.1.7)

Fix t < s. Then, by Corollary 2.1.2 and (ψjψk)(D) = ψj(D)ψk(D),

∣∣∣⟨ n∑

k=m

χk(D)vk, ϕ⟩Rd

∣∣∣E≤ 3

n+1∑

k=m−1

‖2tkvk‖X

∥∥(ψk(D)ϕ

)∥∥`−t∞

. (2.1.8)

By Lemma VI.3.6.3(iii), there exists ` ∈ N such that∥∥(

ψk(D)ϕ)∥∥

`−t∞

≤ c q`,`(ϕ) , ϕ ∈ S(Rd) .

Using this, we deduce from (2.1.8)

∣∣∣⟨ n∑

k=m

χk(D)vk, ϕ⟩Rd

∣∣∣E≤ c q`,`(ϕ)

n+1∑

k=m−1

‖2tkvk‖X , ϕ ∈ S(Rd) . (2.1.9)

Since

`sr(X ) → `t

∞(X ) (2.1.10)

by (VI.2.2.5), we infer from v ∈ `sr(X ) and (2.1.9) that

(⟨∑nk=0 χk(D)vk, ϕ

⟩Rd

)is a

Cauchy sequence in E, uniformly with respect to ϕ in bounded subsets of S(Rd).


Thus(∑n

k=0 χk(D)vk

)is a Cauchy sequence in L(S(Rd), E

)= S ′. By Proposi-

tion VI.1.2.5 we know that S ′ is complete. Hence Rv =∑

k χk(D)vk exists in S ′.From (2.1.9) and (2.1.10) we get

|〈Rv, ϕ〉Rd |E ≤ c q`,`(ϕ) ‖v‖`t∞(X ) ≤ c q`,`(ϕ) ‖v‖`sr(X ) , ϕ ∈ S(Rd) .

This implies

R ∈ L(`sr(X ),S ′) . (2.1.11)

It is obvious that

Rc ∈ L(`s/νr X , `s

r(X ))

. (2.1.12)

If u ∈ S ′, then ψk(D)u ∈ S ′ and χk(D)ψk(D)u ∈ S ′ by Lemma VI.3.6.3(ii). Hencewe get from Corollary 2.1.2

χk(D)ψk(D)u = F−1χkFF−1ψkFu = F−1χkψkFu = ψk(D)u .

Thus, given u ∈ `s/νr X , (2.1.11) and (2.1.12) imply

RRcu = limn→∞

n∑

k=0

χk(D)ψk(D)u = limn→∞

n∑

k=0

ψk(D)u = u in S ′ , (2.1.13)

due to Lemma VI.3.6.2.(2) Note supp(ψj) ∩ supp(χk) = ∅ if |j − k| ≥ 3. Hence, using (2.1.11) and

Lemma VI.3.6.3(ii),

ψj(D)Rv = limn→∞

ψj(D)n∑

k=0

χk(D)vk =j+2∑

k=j−2

ψj(D)χk(D)vk

for j ∈ N and v ∈ `sr(X ). From this and (2.1.1) we get

‖ψj(D)Rv‖X ≤ c

j+2∑

k=j−2

‖vk‖X , j ∈ N , v ∈ `sr(X ) . (2.1.14)

This implies‖Rv‖

`s/νr X =

∥∥(ψk(D)Rv

)∥∥`s

r(X )≤ c ‖v‖`s

r(X ) ,

that is,

R ∈ L(`sr(X ), `s/ν

r X ). (2.1.15)

Together with (2.1.11)–(2.1.13) this shows that (R,Rc) is a retraction-coretractionpair for

(`sr(X ), `s/ν

r X ).


(3) Suppose v ∈ `s∞(X ) and 2sk ‖vk‖X → 0 as k →∞. Then we see from

(2.1.14) that 2sj ‖ψj(D)Rv‖X → 0 as j →∞. Thus R ∈ L(cs0(X ), cs/ν

0 X ), due to

(2.1.15) with r = ∞. By (2.1.12) and the definition of Rc, it is obvious that Rc be-longs to L(

cs/ν0 X , cs

0(X )). Hence, taking (2.1.13) into consideration, we see that

(R,Rc) is a retraction-coretraction pair for(cs0(X ), cs/ν

0 X ).

(4) Since (R, Rc) is a retraction-coretraction pair for(`sr(X ), `s/ν

r X ), we know

from Lemma I.2.3.1 that `s/νr X is isomorphic to a closed linear subspace of the

Banach space `sr(X ). Hence it is complete. Analogously, we see that c

s/ν0 X is

complete.

(5) Clearly, `s/νr BUC is a linear subspace of `

s/νr L∞. Given u ∈ `

s/νr L∞, we

know ψk(D)u ∈ L∞ for k ∈ N. Hence χk(D)u ∈ L∞ for k ∈ N. Thus, by (2.1.2),

ψk(D)u = ψk(D)χk(D)u ∈ BUC , k ∈ N . (2.1.16)

This implies u ∈ `s/νr BUC and proves (ii).

(6) The fact that `s/νr C0 is a closed linear subspace of `

s/νr BUC follows, since

it is complete. The last assertion is obvious. ¥

It should be noted that (R, Rc) is a universal retraction-coretraction pair, thatis, these operators are independent of s and r.

The Final Definition

We define (anisotropic) Besov spaces (of E-valued distributions) on Rd by

Bs/νq,r = Bs/ν

q,r (Rd, E) := `s/νr Lq . (2.1.17)

Thusu ∈ Bs/ν

q,r iff u ∈ S ′ and(2ksψk(D)u

) ∈ `r(Lq) .

Furthermore,

‖u‖B

s/νq,r

=∥∥(

2ks ‖ψk(D)u‖q

)∥∥`r

. (2.1.18)

Observe that, by Lemma 2.1.1, Bs/νq,r (Rd, E) is independent of the particular choice

of (Ω, ψ), except for equivalent norms.

2.2 Embedding Theorems

The following elementary observation is the basis for the proof of the subsequentand many more embedding theorems.


2.2.1 Lemma Let Xi and Yi be LCSs and suppose (r, rc) is an r-c pair for (X1,Y1)

and (X2,Y2). If X1 → X2, then Y1 → Y2, with dense embedding if X1d

→ X2.

Proof Let iX : X1 → X2. Then

iY := r iX rc ∈ L(Y1,Y2) , iY(y) = r(rc(y)

)= y , y ∈ Y1 .

Hence iY : Y1 → Y2. If X1 is dense in X2, then Y1 = r(X1) is dense in Y2 = r(X2)by Lemma VI.1.1.2. ¥

By means of this lemma we can now easily derive most assertions of theadjacent basic embedding theorem for Besov spaces.

2.2.2 Theorem Suppose 1 ≤ r0, r1 ≤ ∞ and −∞ < s0 < s1 < ∞. Then

S → Bs/νq,r1

→ Bs/νq,r0

d→ S ′ , r1 < r0 , (2.2.1)

and

Bs1/νq,r1

→ Bs0/νq,r0

. (2.2.2)

If q0, q1 ∈ [1,∞] satisfy s1 − |ω|/q1 = s0 − |ω|/q0, then6

Bs1/νq1,r → Bs0/ν

q0,r . (2.2.3)

All embeddings (2.2.1) and (2.2.2) are dense if maxq, r0 < ∞. The one in (2.2.3)is dense if maxq0, r < ∞ .

Proof (1) The middle part of (2.2.1) and (2.2.2) follow from Theorem VI.2.2.2(i)and Lemmas 2.1.3 and 2.2.1.

(2) Suppose t > s. We get from (2.2.2) and Lemma VI.3.6.3(iii) that thereexists m ∈ N such that

‖u‖B

s/νq,r

≤ c ‖u‖B

t/νq,∞

= c∥∥(

ψk(D)u)∥∥

`t∞(Lq)≤ c qm,m(u)

for u ∈ S. This proves S → Bs/νq,r .

Assume maxq, r < ∞ and u ∈ Bs/νq,r . Let ε > 0. Since Rcu ∈ `s

r(Lq), by The-orem VI.2.2.2(i) there is v = (vk) ∈ cc(Lq) such that ‖v − Rcu‖`s

r(Lq) < ε/2. Usingthe density of S in Lq, we see that we can find w = (wk) ∈ cc(Lq) with wk ∈ S and‖w − v‖`s

r(Lq) < ε/2. Lemma VI.3.6.3(ii) guarantees that ψk(D)wk ∈ S. HenceRw =

∑k χk(D)wk ∈ S. From this and Lemma 2.1.3(i) we get

‖u− Rw‖B

s/νq,r

≤ ‖R‖ ‖Rcu − w‖`sr(Lq)

≤ ‖R‖ (‖Rcu− v‖`sr(Lq) + ‖v − w‖`s

r(Lq)

) ≤ ε ‖R‖ .

6Corollary 5.6.4 below contains an important complement to this embedding.


This shows that S is dense in Bs/νq,r , if maxq, r < ∞. Thus the first embedding

of (2.2.1) as well as its density if maxq, r1 < ∞ have been shown.

(3) From (2.1.11)–(2.1.13) we get Bs/νq,r0 → S ′. The density of this embedding

follows from S d→ S ′ and S → B

s/νq,r0 .

(4) Suppose q1 < q0. Imagine we have shown that the following anisotropicversion of Nikol′skiı’s inequality applies:

‖ϕ(D)u‖q0 ≤ cρ|ω| (1/q1−1/q0) ‖ϕ(D)u‖q1 (2.2.4)

for ϕ ∈ D(Rd) with supp(ϕ) ⊂ ρ q Ω0 and ρ > 0, and u ∈ S ′ with ϕ(D)u ∈ Lq1 .Then, letting ρ := 2k and ϕ := ψk, we obtain from s1 − s0 = |ω| (1/q1 − 1/q0)

2ks0 ‖ψk(D)u‖q0 = 2ks12k(s0−s1) ‖ψk(D)u‖q0 ≤ c 2ks1 ‖ψk(D)u‖q1

for k ∈ N. This implies

‖u‖B

s0/νq0,r

=∥∥(

ψk(D)u)∥∥

`s0r (Lq0 )

≤ c∥∥(

ψk(D)u)∥∥

`s1r (Lq1 )

= c ‖u‖B

s1/νq1,r

for u ∈ Bs1/νq1,r , that is, (2.2.3). The corresponding density assertion follows then

from the density of S in these spaces if maxq, r < ∞. Hence it remains to prove(2.2.4).

(5) We first assume ρ = 1. We fix λ ∈ D(Rd) with λ |Ω0 = 1. Then ϕ = λϕ.Since ϕ, λ ∈ D(Rd) ⊂ OM (Rd), it follows from (VI.3.4.2), and since F belongs toLaut(S) ∩ Laut(S ′), that

λ(D)ϕ(D)u = (F−1λF)(F−1ϕF)u = F−1λϕFu = ϕ(D)u , u ∈ S ′ .

By the convolution theorem, λ(D)ϕ(D) = F−1λ ∗ ϕ(D)u (cf. Remark 1.9.11(b)in the Appendix). Thus, since F−1λ ∈ S → Lq′1 , we get from Young’s inequality(III.4.2.22)

‖ϕ(D)u‖∞ ≤ ‖F−1λ‖q′1 ‖ϕ(D)u‖q1 = c ‖ϕ(D)u‖q1 .

Consequently,

‖ϕ(D)u‖q0 ≤ ‖ϕ(D)u‖1−q1/q0∞ ‖ϕ(D)u‖q1/q0q1

≤ c ‖ϕ(D)u‖q1 .

This proves (2.2.4) in the present case.(6) Now suppose ρ 6= 1 and supp(ϕ) ⊂ ρ q Ω0. Then supp(σρϕ) ⊂ Ω0. By Pro-

position VI.3.1.3(iii),

(σρϕ)(D)u = F−1((σρϕ)u

)= F−1

(σρ(ϕσ1/ρu)

)= σ1/ρF−1ϕF(σρu) .

Hence, by (ii) of Proposition VI.3.1.3,

‖(σρϕ)(D)u‖q = ρ|ω|/q ‖ϕ(D)(σρu)‖q .


Thus, by step (5),

ρ|ω|/q0 ‖ϕ(D)(σρu)‖q0 = ‖(σρϕ)(D)u‖q0 ≤ c ‖(σρϕ)(D)u‖q1

= cρ|ω|/q1 ‖ϕ(D)(σρu)‖q1

(2.2.5)

for u ∈ S ′ with ϕ(D)(σρu) ∈ Lq1 . Proposition VI.3.1.3 also implies σt ∈ Laut(Lq1).Thus we infer from (2.2.5) that (2.2.4) applies. ¥

Little Besov Spaces

We define very little Besov spaces by

Bs/νq,r = Bs/ν

q,r (Rd, E) is the closure of S in Bs/νq,r . (2.2.6)

We also introduce little Besov spaces by

bs/νq,r = bs/ν

q,r (Rd, E) is the closure of B(s+ν)/νq,r in Bs/ν

q,r . (2.2.7)

The following lemma gives useful characterizations of these Banach spaces.

2.2.3 Lemma It holds:

Bs/νq,r =

Bs/νq,r , if maxq, r < ∞ ,

cs/ν0 Lq , if q < ∞ , r = ∞ ,

`s/νr C0 , if q = ∞ , r < ∞ ,

cs/ν0 C0 , if q = r = ∞ ,

(2.2.8)

and

bs/νq,r =

Bs/νq,r , if r < ∞ ,

Bs/νq,∞ , if q < ∞ , r = ∞ ,

cs/ν0 BUC , if q = r = ∞ .

(2.2.9)

Moreover,

Bt/νq,r

d→ bs/ν

q,r , t > s . (2.2.10)

Proof (1) If maxq, r < ∞, then we get (2.2.8) from Theorem 2.2.2.(2) Lemma 2.1.3 (ii)–(iv) imply that each space on the right side of (2.2.8)

is a closed linear subspace of Bs/νq,r . Hence it suffices to show that S is contained

and dense in these spaces.Set Y := `s

r(X ) if r < ∞, Y := cs0(X ) if r = ∞, where X := Lq if q < ∞

and X := C0 for q = ∞. By Theorem VI.2.2.2 we know that cc(X ) is dense in Y.


Let u ∈ `s/νr X if r < ∞, resp. u ∈ c

s/ν0 X otherwise. Since S is dense in X , we can

approximate Rcu arbitrarily closely in Y by compactly supported sequences whoseelements belong to S. Thus we can find, for any given ε > 0, a finitely supportedsequence w = (wk) in Y ∩ SN satisfying ‖w − Rcu‖Y < ε. From this it follows,as in step (2) of the preceding proof, that Rw ∈ S and ‖u− Rw‖

Bs/νq,r

< ε. Thisproves (2.2.8).

(3) If maxq, r < ∞, then we know from Theorem 2.2.2 that

S d→ Bt/ν

q,r = Bt/νq,r

d→ Bs/ν

q,r , t > s .

This implies bs/νq,r = B

s/νq,r and (2.2.10) in the present case.

(4) By Theorem VI.2.2.2(i), `tr(L∞)

d→ `s

r(L∞) if r < ∞ and t > s. Fromthis and Lemmas 2.1.3 and 2.2.1 we get

Bt/ν∞,r

d→ Bs/ν

∞,r , r < ∞ , t > s .

This proves bs/ν∞,r = B

s/ν∞,r as well as (2.2.10) if r < ∞.

(5) Set X := Lq if q < ∞, and X := BUC if q = ∞. Then Theorem VI.2.2.2(i)and Lemmas 2.1.3 and 2.2.1 imply

`t/ν∞ X d

→ cs/ν0 X , t > s .

Now (2.2.9) and (2.2.10) follow for q < ∞ and r = ∞ from (2.2.8), and fromLemma 2.1.3(ii) if q = r = ∞ . ¥

The following results complement Theorem 2.2.2.

2.2.4 Theorem

(i) Let q < ∞ and 1 ≤ r0, r1 < ∞. If either s0 = s1 and r1 < r0, or s1 > s0, then

S d→ Bs1/ν

q,r1

d→ Bs0/ν

q,r0

d→ Bs0/ν

q,∞ = bs0/νq,∞

d→ S ′ .

(ii) S d→ B

s/ν∞,∞ → b

s/ν∞,∞ → B

s/ν∞,∞

d→ S ′.

(iii) Bs/νq,r1

d→ B

t/νq,r0 , b

s/νq,r1

d→ b

t/νq,r0 for s > t and 1 ≤ r0, r1 ≤ ∞.

(iv) Suppose s1 > s0 and 1 ≤ q0, q1 ≤ ∞ satisfy s1 − |ω|/q1 = s0 − |ω|/q0. Then

Bs1/νq1,r

d→ Bs0/ν

q0,r , bs1/νq1,r

d→ bs0/ν

q0,r .


Proof (1) All assertions (i)–(iii), except the second embedding in (ii), follow di-rectly from Theorem 2.2.2 and the characterization of the little Besov spaces givenin (2.2.9).

(2) Since Bs/ν∞,∞ = c

s/ν0 C0 and b

s/ν∞,∞ = c

s/ν0 BUC, we get B

s/ν∞,∞ → b

s/ν∞,∞ from

Theorem VI.2.2.2(ii) and Lemmas 2.1.3(i) and 2.2.1.

(3) Claim (iv) is a consequence of (2.2.3) and the definition of B and b. ¥

Embeddings With Varying Target Spaces

2.2.5 Theorem Let E1 → E0. Then

Bs/νq,r (Rd, E1) → Bs/ν

q,r (Rd, E0) . (2.2.11)

If E1 is dense in E0, then

Bs/νq,r (Rd, E1)

d→ Bs/ν

q,r (Rd, E0) . (2.2.12)

Proof Since, obviously, Lq(Rd, E1) → Lq(Rd, E0), we obtain (2.2.11) from The-orem VI.2.2.2(ii), and, once more, from Lemmas 2.1.3(i) and 2.2.1. The secondassertion follows now from Lemma 1.6.1. ¥

2.3 Duality

Assuming E to be reflexive or having a separable dual, we prove in this subsectiona duality theorem for vector-valued anisotropic Besov spaces. It is a vast general-ization of the isotropic scalar result. Recall the definition of 〈·, ·〉 in Subsection 1.2and B

s/νq,r = B

s/νq,r if maxq, r < ∞.

2.3.1 Theorem Suppose E is either reflexive or has a separable dual . Then

(Bs/ν

q,r (Rd, E))′ = B

−s/νq′,r′ (Rd, E′) (2.3.1)

with respect to 〈·, ·〉. If 1 < q, r < ∞ and E is reflexive, then Bs/νq,r (Rd, E) is also

reflexive.

Proof As before, we omit (Rd, E). Moreover, (Rd, E′) is replaced by E′.

(1) We set Xq := Lq for q < ∞, and X∞ := C0. It follows from Lemma 2.2.3that

Bs/νq,r =

`s/νr Xq , if r < ∞ ,

cs/ν0 Xq , if r = ∞ .

(2.3.2)


We put X ′q′ := Lq′(E′) if q < ∞, and X ′

1 := MBV (E′). Then, with respect to 〈·, ·〉,X ′

q′ = (Xq)′, due to Theorem 1.2.1 and (1.1.21) (recall (1.1.27)). Thus we get fromTheorem VI.2.2.2(iii)

(`sr(Xq)

)′ = `−sr′ (X ′

q′) if r < ∞ ,(cs0(Xq)

)′ = `−s1 (X ′

q′) , (2.3.3)

with respect to 〈〈·, ·〉〉.(2) Since S d

→ Bs/νq,r , we get from (1.1.20) and Theorem VI.1.3.1

(Bs/νq,r )′ → S ′(E′)

with respect to 〈·, ·〉. Thus

〈u′, u〉B

s/νq,r

= 〈u′, u〉 , u′ ∈ (Bs/νq,r )′ , u ∈ S . (2.3.4)

(3) It follows from (2.3.2) and Lemma 2.1.3(i) that

R ∈L(

`sr(Xq), Bs/ν

q,r

), if r < ∞ ,

L(cs0(Xq), Bs/ν

q,∞)

, if r = ∞ .(2.3.5)

Hence, by (2.3.3),

R′ ∈ L((Bs/ν

q,r )′, `−sr′ (X ′

q′))

. (2.3.6)

We define Su′ :=(χk(D)u′

). It is obvious that

S ∈ L(B−s/νq′,r′ (E′), `−s

r′ (Lq′(E′)))

. (2.3.7)

Let u′ ∈ B−s/νq′,r′ (E′). Then, by Holder’s inequality,

|〈〈Su′, v〉〉| ≤ ‖Su′‖`−s

r′ (Lq′ (E′)) ‖v‖`sr(Lq) (2.3.8)

for v ∈ `sr(Lq). Since C0 is a closed linear subspace of L∞ and cs

0 is one of `s∞, it

follows from (2.3.7) and (2.3.8) that

|〈〈Su′, v〉〉| ≤ c ‖u′‖B−s/ν

q′,r′ (E′) ‖v‖ ,

where v ∈ `sr(Xq) if r < ∞, and v ∈ cs

0(Xq) otherwise. Thus, by (2.3.5) and sinceRc is a coretraction for R,

|〈〈Su′, Rcu〉〉| ≤ c ‖u′‖B−s/ν

q′,r′ (E′) ‖u‖Bs/νq,r

for u ∈ Bs/νq,r . By the evenness of ψk,

〈〈Su′, Rcu〉〉 =∑

k

⟨χk(D)u′, ψk(D)u

⟩=

∑

k

⟨u′, χk(D)ψk(D)u

⟩

=⟨u′,

∑

k

χk(D)ψk(D)u⟩

= 〈u′, RRcu〉 = 〈u′, u〉


for u′ ∈ B−s/νq′,r′ (E′) → S ′(E′) and u ∈ S. From this and step (2) it follows

B−s/νq′,r′ (E′) → (Bs/ν

q,r )′ .

(4) For v′ ∈ `−sr′ (X ′

q′) we set Tv′ :=∑

k ψk(D)v′k. Since T has a similar struc-ture as Rc, we infer from the proof of Lemma 2.1.3 that

T ∈ L(`−sr′ (X ′

q′), `−s/νr′ X ′

q′)

. (2.3.9)

Suppose µk ∈MBV (E′). Then ψk(D)µk = F−1ψk ∗ µk. Hence

‖F−1ψk‖1 = 2|ω| k∥∥σ2kF−1ψ

∥∥1

=∥∥F−1ψ

∥∥1

, k ≥ 1 ,

implies ψk(D)µk ∈ L1(E′) and

‖ψk(D)µk‖L1(E′) ≤ c ‖µk‖MBV (E′) , k ∈ N ,

by Proposition 2.0.2 of the Appendix. Consequently,

T ∈ L(`−sr′ (MBV (E′)), `−s/ν

r′ L1(E′))

. (2.3.10)

Since X ′q′ = Lq′(E′) if 1 < q′ ≤ ∞, it follows from (2.3.9), (2.3.10), and (2.1.17)

that T ∈ L(`−sr′ (X ′

q′), B−s/νq′,r′ (E′)

)for all q and r. Thus, by (2.3.6),

TR′ ∈ L((Bs/ν

q,r )′, B−s/νq′,r′ (E′)

).

Let u′ ∈ (Bs/νq,r )′ and u ∈ S. Then, setting v′ := R′u′, we deduce from step (2)

〈TR′u′, u〉 =⟨∑

k

ψk(D)v′k, u⟩

=∑

k

⟨ψk(D)v′k, u

⟩=

∑

k

⟨v′k, ψk(D)u

⟩

= 〈〈v′, Rcu〉〉 = 〈〈R′u′, Rcu〉〉 = 〈u′, RRcu〉 = 〈u′, u〉 .

This proves (Bs/νq,r )′ → B

−s/νq′,r′ (E′). Together with step (3) we thus obtain (2.3.1).

(5) Suppose 1 < q, r < ∞. Then Xq = Lq is reflexive by Theorem 1.2.1. HenceTheorem VI.2.2.2(iii) guarantees that `s

r(Xq) is reflexive. By Lemmas I.2.3.1 and2.1.3(i), B

s/νq,r = `

s/νr Xq is isomorphic to a closed linear subspace of `s

r(Xq). Henceit is reflexive. ¥

2.3.2 Remark Since there are no restrictions on E in (1.1.21) and (VI.2.2.6), wesee that the equality

(Bs/ν∞,r(Rd, E)

)′ = B−s/ν1,r′ (Rd, E′)

is valid for any Banach space E. ¥


2.4 Fourier Multiplier Theorems

First we prove an important ‘lifting theorem’. Recall definition (VI.3.4.3) of J .

2.4.1 Theorem Let B ∈ B, B, b and t ∈ R. Then J t belongs to Lis(Bs+tq,r ,Bs

q,r)and (J t)−1 = J−t.

Proof By means of Leibniz’ rule, Example VI.3.3.9, and Lemma VI.3.3.2(ii) wefind

|∂α(Λt1ψk)| ≤ c max

β≤α(Λt−β ppppω

1 |∂α−βψk|) . (2.4.1)

Writing ξ = 2k q η ∈ Ωk with η ∈ Ω0, it follows from Λ(ξ) =(1 + (2kN(η)2ν)

)1/2ν

and 1/c ≤ N(η) ≤ c for η ∈ Σ, that

2k/c ≤ Λ(ξ) ≤ c 2k , ξ ∈ Ωk , k ∈ N .

Hence we get from (2.4.1)

Λα ppppω1 |∂α(Λt

1ψk)| ≤ cΛt1 max

β≤αΛ(α−β) ppppω

1 |∂α−βψk|

= cΛt1 max

β≤αΛ(α−β) ppppω

1 2−k(α−β) ppppω ∣∣∂α−βψ∣∣ ≤ c 2kt

for k ≥ 1. Since |∂γψ0| ≤ c for γ ∈ Nd, it follows

2−ktΛt1ψk ∈M(Rd) , ‖2−ktΛt

1ψk‖M ≤ c , k ∈ N . (2.4.2)

Let

X ∈ BUC, C0, Lq . (2.4.3)

We infer from (2.4.2) and Lemma VI.3.6.3

‖(ψj2−ktΛt1ψk)(D)‖L(X ) ≤ c , k, j ∈ N . (2.4.4)

Note that ψk = ψkχk = ψkχ2k implies

ψkΛt1 = 2kt(2−ktΛt

1ψkχk) = 2kt( 1∑

i=−1

(2−ktΛt1ψk)ψk+i

)χk .

Hence we obtain from (2.4.4)

‖ψk(D)J tu‖X ≤ c 2kt ‖χk(D)u‖X ≤ c

1∑

i=−1

2(k+i)t ‖ψk+i(D)u‖X


for k ∈ N, provided u ∈ S ′ is such that ψk(D)u ∈ X. From this we deduce

‖RcJ tu‖`sr(X ) ≤ c ‖Rcu‖`s+t

r (X ) if Rcu ∈ `s+tr (X ) ,

and‖RcJ tu‖cs

0(X ) ≤ c ‖Rcu‖cs+t0 (X ) if Rcu ∈ cs+t

0 (X ) .

Thus we get from Lemma 2.1.3(i)

‖J tu‖`

s/νr X = ‖RRcJ tu‖

`s/νr X ≤ c ‖RcJ tu‖`s

r(X ) ≤ c ‖Rcu‖`s+tr (X )

≤ c ‖u‖`(s+t)/νr X ,

if u ∈ `(s+t)/νr X . Similarly,

‖J tu‖c

s/ν0 X ≤ c ‖u‖

c(s+t)/ν0 X , u ∈ c

(s+t)/ν0 X .

Now the assertion follows from (2.1.17) and Lemmas 2.1.3(ii) and 2.2.3. ¥

The following general Fourier multiplier theorem for Besov spaces is obtained bya suitable modification of the arguments used in the preceding proof.

2.4.2 Theorem Let B ∈ B, B, b. Then:

(i)(m 7→ m(D)

) ∈ L(M(Rd,L(E0, E1)

),L(Bs/ν

q,r (Rd, E0),Bs/νq,r (Rd, E1)

)).

(ii) Assume m1 ∈M(Rd,L(E0, E1)

)and m2 ∈M(

Rd,L(E1, E2)). Then

m2m1 ∈M(Rd,L(E0, E2)

)(2.4.5)

and m2m1(D) = m2(D)m1(D).

Proof (1) Let (2.4.3) be satisfied and write Xi := X (Rd, Ei). Using ψkχk = ψk,Lemma VI.3.6.3(i) gives

‖ψk(D)m(D)u‖X1 = ‖(ψkχk)(D)m(D)u‖X1 = ‖(ψkm)(D)χk(D)u‖X1

≤ c ‖m‖M1∑

i=−1

‖ψk+i(D)u‖X0

for u ∈ S ′ with ψj(D)u ∈ X0 for j ∈ N. This implies for u ∈ S ′

‖Rcm(D)u‖`sr(X1) ≤ c ‖m‖M ‖Rcu‖`s

r(X0) if Rcu ∈ `sr(X0) ,

and‖Rcm(D)u‖cs

0(X1) ≤ c ‖m‖M ‖Rcu‖cs0(X0) if Rcu ∈ cs

0(X0) .

Now we apply Lemma 2.1.3(i), similarly as in the last part of the preceding proof,to arrive at assertion (i).


(2) Since the composition

L(E1, E2)× L(E0, E1) → L(E0, E2) , (m2,m1) 7→ m2m1

is a multiplication, (2.4.5) follows from Lemma VI.3.4.5(ii). Claim (ii) is thenimplied by F−1m2m1F = F−1m2FF−1m1F . ¥

Occasionally, the following simpler Fourier multiplier theorem is also useful.

2.4.3 Theorem Assume B ∈ B, B, b. Then(m 7→ m(D)

) ∈ L(FL1

(Rd,L(E0, E1)

),L(Bs/ν

q,r (Rd, E0),Bs/νq,r (Rd, E1)

)).

Proof Let (2.4.3) be satisfied. Since ψk(D) and m(D) commute, it follows fromTheorem VI.3.4.4 that

‖ψk(D)m(D)u‖X (Rd,E1) ≤ ‖m‖FL1 ‖ψk(D)u‖X (Rd,E0) , k ∈ N ,

provided ψk(D)u ∈ X (Rd, E0). Now the assertion follows by invoking the argu-ments of the preceding proof. ¥

By combining Theorem 2.4.2 with Theorem 2.4.1 we obtain Fourier multipliertheorems involving Besov spaces of different order.

2.4.4 Proposition Let B ∈ B, B, b and s, t ∈ R. Suppose Λ−s1 a is an element of

M(Rd,L(E1, E0)

). Then

(i) a(D) ∈ L(B(s+t)/νq,r (Rd, E1),Bt/ν

q,r (Rd, E0))

and ‖a(D)‖ ≤ c ‖Λ−s1 a‖M, t ∈ R.

(ii) Assume, in addition, that a ∈ C(Rd,Lis(E1, E0)

)and Λs

1a−1 belongs to the

space B(Rd,L(E0, E1)

). Then

a(D) ∈ Lis(B(s+t)/ν

q,r (Rd, E1),Bt/νq,r (Rd, E0)

), a(D)−1 = a−1(D) ,

and ‖a(D)−1‖ ≤ c(‖Λ−s

1 a‖M, ‖Λs1a

−1‖∞).

Proof (1) It follows from Theorem 2.4.2 that

(Λ−s1 a)(D) ∈ L(B(s+t)/ν

q,r (Rd, E1),B(s+t)/νq,r (Rd, E0)

)(2.4.6)

and ‖(Λ−s1 a)(D)‖≤ c ‖Λ−s

1 a‖M. Proposition VI.3.4.2 gives Js ∈ Laut(S ′(Rd, E0)

),

and (2.2.1) guarantees B(s+t)/νq,r (Rd, E0) → S ′(Rd, E0). Hence we infer from (2.4.6)

a(D)u = F−1aFu = F−1Λs1(Λ

−s1 a)F = F−1Λs

1FF−1Λ−s1 aFu

= Js(Λ−s1 a)(D)u

for u ∈ B(s+t)/νq,r (Rd, E1). Now (i) follows from (2.4.6) and Theorem 2.4.1.


(2) Let the additional hypotheses be satisfied. Then Lemma VI.3.4.6(i) im-plies Λs

1a−1 ∈M(

Rd,L(E0, E1)). From this and Theorem 2.4.2 we obtain, simi-

larly as above, that

a−1(D) = J−s(Λs1a

−1)(D) ∈ L(Bt/νq,r (Rd, E0),B(s+t)/ν

q,r (Rd, E1))

and ‖a−1(D)‖ ≤ c(‖Λ−s

1 a‖M, ‖Λs1a

−1‖∞). Furthermore, part (ii) of Theorem 2.4.2

impliesa−1(D)a(D) = (a−1a)(D) = 1B(s+t)/ν

q,r (Rd,E1).

Similarly, a(D)a−1(D) = 1Bt/νq,r (Rd,E0)

. This proves (ii). ¥

2.4.5 Remarks (a) Suppose s ∈ R and a ∈ Hs(Z, E). Then Λ−s1 a1 ∈M(Rd, E)

and ‖Λ−s1 a1‖M ≤ c ‖a‖Hs .

Proof Example VI.3.3.9 and Lemma VI.3.3.6 imply Λ−sa ∈ H0(Z, E) and theestimate ‖Λ−sa‖H0 ≤ c ‖a‖Hs . Hence Λ−s

1 a1 = (Λ−sa)1 ∈M(Rd, E) by LemmaVI.3.4.5, and ‖Λ−s

1 a1‖M ≤ ‖Λ−sa‖H0 . This implies the claim. ¥

(b) Proposition 2.4.4 is also valid if M is replaced by FL1.

Proof Invoke Theorem 2.4.3. ¥

2.5 Operators of Positive Type

As a first application of the multiplier results in the preceding subsection we proveresolvent estimates for Fourier multiplier operators. They are of importance forthe study of elliptic operators. The second main theorem of this subsection isa representation theorem for holomorphic functions of such operators. It is thebasis for kernel representations of semigroups on Besov spaces, for example. Suchapplications are postponed to Volume III.

We use the notation introduced in Subsections VI.3.4 and VI.3.5, the classes(VI.3.4.8) and (VI.3.5.2) in particular.

Resolvent Estimates

2.5.1 Theorem Suppose E1 → E0, B ∈ B, B, b, κ ≥ 1, 0 ≤ ϑ < π, and s ≥ 0.If

a ∈ Ps

(Z,L(E1, E0); κ, ϑ

), (2.5.1)

then

a1(D) ∈ P(L(B(s+t)/νq,r (Rd, E1),Bt/ν

q,r (Rd, E0)); c(κ), ϑ

)(2.5.2)

for t ∈ R.


Proof Let (2.5.1) be satisfied. Then a ∈ Hs

(Z,L(E1, E0)

)and ‖a‖Hs

≤ c(κ).Hence Λ−s

1 a ∈M(Rd,L(E1, E0)

)and ‖Λ−s

1 a‖M ≤ c(κ) by Remark 2.4.5(a). FromExample VI.3.3.9 and Remark VI.3.4.7(a) we infer

Λ−s1 λ ∈M(

Rd,L(E1, E0))

, ‖Λ−s1 λ‖M ≤ c |λ|

for7 λ ∈ C. Now we deduce from Proposition 2.4.4(i) that

λ + a1(D) ∈ L(B(s+t)/νq,r (Rd, E1),Bt/ν

q,r (Rd, E0))

, ‖λ + a1(D)‖ ≤ c(κ)(1 + |λ|)

for λ ∈ C and t ∈ R.(2) We get from Lemma VI.3.5.1 and Proposition 2.4.4(i) that

(λ + a1)−1(D) ∈ L(Bt/νq,r (Rd, E0),B(js+t)/ν

q,r (Rd, Ej))

(2.5.3)

and

(1 + |λ|)1−j ‖(λ + a1)−1(D)‖(j) ≤ c(κ) , λ ∈ Sϑ , j = 0, 1 , t ∈ R ,

where ‖·‖(j) is the norm in the space occurring in (2.5.3).

(3) It follows from Lemma VI.3.5.1 that λ + a1 ∈ C(Rd,Lis(E1, E0)

)and

Λs1(λ + a1)−1 ∈ B

(Rd,L(E0, E1)

)

with ‖Λs1(λ + a1)−1‖∞ ≤ c(κ) for λ ∈ Sϑ. Thus, since

Λs1(λ + a1) =

(Λ−s

1 (λ + a1))−1

, λ ∈ Sϑ ,

we infer from part (ii) of Proposition 2.4.4 and step (1) that

(λ + a1)−1(D) =(λ + a1(D)

)−1, λ ∈ Sϑ . (2.5.4)

From this, step (1) (with λ = 0), and step (2) we get the assertion. ¥

2.5.2 Remark Let κ ≥ 1 and ϑ = π/2. Then definition (VI.3.5.7) shows that ϕequals π/2− arcsin(1/2κ). Recall that Γ stands for the positively oriented bound-ary of W = Sϕ ∩ [ |z| ≥ 1/2κ]. Suppose A ∈ P(L(E); κ

). It follows that

e−tA :=1

2πi

∫

Γ

e−tλ(λ− A)−1 dλ , t > 0 , (2.5.5)

is well-defined in L(E). We put e−0A := 1E . Then the Dunford–Taylor functionalcalculus shows that e−tA ; t ≥ 0 is an analytic semigroup on E, that is, in L(E),which is strongly continuous iff A is densely defined. Hereby, analytic means that

7As usual, we identify λ ∈ C with λiE1,E0 , where iE1,E0 is the injection E1 → E0.


t 7→ e−tA : (0,∞) → L(E) has an analytic extension over the sector Sarcsin(1/2κ).Furthermore, A is uniquely determined by this semigroup and vice versa. −A issaid to be the infinitesimal generator of e−tA ; t ≥ 0 . This is justified, sinceim(e−tA) ⊂ dom(A) and ∂(e−tA) = −Ae−tA for t > 0.

Proof See Section 2.1 in Chapter 2 of A. Lunardi [Lun95]. Based on the Dunford–Taylor representation (2.5.5), there is carried out a detailed study of analyticsemigroups, which are not necessarily strongly continuous. ¥

2.5.3 Corollary Suppose a ∈ Ps

(L(E1, E0); κ). Then −a1(D) generates an analytic

semigroup on Bt/νq,r (Rd, E0).

A Representation Theorem

Now we restrict our considerations to the case where E0 = E1, that is, we assume

a ∈ Ps

(Z,L(E); κ, ϑ

)

and use the notations of Subsection VI.3.5. It follows from Theorem 2.5.1 that

a1(D) ∈ P(L(B(s+t)/νq,r ,Bt/ν

q,r

); c(κ), ϑ

) ⊂ P(L(Bt/νq,r ); c(κ), ϑ

). (2.5.6)

Hence

h(a1(D)

):=

12πi

∫

Γ

h(λ)(λ − a1(D)

)−1dλ (2.5.7)

is a well-defined element of L(Bs/νq,r ). The following theorem gives a natural repre-

sentation formula for h(a1(D)

)in terms of h(a1).

2.5.4 Theorem Let h : Sψ → C be holomorphic for some ψ ∈ (ϕ, π − ϑ) and letthere exist δ > 0 such that |z|δ h(z) → 0 as |z| → ∞. Then

h(a1(D)

)= h(a1)(D) . (2.5.8)

Proof (1) We set X := M(Rd,L(E)

)and A := a1, respectively X := L(Bt/ν

q,r ) andA := a1(D). Furthermore, κ := κ ∨ c(κ), where c(κ) is the constant occurring in(2.5.6), and ϕ := ϕ(κ, ϑ). Then Γ is the positively oriented boundary of W := Wκ,ϑ

(cf. (VI.3.5.7)) and ΓR := Γ ∩ [ |z| ≤ R] for R > 2κ. In (VI.3.5.13) we can replace Γby Γ. Similarly, in (2.5.7) we deform Γc(κ),ϑ into Γ. Then

IR(A) :=1

2πi

∫

ΓR

h(λ)(λ − A)−1 dλ ∈ L(X ) , 2κ < R ≤ ∞ .


In the first case, Proposition VI.3.5.5 implies IR(a1) → h(a1) in M(Rd,L(E)

)as

R →∞. Hence, by Theorem 2.4.2,

IR(a1)(D) → h(a1)(D) in L(Bt/νq,r )

as R →∞. In the second case, we infer from (2.5.6) and the assumption on h thatthere exists R0 such that

|h(λ)| ∣∣(λ− a1(D))−1∣∣

L(Bt/νq,r )

≤ 1/|λ|1+δ , λ ∈ ΓR ∩ [ |z| ≥ R] ,

for R ≥ R0. Consequently,

IR

(a1(D)

) → 12πi

∫

Γ

h(λ)(λ − a1(D)

)−1dλ in L(Bt/ν

q,r )

as R →∞. Thus it suffices to show that

IR(a1)(D) = IR

(a1(D)

), R ≥ R0 .

(2) We fix R ≥ R0 and let λ : [0, 1] → C be a piece-wise smooth parametriza-tion of ΓR. Then

IR(A) = limδ(π)→0

IπR(A) in X , (2.5.9)

where

IπR(A) :=

12πi

k−1∑

j=0

h(λ(tj)

)(λ(tj)− A

)−1λ(tj)(tj+1 − tj) ∈ X

is the Riemannian sum with respect to the partition π := t0, . . . , tk of [0, 1], andδ(π) := max(tj+1 − tj). Using Theorem 2.4.2 once more, we deduce from (2.5.9)(with A = a1) that

IR(a1)(D) = limδ(π)→0

IπR(a1)(D) in L(Bs/ν

q,r ) .

Thus it suffices to show that

IπR(a1)(D) = Iπ

R

(a1(D)

). (2.5.10)

We know from (2.5.4) that(λ(tj) − a1

)−1(D) =(λ(tj) − a1(D)

)−1. This implies(2.5.10), hence the assertion. ¥

Bounded Imaginary Powers

As a first application we consider the power function (VI.3.5.22). Recall the con-cept of operators with bounded imaginary powers discussed in Subsection III.4.7.


Henceforth,

Ps

(Z,L(E1, E0)

):=

⋃

κ≥10≤ϑ<π

Ps

(Z,L(E1, E0); κ, ϑ

)(2.5.11)

if E1 → E0 and s > 0.

2.5.5 Theorem Let B ∈ B, b, s > 0, and a ∈ Ps

(Z,L(E)

). Then

a1(D) ∈ Lis(B(t+s)/νq,r ,Bt/ν

q,r ) ∩ BIP(Bt/νq,r ) , t ∈ R .

Proof First we note that, by Theorem 2.2.4, (B(s+t)/νq,r ,Bt/ν

q,r ) is a densely embed-ded Banach couple.

Suppose a ∈ Ps

(Z,L(E); κ, ϑ

). Theorem 2.5.1 implies that a1(D) is a toplin-

ear isomorphism from B(t+s)/νq,r onto Bt/ν

q,r and

A := a1(D) ∈ P(L(Bt/νq,r ); c(κ), ϑ

).

Thus8 Theorem III.4.6.5 implies that Az is well-defined for z ∈ C and Az = hz(A)if Rez < 0.

Proposition VI.3.5.6 guarantees that az1 ∈M(

Rd,L(E))

if Rez < 0. Hencewe can apply Theorem 2.5.4 to obtain

Az = hz(A) = hz(a1)(D) , Rez < 0 .

Hence we deduce from estimate (VI.3.5.24) and Theorem 2.4.2 that

‖Az‖L(Bt/νq,r )

≤ c(κ, ϑ)e|Imz|ϕ , Rez < 0 ,

where ϕ = ϕ(κ, ϑ). Now the fact that A belongs to BIP(Bt/νq,r ) is a consequence of

Lemma III.4.7.4. ¥

Interpolation-Extrapolation Scales

Lastly, we give a simple application of the preceding theorem which sheds a newlight on the Banach space scales B

s/νq,r and b

s/νq,r for s ∈ R.

2.5.6 Theorem Suppose B ∈ B, b. Then J ∈ BIP(B0/νq,r ) and the interpolation-

extrapolation scale[(Eα, Aα) ; α ≥ −m

], generated by (E0, A) := (B0/ν

q,r , J) and[·, ·]θ, 0 < θ < 1, satisfies Eα

.= Bα/νq,r for m ∈ N.

8Observe the change of notation.


Proof Example VI.3.5.8 guarantees Λ = Λ1E ∈ P1

(Z,L(E)

). Hence we infer from

Theorem 2.5.5 that J = Λ1(D) ∈ BIP(B0/νq,r ). Now the assertion follows from The-

orems V.1.5.4 and 2.4.1. ¥

2.5.7 Remark It is a consequence of this theorem, Theorem V.1.5.4, and the factthat the spaces Bs/ν

q,r are defined for all s ∈ R, that the two-sided fractional powerscale generated by (B0/ν

q,r , J) is well-defined. ¥

2.5.8 Corollary Suppose B ∈ B, b and 0 < θ < 1. Then

[Bs0/νq,r ,Bs1/ν

q,r ]θ.= Bsθ/ν

q,r , −∞ < s0 < s1 < ∞ ,

where sθ = (1 − θ)s0 + θs1.

Proof This follows from Theorems 2.5.6 and V.1.5.4. ¥

In Subsection 2.7 we shall give another proof for these interpolation theorems.

2.6 Renorming by Derivatives

First we show that Besov spaces behave naturally with respect to differentiation.

2.6.1 Theorem Suppose B ∈ B, B, b. Then

∂α ∈ L(B(s+α ppppω)/νq,r ,Bs/ν

q,r ) , α ∈ Nd .

Proof Set a(ξ) := ξα. Then a is smooth, positively α q ω-homogeneous, and ∂βavanishes for β ∈ Nd unless β ≤ α. Hence ∂βa is positively (α − β) q ω-homogeneousby Lemma VI.3.3.2. Thus we deduce from ∂βa(ξ) = ∂βa

(Λ1

q rΛ(ξ))

and the com-pactness of [Λ = 1] ∩ Rd that

|∂βa| ≤ cΛ(α−β) ppppω1 , β ≤ α . (2.6.1)

By Leibniz’ rule and ∂βa = 0 if β α,

∂γ(Λ−α ppppω1 a) =

∑

β≤γβ≤α

(γβ

)∂γ−βΛ−α ppppω

1 ∂βa , γ ∈ Nd .

From this, Example VI.3.3.9, and (2.6.1) we infer

|∂γ(Λ−α ppppω1 a)| ≤ c

∑

β≤γβ≤α

Λ−α ppppω−(γ−β) ppppω1 Λ(α−β) ppppω

1 ≤ cΛ−γ ppppω1

for γ q ω ≤ k(ν). Hence Λ−α ppppω1 a ∈M(Rd). Now the assertion follows from Propo-

sition 2.4.4 and a(D) = Dα. ¥


Equivalent Norms

Next we prove renorming theorems for Besov spaces. The first one is the basis forderiving ‘sandwich theorems’ between Besov spaces and classical function spaces.

2.6.2 Theorem Suppose B ∈ B, B, b and m ∈ νqN. The following assertions are

equivalent:

(α) u ∈ B(s+m)/νq,r .

(β) ∂αu ∈ Bs/νq,r , α q ω ≤ m.

(γ) u, ∂m/ωj

j u ∈ Bs/νq,r , 1 ≤ j ≤ d.

Furthermore, set‖·‖(1)

B(s+m)/νq,r

:= ‖·‖Bs/νq,r

+d∑

j=1

‖∂m/ωj

j · ‖Bs/νq,r

,

‖·‖(2)

B(s+m)/νq,r

:= ‖·‖Bs/νq,r

+∑

i=1

‖∇m/νixi

· ‖Bs/νq,r

,

‖·‖(3)

B(s+m)/νq,r

:=∑

i=1

m/νi∑

j=1

‖∇jxi· ‖Bs/ν

q,r,

‖·‖(4)

B(s+m)/νq,r

:=∑

α ppppω≤m

‖∂α · ‖Bs/νq,r

.

Then ‖·‖(k)

B(s+m)/νq,r

∼ ‖·‖B(s+m)/νq,r

for 1 ≤ k ≤ 4.

Proof For abbreviation, Bs/ν := Bs/νq,r , etc.

(1) By Theorem 2.6.1, ∂α maps B(s+m)/ν continuously into B(s+m−α ppppω)/ν .The latter space is continuously embedded in Bs/ν for α q ω ≤ m. From this weget

∂α ∈ L(B(s+m)/ν ,Bs/ν) , α q ω ≤ m . (2.6.2)

Writing ‖·‖(k) for ‖·‖(k)

Bs/νq,r

we obtain

‖·‖(1) ≤ ‖·‖(2) ≤ ‖·‖(3) ≤ c ‖·‖(4) ≤ c ‖·‖B(s+m)/ν ,

the last inequality being due to (2.6.2).(2) We put H := R+ (cf. Remark VI.3.3.1(b)). Then

a(ζ) := η2m/ν1 +d∑

j=1

(ξj)2m/ωj , ζ = (ξ, η) ∈ Z ,

belongs to H2m

(Z, (0,∞)

). Hence a−1 ∈ H−2m

(Z, (0,∞)

)by Lemma VI.3.3.8.

This, Example VI.3.3.9, and Lemma VI.3.3.6 imply Λ−2ma, Λ2ma−1 ∈ H0(Z).


Consequently, by Remark VI.3.4.7(a), these functions belong to M(Rd). NowProposition 2.4.4 gives ‖·‖B(s+2m)/ν ∼ ‖a1(D) · ‖Bs/ν . Hence, by Theorem 2.4.1,

‖·‖B(s+m)/ν ∼ ‖J−m · ‖B(s+2m)/ν ∼ ‖a1(D)J−m · ‖Bs/ν . (2.6.3)

Since (ξj)m/ωj Λ−m ∈ H0(Z), we infer from Remark VI.3.4.7(a) and Theorem 2.4.2

Dm/ωj

j J−m ∈ L(Bs/ν) , 1 ≤ j ≤ d .

Thus D2m/ωj

j J−m = Dm/ωj

j J−mDm/ωj

j and

a1(D)J−m =(1 +

d∑

j=1

D2m/ωj

j

)J−m = J−m +

d∑

j=1

(Dm/ωj

j J−m)Dm/ωj

j

and (2.6.3) imply

‖u‖B(s+m)/ν ≤ c ‖a1(D)J−mu‖Bs/ν ≤ c(‖J−mu‖Bs/ν +

d∑

j=1

‖Dm/ωj

j u‖Bs/ν

)

≤ c ‖u‖(1)

B(s+m)/ν ,

due to J−m ∈ L(Bs/ν). Now the assertion follows from step (1). ¥

The following theorem is of particular importance if s ≤ 0. It is an analogueof Theorem 1.5.3.

2.6.3 Theorem Suppose B ∈ B, B, b and m ∈ νN. Then u ∈ Bs/νq,r iff there exist

uα ∈ B(s+m)/νq,r for α pppp ω ≤ m such that

u =∑

α ppppω≤m

(−1)|α|∂αuα . (2.6.4)

Furthermore,

u 7→ inf∑

α ppppω≤m

‖uα‖B(s+m)/νq,r

(2.6.5)

is a norm for Bs/νq,r , the infimum being taken over all representations (2.6.4).

Proof It follows from Theorem 2.6.1 that u, defined by the right side of (2.6.4),belongs to Bs/ν

q,r and depends continuously on the uα.

Conversely, let v ∈ Bs/νq,r . Then u := J−2mv ∈ B(s+2m)/ν

q,r by Theorem 2.4.1.Hence

v = J2mu =(1 +

d∑

j=1

D2m/ωj

j

)u = u +

d∑

j=1

(−1)m/ωj ∂m/ωj

j uj ,


where uj := −(−1)m/ωj ∂m/ωj

j u ∈ B(s+m)/νq,r for 1 ≤ j ≤ d, due once more to The-

orem 2.6.1.

Let n :=∑

α ppppω≤m 1. Then the above considerations show that

T : (B(s+m)/νq,r )n → Bs/ν

q,r , (uα) 7→ ∑α ppppω≤m(−1)|α|∂αuα

is a continuous linear surjection. We define T by the commutativity of the diagram

(B(s+m)/νq,r )n Bs/ν

q,r

(B(s+m)/νq,r )n/ ker(T )

T

T -

@@@R ¡

¡¡µ

where the unlabeled arrow denotes the canonical projection. Then T is a toplinearisomorphism by the open mapping theorem. This proves the assertion, since (2.6.5)is the quotient norm of the factor space. ¥

2.6.4 Remark The theorem remains true if (2.6.4) is replaced by

u = u0 +d∑

j=1

∂m/ωj

j uj

or

u = u0 +∑

i=1

∇m/νixi

ui

and (2.6.5) is modified accordingly.

Proof Obvious by the above proof. ¥

Sandwich Theorems

The following ‘sandwich theorem’ clarifies the relation between Besov and classicalfunction spaces.

2.6.5 Theorem Suppose m ∈ νN. Then

Bm/νq,1

d→ Wm/ν

q → Bm/νq,∞ , q < ∞ , (2.6.6)

and

Bm/ν∞,1

d→ BUCm/ν → Bm/ν

∞,∞ , (2.6.7)


and

Bm/ν∞,1

d→ C

m/ν0

d→ Bm/ν

∞,∞ . (2.6.8)

Proof (1) By Lemma VI.3.6.2, u =∑

k ψk(D)u in S ′. Thus we find

‖u‖q ≤∑

k

‖ψk(D)u‖q = ‖u‖B

0/νq,1

, u ∈ B0/νq,1 → S ′ ,

that is,B

0/νq,1 → Lq , q < ∞ .

If q = ∞, then it follows from (2.1.2) that∑

k ψk(D)u converges in BUC if u be-longs to B

0/ν∞,1. This shows that

B0/ν∞,1 → BUC .

From (2.1.1) and (2.1.2) we infer

‖u‖B

0/νq,∞

= supk

‖ψk(D)u‖q ≤ c ‖u‖q , q < ∞ ,

and‖u‖

B0/ν∞,∞

= supk

‖ψk(D)u‖∞ ≤ c ‖u‖BUC ,

that is,Lq → B0/ν

q,∞ , q < ∞ , BUC → B0/ν∞,∞ .

This proves (2.6.6) and (2.6.7) if m = 0.

(2) Let m ∈ νqN. Then (2.6.6) and (2.6.7) follow from step (1), Theorems 2.6.2

and 1.4.1, (1.1.4), and definition (1.1.7).

(3) Since S ⊂ BUCm/ν ⊂ BCm/ν , we obtain from (1.1.10) that Cm/ν0 is the

closure of S in BUCm/ν . Thus (2.6.8) is a consequence of (2.6.7) and the definitionof B

m/ν∞,r . ¥

Sobolev Embeddings

As a first application we can extend the Sobolev type embeddings (2.2.3) to vector-valued Sobolev spaces. For further results of this type we refer to Subsection 5.6.

2.6.6 Theorem (Sobolev embedding theorem) Suppose m ∈ νN.(i) Let 0 ≤ s < m and 1 ≤ q0 ≤ ∞ satisfy m− |ω|/q = s − |ω|/q0. Then

Wm/νq

d→ Bs/ν

q0,∞ .

(ii) Assume k ∈ νN and m > k + |ω|/q. Then Wm/ν

q → Ck/ν0 .


Proof (1) It is immediate from (2.6.6) and (2.2.3) that Wm/ν

q → Bs/νq0,∞. Hence

(i) is a consequence of the density of S in Wm/ν

q .

(2) Set s := m− |ω|/q > k. Then Wm/ν

qd

→ Bs/ν∞,∞ by (i). It follows from

(2.2.2) that Bs/ν∞,∞

d→ B

k/ν∞,1. Now we get claim (ii) from (2.6.8). ¥

2.6.7 Corollary Let 0 ≤ s < m and 1 ≤ q0 < ∞ satisfy m− |ω|/q > s − |ω|/q0.

Then Wm/ν

qd

→ Bs/νq0,1

d→ B

s/νq0,q0 .

Proof Let t := m− |ω| (1/q − 1/q0) > s. Then Wm/ν

qd

→ Bt/νq0,∞

d→ B

s/νq0,1 by (i)

and Theorem 2.2.4(iii). ¥

2.7 Interpolation


• 1 ≤ r0, r1 ≤ ∞ .

• s0, s1 ∈ R with s0 6= s1 .

• 0 < θ < 1 ,

and we use notation (VI.2.3.1). Recall (2.0.1).

Real and Complex Interpolation

The scale of Banach spaces is closed under real and complex interpolation. Inaddition, as we know already, the scales of little and very little Besov spaces arealso invariant under complex interpolation. These and other results are proved inthe next theorem. They are most useful for various applications.

2.7.1 Theorem It holds for 1 ≤ q ≤ ∞:

(i) (Bs0/νq,r0 , B

s1/νq,r1 )θ,r

.= Bsθ/νq,r 1 ≤ r ≤ ∞.

(ii) (Bs0/νq,r0 , B

s1/νq,r1 )0θ,∞

.= (bs0/νq,r0 , b

s1/νq,r1 )0θ,∞ = b

sθ/νq,∞ .

(iii) [Bs0/νq,r0 , B

s1/νq,r1 ]θ

.= Bsθ/νq,r(θ), minr0, r1 < ∞.

(iv) [Bs0/νq,∞ , B

s1/νq,∞ ]θ

.= Bsθ/νq,∞ .

(v) [bs0/ν∞,∞, B

s1/ν∞,∞]θ = [bs0/ν

∞,∞, bs1/ν∞,∞]θ

.= bsθ/ν∞,∞.

Proof (1) By Lemma 2.1.3 we know that (R,Rc) is a retraction-coretraction pairfor

(`sr(X ), `s/ν

r X )and for

(cs0(X ), cs/ν

0 X ), where X ∈ BUC, C0, Lq. Hence it

follows from Proposition I.2.3.2 that the Banach space scales [ `s/νr X ; s ∈ R ] and

[ cs/ν0 X ; s ∈ R ] possess the same interpolation properties as [ `s

r(X ) ; s ∈ R ] and[ cs

0(X ) ; s ∈ R ], respectively.


(2) Assertion (i) follows from step (1), (2.1.17), and Theorem VI.2.3.5.(3) Suppose r1 < ∞. By Theorem 2.2.2,

Bs/νq,r1

→ Bt/νq,∞ → Bs1/ν

q,r1, s > t > s1 .

Theorem 2.2.4 guarantees Bs/νq,r1

d→ B

s1/νq,r1 for s > s1. Hence

Bt/νq,∞

d→ Bs1/ν

q,r1, t > s1 , r1 < ∞ . (2.7.1)

We can assume s1 > s0. The definition of the continuous interpolation functor andassertion (i) guarantee that (Bs0/ν

q,r0 , Bs1/νq,r1 )0θ,∞ is the closure of B

s1/νq,r1 in B

sθ/νq,∞ .

Thus we deduce from (2.2.10) and (2.7.1)

(Bs0/νq,r0

, Bs1/νq,r1

)0θ,∞.= bsθ/ν

q,∞ . (2.7.2)

We fix t0 < s0 < s1 < t1 and put θj := (sj − t0)/(t1 − t0). Then (2.7.2) and thereiteration theorem for the continuous interpolation functor, more specifically(I.2.8.7), give

(bs0/νq,∞ , bs1/ν

q,∞ )0θ,∞.=

((Bt0/ν

q,∞ , Bt1/νq,∞ )0θ0,∞, (Bt0/ν

q,∞ , Bt1/νq,∞ )0θ1,∞

)0

θ,∞.= (Bt0/ν

q,∞ , Bt1/νq,∞ )0(1−θ)θ0+θθ1,∞

.= bsθ/νq,∞ .

This proves (ii).(4) Claim (iii) is a consequence of step (1) and Theorem VI.2.3.4(ii).(5) Assertions (iv) and (v) follow from (2.2.8) and (2.2.9), respectively, and

from step (1) and Theorem VI.2.3.4(iv). For (v) we use in addition the followingproperty of the complex interpolation functor: if E1 → E0 and E0 is the closureof E1 in E0, then

[E0, E1]θ = [E0, E1]θ

(cf. [BeL76, Theorem 4.2.2(b)] or [Tri95, Theorem 1.9.3(g)]). ¥

Interpolation with Different Target Spaces

In the next theorem we complement these interpolation results by consideringsituations in which two Banach spaces E0 and E1 are involved.

2.7.2 Theorem Let (E0, E1) be an interpolation couple and 1 ≤ q0, q1 < ∞. Then

(i)(B

s0/νq0,r0(Rd, E0), B

s1/νq1,r1(Rd, E1)

)θ,r(θ)

.= Bsθ/νq(θ),r(θ)(R

d, Eθ,r(θ)),provided r0, r1 < ∞ and q(θ) = r(θ).

(ii)[B


s1/νq1,r1(Rd, E1)

]θ


d, E[θ]), r0 < ∞.

(iii)[B


s1/νq1,∞(Rd, E1)

]θ


d, E[θ]).

(iv)[B

s0/νq0,∞(Rd, E0), B

s1/νq1,∞(Rd, E1)

]θ

.= Bsθ/νq(θ),∞(Rd, E[θ]).


Proof From interpolation theory it is known that(Lq0(Rd, E0), Lq1(Rd, E1)

)θ

.= Lq(θ)

(Rd, (E0, E1)θ

)(2.7.3)

for (·, ·)θ ∈(·, ·)θ,q(θ), [·, ·]θ

(e.g., [Tri95, Theorem 1.18.4]). Thus the assertions

follow from step (1) of the preceding proof, Lemma 2.2.3, and Theorem VI.2.3.4. ¥

Embeddings of Intersection Spaces

As an application of part (i) of this theorem and of the embedding results ofSubsection 2.2, we can prove an embedding theorem for intersections of Besovspaces. A very simple special case thereof will be of use in Subsection 7.4 below.

2.7.3 Theorem Let (E0, E1) be an interpolation couple. Suppose q, q0, q1 ∈ [1,∞],s, s0, s1 ∈ R with s0 6= s1, and 0 < θ < 1 satisfy

s < sθ and s − |ω|/q < sθ − |ω|/q(θ) . (2.7.4)

ThenBs1/ν

q1,r1(Rd, E1) ∩ Bs0/ν

q0,r0(Rd, E0) → bs/ν

q,r (Rd, Eθ,p) .

for 1 ≤ p ≤ ∞.

Proof It follows from (2.7.4) that we can choose σj < sj and θ < ϑ < 1 sufficientlyclose to sj and θ, respectively, such that s < σϑ and s − |ω|/q < sϑ − |ω|/q(ϑ).Thus, by (2.2.2) and omitting Rd,

Bs1/νq1,r1

(E1) ∩Bs0/νq0,r0

(E0) → Bσ1/νq1,q1

(E1) ∩Bσ0/νq0,q0

(E0) .

The last intersection space embeds continuously into Bσj/νqj ,qj (Ej) for j = 0, 1. From

this we get by interpolation, due to Theorem 2.7.2(i),

Bs1/νq1,r1

(E1) ∩Bs0/νq0,r0

(E0) → (Bσ0/ν

q0,q0(E0), Bσ1/ν

q1,q1(E1)

)ϑ,q(ϑ)

.= Bσϑ/νq(ϑ),q(ϑ)(Eϑ,q(ϑ)) .

Now we choose t0 < t1 in (s, σϑ). Then (2.2.3) and (2.2.2) imply

Bσϑ/νq(ϑ),q(ϑ)(Eϑ,q(ϑ)) → B

t1/νq,q(ϑ)(Eϑ,q(ϑ)) → Bt0/ν

q,r (Eϑ,q(ϑ)) .

Since Eϑ,q(ϑ) → Eθ,p by (I.2.5.2), we infer from Theorem 2.2.5 and (2.2.10)

Bt0/νq,r (Eϑ,q(ϑ)) → Bt0/ν

q,r (Eθ,p) → bs/νq,r (Eθ,p) .

This proves the theorem. ¥

Theorem 5.6.6 and Remark 5.6.7 deal with the case where the equality signsin (2.7.4) are permitted.


Interpolation of Classical Spaces

It is important to know that Besov spaces of positive order can be obtained fromclassical function spaces by interpolating.

2.7.4 Theorem Suppose k < s < m with k,m ∈ νN. If θ := (s − k)/(m− k), then

Bs/νq,r

.= (W k/νq ,Wm/ν

q )θ,r , q < ∞ , (2.7.5)

and

Bs/ν∞,r

.= (BUCk/ν , BUCm/ν)θ,r . (2.7.6)

Furthermore,

bs/ν∞,∞

.= (BUCk/ν , BUCm/ν)0θ,∞ . (2.7.7)

Proof We know from Theorem 2.7.1(i) that

Bs/νq,r

.= (Bk/νq,1 , B

m/νq,1 )θ,r

.= (Bk/νq,∞, Bm/ν

q,∞ )θ,r .

From this and (2.6.6), resp. (2.6.7), we obtain (2.7.5), resp. (2.7.6).

By Theorem 2.7.1(ii),

(Bk/ν∞,1, B

m/ν∞,1 )0θ,∞

.= (Bk/ν∞,∞, Bm/ν

∞,∞)0θ,∞ = bs/ν∞,∞ .

Hence (2.6.7) implies (2.7.7). ¥

2.7.5 Corollary

(i) If m ∈ νN and s > m, then Bs/ν∞,r → BUCm/ν .

(ii) BC∞ d→ b

s/ν∞,∞.

Proof (1) Claim (i) is implied by (2.7.6).

(2) Since BUCm/ν d→ b

s/ν∞,∞ for m > s, and due to (2.7.7), assertion (ii) fol-

lows from (1.1.9). ¥

In Theorem 3.7.6 it is shown that Bs/ν∞,∞ = (Ck/ν

0 , Cm/ν0 )0θ,∞.

2.8 Besov Spaces on Corners

The basis for constructing function spaces on corners is contained in the followingsimple observation about image spaces of retractions.


2.8.1 Lemma Let X0, X , and Y be LCSs with X0 → X . Suppose (r, rc) is an r-cpair for (X ,Y). Then the diagrams

r

r

X

X0

Y

rX0

-

-

6

£¢6

£¢

rc

rc

X

X0

Y

rX0

¾

¾

6

£¢6

£¢

are commuting and (r, rc) is an r-c pair for (X0, rX0). If X0 is a Banach space,then ‖·‖rX0

∼ ‖rc ·‖X0 .

Proof The first part of the statement follows from Remarks VI.2.2.1.Let X0 = (X0, ‖·‖) be a Banach space. Since (r, rc) is an r-c pair for (X0, rX0),

Lemma I.2.3.1 guarantees the direct sum decomposition

X0 = pX ⊕ ker(r) , p := rcr .

Hence we can endow X0 with the equivalent norm

x 7→ ‖y‖+ ‖z‖ , x = y ⊕ z , y = px , rz = 0 .

Then‖y‖rX0 = inf

x∈r−1(y)‖x‖ ∼ inf

z∈ker(r)(‖rcy‖ + ‖z‖) = ‖rcy‖

for y ∈ rX0. ¥

We assume• K is a corner in Rd .

• B = B, B, b .

As before, Bs/νq,r := Bs/ν

q,r (Rd, E).

It follows from (2.2.1) and the definitions of Bs/νq,r and b

s/νq,r that

S → Bs/νq,r → S ′ .

Hence Theorem VI.1.2.3 and Lemma 2.8.1 imply that RKBs/νq,r is a well-defined

Banach space satisfying

S(K, E) → RKBs/νq,r → S ′(K, E) .

Henceforth,

Bs/νq,r (K, E) := RKBs/ν

q,r . (2.8.1)

This defines the Besov spaces Bs/νq,r (K, E), the very little Besov spaces B

s/νq,r (K, E),

and the little Besov spaces bs/νq,r (K, E) on corners.


2.8.2 Theorem Bs/νq,r (K, E) is a Banach space and (RK, EK) is a universal r-e pair

for(Bs/ν

q,r ,Bs/νq,r (K, E)

).

Proof This is a restatement of the preceding considerations. ¥

Suppose m ∈ νN and

X ∈ BCm/ν , BUCm/ν , C

m/ν0 , Wm/ν

q ; q < ∞.

Then X (X, E) has been defined for X ∈ Rd,K in Subsections 1.2 and 1.1. Itfollows from Theorem 1.3.1 that

X (K, E) .= RKX , (2.8.2)

where X = X (Rd, E) on the right side.

2.8.3 Theorem 9 The Besov spaces Bs/νq,r (K, E) possess the same embedding and

interpolation properties as the spaces Bs/νq,r (Rd, E).

Proof This is a consequence of Theorem 2.8.2, Lemma 2.2.1, Proposition I.2.3.2,and (2.8.2). ¥

The next theorem shows that the duality assertions for Bs/νq,r also apply to

Besov spaces on corners.

2.8.4 Theorem Suppose E is reflexive or has a separable dual. Then

(Bs/ν

q,r (K, E))′ = B

−s/νq′,r′ (K∗, E′)

with respect to 〈·, ·〉. If 1 < q, r < ∞ and E is reflexive, then Bs/νq,r (K, E), too, is

reflexive.

Proof Since (RK, EK) is an r-e pair for(B

s/νq,r (Rd), Bs/ν

q,r (K)), we deduce from

Theorem 2.8.2, and Remark VI.1.2.1(c) that (E∗K, R∗

K) is an r-e pair for

((Bs/ν

q,r (Rd, E))′, (Bs/νq,r (K, E))′

).

Thus, by Theorem 2.3.1, (E∗K, R∗

K) is an r-e pair for

(B−s/νq′,r′ (Rd, E′), (Bs/ν

q,r (K, E))′)

.

9For brevity, here we do not list all the embedding and interpolation results derived in thepreceding subsections. However, whenever we refer to this theorem in the following, we alwaysspecify which embedding or interpolation fact is meant.


Lemma VI.1.2.2 implies((EK)∗, (RK)∗

)= (RK∗ , EK∗). Consequently,

(Bs/ν

q,r (K, E))′ .= RK∗B

−s/νq,r (Rd, E′) = B−s/ν

q,r (K∗, E′) .

This proves the first assertion. If 1 < q, r < ∞ and E is reflexive, then Bs/νq,r is

reflexive by Theorem 2.3.1. Hence we infer from Theorem 2.8.2 and Lemma I.2.3.1that B

s/νq,r (K, E) is isomorphic to a closed linear subspace of a reflexive Banach

space. Thus it is reflexive as well. ¥

2.8.5 Remark(B

s/ν∞,∞(K, E)

)′ = B−s/ν1,1 (K∗, E) holds without any restriction on E.

Proof Remark 2.3.2. ¥

In the case where K is closed, distributional derivatives behave naturally andthey can be used to introduce equivalent norms.

2.8.6 Theorem Suppose K is closed. Then:

(i) ∂α ∈ L(B(s+α ppppω)/νq,r (K, E),Bs/ν

q,r (K, E)), α ∈ Nd.

(ii) The assertions of Theorems 2.6.2 and 2.6.3 and Remark 2.6.4 apply also toBs/ν

q,r (K, E).

Proof This follows from Theorems VI.1.2.3(iii) and 2.8.2. ¥

2.9 Notes

As already mentioned in the notes to Section VI.3, in the isotropic case the defi-nition of Besov spaces (2.1.17), (2.1.18) is the standard one originating in J. Pee-tre’s paper [Pee67]. An extension of this approach to anisotropic Besov spaceshas been given by M. Yamazaki [Yam86], who employed the ν-quasinorm E ofExample VI.3.2.1(b). Our technique, being based on ν-dyadic partitions of unityinduced by (Ω, ψ), is an obvious modification thereof and works for distributionswith values in arbitrary Banach spaces. Some results on vector-valued Besov spacesare also found in the work of H.-J. Schmeißer and W. Sickel [SS01], [SS05]. Alsosee R. Denk and M. Kaip [DK13].

It has been pointed out to the author by S.I. Hiltunen that the duality The-orem 2.3.1 can be generalized as follows: in the case q 6= ∞, the assumption thatE has a separable dual can be replaced by the more general condition that E itselfis separable. The corresponding proof relies on Lebesgue spaces whose elementstake their values in certain locally convex spaces (see Remark 2 in [Hil18]).

The fundamental Fourier multiplier theorem 2.4.2 is an anisotropic exten-sion of the corresponding result presented, in the isotropic case, by the authorin [Ama97] (also see the announcement of L. Weis [Wei97]). For the sake of sim-plicity, we do not insist on minimal differentiability requirements for the symbol


but employ the universal multiplier space M. It follows from Remark VI.3.6.4(b)that, everywhere in this section, M can be replaced by M2 |ω|. In the homogeneouscase it suffices to assume that the symbol belongs to class Cd+1 (see [Ama97]),whereas a ∈M2 |ω|(Rd,L(E,F )

)requires a ∈ C2d. The order d + 1 is optimal as

long as no restriction on the Banach space E is imposed. It has been shown byM. Girardi and L. Weis [GiW03] that we can get away with less smoothness of thesymbol if E belongs to suitably restricted classes of Banach spaces.

The Ck(ν) regularity for the symbols is only needed in Subsection 5.3, wherewe show that M is a multiplier space for Triebel–Lizorkin spaces.

The other material of this subsection consists of more or less straightforwardamplifications of the corresponding results in [Ama09].

Of course, most of the theory of Besov spaces developed in this section re-duces, in the classical isotropic scalar case, to well-known results, essentially con-tained in H. Triebel’s books, in particular in [Tri83]. In that book, as well asin [Tri95], there is also sketched the possibility of defining anisotropic Besov (andTriebel–Lizorkin) spaces by Fourier analytic techniques. A noteworthy exceptionis Theorem 2.6.3 whose isotropic scalar counterpart seems to appear for the firsttime in [Ama00b, Theorem 2.1].

Last but not least, we direct the reader’s attention to Section 2.2 of [Tri83]for a survey of the historical evolution of the theory of function spaces. Detailedreferences to the early developments, in particular by the work of S.M. Nikol′skiıand his students, notably O.V. Besov, are found in [Tri95] and in A. Kufner,O. John, and S. Fucik [KuJF77].

VII.3 Slobodeckii and Holder Spaces 133

3 Intrinsic Norms, Slobodeckii and Holder Spaces

The one sided Besov space scales [ Bs/νp,r ; s > 0 ], 1 ≤ p, r ≤ ∞, are of particular

importance. Indeed, they can be characterized intrinsically and contain the (in theisotropic case) widely used Slobodeckii spaces W

s/νr for 1 ≤ r < ∞ and s /∈ νN,

and the Holder spaces BUCs/ν for r = ∞ and s /∈ νN. For these reasons they areinvestigated in the present section in some detail. The main results are containedin Theorems 3.5.1 and 3.5.2. The former contains a significant intersection spacerepresentation and the latter a characterization by a most useful norm.

In order to arrive at these results we have to establish some facts on pair-wisecommuting strongly continuous contraction semigroups, translation semigroups inparticular. This is done in Subsections 3.1–3.3. In the following three subsectionswe introduce (anisotropic) Besov, Besov–Slobodeckii and Besov–Holder spaces andapply the foregoing results to deduce the theorems mentioned above.

In connection with quasilinear parabolic equations, Besov–Holder spaces areoften most useful. However, this scale lacks the property of dense injection whoseimportance is manifest by the considerations in Volume I. In the next to lastsubsection we give an intrinsic characterization of the class of anisotropic littleHolder spaces which do not possess these ‘shortcomings’.

Finally, in the last subsection, we characterize Besov–Slobodeckii and Besov–Holder spaces on open corners.

3.1 Commuting Semigroups

Let X = (X, ‖·‖) be a Banach space. Given linear operators A and B in X, theproduct AB is the linear operator in X defined by

dom(AB) :=

x ∈ dom(B) ; Bx ∈ dom(A)

, ABx := A(Bx) .

Then A and B commute, if

ABx = BAx , x ∈ dom(AB) ∩ dom(BA) .

If A,B ∈ C(X), that is, A and B are closed, then A and B are resolvent commuting

if [(λ + A)−1, (µ + B)−1

]= 0 , λ ∈ ρ(−A) , µ ∈ ρ(−B) ,

where [·, ·] denotes the commutator.

3.1.1 Lemma If 10 A,B ∈ G(X) are resolvent commuting, then they commute.

10Recall that G(X) is the set of all negative infinitesimal generators of strongly continuoussemigroups on X.


Proof Lemma III.4.9.1 guarantees that Ae−tB ⊃ e−tBA for t > 0. Hence, givenx ∈ dom(AB) ∩ dom(BA),

t−1A(e−tB − 1)x = t−1(e−tB − 1)Ax . (3.1.1)

Since Ax ∈ dom(B), the right side converges for t → 0 towards −BAx. Further-more, x ∈ dom(B) so that t−1(e−tB − 1)x → −Bx as t → 0. Now the closednessof A implies that the left side of (3.1.1) converges toward −ABx. This proves theclaim. ¥

Now we suppose that A1, . . . , Ad are pair-wise commuting linear operatorsin X and set A := (A1, . . . , Ad). For α ∈ Nd we define Aα by A0 := 1 = 1E and

dom(Aα) :=d⋂

j=1

dom(Aαj

j ) , Aαx := Aα11 · · ·Aαd

d x .

NoteAα = A

ασ(1)

σ(1) · · ·Aασ(d)

σ(d) = B1 · · ·B|α| ,

where σ ∈ Sd and B1, . . . , B|α| ∈ A1, . . . , Ad with αj of the B1, . . . , B|α| beingequal to Aj .

Suppose

A1, . . . , Aα ∈ −G(X) are pair-wise resolvent commuting. (3.1.2)

Given m ∈ νN, we put

Km/νX (A) :=

( ⋂

α ppppω≤m

dom(Aα), ‖·‖K

m/νX (A)

), (3.1.3)

where‖x‖

Km/νX (A)

:=∑

α ppppω≤m

‖Aαx‖ .

3.1.2 Lemma Let (3.1.2) be satisfied and m ∈ νqN. Then

(i) K = Km/νX (A) is a Banach space such that K

d→ X.

(ii) K is invariant under Uj := etAj ; t ≥ 0 for j = 1, . . . , d.(iii) The K-realizations Uj,K of Uj are pair-wise commuting strongly continuous

semigroups on K, and the infinitesimal generator of Uj,K is the K-realizationAj,K of Aj.

(iv) Suppose n ∈ νqN. Then

K(m+n)/νX (A) = K

n/νK (AK) ,

where AK := (A1,K , . . . , Ad,K).


Proof (1) It is clear that ‖·‖K is a norm on K and K → X. Hence we have toshow that K is complete. For this we proceed by induction on d.

(2) Assume d = 1. Since A = A1 ∈ C(X) and ρ(A) 6= ∅, Theorem 2.16.4 in[HillP57] guarantees that Aj ∈ C(X) for j ∈ N. Thus the assertion is clear in thiscase.

(3) Suppose d ≥ 2 and that it has already been shown that K is com-plete if it is constructed with d − 1 pair-wise resolvent commuting generators.We set A := A1 and B := (A2, . . . , Ad). Then, writing α = (j, β) with j ∈ N andβ ∈ Nd−1, and, analogously, ω = (ω1, ω

′),

K =⋂

jω1+β ppppω′≤m

dom(AjBβ)

and‖x‖K =

∑

jω1+β ppppω′≤m

‖AjBβx‖ .

Let (xk) be a Cauchy sequence in K. Then (Aαxk) = (AjBβxk) is a Cauchysequence in X for each α = (j, β) with α q ω = jω1 + β q ω′ ≤ m. Hence thereexist xα ∈ X such that Aαx = AjBβxk → xα in X as k →∞. In particular,xk → x0 =: x in X. We have to show that x ∈ K and Aαxk → Aαx if α q ω ≤ m.For this we have to verify that x ∈ dom(Aα) and Aαxk → Aαx for each α ∈ Nd

with α q ω ≤ m.We fix β and proceed by induction on j. For abbreviation, y := xα and

C := Bβ . Suppose j = 0. Then the assertion follows from the induction hypothe-sis. Assume the claim is true for 0 ≤ i ≤ j − 1 with jω1 + β q ω′ ≤ m. Choose anyλ ∈ ρ(A) and note that

Aα = (λ − A)p(A) + r(A) , (3.1.4)

where

p(A) := −j−1∑

i=0

λj−1−iAiC , r(A) := λjC . (3.1.5)

Hence we get from xk ∈ K ⊂ dom(Aα) ⊂ dom(AiC), 0 ≤ i ≤ j, and (3.1.5) that

p(A)xk = (λ− A)−1(Aαxk − r(A)xk

).

The induction hypothesis guarantees that x belongs to dom(Aj−1C) and

r(A)xk = λjCxk → r(A)x .

Thusp(A)xk → (λ− A)−1

(xα − r(A)x

)


in X. Using (3.1.5) and once more the induction hypothesis, we find that(p(A)xk

)converges in X towards p(A)x. Hence

p(A)x = (λ− A)−1(xα − r(A)x

).

This shows that p(A)x ∈ dom(A) and (λ − A)p(A)x = xα − r(A)x. Hence

x ∈ dom(Aα) = (λ − A)−1(dom(Aj−1C)

)

and Aαx = xα. Since these arguments apply to each permutation of A1, . . . , Ad,we have shown that K is well-defined and complete.

(4) Assume A ∈ −G(X). Then, by (s 7→ esAx) ∈ C(R+, X),

limt→0

1t

∫ t

0

esAx ds = x , x ∈ X . (3.1.6)

Moreover,∫ t

0

esAx ds ∈ dom(A) , A

∫ t

0

esAx ds =∫ t

0

AesAx ds = etAx− x (3.1.7)

for t > 0 and x ∈ X, as is well-known and readily seen.We set n := m/ω1 + · · · + m/ωd and let Bi ∈ A1, . . . , Ad, 1 ≤ i ≤ n, with

m/ωj of the B1, . . . , Bn being equal to Aj , 1 ≤ j ≤ d. Then we put

xt :=1tn

∫ t

0

· · ·∫ t

0

eni=1 siBix ds1 · · · dsn , t > 0 , x ∈ X ,

where

eni=1 siBi = es1B1 · · · esnBn . (3.1.8)

Since A1, . . . , Ad are pair-wise resolvent commuting, Lemma III.4.9.1 guaranteesthat the semigroups esBi ; s ≥ 0 , 1 ≤ i ≤ n, are pair-wise commuting. Thus(3.1.8) is well-defined. It follows from (3.1.6) that xt → x in X. From (3.1.7) weinfer that

xt ∈d⋂

j=1

dom(Am/ωj

j ) =: Y .

Hence Yd

→ X. If α q ω ≤ m, then αj ≤ m/ωj . Thus Y ⊂ D(Aα). Consequently,Y ⊂ K, which shows that K is dense in X. This proves claim (i).

(5) Assertion (ii) is immediate from Lemma III.4.9.1 and (3.1.7).(6) Let j ∈ 1, . . . , d and set A := Aj . By (ii), the K-realization of U := Uj

is simply the restriction of U to K. Suppose x ∈ K. Then

AαetAx = etAAαx → Aαx in X as t → 0

for α q ω ≤ m. Thus etAx → x in K. This shows that UK is strongly continuous.


Let B be the infinitesimal generator of UK . Suppose x ∈ dom(B). Thent−1(etAx− x) → Bx in K as t → 0. Hence K → X implies that this convergencetakes place in X as well. Thus x ∈ dom(A) and Ax = Bx. This proves AK ⊃ B.

Conversely, assume x ∈ dom(AK). Then x ∈ K, Ax ∈ K, and

t−1(etAx − x) → Ax (3.1.9)

in X. From Lemma 3.1.1 we infer

Aα(t−1(etAx− x) − Ax

)= t−1(etAAαx − Aαx) − A(Aαx) → 0

for α q ω ≤ m. Hence convergence (3.1.9) takes place in K. Consequently, x be-longs to dom(B) and Ax = Bx, that is, AK ⊂ B. Clearly, the semigroups Uj,K ,1 ≤ j ≤ d, are pair-wise commuting. This proves (iii).

(7) It follows from (i) and (iii) that M := Kn/νK (AK) is a well-defined Banach

space. Furthermore,

‖x‖M =∑

β ppppω≤n

‖AβKx‖K =

∑

β ppppω≤n

∑

α ppppω≤m

‖AαAβKx‖

=∑

α ppppω≤m

∑

β ppppω≤n

‖Aα+βx‖ = ‖x‖K

(m+n)/νX (A)

implies assertion (iv). ¥

3.1.3 Remark Let A ∈ C(X). A linear subspace Y of dom(X) is a core for A, ifA equals the closure of its restriction to Y . Suppose A ∈ −G(X). Then the coretheorem guarantees that Y ⊂ dom(A) is a core for A, if Y is dense in X andinvariant under the semigroup etA ; t ≥ 0 (e.g., [Dav80, Theorem 1.9]). ThusLemma 3.1.2 shows that K

m/νX (A) is a core for each Aj . ¥

3.2 Semigroups and Interpolation

In this subsection we present equivalent norms for the real interpolation spaces(X, K)θ,r. To do this, we need some preparation. Thus we assume:

A1, . . . , Ad are pair-wise resolvent commuting generators ofstrongly continuous contraction semigroups

Uj(t) ; t ≥ 0

on X.

(3.2.1)


Preliminary Estimates

3.2.1 Lemma Suppose αj ∈ N and βj ∈qN. Set γ := α1β1 + · · · + αdβd. Then

d∏

j=1

(Uj(tβj )− 1

)αj

= 2−γd∏

j=1

(Uj((2t)βj ) − 1

)αj +d∑

j=1

d∏

`=1

(U`(tβ`) − 1

)α`(Uj(tβj ) − 1

)p(t)

for t ∈ R+, where p(t) := p(U1(tβ1), . . . , Ud(tβd)

)with a polynomial p in d inde-

terminates.

Proof (1) In this step we denote by q polynomials in one indeterminate, notnecessarily the same at different occurrences. Let r ∈

qN. Then

2r =(z + 1 − (z − 1)

)r = (z + 1)r + (z − 1) q(z) , z ∈ C .

By multiplying this relation by 2−r(z − 1)r we arrive at

(z − 1)r = 2−r(z2 − 1)r + (z − 1)r+1 q(z) , z ∈ C . (3.2.2)

Now we claim that, given s ∈qN,

(z − 1)r = 2−rs(z2s − 1)r + (z − 1)r+1 q(z) , z ∈ C . (3.2.3)

Indeed, suppose its validity has been shown for some s ∈qN. Then we replace z in

(3.2.2) by z2s

to get

(z2s − 1)r = 2−r(z2s+1 − 1)r + (z − 1)r+1 q(z) , z ∈ C .

By inserting this expression into (3.2.3) we see that this identity applies with sreplaced by s + 1. This proves the claim.

(2) We apply (3.2.3) with r = αj and s = βj to Uj(tj). This gives(Uj(tj) − 1

)αj = 2−αjβj(Uj(2βj tj) − 1

)αj +(Uj(tj) − 1

)αj+1qj

(Uj(tj)

).

Now we replace tj by tβj and multiply the resulting identities. Then we get theassertion. ¥

3.2.2 Lemma Let αj, βj, and γ be as above and suppose rj ∈ N with rj ≥ αj. Then

‖Aαx‖ ≤ c(‖x‖+

d∑

j=1

∫ ∞

0

t−γ∥∥(

Uj(tβj ) − 1)rj

x∥∥ dt/t

)(3.2.4)

for x ∈ D(Aα).


Proof (1) It follows from (3.1.6) and (3.1.7) that, given x ∈ D(Akj ),

Akj x = lim

t→0t−k(e−tAj − 1)kx , k ∈ N .

Hence, if x ∈ D(Aα),

Aαx = limt→0

t−α(U(t)− 1

)αx , (3.2.5)

where t := (t1, . . . , td) ∈ (0,∞)d and U(t) := U1(t1) · · ·Ud(td). Putting tj := tβj in(3.2.5), it follows

Ax = limt→0

t−γd∏

j=1

(Uj(tβj ) − 1

)αjx .

We write ϕ(t) for the argument of limt→0. Then we get from

ϕ(t) +(ϕ(2−1t)− ϕ(t)

)+ · · · + (

ϕ(2−kt) − ϕ(2−k+1t))

= ϕ(2−kt)

that, given any t > 0,

Aαx = ϕ(t) +∞∑

k=1

(ϕ(2−kt)− ϕ(2−k+1t)

)

= t−γ∏

j

(Uj(tβj ) − 1

)αjx

+∞∑

k=1

[(2−kt)−γ

∏

j

(Uj((2−kt)βj )− 1

)αjx

− (2−k+1t)−γ∏

j

(Uj((2−k+1t)βj ) − 1

)αjx]

.

(3.2.6)

Next we consider the difference between the term on the left side of the identityin Lemma 3.2.1 and the first one on the right side, replace t by 2−kt, and multiplythe outcome by (2−kt)γ . The resulting expression, applied to x, then equals theone in the bracket of (3.2.6). Thus, letting 1/2 ≤ t ≤ 1, we arrive at the estimate

‖Aαx‖≤ ‖x‖ + c

∞∑

k=1

(2−kt)−γd∑

j=1

∥∥∥d∏

`=1

(U`((2−kt)β`) − 1

)α`(Uj((2−kt)βj ) − 1

)x∥∥∥ .

Here we used the commutativity of p(t) and Ui

((2−kt)βi

)and the uniform bound-

edness of p(t). Integration with respect to dt/t over [1/2, 1] yields

‖Aαx‖

≤ c(‖x‖+

∫ 1

0

t−γ∥∥∥

d∑

j=1

d∏

`=1

(U`(tβ`)− 1

)α`(Uj(tβj )− 1

)x∥∥∥ dt/t

),

(3.2.7)

where we also enlarged the range of integration on the right side.


(2) Suppose αj ∈ N satisfy γ := α1β1 + · · · + αdβd > γ. Then we replace αj

in Lemma 3.2.1 by αj , multiply the resulting identity by t−γ , and integrate toarrive at the estimate

∫ 1

0

t−γ∥∥∥

d∏

j=1

(Uj(tβj ) − 1

)αjx∥∥∥ dt/t

≤ 2−γ

∫ 1

0

t−γ∥∥∥

d∏

j=1

(Uj((2t)βj ) − 1

)αjx∥∥∥ dt/t

+ c

d∑

j=1

∫ 1

0

t−γ∥∥∥

d∏

`=1

(U`(tβ`)− 1

)αj(Uj(tβj ) − 1

)x∥∥∥ dt/t .

(3.2.8)

In the first integral on the right side we use the substitution t′ = 2t. Then weget the same term as on the left side, multiplied by 2−(γ−γ) < 1, except that theintegration ranges from 0 to 2. The integral

∫ 2

1. . . dt/t can be estimated by c ‖x‖.

Consequently, if the left side is finite, we find

∫ 1

0

t−γ∥∥∥

d∏

j=1

(Uj(tβj ) − 1

)αjx∥∥∥ dt/t

≤ c(‖x‖ +

d∑

j=1

∫ 1

0

t−γ∥∥∥

d∏

`=1

(U`(tβ`) − 1

)α`(Uj(tβj ) − 1

)x∥∥∥ dt/t

)

= c(‖x‖ +

d∑

j=1

∫ 1

0

t−γ∥∥∥

d∏

`=1

(U`(tβ`) − 1

)αj,`x∥∥∥ dt/t

),

where αj,` := αj if ` 6= j, and αj,j := αj + 1. Iteration leads to

∫ 1

0

t−γ∥∥∥

d∏

j=1

(Uj(tβj ) − 1

)αjx∥∥∥ dt/t

≤ c(‖x‖ +

d∑

j=1

∫ 1

0

t−γ∥∥∥

d∏

`=1

(U`(tβ`)− 1

)α′j,`x

∥∥∥ dt/t)

,

where α′j,j ≥ rj for 1 ≤ j ≤ d. Thus, retaining the factor

(Uj(tβj ) − 1

)rj and ma-jorizing the remaining ones by constants, we obtain

∫ 1

0

t−γ∥∥∥

d∏

j=1

(Uj(tβj )− 1

)αjx∥∥∥ dt/t

≤ c(‖x‖+

d∑

j=1

∫ 1

0

t−γ∥∥(

Uj(tβj ) − 1)rj

x∥∥ dt/t

).


Now we use this estimate to majorize the right side of (3.2.7). More precisely, inthe j-th summand we put α′

` = α` for ` 6= j, and α′j := αj + 1. Then the assertion

follows, provided the left side of (3.2.8) is finite. However, if the left side of (3.2.8)is infinite (with the above choice of αj), then the right side of (3.2.4) is infinite aswell. This proves the lemma. ¥

Renorming Intersections of Interpolation Spaces

After these preparations we can establish the main result of this subsection. Werecall that [t] is the largest integer less than or equal to t ∈ R. For abbreviation,I := (0,∞) and

L∗r(I

n) := Lr(In, dy/|y|n) , n ∈qN , 1 ≤ r ≤ ∞ . (3.2.9)

3.2.3 Proposition Let assumption (3.2.1) be satisfied. Suppose 0 < s < m withm ∈ νN, and 1 ≤ r ≤ ∞. Then:

(i) (X, Km/νX )s/m,r

.=d⋂

j=1

(X,D(Am/ωj

j ))s/m,r

.

(ii) The function

x 7→ ‖x‖+d∑

j=1

∥∥t−s/ωj ‖(Uj(t) − 1)m/ωj x‖ ∥∥L∗

r(I)(3.2.10)

is an equivalent norm for (X, Km/νX )s/m,r.

(iii) Assume ω = ω1 and put U(t) := U1(t1) · · ·Ud(td) with t := (t1, . . . , td). Then

x 7→ ‖x‖ +∥∥ |t|−s/ω ‖(U(t) − 1)m/ωx‖

∥∥L∗

r(Id)(3.2.11)

is also an equivalent norm.

Proof (1) Let n1, . . . , nd ∈ N and 0 < θ < 1. An interpolation result of P. Gris-vard [Gri66, Theoreme 7.1] guarantees that

d⋂

j=1

(X, D(Anj

j ))θ,r

.=(X,

d⋂

j=1

(D(Anj

j )))

θ,r. (3.2.12)

Furthermore, for k ∈ N and 0 < σ < k,

x 7→ ‖x‖X +∥∥t−σ ‖(Uj(t)− 1)kx‖

∥∥L∗

r(I)(3.2.13)

is a norm for(X, D(Ak

j ))σ/k,r

(e.g., [Tri95, Theorems 1.15.5 and 1.13.2]).


(2) We assume α ∈ Nd satisfies α q ω ≤ m. Then we set

a :=m

ω1· · · m

ωd, βj :=

aωj

m, γ := α1β1 + · · · + αdβd .

Moreover, for k ∈qN, rj := km/ωj and σj := γ/βj . Then it follows from (3.2.4)

that

‖Aαx‖ ≤ c(‖x‖ +

d∑

j=1

∫ ∞

0

t−σj∥∥(

Uj(t)− 1)rj

x∥∥ dt/t

),

due to rj ≥ m/ωj ≥ αj . We infer from (3.2.13) that the right side is a norm for

d⋂

j=1

(X, D(Arj )

)σj/rj ,1

.

Note that σj/rj = α q ω/km ≤ 1/k. Hence the monotonicity of the interpolationspaces with respect to θ (cf. (I.2.5.2)) implies

d⋂

j=1

(X, D(Akm/ωj )

)1/k,1

→ D(Aα) .

Since this is true for every α ∈ Nd with α q ω ≤ m, we obtain from (3.1.3)

d⋂

j=1

(X, D(Akm/ωj

j ))1/k,1

→ Km/νX , k ∈

qN .

(3) From interpolation theory it is known that

D(Am/ωj

j ) → (X, D(Akm/ωj

j ))1/k,∞

(e.g., [Tri95, Theorem 1.14.3]). Consequently,

d⋂

j=1

D(Am/ωj

j ) →d⋂

j=1

(X, D(Akm/ωj

j ))1/k,∞ .

From this embedding, the result of step (2), and (3.2.12) we deduce

(X,

d⋂

j=1

D(Akm/ωj

j ))

1/k,1→ K

m/νX →

d⋂

j=1

D(Am/ωj

j )

→(X,

d⋂

j=1

D(Akm/ωj

j ))

1/k,∞

for k ∈qN. Now (i) is a consequence of the reiteration theorem (I.2.8.2).


(4) Assertion (ii) follows from (i), (3.2.12), and (3.2.13).

(5) Suppose ω = ω1 and set σ := s/ω and k := m/ω. Note that

U(t) − 1 =d∑

j=1

U1(t1) · · ·Uj−1(tj−1)(Uj(tj)− 1

),

where the empty product is given the value 1. From this and the multinomialtheorem we estimate

∥∥(U(t)− 1

)kx∥∥ ≤ c

∑

|α|=k

d∏

j=1

∥∥(Uj(tj)− 1

)αjx∥∥ . (3.2.14)

Now we apply Lemma 3.2.1 with β1 = · · · = βd := ω, replace tβ by t, and take theL∗

r(Id) norm. Then

∑

|α|=k

∥∥∥ |t|−σd∏

j=1

∥∥(Uj(tj)− 1

)αjx∥∥

∥∥∥L∗

r(Id)

≤ 2−m∑

|α|=k

∥∥∥ |t|−σd∏

j=1

∥∥(Uj(2ωtj) − 1

)αjx∥∥

∥∥∥L∗

r(Id)

+ c∑

|α|=k

d∑

j=1

∥∥∥ |t|−σd∏

`=1

∥∥(U`(t`) − 1

)αj(Uj(tj) − 1

)x∥∥

∥∥∥L∗

r(Id).

In the first term on the right side we use the substitution 2ωt → τ . Then we getthe same expression as the one on the left, but multiplied by 2−(m−s) < 1. Hencethe left side can be estimated by the second term on the right. By iterating thisestimate we find, similarly as in step (2) of Lemma 3.2.2,

∑

|α|=k

∥∥∥ |t|−σd∏

j=1

∥∥(Uj(tj)− 1

)αjx∥∥

∥∥∥L∗

r(Id)

≤ c

d∑

j=1

∥∥∥ |t|−σ∥∥(

Uj(tj) − 1)k

x∥∥

∥∥∥L∗

r(Id).

(3.2.15)

Assume r < ∞. Fix j and write ξ := tj and η :=(t1, . . . , tj , . . . , td

). Then

∥∥∥ |t|−σ∥∥(

Uj(tj)− 1)k

x∥∥

∥∥∥L∗

r(Id)

=(∫ ∞

0

∫

Id−1(ξ2 + |η|2)−(σr+d)/2 dη

∥∥(Uj(ξ) − 1

)kx∥∥r

dξ)1/r


and∫

Id−1(ξ2 + |η|2)−(σr+d)/2 dη = ξ−(σr+d)+d−1

∫

Id

(1 + |ζ|2)−(σr+d)/2 dζ

= c ξ−σr−1 .

Consequently,∥∥∥ |t|−σ

∥∥(Uj(tj) − 1

)kx∥∥

∥∥∥L∗

r(Id)≤ c

∥∥∥t−σ∥∥(

Uj(t) − 1)k

x∥∥

∥∥∥L∗

r(I). (3.2.16)

If r = ∞, then it is obvious that (3.2.16) applies as well.By combining estimates (3.2.14)–(3.2.16), we get

∥∥∥ |t|−s/ω∥∥(

U(t) − 1)m/ω

x∥∥

∥∥∥L∗

r(Id)≤ c

d∑

j=1

∥∥∥t−s/ω∥∥(

Uj(t) − 1)m/ω

x∥∥

∥∥∥L∗

r(I).

(6) Retaining the notations of step (5), we find

k∑

j=0

(−1)j(

kj

)(zj − w)k =

k∑

j=0

(−1)j(

kj

) k∑

`=0

(−1)k−`(

k`

)zj`wk−`

=k∑

`=0

(−1)k−`(

k`

)wk−`

k∑

j=0

(−1)j(

kj

)zj`

=k∑

`=0

(−1)k−`(

k`

)wk−`(1 − z`)k

for z, w ∈ C. Leaving only the term with j = 0 on the left, we obtain

(1− w)k =k∑

j=0

(−1)j(

kj

)((zj − 1)kwk−j − (zj − w)k

).

Here we replace z by U1(t)U(t) and w by U1(t). Then

(1− U1(t)

)k =k∑

j=1

(−1)j(

kj

)((U1(jt)U(jt) − 1

)kU1

((k − j)t

)

− (U1((j − 1)t)U(jt)− 1

)kU1(kt)

).

Hence

∥∥(U1(t)− 1

)kx∥∥ ≤ c

k∑

j=1

(∥∥(U(j(t + te1) − 1)k

)x∥∥

+∥∥(

U(jt + (j − 1)te1)− 1)k

x∥∥)

,


where e1 := (1, 0, . . . , 0) ∈ Rd. We integrate this inequality over [0, t]d to obtain

∥∥(U1(t) − 1

)kx∥∥ ≤ c

td

k∑

j=1

∫

[0,t]d

(∥∥(U(jt + jte1)− 1

)kx∥∥

+∥∥(

U(jt + (j − 1)te1) − 1)k

x∥∥)

dt .

(3.2.17)

We set U ′(t′) := U2(t2) · · ·Ud(td) for t′ = (t2, . . . , td). Then a substitution of vari-ables gives

∫

[0,t]d

∥∥(U(jt + jte1) − 1

)kx∥∥ dt

=∫ t

0

∫

[0,t]d−1

∥∥(U1(j(t1 + t))U ′(jt′) − 1

)kx∥∥ dt′ dt1

≤∫ 2kt

0

∫ kt

0

· · ·∫ kt

0

∥∥(U(t) − 1

)kx∥∥ dt .

From this we get

t−σ∥∥(

U1(t)− 1)k

x∥∥

≤ c

∫ 2kt

0

∫ kt

0

· · ·∫ kt

0

( |t|t

)d+σ

|t|−σ∥∥(

U(t) − 1)k

x∥∥ dt

|t|d .(3.2.18)

Consider the integral operator Kf , defined for measurable functions f : R→ Cby

Kf(t) :=∫ 2kt

0

∫ kt

0

· · ·∫ kt

0

( |t|t

)d+σ

f(t)dt

|t|d .

If f ∈ L∗∞(Id), then

‖Kf‖L∗∞(I) ≤ c ‖f‖L∗∞(Id) .

Suppose f ∈ L∗1(I

d). Then∫ ∞

0

|Kf(t)| dt

t≤

∫ ∞

0

∫ 2kt

0

∫ ∞

0

· · ·∫ ∞

0

1|t|

( |t|t

)d+σ+1

|f(t)| dt

|t|d

=∫

Id

∫ ∞

|t|/2k

1|t|

( |t|t

)d+σ+1

dt |f(t)| dt

|t|d

=∫

Id

∫ ∞

1/2k

τ−(d+σ+1) dτ |f(t)| dt

|t|d

= c

∫

Id

|f(t)| dt

|t|d .

This shows that K is a continuous linear map from L∗1(I

d) into L∗1(I), and from

L∗∞(Id) into L∗

∞(I). Thus K ∈ L(L∗

r(Id), L∗

r(I))

by the Riesz–Thorin interpolation


theorem (e.g., [BeL76, Theorem 1.1.1] or [Tri95, Theorem 1.18.7]). Consequently,letting f(t) := t−σ

∥∥(U1(t) − 1

)kx∥∥, we obtain from (3.2.18)

∥∥∥t−σ∥∥(

U1(t) − 1)k

x∥∥

∥∥∥L∗

r(I)≤ c

∥∥∥ |t|−σ∥∥(

U(t) − 1)k

x∥∥

∥∥∥L∗

r(Id).

Of course, analogous estimates apply to U2, . . . , Ud. This implies

d∑

j=1

∥∥∥t−s/ω∥∥(

Uj(t)− 1)m/ω

x∥∥

∥∥∥L∗

r(I)≤ c

∥∥∥ |t|−σ∥∥(

U(t) − 1)m/ω

x∥∥

∥∥∥L∗

r(Id).

Together with step (5), this proves assertion (iii). ¥

3.3 Translation Semigroups

LetK be a closed corner in Rd and X ∈ Rd,K. Recall that assumption (VI.3.1.20)applies.

Since X is a convex cone, it is invariant under addition: X+ X ⊂ X. Hence the(d-parameter) left translation semigroup λy ; y ∈ X is well-defined on L1,loc(X, E)by

λyu(x) := u(x + y) , a.a. x, y ∈ X .

As usual, we employ the same symbol for its restriction to any of its invariantsubspaces.

For a ∈qX we consider the one-parameter semigroup La := λta ; t ≥ 0 . We

also denote by

∂a :=d∑

j=1

aj∂j ∈ L(S ′(X, E))

the distributional directional derivative in the direction of a = (a1, . . . , ad).

3.3.1 Theorem Let K be a closed corner in Rd and X ∈ Rd,K.(i) La is a strongly continuous semigroup on S(X, E) and on S ′(X, E).

(ii) SupposeX ∈

C0(X, E), BUC(X, E), Lq(X, E)

.

Then La is a strongly continuous contraction semigroup on X . Its infinites-imal generator is the X -realization of the distributional directional deriva-tive ∂a.

(iii) In either case La is a strongly continuous group if X = Rd.


Proof (1) Let u ∈ S = S(X, E) and k, m ∈ N. Then, given y ∈ X,

qk,m(λyu) = supx∈X

max|α|≤m

〈x〉k |∂α(λyu)(x)|E

≤ c(k, y) supx∈X

max|α|≤m

〈x + y〉k |∂αu(x + y)|E = c(k, y) qk,m(u) ,

where c(k, y) := supx∈X(〈x〉/〈x + y〉)k. Since K is closed, it follows from the defi-

nition of S that λy ∈ L(S).Suppose |y| ≤ 1. Then we infer from

〈x〉〈x + ty〉 ≤

〈x〉(1 + (|x| − t |y|)2)1/2

≤( 1 + |x|2

1 + |x|2 − 2 |x| + 1

)1/2

≤ c

for x ∈ X and 0 ≤ t ≤ 1, and the mean-value theorem,

((λy − 1)u

)(x) = u(x + y)− u(x) =

∫ 1

0

∂u(x + ty)y dt , (3.3.1)

that

〈x〉k∣∣∂α

((λy − 1)u

)(x)

∣∣E≤ c |y| qk,m+1(u) , x ∈ X , k,m ∈ N .

Hence

qk,m

((λy − 1)u

) ≤ c |y| qk,m+1(u) , y ∈ X , |y| ≤ 1 , k, m ∈ N . (3.3.2)

This implies that La is a strongly continuous semigroup on S.For u ∈ S and x ∈ X,

t−1(λta − 1)u(x) = t−1(u(x + ta) − u(x)

) → ∂au(x)

in E as t → 0. Hence, using the mean-value theorem once more,

(t−1(λta − 1)− ∂a

)u =

∫ 1

0

(λτta − 1)∂au dτ , t > 0 . (3.3.3)

From this and (3.3.2) we get, for k,m ∈ N,

qk,m

((t−1(λta − 1) − ∂a)u

) ≤ ctqk,m+1(u) , t > 0 .

Thus t−1(λta − 1)u → ∂au in S as t → 0. Hence ∂a ∈ L(S) is the infinitesimalgenerator of La on S.

(2) Suppose X ∈ C0(X, E), Lq(X, E)

and u ∈ X . It is obvious, due to the

closedness of K, that λtau ∈ X and ‖λtau‖X ≤ ‖u‖X for t ≥ 0. Thus La is a con-traction semigroup on X . Hence, given u ∈ X and v ∈ S,

‖λtau− u‖X ≤ ‖λta(u − v)‖X + ‖λtav − v‖X + ‖v − u‖X≤ 2 ‖u − v‖X + q0,0

((λta − 1)v

)


for t > 0. Let ε > 0. Since S d→ X , there exists v ∈ S with ‖u − v‖X < ε/3. By

step (1) there exists tε > 0 such that q0,0

((λta − 1)v

)< ε/3 for 0 < t < tε. Hence

‖(λta − 1)u‖X < ε for 0 < t < tε. This proves that La is strongly continuous on X .By step (1), S is invariant under La and contained in the domain of its

infinitesimal generator A. Hence, by the core theorem, S is a core for A. Thus,given u ∈ dom(A), there exists a sequence (uk) in S such that

uk → u , ∂auk = Auk → Au in X .

From (VI.1.2.6), which holds for X=Rd also, we get ∂a ∈L(S ′) with S ′ := S ′(X, E).Theorem VI.1.2.3(i) guarantees that S is dense in S ′. From this and (1.2.1) weinfer that ∂auk → ∂au and ∂auk → Au in S ′. Hence Au = ∂au for u ∈ dom(A),which shows that A is the X -realization of ∂a.

(3) Suppose X = BUC(X, E). Set Y := BUC2(X, E). It is again obvious thatY is an invariant linear subspace for La. Given u ∈ Y , it follows from (3.3.1) and(3.3.3) that

∥∥(t−1(λta − 1)− ∂a

)u‖∞ ≤ t ‖∂2

au‖∞ , t > 0 .

Thus Y ⊂ dom(A) and Au = ∂au for y ∈ Y . Theorem 1.6.4 implies that Y is densein X . Thus Y is a core for A. Similarly as in the preceding step, we now deducefrom the core theorem that A is the X -realization of ∂a. ¥

3.3.2 Remark IfK is not closed, then neither S(K, E) nor C0(K, E) nor BUC(K, E)is invariant under La since the ‘boundary conditions’ u |∂jK = 0 for j ∈ J∗ are notpreserved. ¥

We denote by e1, . . . , ed the standard basis of Rd. Then Uj := Lej, 1 ≤ j ≤ d,

are pair-wise commuting strongly continuous contraction semigroups on X . Theinfinitesimal generator of Uj is the X -realization of the distributional derivative ∂j

on X.

3.3.3 Lemma Suppose

X ∈ C0(X, E), BUC(X, E), Lq(X, E)

.

Set ∂ := (∂1, . . . , ∂d). Then, given m ∈ νqN,

Km/νX (∂) =

Cm/ν0 (X, E) , if X = C0(X, E) ,

BUCm/ν(X, E) , if X = BUC(X, E) ,

Wm/νq (X, E) , if X = Lq(X, E) .

Proof It follows from Theorem 3.3.1 and Lemma III.4.9.1 that −∂1, . . . ,−∂d arepair-wise resolvent commuting elements of G(X ). Hence the assertion is a conse-quence of (3.1.3) and definitions (1.1.4), (1.1.7) and Theorem 1.1.4(i), due to thefact that S(X, E) is a core for ∂j if X = C0(X, E). ¥


3.4 Renorming Besov Spaces

Now we apply the general results of the preceding subsections to Besov spaces ofpositive order. This leads to important characterizations by intrinsic norms.

Recall definition (3.2.9) of L∗q . Suppose K is a corner in Rd and X ∈ Rd,K.

For u ∈ L1,loc(X, E) we put

4yu(x) := u(x + y) − u(y) = (λy − 1)u(x) , a.a. x, y ∈ X . (3.4.1)

Moreover, 4k+1y := 4y4k

y for k ∈ N with 40y = id. Given s > 0 and p, r ∈ [1,∞],

[u]s,p,r,j = [u]Es,p,r,j :=∥∥t−s ‖4[s]+1

teju‖Lp(X,E)

∥∥L∗

r(I)(3.4.2)

and

[u]s,p,r = [u]Es,p,r :=∥∥ |y|−s ‖4[s]+1

y u‖Lp(X,E)

∥∥L∗

r(Id). (3.4.3)

Furthermore,[·]s,p,j := [·]s,p,p,j , [·]s,p := [·]s,p,p .

By means of these seminorms we can characterize Bs/νp,r (X, E) intrinsically.

In the following theorem we set, for m ∈ νN,

Xm/νp :=

Wm/ν

p (X, E) , if p < ∞ ,

BUCm/ν(X, E) , if p = ∞ ,

and Xp := X 0/νp . Moreover, B

s/νp,r = B

s/νp,r (X, E).

3.4.1 Theorem Let K be a closed corner in Rd and X ∈ Rd,K. Suppose s > 0and 1 ≤ p, r ≤ ∞.

(i) It holds

u ∈ Bs/νp,r iff u ∈ Xp and [u]s/ωj ,p,r,j < ∞ for 1 ≤ j ≤ d .

Moreover,

‖·‖∗s/ν,p,r := ‖·‖p +d∑

j=1

[·]s/ωj ,p,r,j (3.4.4)

is a norm for Bs/νp,r .

(ii) Let the weight system be homogeneous, that is, ω = ν1, and assume m ∈ νN.(α) If m < s, then

u ∈ Bs/νp,r iff u ∈ Xm/ν

p and [∂αu](s−m)/ν,p,r < ∞ for |α| ≤ m/ν .


Furthermore,

u 7→ |||u|||(m)s/ν,p,r := |||u|||m/ν,p +

∑

|α|≤m/ν

[∂αu](s−m)/ν,p,r


(β) If m ∈ νN and m < s < m + ν, then

u ∈ Bs/νp,r iff u ∈ Xm/ν

p and [∂αu](s−m)/ν,p,r < ∞ for |α| = m/ν ,

andu 7→ |||u|||s/ν,p,r := |||u|||m/ν,p +

∑

|α|=m/ν

[∂αu](s−m)/ν,p,r


Proof (1) We fix m ∈ νN with s < m. Then Lemma 3.3.3 and Theorems 2.7.4and 2.8.3 imply

Bs/νp,r

.=(Xp,K

m/νXp

(∂))s/m,r

. (3.4.5)

Thus it follows from Proposition 3.2.3(ii) that

u 7→ ‖u‖p +d∑

j=1

∥∥t−s/ωj ‖4m/ωj

teju‖Lp

∥∥L∗

r(I)

is a norm for Bs/νp,r . Due to (3.2.13), we can replace m/ωj by [s/ωj ] + 1. This

proves (i).(2) Suppose ω = ν1 and m ∈ νN satisfies m < s. Assume n− ν ≤ s < n with

n ∈ νN. Set θ := (s − m)/(n− m) and Yp := Km/νXp

(∂). Then Lemma 3.1.2(iv) im-plies (Yp,K

(n−m)/νYp

(∂))θ,r

=(K

m/νXp

(∂),Kn/νXp

(∂))θ,r

.

By Lemma 3.3.3, the space on the right equals (Xm/νp ,Xn/ν

p )θ,r. Thus we obtainfrom Theorems 2.7.4 and 2.8.3

Bs/νp,r

.=(Yp,K

(n−m)/νYp

(∂))θ,r

.

Now Proposition 3.2.3(iii) implies that

u 7→ |||u|||m/ν,p +∥∥ |y|−(s−m)/ν ‖4(n−m)/ν

y u‖Yp

∥∥L∗

r(Id)(3.4.6)

is an equivalent norm for Bs/νp,r . Note that

‖4(n−m)/νy u‖Yp = ‖4[(s−m)/ν]+1

y u‖Xm/νp

=∑

|α|≤m/ν

‖∂α4[(s−m)/ν]+1y u‖p . (3.4.7)

Since ∂α and 4y commute, we thus see that (3.4.6) equals |||·|||(m)s/ν,p,r. This shows

that (ii.α) is true.


(3) Assume that m < s < m + ν = n. Then[(s − m)/ν

]= 0. Hence (3.4.7)

reads as‖4yu‖Yp =

∑

|α|≤m/ν

‖∂α4yu‖p .

If u ∈ Xp ∩ C1, then the mean-value theorem implies

4yu(x) =∫ 1

0

∇u(x + ty)y dt =∫ 1

0

λty∇u(x) dt y , a.a. x, y ∈ X . (3.4.8)

Thus ‖λzu‖L(Xp) ≤ 1 for z ∈ X gives

‖4yu‖p ≤ |y| ‖∇u‖p and ‖4yu‖p ≤ 2 ‖u‖p

for y ∈ Id. Set M := y ∈ Id ; |y| ≤ 1 . Then we deduce from these estimates

[∂αu](s−m)/ν,p,r =((∫

M

+∫

Id\M

)(|y|−(s−m)/ν ‖4y∂αu‖p

)r dy

|y|d)1/r

≤ c(‖∇∂αu‖p + ‖∂αu‖p

)

if r < ∞, and, similarly,

[∂αu](s−m)/ν,p,∞ ≤ c(‖∇∂αu‖p + ‖∂αu‖p

),

provided u ∈ Xm/νp and |α| < m/ν. Consequently,

∑

|α|<m/ν

[∂αu](s−m)/ν,p,r ≤ c |||u|||m/ν,p,r .

This implies that |||·|||s/ν,p,r is a stronger norm than |||·|||(m)s/ν,p,r. Since it is obviously

weaker also, we see that these two norms are equivalent. This proves (ii.β), hencethe theorem. ¥

In the next subsection we consider non-homogeneous weight systems andprove an analogue of part (ii.β).

3.5 Intersection-Space Characterizations

In accordance with (VI.3.1.1) we decompose X as follows:

X = X1 × · · · × X` , Xi := Id(i−1)+1 × · · · × Id(i) , (3.5.1)

where d(i) := d1 + · · · + di for 1 ≤ i ≤ `, and d(0) := 0. Then either Xi = Rdi , orXi is a closed corner in Rdi . We also set11

Xı := X1 × · · · × Xi × · · · × Xd

and write (xi; xı) for x ∈ X and 1 ≤ i ≤ ` with xı ∈ Xı.

11As usual, the hat over a factor (or a component), the ‘omission symbol’, means that thecorresponding entry is absent.


Fubini’s theorem implies that

u 7→ (xi 7→ u(xi; ·)

)(3.5.2)

is an isomorphic isomorphism

from Lq(X, E) onto Lq

(Xi, Lq(Xı, E)

), 1 ≤ q < ∞ , (3.5.3)

(cf. [AmE08, Theorem X.6.2.2]). Furthermore, it is also an isometric isomorphism

from B(X, E) onto B(Xi, B(Xı, E)

). (3.5.4)

By means of these isomorphisms we often identify the spaces occurring in (3.5.3),resp. in (3.5.4), with each other. Thus we write, somewhat loosely,

Lq(X, E) = Lq

(Xi, Lq(Xı, E)

), 1 ≤ q < ∞ , (3.5.5)

and

B(X, E) = B(Xi, B(Xı, E)

). (3.5.6)

Next, we look at BUC(X, E). From

u(x)− u(y) = u(xi; xı) − u(yi; xı) + u(yi; xı)− u(yi; yı)

we infer that u ∈ BUC(X, E) iff u(·; xı) ∈ BUC(Xi, E), uniformly with respect toxı ∈ Xı, and u(xi; ·) ∈ BUC(Xı, E), uniformly with respect to xi ∈ Xi. Thus

BUC(X, E) .= BUC(Xi, B(Xı, E)

) ∩ BUC(Xı, B(Xi, E)

).

Suppose u belongs to the intersection space on the right. Then it follows fromu ∈ BUC

(Xı, B(Xi, E)

)that u(xi; ·) is uniformly continuous for each xi ∈ Xi.

Since BUC(Xı, E) is closed in B(Xı, E), we get from u(·; xı) ∈ BUC(Xi, B(Xı, E)

)

that BUC(X, E) → BUC(Xi, BUC(Xı, E)

). The converse inclusion being obvious,

we arrive at

BUC(X, E) = BUC(Xi, BUC(Xı, E)

). (3.5.7)

Intersection Space Representations

Now we can prove a fundamental intersection space characterization of Besovspaces.

3.5.1 Theorem Suppose K is a closed corner in Rd and X equals either Rd or K.Let 1 ≤ q < ∞ and 1 ≤ r ≤ ∞. Then

Bs/νq,r (X, E) .=

⋂

i=1

Bs/νiq,r

(Xi, Lq(Xı, E)

)(3.5.8)


and

Bs/ν∞,r(X, E) .=

⋂

i=1

Bs/νi∞,r

(Xi, BUC(Xı, E)

). (3.5.9)

Proof Put Xj := Lq(∂jX, E) for 1 ≤ j ≤ d. By Theorem 3.3.1, Lejis a strongly

continuous contraction semigroup on Y := Lq(Ij ,Xj) whose infinitesimal gener-ator, Aj , is the Y-realization of the Xj-valued distributional derivative ∂j on Ij .Hence D(Am/ωj

j ) .= Wm/ωj

q (Ij ,Xj) for m ∈ ωjN by Theorem 1.4.1 and (3.1.3) withA := ∂j.

Suppose m ∈ ωN with m > s. Then we get from this observation, Proposi-tion 3.2.3(i), Lemma 3.3.3, and (3.5.5) that

(Lq(X, E), Wm/ν

q (X, E))s/m,r

.=d⋂

j=1

(Lq

(Ij , Lq(∂jX, E)

),Wm/ωj

q

(Ij , Lq(∂jX, E)

))s/m,r

.

Thus, by Theorems 2.7.4 and 2.8.3,

Bs/νq,r (X, E) .=

d⋂

j=1

Bs/ωjq,r

(Ij , Lq(∂jX, E)

). (3.5.10)

Similarly,

Bs/νiq,r

(Xi, Lq(Xı, E)

) .=d(i)⋂

j=d(i−1)+1

Bs/νiq,r

(Ij , Lq(∂jXi, Lq(Xı, E))

).

Note that, once more by (3.5.5), Lq

(∂jXi, Lq(Xı, E)

)= Lq(∂jX, E). Hence

Bs/νiq,r

(Xi, Lq(Xı, E)

) .=d(i)⋂

j=d(i−1)+1

Bs/νiq,r

(Ij , Lq(∂jX, E)

).

Since νi = ωj for d(i− 1) + 1 ≤ j ≤ d(i), (3.5.8) is thus a consequence of (3.5.10).By replacing in this line of arguments (3.5.5) by (3.5.7) and Lq(Ij , ∂jX) by

BUC(Ij , ∂jX) etc., we get the second claim. ¥

Equivalent Norms

As a consequence of this theorem we obtain a very useful and important equivalentnorm for B

s/νp,r (X, E). In the generic case it involves seminorms containing only the

highest derivatives.


For this we set, for 0 < θ < 1,

[[u]]θ,p,r;i :=∥∥ |yi|−θ ‖4(yi;0)u‖Lp(X,E)

∥∥L∗

r(Idi )

for 1 ≤ i ≤ `, and[[·]]θ,p;i := [[·]]θ,p,p;i .

3.5.2 Theorem Let K be a closed corner in Rd and X ∈ Rd,K. Suppose mi ∈ νiNsatisfy mi < s < mi + νi for 1 ≤ i ≤ `, and p, r ∈ [1,∞]. Set

|||u|||∗s/ν,p,r :=∑

i=1

(mi/νi∑

j=0

‖∇jxi

u‖p + [[∇mi/νixi u]](s−mi)/νi,p,r;i

).

Then |||·|||∗s/ν,p,r is an equivalent norm for Bs/νp,r (X, E).

Proof It follows from (3.5.5) that, given yi ∈ Xi and 1 ≤ q < ∞,

‖4(yi;0)u‖Lq(X,E) =∥∥xi 7→ ‖4yiu(xi; ·)‖Lq(Xı,E)

∥∥Lq(Xi)

.

Similarly, (3.5.6) implies

‖4(yi;0)u‖B(X,E) =∥∥xi 7→ ‖4yiu(xi; ·)‖B(Xı,E)

∥∥B(Xi)

.

From this and Theorem 3.4.1(ii.β) we deduce

u 7→mi/νi∑

j=0

‖∇jxi

u‖p + [[∇mi/νixi u]](s−mi)/νi,p,r;i

is a norm for Bs/νip,r

(Xi, Lp(Xı, E)

)if p < ∞, and for B

s/νi∞,r

(Xi, BUC(Xı, E)

)oth-

erwise. Now Theorem 3.5.1 implies the assertion. ¥

3.5.3 Remark Suppose we drop the assumption that s /∈ νiN for 1 ≤ i ≤ `. Forexample, assume s = m1 + ν1. Then we have to replace [[∇m1/ν1

x1 u]](s−m1)/ν1,r;1 by∑m1/ν1

j=0 [[∇jx1

u]](s−m1)/ν1,r;1

.

Proof This follows from Theorem 3.4.1(ii.α). ¥

Nikol′skiı Spaces

The Nikol′skiı spaces Bs/νq,∞(X, E), 1 ≤ q < ∞, are distinguished by the fact that the

seminorms [[·]](s−mi)/νi,q,∞ are particularly simple. More precisely, the followingcharacterization applies.


3.5.4 Proposition Let K be a closed corner in Rd and X ∈ Rd,K. Supposemi ∈ νiN satisfy mi < si < mi + νi for 1 ≤ i ≤ `, and 1 ≤ q < ∞. Then u belongsto B

s/νq,∞(X, E) iff

∇jxi

u ∈ Lq(X, E) , 0 ≤ j ≤ mi/νi , 1 ≤ i ≤ ` ,

and

supy∈Xi

‖4(y;0)∇mi/νixi u‖q

|y|(s−mi)/νi< ∞ , 1 ≤ i ≤ ` .

Proof Obvious. ¥

3.6 Besov–Slobodeckii and Besov–Holder Spaces

Let M be a corner in Rd and Y ∈ Rd,M. Then we set

Bt/νr (Y, E) := Bt/ν

r,r (Y, E) , t ∈ R , 1 ≤ r ≤ ∞ .

These spaces are called Besov–Slobodeckii spaces if r < ∞, and Besov–Holder spaces

for r = ∞.

Mixed Intersections

In the case of these spaces we can replace (some of) the spaces Bs/νiq

(Xi, Lq(Xı, E)

)

in (3.5.8) by Lq

(Xı, B

s/νiq (Xi, E)

). A similar statement applies to the Besov–Holder

spaces in (3.5.9). The proofs are based on the following facts.

3.6.1 Lemma Let 1 ≤ i ≤ `. Then

Lq

(Xı, B

s/νiq (Xi, E)

) .= Bs/νiq

(Xi, Lq(Xı, E)

), 1 ≤ q < ∞ ,

andB

(Xı, B

s/νi∞ (Xi, E)) .= Bs/νi∞

(Xi, B(Xı, E)

).

Proof (1) Recall I = (0,∞). Let t > 0. By Fubini’s theorem,∫

Xı

∫

Idi

|y|−tq ‖4[t]+1y u(·; xı)‖q

Lq(Xi,E)

dy

|y|didxı

=∫

Idi

|y|−tq

∫

Xi

∫

Xı

|4[t]+1y u(xi; xı)|qE dxı dxi

dy

|y|di

=∫

Xi

∫

Idi

|y|−tq ‖4[t]+1y u(xi; ·)‖q

Lq(Xı,E)

dy

|y|didxi .


Consequently,

∥∥xı 7→[u(·; xı)

]E

t,q

∥∥Lq(Xı)

=∥∥[

xi 7→ u(xi; ·)]t,q

∥∥Lq(Xi,E)

. (3.6.1)

From Theorem 3.4.1(ii) we infer that, given xı ∈ Xı,

u(·; xı) 7→ ‖u(·; xı)‖Lq(Xi,E) +[u(·; xı)

]E

s/νi,q

is a norm for u(·; xı) ∈ Bs/νiq (Xi, E). Hence

u 7→ ‖u‖Lq(Xı,Lq(Xi,E)) +∥∥xı 7→

[u(·; xı)

]E

s/νi,q

∥∥Lq(Xı)

is a norm for Lq

(Xı, B

s/νiq (Xi, E)

). Thus (3.5.5) and (3.6.1) give the first claim.

(2) We calculate

∥∥xı 7→[u(·; xı)

]E

t,∞∥∥

B(Xı)= sup

xı∈Xı

supy∈Idi

|y|−t supxi∈Xi

|4[t]+1y u(xi; xı)|E

= supxi∈Xi

supy∈Idi

|y|−t supxı∈Xı

|4[t]+1y u(xi; xı)|E

=∥∥[

xi 7→ u(xi; ·)]E

t,∞∥∥

B(Xi),

where it is understood that 4y operates on the first argument of u. From this and(3.5.6) we obtain the second claim. ¥

3.6.2 Theorem Let K be a closed corner in Rd, X ∈ Rd,K, and s > 0. SupposeM is a (possibly empty) subset of L := 1, . . . , `. Then

Bs/νq (X, E) .=

⋂

µ∈M

Lq

(Xµ, Bs/νµ

q (Xµ, E)) ∩

⋂

λ∈L\MBs/νλ

q

(Xλ, Lq(Xλ, E)

)(3.6.2)

for 1 ≤ q < ∞, and

Bs/ν∞ (X, E) .=

⋂

µ∈M

B(Xµ, Bs/νµ∞ (Xµ, E)

) ∩⋂

λ∈L\MBs/νλ∞

(Xλ, BUC(Xλ, E)

).

Proof This is immediate from Theorem 3.5.1 and Lemma 3.6.1. ¥

3.6.3 Corollary We write νq= (ν2, . . . , ν`) and X q

:= X2 × · · · × X`.Then

Bs/νq (X, E) .= Lq

(X1, B

s/νpppp

q (X q, E)

) ∩ Bs/ν1q

(X1, Lq(X

q, E)

)


for 1 ≤ q < ∞, and

Bs/ν∞ (X, E) .= B

(X1, B

s/νpppp

∞ (X q, E)

) ∩Bs/ν1∞(X1, BUC(X q

, E))

.

Proof Note that, for 2 ≤ i ≤ `,

Lq

(Xı, B

s/νiq (Xi, E)

)= Lq

(X1, Lq

(X q

ı, Bs/νiq (Xi, E)

)).

Hence

⋂

i=2

Lq

(Xı, B

s/νiq (Xi, E)

) .= Lq

(X1,

⋂

i=2

Lq

(X q

ı, Bs/νiq (Xi, E)

)).

Using (3.6.2) with L := 2, . . . , ` and M := L, we get

⋂

i=2

Lq

(X q

ı, Bs/νiq (Xi, E)

) .= Bs/νpppp

q (X q, E) .

Now the first assertion follows by applying (3.6.2) once more, this time withL := 1, . . . , ` and M := 2, . . . , `.

The second assertion follows similarly by invoking (3.5.6). ¥

Slobodeckii, Holder, and Little Holder Spaces

We define Slobodeckii spaces by

W s/νq (X, E) := Bs/ν

q (X, E) , s ∈ R+\νN , (3.6.3)

and Holder spaces by

BUCs/ν(X, E) := Bs/ν∞ (X, E) , s ∈ R+\νN . (3.6.4)

Little Holder spaces are introduced by

bucs/ν(X, E) :=

BUCs/ν(X, E) , s ∈ νN ,

bs/ν∞ (X, E) , s ∈ R+\νN .

(3.6.5)

In addition, we set

Cs/ν0 (X, E) := Bs/ν

∞ (X, E) , s ∈ R+\νN , (3.6.6)

and call Cs/ν0 (X, E) for s ≥ 0 very little Holder spaces (recall (1.1.10)). Then

[W t/ν

q (X, E) ; t ≥ 0]

is the Sobolev–Slobodeckii space scale (over Lq), and[BUCt/ν(X, E) ; t ≥ 0

],

[buct/ν(X, E) ; t ≥ 0

],

[C

t/ν0 (X, E) ; t ≥ 0

]

are the Holder space, the little Holder space, and the very little Holder space scales,respectively.


3.6.4 Remark Suppose ω = 1, X = Rd, and 0 < s < 1. Then it is clear that

[u]s,q.=

(∫

Rd×Rd

( |u(x) − u(y)|E|x − y|s

)q dx dy

|x− y|d)1/q

and

[u]s,∞.= sup

x 6=y

|u(x) − u(y)|E|x − y|s .

Hence Theorem 3.4.1 shows that Bsq(Rd, E) is the usual Slobodeckii space, and

Bs∞(Rd, E) is the standard Holder space, of order s. ¥

3.6.5 Examples (Parabolic weight systems) Let [m, ν] be a parabolic weightsystem. Suppose Y is either a closed corner in Rm or Y = Rm, and J ∈ R,R+.Denote the general point of Y× J by x = (y, t) with t ∈ J . We set

B(s,s/ν)r (Y× J,E) := Bs/ν

r (X, E) , s > 0 , 1 ≤ r ≤ ∞ .

(a) It holds, for 1 ≤ q < ∞,

B(s,s/ν)q (Y× J,E) .= Lq

(J,Bs

q(Y, E)) ∩Bs/ν

q

(J, Lq(Y, E)

).= Bs

q

(Y, Lq(J,E)

) ∩Bs/νq

(J, Lq(Y, E)

)

and

B(s,s/ν)∞ (Y× J,E) .= B

(J,Bs

∞(Y, E)) ∩Bs/ν

∞(J,BUC(Y, E)

).= Bs

∞(Y, BUC(J,E)

) ∩Bs/ν∞

(J,BUC(Y, E)

).

(b) Assume n ∈ N and k ∈ νN satisfy

n < s < n + 1 , k < s < k + ν .

Set [[·]]ϑ,r;t := [[·]]ϑ,r;1 and [[·]]ϑ,r;y := [[·]]ϑ,r;2. Thus, in particular,

[[u]]ϑ,∞;t = supt∈J

[u(·, t)]

ϑ,∞ , [[u]]ϑ,∞;y = supy∈Y

[u(y, ·)]

ϑ,∞ .

Then

|||u|||∗s/ν,r ∼n∑

j=0

‖∇jyu‖r +

k/ν∑

j=0

‖∂jt u‖r + [[∇n

y u]]s−n,r;t

+ [[∂k/νt u]](s−k)/ν,r;y

for u ∈ B(s,s/ν)r (X, E).


(c) Suppose s /∈ N. Then

W (s,s/ν)q (Y× J,E) .= Lq

(J,W s

q (Y, E)) ∩W s/ν

q

(J, Lq(Y, E)

).= W s

q

(Y, Lq(J,E)

) ∩W s/νq

(J, Lq(Y, E)

)

and

BUC(s,s/ν)(Y× J,E) .= B(J,BUCs(Y, E)

) ∩BUCs/ν(J,BUC(Y, E)

).= BUCs

(Y, BUC(J,E)

) ∩BUCs/ν(J,BUC(Y, E)

).

Proof This follows from (a) and definitions (3.6.3) and (3.6.4). ¥

3.7 Little Holder Spaces

The definition of little Holder spaces, Corollary 2.7.5, and Theorem 2.8.3 implythat b

s/ν∞ (X, E) is the closure of BC∞(X, E) in B

s/ν∞ (X, E). In the next theorem

we give an intrinsic characterization of bs/ν∞ (X, E).

Let 0 < ϑ ≤ 1 and δ > 0. Suppose either Y = Rn or Y is a closed corner in Rn,and F is a Banach space. We set, for v : Y→ F ,

[v]δϑ,∞ := sup0<|h|≤δ

h∈Y

‖4[ϑ]+1h v‖∞|h|ϑ .

Then

[·]ϑ,∞ ≤ [·]δϑ,∞ + 4δ−ϑ ‖·‖∞ . (3.7.1)

Suppose v ∈ C1+[ϑ](Y, F ). Then (see (3.4.8))

4hv(y) =∫ 1

0

∇v(y + τh)h dτ

and, if ϑ = 1,

42hv(y) =

∫ 1

0

∫ 1

0

∇2v(y + (τ1 + τ2)h

)[h, h] dτ1 dτ2 .

From this we get

[v]δϑ,∞ ≤ δ1+[ϑ]−ϑ ‖∇1+[ϑ]v‖∞ , δ > 0 , 0 < ϑ ≤ 1 . (3.7.2)

Now we consider the isotropic case first.

3.7.1 Theorem Let K be a closed corner in Rd and X ∈ Rd,K. Suppose k ∈ Nand k < s ≤ k + 1 with s < k + 1 if X = K. Then u ∈ bs

∞(X, E) iff

u ∈ BUCk(X, E) and limδ→0

[∇ku]δs−k,∞ = 0 . (3.7.3)


Proof (1) Let u ∈ bs∞ = bs

∞(X, E). Since bs∞ → BUCk, the first part of (3.7.3) is

clear.Assume ε > 0 and set ϑ := s − k ∈ (0, 1]. We can find v ∈ BC∞ such that

|||u − v|||∗s,∞ = |||u− v|||k,∞ + [∇ku]ϑ,∞ < ε/2 . (3.7.4)

We infer from (3.7.2) that

[∇kv]δϑ,∞ ≤ δ1+[ϑ]−ϑ |||v|||k+2,∞ , δ > 0 .

Hence there exists δε > 0 such that

[∇kv]δϑ,∞ < ε/2 , δ ≤ δε . (3.7.5)

Since[∇ku]δϑ,∞ ≤ [∇k(u − v)

]ϑ,∞ + [∇kv]δϑ,∞ ,

it follows from (3.7.4) and (3.7.5) that

[∇ku]δϑ,∞ ≤ |||u− v|||∗s,∞ + ε/2 < ε , δ ≤ δε .

Thus u satisfies (3.7.3).(2) We denote by X = X (X, E) the linear subspace of BUC of all u satisfying

(3.7.3). Then we suppose that u ∈ X .First we assume that X = Rd. We fix a mollifier ϕε ; ε > 0 . Then we find

(cf. (III.4.2.10) and (III.4.2.25))

ϕε ∗ u ∈ BC∞ , ‖∂β(ϕε ∗ u)‖∞ ≤ ‖∂βu‖∞ (3.7.6)

for β ∈ Nd with |β| ≤ k, and

limε→0

ϕε ∗ u = u in BUCk . (3.7.7)

Given v ∈ L1,loc, we obtain from

ϕε ∗ v(x) =∫

Xϕε(y)v(x − y) dy

that4[ϑ]+1

h (ϕε ∗ v) = ϕε ∗ 4[ϑ]+1h v .

Hence, by (3.7.6),[∇k(ϕε ∗ u − u)

]δ

ϑ,∞ ≤ [∇k(ϕε ∗ u)]δ

ϑ,∞ + [∇ku]δϑ,∞ ≤ 2 [∇ku]δϑ,∞ . (3.7.8)

Let ξ > 0. By (3.7.3) and (3.7.8) we find δξ > 0 such that[∇k(ϕε ∗ u− u)

]δξ

ϑ,∞ ≤ ξ/2 , ε > 0 .


Thus, by (3.7.1),[∇k(ϕε ∗ u − u)

]ϑ,∞ ≤ ξ/2 + 4δ−ϑ

ξ ‖∇k(ϕε ∗ u− u)∥∥∞

for ε > 0. Now it follows from (3.7.7) that there exists εξ > 0 such that[∇k(ϕε ∗ u− u)

]ϑ,∞ < ξ , 0 < ε ≤ εξ .

From this and (3.7.7) we see that ϕε ∗ u converges in BUCs towards u. This provesX ⊂ bs

∞, that is, condition (3.7.3) is sufficient if X = Rd.

(3) Suppose X = K, s 6= k + 1, and u ∈ X (K, E). Then (3.7.1) implies

[∇ku]ϑ,∞ ≤ [∇ku]1ϑ,∞ + 4 ‖∇ku‖∞ .

Thus u ∈ BUCs(K, F ). Hence v := EKu ∈ BUCs(Rd, E) by Theorem 2.8.2.

We write K = X1 × · · · × Xd with Xi ∈ R+,R and set Fi := BUC(Xı, E)and

wi :=(xi 7→ w(xi; ·)

) ∈ BUC(Xi, Fi)

for w ∈ BUC(X, E).

Suppose i is such that Xi = R+. If xi < yi ≤ 0, then, recalling the notationsof Subsection VI.1.1,

|vi(xi)− vi(yi)|Fi

|xi − yi|ϑ ≤∫ ∞

0

tϑ |h(t)| |ui(−txi) − ui(−tyi)|Fi

|(−txi) − (−tyi)|ϑ dt . (3.7.9)

Let ε > 0 be given and suppose s < 1. We fix τi > 0 such that∫ ∞

τi

tϑ |h(t)| dt [ui]ϑ,∞ < ε/2 .

Then we choose δi = δi(ε) > 0 such that∫ ∞

0

tϑ |h(t)| dt [ui]τiδϑ,∞ < ε/2 , 0 < δ < δi .

Now we infer from (3.7.9) that

|vi(xi)− vi(yi)|Fi

|xi − yi|ϑ < ε , xi, yi < 0 , |xi − yi| < δ , 0 < δ < δi .

Let xi < 0 < yi. Then |xi|, |yi| < |xi − yi| imply

|vi(xi) − vi(yi)|Fi

|xi − yi|ϑ ≤ |vi(xi) − vi(0)|Fi

|xi|ϑ +|ui(0) − ui(yi)|Fi

|yi|ϑ .


From this and (3.7.9) we obtain[(EKu)i

]δ

ϑ,∞ → 0 as δ → 0. Using (VI.1.1.6) and(VI.1.1.7), we thus find

limδ→0

[∇k(EKu)i

]δ

ϑ,∞ = 0 , 1 ≤ i ≤ d . (3.7.10)

Writing

w(x) − w(y)

=d∑

i=1

(w(y1, . . . , yi−1, xi, xi+1, . . . , xd) − w(y1, . . . , yi−1, yi, xi+1, . . . , xd)

),

we get|w(x)− w(y)|

|x − y|ϑ ≤d∑

i=1

‖w(xi; ·) − w(yi; ·)‖∞|xi − yi|ϑ .

Thus, by (3.7.10),limδ→0

[∇k(EKu)]δ

ϑ,∞ = 0 .

Theorem 1.3.1 implies EKu ∈ BUCk(Rd, E). This proves that EKu ∈ X (Rd, E).Hence step (2) guarantees the existence of a sequence (wj) in BC∞(Rd, E) con-verging in B

s/ν∞ (Rd, E) towards EKu. Now we invoke Theorem 2.8.2 and Corol-

lary 1.3.2(ii) to infer that uj := RKwj belongs to BC∞(K, E) and uj convergestowards RKEKu = u in B

s/ν∞ (K, E). This proves X (K, E) ⊂ b

s/ν∞ (K, E), hence the

theorem. ¥

Now we turn to anisotropic spaces and use the notations and conventions ofSubsection 3.5.

3.7.2 Theorem Let K be a closed corner in Rd and X ∈ Rd,K. Suppose s belongsto R+\N. Then

bucs/ν(X, E) .=⋂

i=1

bucs/νi(Xi, BUC(Xı, E)

). (3.7.11)

Proof We omit E, set Fi := BUC(Xı), and denote the above intersection spaceby X (X).

(1) It follows From Theorem 3.5.1 that

bucs/ν(X) → BUCs/ν(X) .=⋂

i=1

BUCs/νi(Xi, Fi) .

We write vi :=(xi 7→ v(xi; ·)

)for v ∈ BUCs/ν(X). Let (u(j)) be a sequence in

BC∞(X) converging in BUCs/ν(X) towards some u. Then u(j),i ∈ BC∞(Xi, Fi),


and (u(j),i) converges in BUCs/νi(Xi, Fi) towards some ui. Hence ui belongs tobucs/νi(Xi, Fi) for 1 ≤ i ≤ `. This proves bucs/ν(X) → X (X).

(2) By bucs/νi → BUCs/νi and Theorem 3.5.1,

X (X) →⋂

i=1

BUCs/νi(Xi, Fi).= BUCs/ν(X) .

Suppose X = Rd and u ∈ X (X). Let ϕε ; ε > 0 be a mollifier on Rd. Thenthe arguments of step (2) of the preceding proof show that (ϕε ∗ u)i → ui inBUCs/νi(Xi, Fi) for 1 ≤ i ≤ ` as ε → 0. Thus, by Theorem 3.5.1, ϕε ∗ u → u inBUCs/ν(X) as ε → 0. Hence u ∈ bucs/ν(X). This proves X (Rd) → bucs/ν(Rd).

If X = K, then we find similarly X (K) → bucs/ν(K) by appropriately modi-fying the extension and restriction arguments of step (3) of the preceding proof. ¥

3.7.3 Corollary Define ki ∈ N by νiki < s < νi(ki + 1) for 1 ≤ i ≤ `. Then u be-longs to bucs/ν(X, E) iff

u ∈⋂

i=1

BUCki(Xi, BUC(Xı, E)

)

and

limδ→0

∑

i=1

[∇kiui]δ(s−νiki)/νi,∞ = 0 .

Proof This follows from (3.7.11) and Theorem (3.7.1). ¥

Lastly, we prove a partial analogue of Corollary 3.6.3.

3.7.4 Theorem Let K be a closed corner in Rd and X ∈ Rd,K. Suppose s belongsto R+\N. Using (3.5.1), write

νq:= (ν2, . . . , ν`) and X q

:= X2 × · · · × X` .

Then

bucs/ν(X, E) → BUC(X1, bucs/ν

pppp(X q

, E)) ∩ bucs/ν1

(X1, BUC(X q

, E))

.

Proof Again we omit E and write ui :=(xi 7→ u(xi; ·)

). Theorem 3.7.2 implies

bucs/ν(X) .= bucs/ν1(X1, BUC(X q

)) ∩ bucs/ν

pppp(X q

, BUC(X1))

. (3.7.12)


Let ε > 0 and u ∈ bucs/ν(X) be given. There exists v ∈ BC∞(X) satisfying|||u− v|||∗s/ν,∞ < ε/3. Consequently,

|||u1(x1) − v1(x1)|||∗s/ν pppp,∞ < ε/3 , x1 ∈ X1 .

Thus we deduce from

|||u1(x1) − u1(y1)|||∗s/ν pppp,∞ ≤ 2ε/3 + |||v1(x1)− v1(y1)|||∗s/ν pppp,∞and the smoothness of v that

u1 ∈ BUC(X1, BUCs/ν

pppp(X q

))

. (3.7.13)

Assume ki is as in the preceding corollary and set ϑi := (s − νiki)/νi. Then

[∇kiui]δϑi,∞ ≤ [∇ki(ui − vi)]ϑi,∞ + [∇kivi]δϑi,∞ ≤ ε/3 + [∇kivi]δϑi,∞

for 2 ≤ i ≤ `. Hence there exists δε > 0 such that [∇kiui]δϑi,∞ < ε for δ ≤ δε and2 ≤ i ≤ `. From this, (3.7.13), and Corollary 3.7.3 we deduce that u1 belongs toBUC

(X1, bucs/ν

pppp(X q

)). Now the assertion follows from (3.7.12). ¥

3.7.5 Example (Parabolic weight systems) Let [m, ν] be a parabolic weight systemand use the notation of Example 3.6.5. Then

buc(s,s/ν)(Y× J,E) .= bucs(Y, BUC(J,E)

) ∩ bucs/ν(J,BUC(Y, E)

)

→ BUC(J, bucs(Y, E)

) ∩ bucs/ν(J,BUC(Y, E)

).

Let k, n ∈ N satisfy n < s < n + 1 and kν < s < (k + 1)ν. It follows that u belongsto buc(s,s/ν)(Y× J,E) iff

u ∈ BUCn(Y, BUC(J,E)

) ∩BUCk(J,BUC(Y, E)

)

and [∇nu(·, t)]δ

s−n,∞ +[∂ku(y, ·)]δ

(s−kν)/ν,∞ → 0 as δ → 0 ,

uniformly with respect to (y, t) ∈ Y× J .

Proof Theorems 3.7.2 and 3.7.4 and Corollary 3.7.3. ¥

Very Little Holder Spaces

Finally, we characterize the very little Besov–Holder spaces by continuous inter-polation.


3.7.6 Theorem Let K be a closed corner in Rd and X ∈ Rd,K.(i) Suppose k < s < m with k,m ∈ νN and set θ := (s − k)/(m− k). Then

Bs/ν∞ (X, E) .=

(C

k/ν0 (X, E), Cm/ν

0 (X, E))0

θ,∞ .

(ii) Let 0 < s0 < s1 and 0 < θ < 1. Then

Bsθ/ν∞ (X, E) .=

(Bs0/ν∞ (X, E), Bs1/ν

∞ (X, E))0

θ,∞ .

Proof We write Bs/ν∞ for B

s/ν∞ (X, E), etc.

(1) Set Xθ := (Ck/ν0 , C

m/ν0 )θ,∞. It follows from C

j/ν0 → BUCj/ν , (2.7.6), and

Theorem 2.8.3 that Xθ → Bs/ν∞ . From Proposition 3.2.3 and Lemma 3.3.3 we infer

that the norm of Bs/ν∞ induces the one of Xθ. Thus, by the completeness of Xθ, we

see that Xθ is a closed linear subspace of Bs/ν∞ .

Let X 0θ := (Ck/ν

0 , Cm/ν0 )0θ,∞. Then X 0

θ is the closure of Cm/ν0 in Xθ. By the

density of S in Cm/ν0 , it follows that X 0

θ is the closure of S in Xθ, hence in Bs/ν∞ .

This proves X 0θ = B

s/ν∞ , that is, assertion (i).

(2) Fix 0 < s0 < s1 < m with m ∈ νN. Then Bsj/ν∞

.= (C0, Cm/ν0 )0sj/m,∞ for

j = 0, 1. Thus claim (ii) follows from the reiteration theorem (I.2.8.7). ¥

Lastly, we briefly consider corners which are not necessarily closed.

3.7.7 Theorem Let M be a corner in Rd, s > 0, and 1 ≤ r ≤ ∞. Then

Bs/νr (M, E) → Bs/ν

r (M, E) and bs/ν∞ (M, E) → bs/ν

∞ (M, E) .

Proof We omit E.

(1) Suppose r < ∞. Let mi ∈ νN satisfy m0 < s < m1. By definition (1.2.2)and S(M) → S(M) we see that W

mi/νr (M) is a closed subspace of W

mi/νr (M).

From this, (2.7.5), and Theorem 2.8.3 we obtain Bs/νr (M) → B

s/νr (M).

(2) Definitions (1.1.6) and (1.1.7) imply BUCmi/ν(M) → BUCmi/ν(M). Thuswe get the assertions by invoking (2.7.6) and (2.7.7), similarly as in step (1). ¥

3.8 Notes

In the scalar-valued case, S.M. Nikol′skiı presents in his book [Nik75] (also see[BesIN78], [BesIN79]) a detailed study of anisotropic Banach spaces Bs

p,r on Rd


(and subsets thereof). Here s = (s1, . . . , sd) with si ∈ (0,∞) and p = (p1, . . . , pd)with 1 ≤ pi ≤ ∞. Using our notations, these spaces are defined by

Bsp,r(Rd) :=

d⋂

i=1

Bsipi,r

(Xi, Lpi(Xı)

),

with Xi := R for 1 ≤ i ≤ d. In particular, he establishes optimal embedding theo-rems of the form

Bsp,r → Bs′

p′,r (3.8.1)

with s > s′.

The spaces Bsp,r are much more general than the spaces B

s/νp,r , considered here.

Due to this great generality, the conditions guaranteeing the validity of embedding(3.8.1) are rather complicated. On the other hand, the spaces B

s/νp,r can be studied

systematically, as is demonstrated in this and the preceding section. Moreover,besides of being defined for all s ∈ R, they are closed under interpolation. This isnot true for the more general spaces Bs

p,r. In fact, let 0 < σ < τ and set s := σ/ν

and t := τ/ν. Then t = (τ/σ)s. In Theorem 4 of [SchT76] it is shown that, given si

with sij > 0 for 1 ≤ j ≤ d and i = 0, 1,

(Bs0

2·1, 2(Rd), Bs1

2·1, 2(Rd))θ,2

= B(1−θ)s0+θs1

2·1, 2 (Rd)

iff there exists µ > 0 such that s0 = µs1. This indicates that the restriction to theanisotropic Besov spaces B

s/νp,r is a natural one. Fortunately, this class is broad

enough to cover the needs of most relevant applications.

Although it is possible to extend much of the theory expounded here andin the preceding section to ‘mixed norm spaces’ B

s/νp,r , we refrain from doing this

(see, for example, [JoS07], [JoMHS14]).

In the literature known to us, it is always claimed that the closedness ofA1, . . . , Ad obviously implies that Km/ν(A) is complete. Since we could not seethis, we included a demonstration of this fact in steps (2) and (3) of the proof ofLemma 3.1.2. It is inspired by the argument for the case d = 1 given in [HillP57].

Proposition 3.2.3, the basic result of Subsection 3.2, is due to H.-J. Schmeißerand H. Triebel [SchT76]. Our exposition of assertions (i) and (ii) is an elaborationof their proof. In step (5) of its proof we follow the proof of [Tri95, Lemma 1.13.4].The reasoning in step (6) is taken from S.G. Kreın, J.U. Petunin, and E.M. Se-menov [KrPS82, Lemma V.1.11]. (The latter authors attribute it to T. Mura-matu [Mur70].)

A fundamental consequence of Proposition 3.2.3 are the renorming state-ments and intersection space characterizations of Subsections 3.4–3.6. They extendcorresponding statements given in [Ama09].


In the isotropic case, the characterization of little Holder spaces, containedin Theorem 3.7.1, is well-known (see, for example, A. Lunardi [Lun95] for theone-dimensional case). A similar characterization holds for the little Nikol′skiıspaces b

s/νq,∞ (see G. Simonett [Simo92]).


4 Bessel Potential Spaces

In the preceding sections we have shown that the Slobodeckii spaces Ws/ν

p canbe obtained by real interpolation from the Sobolev spaces W

m/νp with m ∈ νN.

In this section we investigate the scale of Bessel potential spaces which can beobtained from the Sobolev spaces by complex interpolation. Unlike in the case ofBesov spaces, we now have to restrict the class of Banach spaces to (subclassesof) UMD spaces and q to the interval (1,∞) in order to get interesting and usefulresults. Given these restrictions, we are then rewarded by the fact that the Sobolevspaces are included in the scale of Bessel potential spaces, which is not true, ingeneral, for Besov space scales.

The first subsection contains the definition of Bessel potential spaces as wellas immediate consequences thereof for embeddings and real interpolation.

As in the foregoing sections, a fundamental role is played by Fourier multi-plier theorems. It is this place where the restrictions on E and q, alluded to above,come into play. In Subsection 4.2 we present suitable extensions of the well-knownclassical Fourier multiplier theorems due to Mikhlin and Marcinkiewicz, respec-tively, where we restrict ourselves to scalar symbols. The main result is the factthat M(Rd) is an admissible symbol class in either case. As a consequence, obvi-ous modifications of the proofs of Section 2, which are based on the scalar-valuedversion of the Fourier multiplier theorem 2.4.2, lead to corresponding results forBessel potential spaces. This applies, in particular, in Subsection 4.3 to the theo-rem about renorming by derivatives.

In Subsection 4.4 we prove a duality theorem. Subsection 4.5 is devoted tocomplex interpolation. Notably, we establish a complex interpolation theorem forBessel potential spaces with different target spaces.

Lastly, using the Fourier multiplier theorems once more, we derive in thelast subsection an intersection space characterization similar to the one for Besov-Slobodeckii spaces.

Once more, we remind the reader of assumption (VI.3.1.20).

4.1 Basic Facts, Embeddings, and Real Interpolation

Let s ∈ R and 1 ≤ q < ∞. It follows from Lq = Lq(Rd, E) → S ′ = S ′(Rd, E) andProposition VI.3.4.2 that J−s ∈ L(Lq,S ′). Hence Remarks VI.2.2.1 imply that

Hs/νq = Hs/ν

q (Rd, E) := J−sLq(Rd, E) , (4.1.1)

the image space of Lq under J−s in S ′, is a well-defined Banach space, an E-valuedanisotropic Bessel potential space on Rd. Note that

Hs/νq =

(u ∈ S ′ ; Jsu ∈ Lq , ‖·‖Hs/νq

), (4.1.2)

VII.4 Bessel Potential Spaces 169

where

‖u‖H

s/νq

= ‖Jsu‖q . (4.1.3)

Proposition VI.3.4.2 and Remarks VI.2.2.1 imply

S d→ Hs/ν

q

d→ S ′ . (4.1.4)

Furthermore, H0/νq = Lq and

Js ∈ Lis(H(s+t)/νq ,Ht/ν

q ) , s, t ∈ R , (4.1.5)

with (Js)−1 = J−s. In fact, the isomorphism (4.1.5) is isometric.

Let K be a corner in Rd. It follows from (4.1.4) that we can define

Hs/νq (K, E) := RKHs/ν

q , s ∈ R , 1 ≤ q < ∞ , (4.1.6)

in analogy with (2.8.1).

4.1.1 Theorem Hs/νq (K, E) is a Banach space and (RK, EK) is a universal r-e pair

for(H

s/νq ,H

s/νq (K, E)

).

Proof Recall the considerations at the beginning of Subsection 2.8. ¥

4.1.2 Corollary The Bessel potential spaces Hs/νq (K, E) enjoy the same embedding

and interpolation properties as the spaces Hs/νq (Rd, E).

Proof See the demonstration of Theorem 2.8.3. ¥

4.1.3 Theorem Let K be a corner in Rd, X ∈ Rd,K, and 1 ≤ q < ∞.

(i) Suppose −∞ < s0 < s < s1 < ∞. Then

Hs1/νq (X, E)

d→ B

s/νq,1 (X, E)

d→ Hs/ν

q (X, E)d

→ Bs/νq,∞(X, E)

d→ Hs0/ν

q (X, E) .

(ii) Let s0, s1 ∈ R with s0 6= s1, 0 < θ < 1, and sθ := (1 − θ)s0 + θs1. Then(Hs0/ν

q (X, E),Hs1/νq (X, E)

)θ,r

.= Bsθ/νq,r (X, E) , 1 ≤ r ≤ ∞ .

Proof By Corollary 4.1.2 we can assume that X = Rd.

(1) From Theorem 2.6.5 we know that B0/νq,1 → Lq → B

0/νq,∞. Theorem 2.4.1

guarantees thatJs ∈ Lis(Bs/ν

q,r , B0/νq,r ) , r ∈ 1,∞ .


This and (4.1.5) give

Bs/νq,1 → Hs/ν

q → Bs/νq,∞ . (4.1.7)

Now we apply (2.2.2) to obtain

Bs1/νq,∞ → B

s/νq,1 and Bs/ν

q,∞ → Bs0/νq,1 .

Together with (4.1.7) we thus find

Hs1/νq → B

s/νq,1 → Hs/ν

q → Bs/νq,∞ → Hs0/ν

q . (4.1.8)

Claim (1) follows now from the density of S in Hs/νq , B

s/νq,1 , and B

s/νq,∞.

(2) Assertion (ii) is a consequence of the middle part of (4.1.8) and Theo-rem 2.7.1(i). ¥

By combining the embedding theorems for Besov spaces with the sandwichingproperty of Theorem 4.1.3(i) we obtain inclusion theorems for Bessel potentialspaces as well. In particular, this method leads to the following ‘Sobolev-typeembedding’ result.

4.1.4 Theorem Let K be a corner in Rd and X ∈ Rd,K.(i) Let s0, s1 ∈ R and 1 ≤ q0, q1 < ∞. If

s1 − |ω|/q1 > s0 − |ω|/q0 , 1/q1 > 1/q0 , (4.1.9)

then Hs1/νq1 (X, E)

d→ H

s0/νq0 (X, E).

(ii) Suppose either s ≥ t + |ω|/q and t ∈ R+\νN, or s > t + |ω|/q. Then

Hs/νq (X, E)

d→ Ct(X, E) .

Proof We can assume that X = Rd.(1) Set t := s0 + |ω| (1/q1 − 1/q0). Then s1 > t > s0. Hence we infer from

Theorems 4.1.3(i) and 2.2.2

Hs1/νq1

→ Hs/νq → Hs0/ν

q0.

This proves (i), since the density assertion follows from the density of S in Hs/νp .

(2) Similarly, using also (2.2.3),

Hs/νq → Bs/ν

q,∞ → Bt/ν∞,∞ ,

where Bt/ν∞,∞ can be replaced by B

t/ν∞,1 if t ∈ N. Now the second assertion follows

from (2.6.6), (3.6.6), and the density of S in Hs/νq . ¥

Below, in Theorem 5.6.5(ii), it is shown that in the first part of (4.1.9) theequality sign can be admitted if q1 > 1.


4.2 A Marcinkiewicz Multiplier Theorem

In Section III.4 we have presented an extension of the Mikhlin multiplier theoremto vector-valued Banach spaces. In connection with anisotropic Bessel potentialspaces we need a variant of the Marcinkiewicz multiplier theorem. For this purposewe have to consider a subclass of UMD spaces which satisfy the following additionalrestriction.

The Banach space E has property (α) if

∫ 1

0

∫ 1

0

∣∣∣n∑

i,j=1

ri(s)rj(t)αijeij

∣∣∣E

ds dt

≤ c

∫ 1

0

∫ 1

0

∣∣∣n∑

i,j=1

ri(s)rj(t)eij

∣∣∣E

ds dt

(4.2.1)

for each n ∈qN and (eij , αij) ∈ E × C with |αij | ≤ 1, where (rj) is the sequence of

Rademacher functions rj(t) := sign(sin 2jπt) for 0 < t < 1.

4.2.1 Examples (a) Every finite-dimensional Banach space has property (α).

(b) If E has property (α), then each closed linear subspace thereof has prop-erty (α).

(c) Every Banach space isomorphic to one with property (α) possesses property (α)also.

(d) If E is a UMD space with property (α), then E′ too is a UMD space withproperty (α).

(e) Let (X, µ) be a σ-finite measure space and 1 ≤ q < ∞. If E has property (α),then Lq(X, µ; E) too possesses property (α).

Proof See Section 4 in P.Ch. Kunstmann and L. Weis [KW04]. ¥

Henceforth, we write Mi(Rd, E), resp. Ma(Rd, E), for the Banach space ofall m ∈ Cd

((Rd)

q, E

)satisfying

‖m‖Mi := maxα≤1

supξ∈(Rd)

pppp |ξ||α| |∂αm(ξ)|E < ∞ ,

resp.‖m‖Ma := max

α≤1|ξα∂αm(ξ)|E < ∞ ,

endowed with the norm ‖·‖Mi, resp. ‖·‖Ma. (Cf. Section III.4; Mi and Ma shouldremind the reader of Mikhlin and Marcinkiewicz, respectively. Also note thatMi(Rd) has been denoted in Section III.4 by MM (Rd).)


4.2.2 Lemma

(i) M(Rd, E) →Ma(Rd, E).(ii) If ν = ν1, then M(Rd, E) →Mi(Rd, E).

Proof (1) Note

|(ξj)αj | = (|ξj |2ω/ωj )αjωj/2ω ≤ (|ξj |2ω/ωj + 1)αjωj/2ω ≤ Λ1(ξ)αjωj .

Hence |ξα| ≤ Λα ppppω1 (ξ). Moreover, α ≤ 1 implies α q ω ≤ |ω| ≤ k(ν). From this and

(VI.3.4.7) we get ‖·‖Ma ≤ ‖·‖M, hence (i).(2) Let ν = ν1. Then

|ξ||α| =(∑

i=1

|ξi|2)|α|/2

≤(∑

i=1

|ξi|2 + 1)|α| ν/2ν

= Λα ppppω1 (ξ) .

Consequently, as above, ‖·‖Mi ≤ ‖·‖M. This proves (ii). ¥

4.2.3 Lemma If E1 × E2 → E3 is a multiplication, then its point-wise extensionis a multiplication

F(Rd, E1) × F(Rd, E2) → F(Rd, E3)

for F ∈ Ma,Mi.Proof Leibniz’ rule. ¥

After these preparations we can formulate the following vector-valued exten-sion of the Marcinkiewicz multiplier theorem.

4.2.4 Theorem Let E be a UMD space with property (α). Then m 7→ m(D) isa continuous algebra homomorphism from Ma(Rd) into L(

Lp(Rd, E)), provided

1 < p < ∞.

Proof It follows from [KW04, 5.2.a] that(m 7→ m(D)

) ∈ L(Ma(Rd),L(Lp(Rd, E)))

.

Hence the assertion is a consequence of

F−1m1m2F = F−1m1FF−1m2F

for m1,m2 ∈Ma(Rd). ¥

Using the definition of Hsp(Rd, E), it is now easy to extend this theorem to a

Fourier multiplier theorem for Bessel potential spaces. Since in the homogeneouscase, that is, if ν = ν1, property (α) is not required, it is convenient to introduce


the following definition: The Banach space E is ν-admissible if it is a UMD spacewhich has property (α) if ν 6= ν1.

4.2.5 Theorem Let E be ν-admissible, 1 < p < ∞, and s ∈ R. Then m 7→ m(D)is a continuous algebra homomorphism from M(Rd) into L(

Hs/νp (Rd, E)

).

Proof (1) Suppose s = 0. Then the assertion follows from Lemma 4.2.2 and The-orem III.4.4.3 if ν = ν1, resp. Theorem 4.2.4 otherwise.

(2) It is a consequence of

m(D) = F−1mF = F−1Λ−s1 mΛs

1F = J−sm(D)Js

and (4.1.5) that the diagram

Hs/νp (Rd, E) H

s/νp (Rd, E)

Lp(Rd, E) Lp(Rd, E)

m(D)

m(D)

∼=Js ∼= Js

-

-? ?

is commuting. Hence the assertion follows from step (1) and (4.1.5). ¥


Due to Theorem 4.2.5, the proofs of the preceding sections, which make use ofthe Fourier multiplier theorem 2.4.2 with scalar symbols, apply verbatim to Besselpotential spaces too. As a first application of this observation we get the followinganalogues of the theorems of Subsection 2.6.

4.3.1 Theorem Let K be a closed corner in Rd and X ∈ Rd,K. Suppose E isν-admissible, 1 < p < ∞, and s ∈ R.

(i) If α ∈ Nd, then ∂α ∈ L(H

(s+α ppppω)/νp (X, E),Hs/ν

p (X, E)).

(ii) Suppose m ∈ νqN. The following assertions are equivalent.

(α) u ∈ H(s+m)/νp (X, E).

(β) ∂αu ∈ Hs/νp (X, E), α q ω ≤ m.

(γ) u, ∂m/ωj

j u ∈ Hs/νp (X, E), 1 ≤ j ≤ d.


Furthermore, set

‖·‖(1)

H(s+m)/νp

:= ‖·‖H

s/νp

+d∑

j=1

‖∂m/ωj

j · ‖H

s/νp

,

‖·‖(2)

H(s+m)/νp

:= ‖·‖H

s/νp

+∑

i=1

‖∇m/νixi

· ‖H

s/νp

,

‖·‖(3)

H(s+m)/νp

:=∑

i=1

m/νi∑

j=1

‖∇jxi· ‖

Hs/νp

,

‖·‖(4)

H(s+m)/νp

:=∑

α ppppω≤m

‖∂α · ‖H

s/νp

.

Then ‖·‖(k)

H(s+m)/νp

∼ ‖·‖H

(s+m)/νp

for 1 ≤ k ≤ 4.

Proof (1) Let X = Rd. Then the assertions follow from the proofs of Theorems2.6.1 and 2.6.2 and the preceding remarks.

(2) If X = K, then we get the claims from Theorems VI.1.2.3(iii) and 4.1.1. ¥

As an immediate consequence of these renorming results we obtain the fol-lowing fundamental fact.

4.3.2 Theorem Let K be a corner in Rd and X ∈ Rd,K. Suppose that E isν-admissible and 1 < p < ∞. If m ∈ νN, then H

m/νp (X, E) .= W

m/νp (X, E).

Proof Note that ‖·‖(4)

Hm/νp

= |||·|||m/ν,p. Hence the assertion is implied by Theo-

rems 1.4.1(ii) and 4.3.1. ¥

4.4 Duality

First we show that the Bessel potential spaces on Rd form an interpolation-extrapolation scale in the sense of Section V.1.

4.4.1 Theorem Let E be ν-admissible and 1 < p < ∞.

The interpolation-extrapolation scale[(Xα, Aα) ; α ∈ R ]

, generated by

(X0, A) :=(Lp(Rd, E), J

)and [·, ·]θ, 0 < θ < 1,

satisfies Xα.= H

α/νp (Rd, E).


Proof An obvious modification of the proof of Theorem 2.5.5, by invoking The-orem 4.2.4 instead of Theorem 2.4.2, shows that

J ∈ BIP(Hs/νp ) . (4.4.1)

The assertion follows by changing the proof of Theorem 2.5.6 analogously, takingalso Remark 2.5.7 into account. ¥

A ν-admissible Banach space is reflexive. Using this fact and Theorem 4.4.1,we can easily prove the following analogue of Theorems 2.3.1 and 2.8.4.

4.4.2 Theorem Let K be a corner in Rd and X ∈ Rd,K. Suppose 1 < p < ∞ andE is ν-admissible. Then H

s/νp (X, E) is reflexive and

(Hs/ν

p (X, E))′ = H

−s/νp′ (X∗, E′)

with respect to the Lp duality pairing 〈·, ·〉.Proof (1) First we assume that X = Rd.

We set E0 := Lp(Rd, E) and E]0 := Lp′(Rd, E′). Then E0 is reflexive and

E′0 = E]

0 with respect to 〈·, ·〉. We also denote by A0, resp. A]0, the E0-, resp. E]

0-,realization of J . Then A0 ∈ C(E0) and A]

0 ∈ C(E]0) by Theorem 4.4.1. Moreover,

since Λ1 is even, that is, Λ

1 = Λ1, we get from (III.4.2.2)

〈v, Ju〉 = 〈v,F−1Λ1Fu〉 = (2π)−d⟨v,F

(Λ1u)

⟩

= (2π)−d⟨v,F(Λ1

u)⟩

= (2π)−d〈Λ1v, u

〉 = (2π)−d⟨(

Λ1v), u

⟩

= (2π)−d⟨(

(Λ1v))

, u⟩

= 〈F−1Λ1Fv, u〉 = 〈Jv, u〉

for u ∈ S(Rd, E) and v ∈ S(Rd, E′). This and the density of S in Hs/νp imply

〈v, A0u〉 = 〈A]0v, u〉 , (v, u) ∈ D(A]

0) × D(A0) , (4.4.2)

since D(A0) = H1/νp (Rd, E) and D(A]

0) = H1/νp (Rd, E′) by Theorem 4.4.1. It fol-

lows from (4.4.2) that A′0 := (A0)′ ⊃ A]

0.

(2) Suppose u′ ∈ dom(A′0). Since A]

0 ∈ Lis(E]1, E

]0), there exists u] ∈ dom(A]

0)such that

A′0u

′ = A]0u

] = A′0u

] . (4.4.3)

From dom(A0) = H1/νp (Rd, E) and (4.1.4) it follows that A0 has a dense range

in E0. Hence A′0 is injective, so that we get u′ = u] from (4.4.3). Together with

step (1), this proves dom(A′0) = dom(A]

0). Hence A′0 = A]

0. Now Theorem V.1.5.12implies the assertion if X = Rd.

(3) If X = K, the claim follows by an obvious modification of the proof ofTheorem 2.8.4. ¥


4.4.3 Corollary If m ∈ νN, then H−m/νp (X, E) .= W

−m/νp (X, E).

Proof Theorem 4.3.2 and (1.2.4). ¥

4.5 Complex Interpolation

Theorem 4.1.3 shows that real interpolation of Bessel potential spaces results inBesov spaces. By contrast, Bessel potential spaces are stable under complex inter-polation, as the next theorem shows. Recall that sθ := (1 − θ)s0 + θs1 for s0 6= s1

and 0 < θ < 1.

4.5.1 Theorem Let K be a corner in Rd and X ∈ Rd,K. Suppose 1 < p < ∞ andE is ν-admissible. Assume −∞ < s0 < s1 < ∞ and 0 < θ < 1. Then

[Hs0/νp (X, E), Hs1/ν

p (X, E)]θ.= Hsθ/ν

p (X, E) .

Proof This is a consequence of (4.4.1) and Theorem V.1.5.4. ¥

It is the purpose of the following considerations to extend this theorem tothe case of different target spaces. For this we need some preparation.

A Holomorphic Semigroup

Recall definition (VI.3.4.7) of the multiplier space M(Rd, E) and that J = Λ1(D).

4.5.2 Lemma The map

(z 7→ Λz1) : [Rez < 0] →M(Rd) = M(Rd,C)

is holomorphic, M(z) := Λz1 log Λ1 ∈M(Rd), and ∂Λz

1 = M(z), where ∂ := ∂z.

Proof We write M := M(Rd).(1) Example VI.3.3.9 and Remark VI.3.4.7(a) imply

Λz1 ∈M , ‖Λz

1‖M ≤ c (1 + |z|k) , Rez ≤ 0 , (4.5.1)

where k := k(ν).(2) In the point-wise sense,

∂Λz1 = Λz

1 log Λ1 = M(z) , z ∈ C , (4.5.2)

and

|M(z)| = ΛRez1 log Λ1 . (4.5.3)


From (VI.3.3.6) we infer

|∂αξ M(z)| ≤ c(|z|)ΛRez−α ppppω

1 (1 + log Λ1)

for 0 < α q ω ≤ k and z ∈ C. Let Rez ≤ −r < 0. Then it follows from (4.5.3) andthis estimate that

Λα ppppω1 |∂α

ξ M(z)| ≤ c(|z|)Λ−r1 (1 + log Λ1) ≤ c(|z|, r) , α q ω ≤ k .

In other words,

M(z) ∈M , |M(z)|M ≤ c(|z|, r) , Rez ≤ −r < 0 . (4.5.4)

(3) Now suppose z, h ∈ C satisfy Rez ≤ −r < 0, Re(z + h) ≤ −r, and |h| ≤ 1.The mean-value theorem gives, in the point-wise sense,

Λz+h1 − Λz

1 = Λz1(Λ

h1 − 1) = Λz

1

∫ 1

0

∂Λth1 dth .

We differentiate this equation with respect to z and use (4.5.2) to obtain

M(z + h) − M(z) = M(z)∫ 1

0

M(th) dth .

Thus, by (4.5.4),

‖M(z + h)− M(z)‖M ≤ c(|z|, r) |h| . (4.5.5)

In the point-wise sense,

Λz+h1 − Λz

1 − ∂Λz1h =

∫ 1

0

(∂Λz+th1 − ∂Λz

1) dth ,

that is,

Λz+h1 − Λz

1 − M(z)h =∫ 1

0

(M(z + th) − M(z)

)dth .

Hence we infer from step (1) and (4.5.5) that

‖Λz+h1 − Λz

1 − M(z)h‖M ≤ c(|z|, r) |h|2 . (4.5.6)

This shows that z 7→ Λz1 is differentiable on [Rez < 0] in the topology of M, hence

analytic. Furthermore, its derivative equals M(z). The lemma is proved. ¥


4.5.3 Theorem Suppose E is ν-admissible , 1 < p < ∞, and s ∈ R. Then:(i) The map

[Rez ≤ 0] → L(Hs/ν

p (Rd, E))

, z 7→ Jz

is strongly continuous and its restriction to [Rez < 0] is holomorphic.

(ii) J−t ; t ≥ 0 is a strongly continuous analytic semigroup on Hs/νp (Rd, E).

Its infinitesimal generator is − log J = − log Λ1(D).

(iii) J i t ; t ∈ R is a strongly continuous group on Hs/νp (Rd, E) satisfying

‖J i t‖L(Hs/νp )

≤ ce|t| , t ∈ R .(4.5.7)

Proof (1) We know from (4.4.1) that J ∈ BIP(Hs/νp ). Since H

(s+1)/νp

d→ H

s/νp by

Theorem 4.1.3(i), J is densely densely defined. Theorem III.4.7.1 guarantees that J−t ; t ≥ 0 is a strongly continuous semigroup on L(Hs/ν

p ), and J i t ; t ∈ R is a strongly continuous group on L(Hs/ν

p ).(2) We deduce from (4.5.1), (4.5.6), Lemma VI.3.4.5, and Theorems 2.4.2

and 4.2.5 that (4.5.7) is true, M(z)(D) ∈ L(Hs/νp ), and

‖h−1(Jz+h − Jz)− M(z)(D)‖L(Hs/νp )

≤ c(|z|, r) |h| (4.5.8)

for Rez, Re(z + h) ≤ r < 0. From (4.5.8) we infer that z 7→ Jz is differentiableon [Rez < 0], hence holomorphic, and that its derivative equals M(z)(D). Thetheorem is proved. ¥

4.5.4 Remark Let E be an arbitrary Banach space, 1 ≤ q ≤ ∞, and s ∈ R. Thenassertions (i)–(iii) of this theorem are valid if we replace H

s/νp (Rd, E) everywhere

by the little Besov spaces bs/νp (Rd, E).

Proof It suffices to use Theorems 2.5.6, 2.2.4(iii), and 2.4.2 instead of Theorems4.4.1, 4.1.3(i), and 4.2.5, respectively. ¥

Interpolation with Different Target Spaces

We need a slightly more general version of the complex interpolation method thanthe one presented in Example I.2.4.2. We use the notation introduced there. Inparticular, S = [0 < Rez < 1] and Sx = [Rez = x] for 0 ≤ x ≤ 1.

Let (E0, E1) be an interpolation couple and γ ∈ R. We denote by F(E0, E1, γ)the set of all f ∈ C(S, E0 + E1) such that f |S is holomorphic and

(z 7→ e−γ |Imz|f(z)

) ∈ C0(Sj , Ej) , j = 0, 1 .

It is a Banach space with the norm

‖f‖F(E0,E1,γ) := maxj=0,1

supt∈R

|e−γ |t|f(j + it)|Ej .

Note that F(E0, E1, 0) = F(E0, E1) in the notation of Example I.2.4.2.


Given θ ∈ (0, 1), put

[E0, E1]θ,γ :=

e ∈ E0 + E1 ; e = f(θ) for some f ∈ F(E0, E1, γ)

.

It is a Banach space with the norm

‖e‖θ,γ := inf ‖f‖F(E0,E1,γ) ; f ∈ F(E0, E1, γ), f(θ) = e

.

In fact,

[E0, E1]θ,γ.= [E0, E1]θ,0 = E[θ] , γ ∈ R . (4.5.9)

In other words, [E0, E1]θ,γ is the complex interpolation space [E0, E1]θ, exceptfor equivalent norms. For a proof of these facts we refer to H. Triebel [Tri95,Subsections 1.9.1 and 1.9.2].

The following theorem is the counterpart of Theorem 2.7.2(ii). Recall that1/p(θ) = (1 − θ)/p0 + θ/p1.

4.5.5 Theorem Let K be a corner and X ∈ Rd,K. Suppose (E0, E1) is an in-terpolation couple of ν-admissible Banach spaces. If s0, s1 ∈ R with s0 6= s1 andp0, p1 ∈ (1,∞), then

[Hs0/ν

p0(X, E0),Hs1/ν

p1(X, E1)

]θ

.= Hsθ/νp(θ) (X, E[θ])

for 0 < θ < 1.

Proof (1) Due to Corollary 4.1.2 we can assume that X = Rd. For abbreviation,

X tp(E) := Ht/ν

p (Rd, E) for t ∈ R and 1 < p < ∞ .

It follows from (2.7.3) that[X 0

p0(E0),X 0

p1(E1)

]θ

.= X 0p(θ)(E[θ]) . (4.5.10)

From Ej → E0 + E1 we get

X sjpj

(Ej) → S ′(Rd, E0 + E1) , j = 0, 1 .

Hence(X s0

p0(E0),X s1

p1(E1)

)is an interpolation couple. Without loss of generality,

s1 > s0.(2) We put

F0 := F(X 0p0

(E0),X 0p1

(E1), 0)

, F1 := F(X s0p0

(E0),X s1p1

(E1), 1)

.

Suppose f ∈ F0 and set

g(z) := (Ff)(z) := J−s0−z(s1−s0)f(z) , z ∈ S .


Theng(j + it) = J−i t(s1−s0)J−sj f(j + it) , j = 0, 1 ∈ R .

Since f |Sj ∈ C0

(R,X 0

j (Ej)), it follows from (4.1.3) and (4.5.7) that g |Sj belongs

to C0

(R,X sj

pj (Ej))

and

e−|t| ‖g(j + it)‖X sjpj

(Ej)= e−|t| ‖Jsj g(j + it)‖X 0

pj(Ej)

≤ c ‖f(j + it)‖X 0pj

(Ej) ≤ c supt∈R

‖f(j + it)‖X 0pj

(Ej)

for j = 0, 1 and t ∈ R. Consequently,

supt∈R

e−|t| ‖Ff(j + it)‖X sjpj

(Ej)≤ c sup

t∈R‖f(j + it)‖X 0

pj(Ej) , j = 0, 1 . (4.5.11)

We can write f = f0 + f1, where

fj ∈ B(S,X 0

pj(E0)

)with fj(j + i·) ∈ C0

(R,X 0

pj(Ej)

)

and fj being holomorphic from S into X 0pj

(Ej), j = 0, 1 (see the proof of Theorem1.10.3.1 in [Tri95]). From this,

gj(z) := Ffj(z) = J−s0−z(s1−s0)fj(z) , z ∈ S ,

and Theorem 4.5.3(i) we deduce that f : S → X s0p0

(E0) + X s1p1

(E1) is holomorphic.Together with (4.5.11) it thus follows that

F ∈ L(F0,F1) . (4.5.12)

(3) Suppose u ∈ X sθ

p(θ)(E[θ]). Then

Jsθu ∈ X 0p(θ)(E[θ])

.=[X 0

p0(E0),X 1

p1(E1)

]θ,1

by (4.5.9) and (4.5.10). Hence there exists f ∈ F0 with f(θ) = Js0u. Moreover,

‖u‖X sθp(θ)(E[θ])

= ‖Jsθu‖X 0p(θ)(E[θ])

= inf ‖f‖F0 ; f ∈ F0, f(θ) = Jsθu

.

(4.5.13)


‖u‖[X s0p0 (E0),X s1

p1 (E1)]θ,1= inf

‖g‖F1 ; g ∈ F1, g(θ) = u

≤ inf ‖Ff‖F1 ; f ∈ F0, Ff(θ) = J−sθf(θ) = u

≤ c inf ‖f‖F0 ; f ∈ F0, f(θ) = Jsθu

.

From this and (4.5.13) we obtain

‖u‖[X s0p0 (E0),X s1

p1 (E1)]θ,1≤ c ‖u‖X sθ

p(θ)(E[θ]), u ∈ X sθ

p(θ)(E[θ]) .


Thus, due to (4.5.9),

X sθ

p(θ)(E[θ]) → [X s0p0

(E0),X s1p1

(E1)]θ

. (4.5.14)

(4) Assume g ∈ F(X s0p0

(E0),X s1p1

(E1), 0)

=: F0. Put

f(z) := Gg(z) := Js0+z(s1−s0)g(z) , z ∈ S .

Thenf(j + it) = J i t(s1−s0)Jsj g(j + it) , j = 0, 1 , t ∈ R .

Using (4.1.3) and (4.5.7), we get

e−|t| ‖Gg(j + it)‖X 0pj

(Ej) ≤ c ‖g(j + it)‖X sjpj

(Ej), j = 0, 1 , t ∈ R .

We fix r > s1. Then

Re(−r + s0 + z(s1 − s0)

)< 0 , z ∈ S .

We writeGg(z) = J−r+s0+z(s1−s0)Jrg(z) , z ∈ S .

Note that Jr is a continuous linear map

from F(X s0p0

(E0),X s1p1

(E1), 0)

into F(X s0−rp0

(E0),X s1−rp1

(E1), 0)

.

Hence we deduce, similarly as in step (2), that

Gg ∈ F(X 0p0

(E0),X 0p1

(E1), 1)

=: F1

and

G ∈ L(F0, F1) . (4.5.15)

(5) Assume u ∈ X sθ

p(θ)(E[θ]). Then

Jsθu ∈ X 0p(θ)(E[θ])

.=[X 0

p0(E0),X 0

p1(E1)

]θ,1

by (4.1.1), (4.5.9), and (4.5.10). Hence we find an f ∈ F1 with f(θ) = Jsθu. Fur-thermore, similarly as in the preceding step, we deduce from Gg(θ) = Jsθg(θ) and(4.5.15) that

‖u‖X sθp(θ)(E[θ])

= ‖Jsθu‖X 0p(θ)(E[θ])

≤ c inf ‖f‖F1

; f ∈ F1, f(θ) = Jsθu

≤ c inf ‖Gg‖F1

; g ∈ F0, Gg(θ) = Jsθu

≤ c inf ‖g‖F0

; g ∈ F0, g(θ) = u

= c ‖u‖[X s0p0 (E0),X s1

p1 (E1)]θ,1.

This implies [X s0p0

(E0),X s1p1

(E1)]θ

→ X sθ

p(θ)(E[θ]) ,

which, due to (4.5.14), proves the theorem. ¥


Note that this theorem contains Theorem 4.5.1 as a particular case. Thus itprovides an alternative proof for that theorem.


Let K be a corner in Rd and X ∈ Rd,K. Recall that (VI.3.3.2) applies. Weconsider the decomposition X = X1 × · · · × X`, associated with the weight system[`, d, ν] and use the notation introduced in the beginning of Subsection 3.5.

In the following, we show that Hs/νp (X, E) possesses intersection space char-

acterizations analogous to the ones for Bs/νp (X, E).

4.6.1 Theorem Suppose K is a corner in Rd and X ∈ Rd,K. Assume E isν-admissible and 1 0.

Proof (1) Assume X = Rd. Given 1 ≤ i ≤ `,

Λ(i)(ξi, ηi) := Λ((ξi; 0), (ηi; 0)

), (ξi, ηi) ∈ Xi × Hi , (4.6.1)

is the natural νi-quasinorm on Zi := Xi × Hi (see Remark VI.3.3.1(a)). Hence Λs(i)

belongs to H(Zi) by Example VI.3.3.9. We set

ai(ξi) := Λs(i)(ξi, 1) , ξi ∈ Xi .

Thenai(Dxi) ∈ Lis

(Hs/νi

p (Xi, E), Lp(Xi, E))

by (4.1.5). Denoting by Ai the point-wise extension of ai(Dxi) over Xı, defined by(

ai(Dxi)u)(xı) := ai(Dxi)u(·; xı), it follows that

Ai ∈ Lis(Lp(Xı,H

s/νip (Xi, E)

), Lp(Xı, Lp(Xi, E))

). (4.6.2)

(2) We assume X = Rd and set

Λs(1) + · · · + Λs

(`)(ξ, η) := Λs(1)(ξ1, η1) + · · · + Λs

(`)(ξ`, η`)

for (ξ, η) ∈ Z := Rd × H. Then Λs(1) + · · · + Λs

(`) belongs to Hs(Z) and is strictly

positive onqZ. From this and Lemmas VI.3.3.6 and VI.3.3.8 it follows

(Λs(1) + · · · + Λs

(`))/Λs, Λs/(Λs(1) + · · · + Λs

(`)) ∈ H0(Z) . (4.6.3)


Note that

a(ξ) := a1(ξ1) + · · · + a`(ξ`) = (Λs(1) + · · · + Λs

(`))(ξ,1)

for ξ ∈ X and 1 = 1`. Hence we infer from (4.6.3) and Remark VI.3.4.7(a) thata/Λs

1, Λs1/a ∈M(Rd). Thus

(a/Λs1)(D), (Λs

1/a)(D) ∈ L(Lp(X, E)

)

by Theorem 4.2.5. Consequently, by that theorem, (3.5.5), and step (1),

‖u‖H

s/νp (X,E)

= ‖Jsu ‖Lp(X,E) ≤ c ‖Λs1/a‖M ‖a(D)u‖Lp(X,E)

≤ c∑

i=1

‖Aiu‖Lp(X,E) ≤ c∑

i=1

‖u‖Lp(Xı,H

s/νip (Xi,E))

for

u ∈ X (X, E) :=⋂

i=1

Lp

(Xı,H

s/νip (Xi, E)

).

This proves X (X, E) → Hs/νp (X, E).

(3) Let X = Rd. Similarly as above, we find ai/Λs1 ∈M(Rd) for 1 ≤ i ≤ `.

Thus the arguments of the last step imply

‖Aiu‖Lp(X,E) ≤ c ‖Jsu‖Lp(X,E) = c ‖u‖H

s/νp (X,E)

for u ∈ Hs/νp (X, E) and 1 ≤ i ≤ `. Hence, by (4.6.2) and (3.5.5),

∑

i=1

‖u‖Lp((Xı,E),H

s/νip (Xi,E))

≤ c∑

i=1

‖Aiu‖Lp(X,E) ≤ c ‖u‖H

s/νp (X,E)

for u ∈ Hs/νp (X, E). Consequently, H

s/νp (X, E) → X (X, E). This proves

Hs/νp (X, E) .=

⋂

i=1

Lp

(Xı, H

s/νip (Xi, E)

)(4.6.4)

in this case.(4) Still assuming X = Rd, Fubini’s theorem gives

‖u‖p

Lp(Xı,Hs/νip (Xi,E))

=∫

Xı

∫

Xi

|ai(Dxi)u(·; xı)|pE dxi dxı

=∫

Xi

∫

Xı

|ai(Dxi)u(·; xı)|pE dxı dxi

= ‖u‖H

s/νip (Xi,Lp(Xı,E))

.


Thus we can replace Lp

(Xı,H

s/νip (Xi, E)

)in (4.6.4) by H

s/νip

(Xi, Lp(Xı, E)

)if

i ∈ L\M. This proves the assertion if X = Rd.(5) We write K = I1 × · · · × Id, where Ii ∈ R,R+. Using the notation of

Subsection VI.1.1, we set Ri := Ei := idR if Ii ∈ R. Then RK = Rd · · · R1 andEK = E1 · · · Ed. We also set dı := d − di for 1 ≤ i ≤ `. We put

Ki := Id(i−1)+1 × · · · × Id(i) , Kı := I1 × · · · × Ki × · · · × Id .

Then, by Theorem 4.1.1, (RKi , EKi) is an r-e pair for(H

t/νip (Rdi , E),Ht/νi

p (Ki, E))

for t ∈ 0, s. Theorem 4.1.1 implies also that (RKı, EKı

) is an r-e pair for

(Lp(R

dı ,Ht/νip (Ki, E)), Lp(Kı,H

t/νip (Ki, E))

)

for t ∈ 0, s. From this we infer that (RKıRKi , EKi EKı

) is an r-e pair for

(Lp(R

dı , Hs/νip (Rdi , E)), Lp(Kı,H

s/νip (Ki, E))

).

Similarly, (RKi RKı, EKı

EKi) is an r-e pair for

(Hs/νi

p (Rdi , Lp(Rdı , E)),Hs/νi

p (Ki, Lp(Kı, E)))

.

We can identify (RKiRKı

, EKı EKi

) and (RKıRKi

, EKi EKı

) with (RK, EK)by means of the canonical identifications based on (3.5.3). This implies that(RK, EK) is an r-e pair for

(F(Rd, E), F(K, E)

), where F(X, E) is the intersection

space on the right side of (4.6.4). Hence it follows from the validity of the assertionfor X = Rd and u = RKEKu that

Hs/νp (K, E) EK−→ Hs/ν

p (Rd, E) .= Fs(Rd, E) RK−→ Fs(K, E) , u 7→ u .

Thus Hs/νp (K, E) → Fs(K, E). Similarly,

Fs(K, E) EK−→ Fs(Rd, E) .= Hs/νp (Rd, E) RK−→ Hs/ν

p (K, E) , v 7→ v ,

implies Fs(K, E) → Hs/νp (K, E). Together with the analogue of step (4), this

proves the assertion for X = K. ¥

4.6.2 Corollary We write νq:= (ν2, . . . , ν`) and X q

:= X2 × · · · × X`. Then

Hs/νp (X, E) .= Lp

(X1, H

s/νpppp

p (X q, E)

) ∩Hs/ν1p

(X1, Lp(X

q, E)

), s > 0 .

If m ∈ νq qN, then

Wm/νp

(X, E) .= Lp

(X1,W

m/νpppp

p(X q

, E)) ∩ Wm/ν1

p

(X1, Lp(X

q, E)

).


Proof (1) Note that

Lp

(Xı,H

s/νip (Xi, E)

)= Lp

(X1, Lp(X

qı,H

s/νip (Xi, E))

)

for 2 ≤ i ≤ `. Hence

⋂

i=2

Lp

(Xı,H

s/νip (Xi, E)

) .= Lp

(X1,

⋂

i=2

Lp

(X q

ı,Hs/νip (Xi, E)

)).

Using the theorem with L := 2, . . . , ` and M := L, we get

⋂

i=2

Lp

(X q

ı, Hs/νip (Xi, E)

) .= Hs/νpppp

p (X q, E) .

From this we get the first assertion by applying the theorem once more, this timewith L := 1, . . . , ` and M := 2, . . . , `.

(2) The second claim is now a consequence of Theorem 4.4.2. ¥

4.6.3 Example (Parabolic weight systems) Suppose E is a UMD space withproperty (α), K a corner in Rd, and X ∈ Rd,K. Let [m, ν] be a parabolic weightsystem with ν > 1 and write X = Y× J with J ∈ R,R+. Then

H(s,s/ν)p (Y× J,E) .= Lp

(J,Hs

p(Y, E)) ∩Hs/ν

p

(J, Lp(Y, E)

), s > 0 .

In particular,

W (ν,1)p (Y× J) .= Lp

(J,W ν

p (Y, E)) ∩ W 1

p

(J, Lp(Y, E)

).

Proof This follows from the corollary by relabeling coordinates. ¥

4.7 Notes

In contrast to the theory of Besov spaces, where no restrictions on the Banachspace E have to be imposed, for most of the above theory of Bessel potentialspaces we require that E be ν-admissible. This is due to the fact that this con-dition guarantees the validity of the Mikhlin, respectively Marcinkiewicz, Fouriermultiplier theorem on Lp(Rd, E). Indeed, it is known that Mikhlin’s theorem is nottrue for Banach spaces E not possessing the UMD property, and the Marcinkiewicztheorem may fail to hold if the UMD space E does not possess property (α).

The attentive reader will have noticed that we restricted ourselves in theFourier multiplier theorems to the case of scalar symbols. In fact, TheoremsIII.4.4.3 and 4.2.4 are not valid, in general, if m is an operator-valued symbol.In this case we have to impose an additional condition on the symbol, namelyR-boundedness.


A subset T of L(E, F ) is R-bounded if there exists a constant c such that,given any n ∈ N, T0, . . . , Tn ∈ T , x0, . . . , xn ∈ E,

∥∥∥n∑

j=0

rjTjxj

∥∥∥L2([0,1],F )

≤ c∥∥∥

n∑

j=0

rjxj

∥∥∥L2([0,1],E)

, (4.7.1)

where the rj are the Rademacher functions on [0, 1]. The smallest constant c isthe R-bound, R(T ), of T .

4.7.1 Remarks (a) The L2 spaces in definition (4.7.1) can be replaced by Lq spacesfor any q ∈ [1,∞).

(b) If Ti ⊂ L(E0, E1) are R-bounded for i = 1, 2, then T1 + T2, too, is R-boundedand R(T1 + T2) ≤ R(T1) + R(T2).

(c) Let T ∈ L(E0, E1) and S ∈ L(E1, E2) be R-bounded. Then

ST := ST ; S ∈ S, T ∈ T

is R-bounded and R(ST ) ≤ R(S)R(T ).

(d) If T ⊂ L(E0, E1) is R-bounded, then it is bounded by R(T ).

(e) If E0 and E1 are Hilbert spaces, then T ⊂ L(E0, E1) is R-bounded iff it isbounded.

(f ) Let E0 and E1 be finite-dimensional. Then T ⊂ L(E0, E1) is R-bounded iff itis bounded.

Proof For (a) see P.Ch. Kunstmann and L. Weis [KW04, Remark 2.6]. (b) and (c)follow directly from the definition. By choosing n = 1 in (4.7.1), we get (d). Fora demonstration of (e) we refer to R. Denk, M. Hieber, and J. Pruss [DHP03,Remark 3.2(3)]. Lastly, (f) is obvious by (e), since E0 and E1 can be identifiedwith Euclidean spaces. ¥

Now we can formulate operator-valued extensions of the multiplier theoremsof Mikhlin and Marcinkiewicz.

4.7.2 Theorem Let E and F be UMD spaces and 1 < p < ∞. Suppose m belongsto Cd

((Rd)

q,L(E, F )

).

(i) If Mi(m) := |ξ|α ∂αm(ξ) ; ξ ∈ (Rd)

q, α ≤ 1

is R-bounded in L(E,F ),

then

m(D) ∈ L(Lp(Rd, E), Lp(Rd, F )

)(4.7.2)

and ‖m(D)‖ ≤ cR(Mi(m)).


(ii) Let E and F possess property (α) and

Ma(m) :=

ξα∂αm(ξ) ; ξ ∈ (Rd)q, α ≤ 1

be R-bounded. Then (4.7.2) applies and ‖m(D)‖ ≤ cR(Ma(m)).

This theorem is due to L. Weis [Wei01], if d = 1. The extension to d ≥ 2has been obtained independently by Z. Strkalj and L. Weis [SW07] and R. Haller,H. Heck, and A. Noll [HHN02]. Detailed expositions of these and related results,historical remarks, and alternative proofs can be found in [KW04], [DHP03], and[PS16]. We refer to these works for discussions of the necessity of assumptions. Asfor the necessity of condition (α), see T. Hytonen and L. Weis [HW08]. A rathercomprehensive historical overview of the development of vector-valued Fourier mul-tiplier theorems, as well as extensions and sharpenings, can be found in Hytonen’sthesis [Hyt03].

Although condition (α) cannot be dispensed with, in general, in the multipliertheorem of Marcinkiewicz type, it follows from a result of T. Hytonen [Hyt07,Corollary 1] that m ∈ H0

(Rd,L(E)

)implies m(D) ∈ L(

Lp(Rd, E))

under the soleassumption that E is a UMD space. Unfortunately, most symbols of interest tous are not homogeneous. Thus that nice result is of no use for us. Consequently,we cannot avoid to require condition (α), even if we restrict ourselves to scalarsymbols.

A detailed and comprehensive presentation of the theory of R-boundednessis contained in the recent books by T. Hytonen, J. van Neerven, M. Veraar, andL. Weis ([HvNVW16] and [HvNVW17]).

Theorem 4.5.5 is new. Subsection 4.6 generalizes the corresponding resultsof [Ama09] to corners.


5 Triebel–Lizorkin Spaces

If we interchange the roles of Lq and `r in the definition of Besov spaces, thenwe arrive at Triebel–Lizorkin spaces. They are closely related to Besov spacesand possess similar properties. In this section we introduce these spaces in theanisotropic vector-valued setting and discuss some of their properties.

In our setup, Triebel–Lizorkin spaces play an auxiliary role only. Hence werestrict our considerations essentially to those properties which we need for our pur-poses. In the present section we employ these spaces and their properties to provesharp Sobolev type embedding theorems for Sobolev and Bessel potential spacesand anisotropic vector-valued extensions of the classical Gagliardo–Nirenberg in-equalities. We stress the fact –– and this is the main reason for introducing Triebel–Lizorkin spaces –– that these results apply to arbitrary Banach spaces E.

Unless explicitly stated otherwise, all spaces of distributions are over Rd andsequence spaces over N. Thus we write Lq(E) := Lq(Rd, E) and `q(E) := `q(N, E),where we omit E if it equals C. Similar conventions apply to S ′(E), etc. Recallthe standing assumption (VI.3.1.20).

5.1 Maximal Inequalities

Let Q be a ν-quasinorm on Rd. Then

BQ(x, t) :=

y ∈ Rd ; Q(x− y) ≤ t

is the Q-ball of radius t centered at x ∈ Rd. We write |BQ(x, t)| for its volume.

Let v be a measurable function on Rd. Then the anisotropic Q-maximal func-

tion MQv of v (of Hardy–Littlewood type) is defined by

MQv(x) := supt>0

1|BQ(x, t)|

∫

BQ(x,t)

|v| dy ∈ [0,∞]

for x ∈ Rd.

The following lemma shows that the particular choice of the ν-quasinorm isunimportant.

5.1.1 Lemma Let P be a second ν-quasinorm on Rd. Then MPu(x) ∼ MQu(x),uniformly with respect to u and x.

Proof We set BQ := BQ(0, 1). Then

|BQ(x, t)| = |BQ(0, t)| =∫

[Q≤t]

dx = t|ω| |BQ| (5.1.1)

VII.5 Triebel–Lizorkin Spaces 189

for x ∈ Rd and t > 0. By Remark VI.3.3.3 there exists a constant β ≥ 1 such thatβ−1P ≤ Q ≤ βP. Hence

BQ(0, 1/β) ⊂ BP ⊂ BQ(0, β) .

Thusβ−|ω| |BQ| = |BQ(0, 1/β)| ≤ |BP| ≤ |BQ(0, β)| = β|ω| |BQ| .

From this we deduce

MPu(x) ≤ supt>0

( β|ω|

|BQ|∫

BQ(x,βt)

|u(y)| dy)

= β2 |ω|MQu(x) .

Similarly, β−2 |ω|MQu(x) ≤ MPu(x) for x ∈ Rd. ¥

The next theorem is an anisotropic version of the classical Hardy–Littlewoodmaximal inequality.

5.1.2 Theorem Suppose 1 < q ≤ ∞. Then∥∥MQ|u|

∥∥q≤ c ‖u‖q for u ∈ Lq.

Proof By the above lemma we can assume that Q is the Euclidean ν-quasinorm.Then the assertion follows from Corollary 1.8 in [CaT75] by choosing in Theo-rem 1.7 for ϕ the characteristic function of [ |x| < 1] and setting f := |u|. Thischoice is possible by Corollary 1.9 of [CaT75]. ¥

The next theorem is an anisotropic version of the vector-valued maximalinequality of C. Fefferman and E.M. Stein [FS71].

5.1.3 Theorem Suppose 1 < p < ∞ and 1 < r ≤ ∞. Then∥∥(MQ|uk|)

∥∥Lp(`r)

≤ c ‖(uk)‖Lp(`r)

for (uk) ∈ Lp(`r).

Proof If Q = E, then this has been shown by M. Yamazaki [Yam86, Theorem 2.2].Hence the claim follows from Lemma 5.1.1. ¥

Preliminary Estimates for Sequences

In the following, we prove some technical results which will be needed in subsequentsubsections. For this

we fix a ν-quasinorm Q and set Ω := [Q < 1]. (5.1.2)

By Example VI.3.6.1, Ω is a ν-admissible 0-neighborhood of Rd. Moreover,

we fix an Ω-adapted smooth function ψ. Then((Ωk), (ψk)

)is the

ν-dyadic partition of unity on Rd induced by (Ω, ψ).(5.1.3)


Recall that

Ω0 = [Q < 2] and Ωk = [2k−1 < Q < 2k+1] for k ≥ 1 .

Henceforth, M := MQ. Furthermore, given v : Rd → E,

Tkv(x) := supy∈Rd

|v(x − y)|EΛ2 |ω|

1 (2k q y), x ∈ Rd , k ∈ N . (5.1.4)

5.1.4 Lemma Let m ∈ N. Then

Tk(F−1uk)(x) ≤ c(M |F−1uk|1/2

E (x))2

, x ∈ Rd ,

for uk ∈ D′(Rd, E) with supp(uk) ⊂ 2k+m q Ω0 and k ∈ N.

Proof (1) We set vk := F−1uk and fix ϕ ∈ S with ϕ(x) = 1 for x ∈ 2m q Ω0. Thenϕk := σ2kϕ belongs to S and ϕk = 2−k |ω|σ2−k ϕ by Proposition VI.3.1.3(iii). Hence2k |ω|ϕk(x) = 1 for x ∈ 2m+k q Ω0. Thus, see Theorem 1.9.1 in the Appendix,

vk = 2k |ω|F−1(ukϕk) = 2k |ω|vk ∗ ϕk

and, by Remark 1.9.6(g) of the Appendix,

∂αvk = 2k |ω|vk ∗ ∂αϕk = 2k |ω|+α ppppωvk ∗ σ2k∂αϕ

for α ∈ Nd. From this we get (cf. Proposition 1.9.3 of the Appendix)

|∂αvk(x− z)|E ≤ 2k |ω|+α ppppω∫

|vk(y)|E |σ2k∂αϕ|(x − y − z) dy

for x, y ∈ Rd. Here and below, integration and suprema are evaluated over Rd.We set t := 2 |ω| + d + 1. Since |Λt

1∂αϕ| ≤ c for |α| = 1,

|∂αvk(x − z)|E ≤ c 2k |ω|∫

|vk(y)|E σ2kΛ−t1 (x − y − z) dy (5.1.5)

for x, z ∈ Rd, |α| = 1, and k ∈ N.It follows from Λ1 = (1 + N2ν)1/2ν and N(ξ + η) ≤ N(ξ) + N(η) that

Λ2 |ω|1 (ξ) ≤ cΛ2 |ω|

1 (ξ − η)Λ2 |ω|1 (η) , ξ, η ∈ Rd . (5.1.6)

By means of this estimate, setting ξ := 2k q (x − y) and η := 2k q z, we obtain from(5.1.5)

|∂αvk(x − z)|EΛ2 |ω|

1 (2k q z)≤ c 2k |ω|

∫ |vk(y)|EΛ2 |ω|

1 (2k q (x − y))σ2kΛ−d−1

1 (x− y − z) dy

≤ c Tkvk(x)(5.1.7)


for x, z ∈ Rd, |α| = 1, and k ∈ N. Here we used the transformation of variablesξ = x− y and, recalling Proposition VI.3.1.3(ii),

2k |ω|∫

σ2kΛ−d−11 (ξ − z) dξ =

∫Λ−d−1

1 (y) dy < ∞ .

Since (5.1.7) holds for all z ∈ Rd,

Tk(∂αvk)(x) ≤ c Tkvk(x) , x ∈ Rd , (5.1.8)

for |α| = 1 and k ∈ N.(2) By Lemma 5.1.1, we can assume that Ω is convex (cf. Lemma VIII.1.2.2

below). Let g ∈ C1(Rd, E) and B := BQ. Then the mean-value theorem

g(y) = g(z) +∫ 1

0

∂g(z + t(y − z)

)(y − z) dt , y, z ∈ B ,

implies, by choosing z such that |g(z)|E = minB |g|E ,

|g(y)|E ≤ c(min

B|g|E + max

|α|=1supB

|∂αg|E)

≤ c(( 1

|2−k q B|∫

2−k ppppB|g|1/2

E dx)2

+ max|α|=1

supB

|∂αg|E)

for y ∈ B and k ∈ N. Here we replace g by σtg for t > 0. Then

|g(x)|E ≤ c max|α|=1

tαppppω sup

t ppppB |∂αg|E + ct−2 |ω|( 1|2−k q B|

∫

t pppp2−k ppppB|g|1/2

E dx)2

for x ∈ t q B and k ∈ N. We substitute g(y) := vk(x − y − z) to obtain

|vk(x − z)|E ≤ c max|α|=1

tαppppω sup

y∈t ppppB |∂αvk(x − y − z)|E

+ ct−2 |ω|( 1|2−k q B|

∫

t pppp2−k ppppB|vk(x − y − z)|1/2

E dy)2 (5.1.9)

for x, z ∈ Rd and k ∈ N.Now we assume 0 < t ≤ 1. Observe that y ∈ t2−k q B iff Q(y) ≤ t2−k if and

only if y ∈ B(0, t2−k). Hence

Q(y + z) ≤ Q(y) + Q(z) ≤ 2−k(1 + 2kQ(z)

)= 2−k

(1 + Q(2k q z)

)

for y ∈ t2−k q B. Thus, setting R := R(k, z) := 1 + Q(2k q z),∫

t2−k ppppB|vk(x− y − z)|1/2

E dy ≤∫

R2−k ppppB|vk(x − ξ)|1/2

E dξ

=∫

B(x,R2−k)

|vk(y)|1/2E dy ≤ |B(x, R2−k)|M(|vk|1/2

E )(x) .


Since |B(x,R2−k)| = |B(0, R2−k)| = R|ω| |2−k q B|, we obtain from (5.1.9)

|vk(x− z)|E ≤ ctmin ω max|α|=1

supy∈B

|∂αvk(x − z − y)|E

+ ct−2 |ω|(1 + Q(2k q z))2 |ω|(

M |vk|1/2E (x)

)2

for x, z ∈ Rd, k ∈ N, and 0 < t ≤ 1. Using 1 + Q ∼ Λ1, we can divide this inequal-ity by Λ2 |ω|(2k q z) and then take the supremum over z ∈ Rd to arrive at

Tkvk(x) ≤ ctmin ω max|α|=1

Tk(∂αvk)(x) + ct−2 |ω|(M |vk|1/2E (x)

)2

for x ∈ Rd, k ∈ N, and 0 < t ≤ 1.Due to (5.1.8) we can find a constant c1 ≥ 1 such that the first term on the

right side of the last inequality is majorized by

c1tmin ωTkvk(x) , x ∈ Rd , k ∈ N , 0 < t ≤ 1 .

Now the assertion follows by setting t := (2c1)−1/ min ω. ¥

We letE0 × E1 → E2 , (e0, e1) 7→ e0e1

be a multiplication and denote its point-wise extension by juxtaposition. Recalldefinition (VI.3.4.7) of the Fourier multiplier space M.

5.1.5 Lemma Suppose 1 ≤ q ≤ ∞ and 1 ≤ r ≤ ∞. Let (wk) be a sequence in Dsatisfying

∥∥Λ2 |ω|1 F−1

(σ2k(wkak)

)∥∥1≤ c ‖ak‖M (5.1.10)

for ak ∈M(E0) and k ∈ N. Then, given m ∈ N,

‖(wkak)(D)uk‖Lq(`r(E2)) ≤ c supk

‖ak‖M(E0) ‖(uk)‖Lq(`r(E1))

for all (uk) ∈ Lq

(`r(E1)

)with supp(uk) ⊂ 2m q Ωk, and all ak ∈M(E0), k ∈ N.

Proof We set

Wk(ak)uk := F−1(wkak) ∗ uk = (wkak)(D)uk , uk ∈ Lq(E1) .

Proposition VI.3.1.3 implies∥∥Λ2 |ω|

1 F−1(σ2k(wkak)

)∥∥1

= ‖(σ2kΛ2 |ω|1 )F−1(wkak)‖1

≥ ‖F−1(wkak)‖1 .(5.1.11)


Hence we get from (5.1.10) that F−1(wkak) is integrable. Thus uk ∈ Lq(E1) andTheorem 1.9.9 of the Appendix guarantee that Wk(ak)uk ∈ Lq(E2) and

Wk(ak)uk(x) =∫

F−1(wkak)(y)uk(x − y) dy , a.a. x ∈ Rd .

Consequently,

|Wk(ak)uk(x− z)|E ≤∫

|F−1(wkak)(y)|E0 |uk(x− y − z)|E1 dy

≤∫

|F−1(wkak)(y)|E0 σ2kΛ2 |ω|1 (y + z)| dy Tkuk(x)

(5.1.12)

for a.a. x, z ∈ Rd. By means of (5.1.6) we majorize the last integral by

Λ2 |ω|1 (2k q z)

∫(σ2kΛ2 |ω|

1 ) |F−1(wkak)|E0 dy ≤ cΛ2 |ω|1 (2k q z) ‖ak‖M

for z ∈ Rd, using (5.1.11) and assumption (5.1.10). Now we divide both sides ofthe inequality resulting from (5.1.12) by Λ2 |ω|

1 (2k q z) and take the supremum withrespect to z ∈ Rd. This gives the estimate

Tk

(Wk(ak)uk

)(x) ≤ c ‖ak‖M Tkuk(x) (5.1.13)

for a.a. x ∈ Rd and k ∈ N. Then, by Lemma 5.1.4 and uk = F−1uk,

Tkuk(x) ≤ c(M |uk|1/2

E1(x)

)2, x ∈ Rd ,

for uk ∈ L1(E1), supp(uk) ⊂ 2m q Ω, and k ∈ N. Thus, since the left side of (5.1.13)is bounded below by |Wk(ak)uk(x)|E , we deduce

∥∥(Wk(ak)uk

)∥∥Lq(`r(E2))

≤ c supk

‖ak‖M∥∥(

(M |uk|1/2E1

)2)∥∥

Lq(`r)

≤ c supk

‖ak‖M ‖(M |uk|1/2E1

)‖2L2q(`2r)

for (uk) ∈ Lq(`r) with supp(uk) ⊂ 2m q Ω. Finally, we apply Theorem 5.1.3 to inferthat

‖(M |uk|1/2E1

)‖2L2q(`2r) ≤ c ‖(|uk|1/2

E1)‖2

L2q(`2r) = c ‖(uk)‖Lq(`r(E1)) .

This proves the assertion. ¥

Estimates for a Single Function

Whereas the preceding estimates concern sequences (uk), we now turn our atten-tion to a single function u.

It is convenient to use the non-reduced weight system [d,1, ω]. Then Q := Nω,the natural ω-seminorm. Recall that SQ = [Q = 1] is the Q-quasisphere.


5.1.6 Lemma vol(St−1Q) = t|ω| (1−1/d) vol(SQ).

Proof (1) We write x = (y, xq) with y ∈ R and define Q

qon Rd−1 by

Q(x) =(y2ω/ω1 + Q

q(x

q)2ω

)1/2ω.

Moreover,ht(x

q) :=

(t2ω − Q

q(x

q)2ω

)ω1/2ω, Q

q(x

q) < t .

Then ±ht is a smooth parametrization of St−1Q ∩ [±y > 0].

We compute

∂jht(xq) = −ω1

ωjht(x

q)(ω1−2ω)/ω1(xj)2(ω−ωj)/ωj xj , 2 ≤ j ≤ d .

Since ht(xq) = tω1h1(t−1 q x q

), it follows ∂jht(xq) = tω1−ωj ∂jh1(t−1 q x q

). Hence

gt,jk(xq) :=

(∂jht(x

q)∣∣∂kht(x

q))

= t2ω1−ωj−ωkg1,jk(t−1 q x q)

for 2 ≤ j, k ≤ d. Set ajk := g1,jk(t−1 q x q), so that gt,jk(x

q) = t2ω1−ωj−ωkajk. Then,

denoting by Sd−1 the group of permutations of d − 1 elements,

det[gt,jk(x

q)]

= t2ω1(d−1)∑

σ∈Sd−1

sign(σ)t−ωj−ωσ(j)ajσ(j)

= t2ω1(d−1)−2 |ω pppp| det[ajk] .

Consequently,

√gt(x

q) = tω1(d−1)−|ω pppp|√g1(t−1 q x q

) , t > 0 , Qq(x

q) < t .

From this we get

vol(St−1Q) = 2∫

[Qpppp<t]

√gt(x

q) dx

q= 2tω1(d−1)

∫

[Qpppp<t]

t−|ωpppp|√g1(t−1 q x q

) dxq

= 2tω1(d−1)

∫

[Qpppp<t]

√g1(y

q) dy

q= tω1(d−1) vol(SQ)

for t > 0.

(2) By solving the equation Q(x) = t for xj instead of x1, we obtain analo-gously vol(tSQ) = tωj(d−1) vol(SQ) for 2 ≤ j ≤ d. Thus, by multiplying these d ex-pressions and taking the d-th root, we arrive at the asserted formula. ¥

Note that this lemma generalizes the well-known result vol(rS) = rd−1 vol(S)for the Euclidean unit-sphere S.


5.1.7 Lemma Suppose f : R+ → R+ is a nonzero decreasing function such thats 7→ s|ω| (1−1/d)f(s) is integrable. Set ϕ := f Q and ϕt(x) := t|ω|ϕ(t q x) for t > 0and x ∈ Rd. Then

supt>0

|ϕt ∗ u(x)| ≤ ‖ϕ‖1 M |u|(x) , a.a. x ∈ Rd ,

for all measurable u : Rd → E for which ϕ ∗ |u| exists a.e.

Proof (1) By Fubini’s theorem and the preceding lemma,∫

ϕdx =∫ ∞

0

f(s) vol(Ss−1Q) ds =∫ ∞

0

s|ω| (1−1/d)f(s) ds vol(SQ) < ∞ .

Hence ϕ is integrable and

‖ϕt‖1 = t|ω| ‖σtϕ‖1 = ‖ϕ‖1 , t > 0 .

(2) First we assume that f is of the form∑n

i=1 aiχ[0,ξi] with ai ≥ 0 andξi > 0. Then ϕ =

∑aiχ[Q≤ξi] =

∑aiχBQ(0,ξi) and

‖ϕ‖1 =∑

ai |BQ(0, ξi)| .

Furthermore,

|χ[Q(t ppppx)<ξi] ∗ u(x)| ≤ χ[Q<ξi/t] ∗ |u| (x) ≤ |BQ(0, ξi/t)|MQ|u|(x)

= t−|ω| |BQ(0, ξi)|M |u|(x) ,

where the last equality follows from (5.1.1). This implies

|ϕt ∗ u(x)| ≤∑

ai |BQ(0, ξi)|M |u|(x) = ‖ϕ‖1 M |u|(x)

for a.a. x ∈ Rd and t > 0.(3) Now let f be arbitrary. We consider the step function hm : R+ → R+,

defined, for m ∈ N, by

hm(ξ) := f(j2−m)χ[(j−1)2−m,j2−m) , 1 ≤ j ≤ 2m .

Observe

hm(ξ) =2m∑

j=1

(f((j − 1)2−m) − f(j2−m)

)χ[0,j2m] , ξ ∈ R .

Hence, by step (2),

|(hm N)t ∗ u(x)| ≤ ‖hm N‖1 M |u|(x) , m ∈ N , t > 0 . (5.1.14)

Observe that hm ≤ hm+1 ≤ f and the sequence converges point-wise towards f .Thus, if u is as in the assertion, by Lebesgue’s dominated convergence theorem


we can pass to the limit in (5.1.14) to obtain |ϕt ∗ u(x)| ≤ ‖ϕ‖1 M |u|(x) for eacht > 0 and a.a. x ∈ Rd. This proves the lemma. ¥

Now we can derive the desired estimate for a single Lp function.

5.1.8 Lemma Suppose 1 0. Hence f satisfies the hypotheses of thepreceding lemma, and ϕ := f N = Λ−2 |ω|

1 .

(2) Recall that ψk = σ2−k ψ for k ≥ 1 and supp(ψ) ⊂ Ω0. Hence we infer fromProposition VI.3.1.3 that

‖(σ2kΛ2 |ω|1 )F−1ψk‖∞ = 2k |ω| ‖Λ2 |ω|

1 F−1ψ‖∞ = c 2k |ω|

for k ≥ 1. Consequently,

|ψk(D)u(x)| ≤∫

(σ2kΛ2 |ω|1 |F−1ψk|)(y)(σ2kΛ−2 |ω|

1 )(y) |u(x− y)| dy

≤ c

∫2k |ω|(σ2kΛ−2 |ω|

1 )(y) |u(x − y)| dy

= c ϕ2k ∗ |u|(x) ≤ cM |u|(x)

for k ∈ N, due to Lemma 5.1.7. Now Theorem 5.1.2 implies the assertion. ¥

5.2 Definition and Basic Embeddings

Throughout the rest of this section

• 1 ≤ q < ∞, 1 ≤ r ≤ ∞ ,

• s ∈ R ,(5.2.1)

unless explicit restrictions are imposed.Let conditions (5.1.2) and (5.1.3) be satisfied. Since ψk(D)u is an element

of OM (E),(2sj ψj(D)u(x)

)is for each x ∈ Rd a sequence in E. Thus the following

definition is meaningful.

The anisotropic Triebel–Lizorkin space Fs/νq,r (E) = F

s/νq,r (Rd, E) consists of all

u ∈ S ′(E) for which(ψk(D)u

)k∈N ∈ Lq

(`sr(E)

). We endow it with the norm

u 7→∥∥(

ψk(D)u)∥∥

Lq(`sr(E))

=∥∥∥∥∥(

2ksψk(D)u)∥∥

`r(E)

∥∥∥q

. (5.2.2)

Clearly, this definition depends on the choice of Q and ψ. The next lemma shows,however, that the topology of F

s/νq,r (E) is independent of (Q, ψ).


Equivalent Norms

5.2.1 Lemma Let Q1 and Q2 be ν-quasinorms, define Ωj := [Qj < 1], and chooseΩj-adapted smooth functions ψj for j = 1, 2. Denote by ‖·‖j the norm (5.2.2) withψk replaced by ψj

k. Then ‖·‖1 ∼ ‖·‖2.

Proof We use the notations of the proof of Lemma 2.1.1. In particular, ψ−i := 0for i ≥ 0. Then we can find m ∈ N such that ψ2

k = ψ2kχ1

k,m for k ∈ N. Thus

ψ2k(D)u = F−1(ψ2

kχ1k,mu) = F−1ψ2

k ∗ χ1k,m(D)u

=m+1∑

i=−m−1

F−1ψ2k ∗ ψ1

k+i(D)u

for u ∈ S(E). Consequently,

2ksψ2k(D)u =

m+1∑

i=−m−1

2−isF−1ψ2k ∗ 2(k+i)sψ1

k+i(D)u . (5.2.3)

It follows from Remark VI.3.6.4(a) that

‖Λ2 |ω|1 F−1(σ2kψ2

k)‖1 ≤ c , k ∈ N . (5.2.4)

Since supp(F−1(ψ1

k+i(D)u))

= supp(ψ1k+iu) ⊂ 2m+1 q Ω for k ∈ N and |i| ≤ m + 1,

we can apply Lemma 5.1.5 with E0 = E1 = E2 = E and ak = 1 to the convolution2−isF−1ψ2

k ∗ ψ1k+i(D)u to obtain

∥∥(2−isF−1ψ2

k ∗ 2(k+i)sψ1k+i(D)u

)k

∥∥Lq(`r(E))

≤ c∥∥(

ψ1k+i(D)u

)k

∥∥Lq(`s

r(E))

≤ c ‖u‖1

for |i| ≤ m + 1. Thus (5.2.3) implies

‖u‖2 =∥∥(

2ksψ2k(D)u

)∥∥Lq(`r(E))

≤m+1∑

i=−m−1

∥∥(2−isF−1ψ2

k ∗ 2(k+i)sψ1k+i(D)u

)k

∥∥Lq(`r(E))

≤ c ‖u‖1 .

This proves that ‖·‖1 is stronger than ‖·‖2. The assertion follows by interchangingthe roles of (Q1, ψ1) and (Q2, ψ2). ¥

We know from Lemma VI.3.6.2 that u =∑∞

k=0 ψk(D)u in S ′(E). The follow-ing lemma shows that this is also true in F

s/νq,r (E) if r < ∞.

5.2.2 Lemma If u ∈ Fs/νq,r (E) and r < ∞, then u =

∑∞k=0 ψk(D)u in F

s/νq,r (E).


Proof Let u ∈ Fs/νq,r (E). Set un := u− ∑n

j=0 ψj(D)u for n ∈ N. Since u ∈ S ′(E),it follows

ψk(D)un = ψk(D)∞∑

j=n+1

ψj(D)u =1∑

i=−1

ψk+i(D)ψk(D)u (5.2.5)

for k ≥ n, and ψk(D)un = 0 for k ≤ n− 1. From (5.2.4), (5.2.5), and Lemma 5.1.5we infer (setting wk := ψk+i, uk := ψk(D)u for k ≥ n, and uk := 0 otherwise) that

‖un‖F

s/νq,r

=∥∥(

ψk(D)un)k

∥∥Lq(`s

r(E))≤ c

∥∥∥( ∞∑

k=n

(2ks |ψk(D)u|E

)r)1/r∥∥∥

q.

Since u ∈ Fs/νq,r (E), it follows that ‖un‖

Fs/νq,r

→ 0 as n →∞. This shows that theclaim is true. ¥

Embeddings

In the following theorem we collect the basic embedding properties of Triebel–Lizorkin spaces.

5.2.3 Theorem We write Fs/νq,r for F

s/νq,r (E), etc. Suppose 1 ≤ q < ∞.

(i) If 1 ≤ r0 < r1 ≤ ∞, then

S d→ B

s/νq,1

d→ F s/ν

q,r0→ F s/ν

q,r1→ Bs/ν

q,∞d

→ S ′ .

Moreover, Fs/νq,r0 is dense in F

s/νq,r1 if r1 < ∞.

(ii) Let 1 ≤ r0 ≤ q ≤ r1 ≤ ∞. Then Fs/νq,r0 → F

s/νq,q = B

s/νq → F

s/νq,r1 .

(iii) Suppose s > t and r0, r1 ∈ [1,∞]. Then Fs/νq,r0 → F

t/νq,r1 .

(iv) If E1 → E0 and 1 ≤ r ≤ ∞, then

F s/νq,r (E1) → F s/ν

q,r (E0) . (5.2.6)

This embedding is dense if E1d

→ E0.

Proof (1) The first and the last embedding of (i) follow from Theorem 2.2.2.The middle one is an immediate consequence of (5.2.2), Theorem VI.2.2.2(i), and(1.6.2).

In order to show the second embedding (without the density assertion), itthus suffices to prove B

s/νq,1 → F

s/νq,1 . This follows from

‖u‖F

s/νq,1 (E)

=∥∥∥

∑

k

|2ksψk(D)u|E∥∥∥

q≤

∑

k

2ks ‖ψk(D)u‖Lq(E) = ‖u‖B

s/νq (E)

.


Similarly,

‖u‖B

s/νq,∞(E)

= supk

2ks ‖ψk(D)u‖Lq(E)

≤∥∥∥ sup

k2ks |ψk(D)u|E

∥∥q

= ‖u‖F

s/νq,∞

≤ c ‖u‖F

s/νq,r1

.

This proves the validity of the next to last embedding.

(2) The Fubini–Tonelli theorem guarantees Fs/νq,q

.= Bs/νq,q = B

s/νq , due to the

fact that, by Lemma 2.1.1, we can assume that Bs/νq,q is given the norm based on

the ν-dyadic partition of unity induced by (Q, ψ). Now (ii) follows from (i).(3) Assertion (iii) is immediate by (VI.2.2.5) and (1.6.2).(4) Claim (5.2.6) is a consequence of Theorem VI.2.2.2(ii) and the trivial

observation (1.6.2).

(5) We show that S is dense in Fs/νq,r if r < ∞. Let u ∈ F

s/νq,r . By Lemma 5.2.2,

it suffices to prove that, given n ∈ N, un :=∑n

k=0 ψk(D)u can be approximatedin F

s/νq,r by elements of S.Let v ∈ D. Then vn :=

∑nk=0 ψk(D)v ∈ S. Note that ψk(D)(un − vn) = 0 for

k ≥ n + 1 implies

‖un − vn‖F

s/νq,r

=∥∥∥(n+1∑

j=0

(2js |ψj(D)(un − vn)|E

)r)1/r∥∥∥

q

≤ 2(n+1)s∥∥∥(n+1∑

j=0

(2j(s−1) |ψj(D)(un − vn)|E

)r)1/r∥∥∥

q

= 2(n+1)s ‖un − vn‖F

(s−1)/νq,r

≤ c ‖un − vn‖B

(s−1)/νq,1

,

due to B(s−1)/νq,1 → F

(s−1)/νq,r . Since D is dense in S and S is dense in B

(s−1)/νq,1 , the

claim follows.

(6) The density assertions of (i) are now immediate by step (5). If E1d

→ E0,

then S(E1)d

→ S(E0) by Lemma 1.6.1. Thus the density statement for (5.2.6) isalso a consequence of step (5). ¥

Completeness

The next theorem guarantees that the Triebel–Lizorkin spaces are complete.

5.2.4 Theorem Let assumption (5.2.1) be satisfied. Then Fs/νq,r (E) is a Banach

space whose topology is independent of the particular choice of (Q, ψ).

Proof (1) Due to Lemma 5.2.1, it remains to show the completeness of Fs/νq,r (E).

Thus let (uj) be a Cauchy sequence in Fs/νq,r (E). It follows from Theorem 5.2.3(i)


that (uj) is a Cauchy sequence in S ′(E). Hence there exist u ∈ S ′(E) such thatuj → u in S ′(E). We claim that u ∈ F

s/νq,r (E) and uj → u in F

s/νq,r (E).

(2) Let (vj) be a sequence in Fs/νq,r (E) such that ‖vj‖F

s/νq,r

≤ c0 and vj → v

in S ′(E). Observe, cf. Subsection III.4.2, that

ψk(D)vj(x) = F−1ψk ∗ vj(x) = τxv

j(F−1ψk)

= vj

(τx(F−1ψk)

)= (2π)−dvj(τxψk)

(5.2.7)

for x ∈ Rd and j, k ∈ N. We set, for j,m ∈ N,

wmj :=

( m∑

k=0

(2ks |ψk(D)vj |E

)r)1/r

if r < ∞, respectively wmj := maxk≤m 2ks |ψk(D)vj |E otherwise. Then

‖wmj ‖q ≤ ‖vj‖F

s/νq,r

≤ c0 , j, m ∈ N .

Since vj → v in S ′(E), it follows from (5.2.7) that

wmj (x) →

( m∑

k=0

(2ks |ψk(D)v(x)|E

)r)1/r

=: wm(x)

if r < ∞, respectively

wmj (x) → max

k2ks |ψk(D)v(x)|E =: wm(x)

if r = ∞. Hence, by Fatou’s lemma, ‖wm‖q ≤ c0. Letting m →∞, we get from themonotone convergence theorem

‖v‖F

s/νq,r

= limm→∞

‖wm‖q ≤ c0 .

(3) Let ε > 0. There exists nε such that ‖uk − uj‖Fs/νq,r

≤ ε for k, j ≥ nε. It fol-

lows from step (2) with vj := uk − uj , where k ≥ nε is fixed, that uk − u ∈ Fs/νq,r (E)

and

‖uk − u‖F

s/νq,r

≤ ε . (5.2.8)

Thus u = uk + (u− uk) ∈ Fs/νq,r (E) and, (5.2.8) being true for each k ≥ nε, we see

that uk → u in Fs/νq,r (E). This proves the theorem. ¥

5.2.5 Remark In step (2) of this proof we have shown that Fs/νq,r (E) enjoys the

Fatou property : If (vj) is a sequence in Fs/νq,r (E) such that ‖vj‖F

s/νq,r

≤ c0 and vj → v

in S ′(E), then v ∈ Fs/νq,r (E) and ‖v‖

Fs/νq,r

≤ c c0. ¥


5.3 Fourier Multiplier Theorems

For Triebel–Lizorkin spaces we can prove a Fourier multiplier theorem which isanalogous to Theorem 2.4.2.

5.3.1 Theorem Let (5.2.1) be true. Suppose

E0 × E1 → E2 , (e0, e1) 7→ e0e1

is a multiplication and denote its point-wise extension again by juxtaposition. Then(a 7→ a(D)

) ∈ L(M(E0),L(F s/ν

q,r (E1), F s/νq,r (E2)

)).

Proof Given u ∈ Fs/νq,r (E1) and a ∈M(E0), we infer from ψk = ψkχk that

2skψk(D)(a(D)u

)= 2skF−1(ψkau) = 2skF−1(ψka) ∗ χku

=1∑

i=−1

2−isF−1(ψka) ∗ 2(k+i)sψk+i(D)u .

Remark VI.3.6.4(a) guarantees that (5.1.10) is satisfied with wk := ψk and ak = a.Since ψk+iu has its support in 2 q Ωk, we can apply Lemma 5.1.5 to obtain

‖a(D)u‖F

s/νq,r (E2)

=∥∥(

ψk(D)(a(D)u))∥∥

Lq(`sr(E2))

≤ c(s) ‖a‖M∥∥(

ψk(D)u)∥∥

Lq(`sr(E1))

= c(s) ‖a‖M ‖u‖F

s/νq,r (E1)

.

This proves the theorem. ¥

It is obvious that the analogue of Theorem 2.4.2(ii) holds in this case also.The next theorem shows that, similarly as for Besov spaces, J t is an isomor-

phism between Triebel–Lizorkin spaces.

5.3.2 Theorem Let assumption (5.2.1) be satisfied and t ∈ R. Then

J t ∈ Lis(F (s+t)/ν

q,r (E), F s/νq,r (E)

)and (J t)−1 = J−t .

Proof The proof of (2.4.2) shows that

ak := 2−ktΛt1ψk ∈M , ‖ak‖M ≤ c , k ∈ N . (5.3.1)

Moreover, using ψk = ψkχ2k,

2ksψk(D)J tu = 2ksF−1(ψkΛt1u) = 2(s+t)kF−1(χkψk2−ktΛt

1χku)

=1∑

i=−1

1∑

j=−1

2−(s+t)jF−1(ψk+iak) ∗ 2(s+t)(k+j)ψk+j(D)u .


From this, (5.3.1), Remark VI.3.6.4(a), and Lemma 5.1.5 we infer that

‖J tu‖F

s/νq,r

≤ c ‖u‖F

(s+t)/νq,r

for u ∈ F(s+t)/νq,r . Now the assertion is obvious. ¥

The following analogue of Proposition 2.4.4 is a consequence of the two pre-ceding theorems.

5.3.3 Proposition Let t ∈ R. Suppose Λ−t1 a ∈M(L(E0, E1)

). Then

(i) a(D) ∈ L(F

(s+t)/νq,r (E0), F

s/νq,r (E1)

)and ‖a(D)‖ ≤ c ‖Λ−t

1 a‖M.

(ii) If, in addition, a ∈ C(L(E0, E1)

)and Λt

1a−1 ∈ B

(L(E1, E0)), then

a(D) ∈ Lis(F (s+t)/ν

q,r (E0), F s/νq,r (E1)

), a(D)−1 = a−1(D) ,

and ‖a(D)−1‖ ≤ c(‖Λ−t

1 a‖M, ‖Λt1a

−1‖∞).

Proof Use obvious substitutions in the proof of Proposition 2.4.4. ¥

5.3.4 Remark It is clear that, by the results of this subsection, we can carry overto Triebel–Lizorkin spaces all results which we proved for Besov spaces by usingTheorems 2.4.1 and 2.4.2 and Proposition 2.4.4. ¥

5.4 Interpolation

As a first application of Remark 5.3.4 we show that Triebel–Lizorkin spaces arecompatible with complex interpolation.

5.4.1 Theorem Suppose 1 ≤ q < ∞ and r < ∞. Then

J ∈ BIP(F 0/ν

q,r (E))

,

and the interpolation-extrapolation scale[(Xα, Aα) ; α ∈ R ]

generated by

(X0, A0) :=(F 0/ν

q,r (E), J)

and [·, ·]θ, 0 < θ < 1 ,

satisfies Xα.= F

α/νq,r (E).

Proof See the proof of Theorem 2.5.6 and Remark 2.5.7. ¥

5.4.2 Corollary Assume 1 ≤ q < ∞, r < ∞, and 0 < θ < 1. Then[F s0/ν

q,r (E), F s1/νq,r (E)

]θ

.= F sθ/νq,r (E) , −∞ < s0 < s1 < ∞ .

Proof Theorem V.1.5.4. ¥



A further implementation of Remark 5.3.4 proves the following analogue of The-orem 2.6.2.

5.5.1 Theorem Let (5.2.1) be satisfied.

(i) ∂α ∈ L(F

(s+α ppppω)/νq,r (E), F s/ν

q,r (E)).

(ii) Let m ∈ νqN. The following assertions are equivalent:

(α) u ∈ F(s+m)/νq,r (E).

(β) ∂αu ∈ Fs/νq,r (E), α q ω ≤ m.

(γ) u, ∂m/ωj

j u ∈ Fs/νq,r (E), 1 ≤ j ≤ d.

Furthermore, define ‖·‖(k)

F(s+m)/νq,r

by replacing B ·q,r in Theorem 2.6.2 by F ·

q,r. Then

‖·‖(k)

F(s+m)/νq,r

∼ ‖·‖F

(s+m)/νq,r

for 1 ≤ k ≤ 4.

Proof Cf. the proof of Theorem 2.6.2. ¥

Sandwich Theorems

Due to Theorem 5.2.3(i), the following ‘sandwich theorem’ sharpens (2.6.6) andputs Bessel potential spaces in relation to Triebel–Lizorkin spaces.

5.5.2 Theorem Suppose 1 < p < ∞.

(i) If m ∈ νN, then Fm/νp,1 (E) → W

m/νp (E) → F

m/νp,∞ (E).

(ii) Fs/νp,1 (E) → H

s/νp (E) → F

s/νp,∞(E) for s ∈ R.

Proof (1) We assume m = 0. Let u ∈ F0/νp,1 (E) → S ′(E). Then u =

∑k ψk(D)u

in S ′(E) by Lemma VI.3.6.2. Hence, denoting by δx the Dirac distribution sup-ported at x,

∣∣∣δx

∑

k

ψk(D)u∣∣∣E

=∣∣∣∑

k

δx

(ψk(D)u

)∣∣∣E

=∣∣∣∑

k

ψk(D)u(x)∣∣∣E

≤∑

k

|ψk(D)u(x)|E

for x ∈ Rd. From this we get

‖u‖p ≤∥∥∥

∑

k

|ψk(D)u|E∥∥∥

Lp

= ‖u‖F

0/νp,1

,

that is, F0/νp,1 (E) → Lp(E).

On the other hand, Lp(E) → F0/νp,∞(E) follows from Lemma 5.1.8. Thus (i) ap-

plies if m = 0.


(2) Assume m ∈ νqN. It follows from step (1) and Theorem 5.5.1 that

‖u‖F

m/νp,∞

≤ c∑

α ppppω≤m

‖∂αu‖Lp(E) ≤ c∑

α ppppω≤m

‖∂αu‖F

0/νp,1

≤ c ‖u‖F

m/νp,1

.

Now (i) is implied by (1.1.4) and Theorem 1.4.1.

(3) Assertion (ii) is immediate from (4.1.5), Theorem 5.3.2, and step (1). ¥

5.6 Sobolev Embeddings and Related Results

With the help of Triebel–Lizorkin spaces we can extend the classical Sobolev em-bedding theorem to the anisotropic vector-valued setting. In addition, we can showthat the equality signs in (4.1.9) can be admitted too. For this we need some prepa-ration.

Multiplicative Inequalities

Henceforth, we use the notation introduced in (VI.2.3.1). Recall that

sθ := (1 − θ)s0 + θs1 and 1/q(θ) = (1 − θ)/q0 + θ/q1 .

5.6.1 Lemma Let s0, s1 ∈ R and 0 < θ < 1. Then

‖(2ksθak)‖`r≤ c ‖(2ks0ak)‖1−θ

`∞ ‖(2ks1ak)‖θ`∞

for (ak) ∈ CN.Proof Without loss of generality, s0 < s1. Set cj := ‖(2ksj ak)k‖`∞ for j = 0, 1.Then c0 ≤ c1. We can assume c1 > 0. Since s0 < s1, there is some j0 > 0 such that

minc02−s0j , c12−s1j =

c02−s0j , if j ≤ j0 ,

c12−s1j , if j > j0 .

From c02−s0j0 ≤ c12−s1j0 and c12−s1(j0+1) ≤ c02−s0(j0+1) we get c1 ∼ c02(s1−s0)j0 .Therefore

‖(2s0kak)‖1−θ`∞ ‖(2s1kak)‖θ

`∞ ∼ c02(s1−s0)j0θ . (5.6.1)

On the other hand, |ak| ≤ minc02−s0k, c12−s1k so that

|ak| ≤

c02−s0k for 0 ≤ k ≤ j0 ,

c12−s1k for j0 < k < ∞ .


Hence, if r < ∞,

‖(2sθkak)‖`r≤

( ∑

k≤j0

cr02

(sθ−s0)kr +∑

k>j0

cr12

(sθ−s1)kr)1/r

≤( ∑

k≤j0

cr02

θ(s1−s0)kr +∑

k>j0

cr02

−(1−θ)(s1−s0)kr+(s1−s0)j0r)1/r

≤ c02(s1−s0)θj0( ∑

k≤j0

2θ(s1−s0)(k−j0)r +∑

k>j0

2−(1−θ)(s1−s0)(k−j0)r)1/r

≤ c c02(s1−s0)θj0 .

Similarly, ‖(2sθkak)‖`∞ ≤ c02(s1−s0)θj0 . Now the assertion follows from (5.6.1). ¥

5.6.2 Lemma Suppose s0, s1 ∈ R, 0 < θ < 1, and r0, r1 ∈ [1,∞].(i) Let 1 ≤ q0, q1 < ∞. Then

‖u‖F

sθ/ν

q(θ),r

≤ c ‖u‖1−θ

Fs0/νq0,r0

‖u‖θ

Fs1/νq1,r1

for u ∈ Fs0/νq0,r0 ∩ F

s1/νq1,r1 (E).

(ii) If 1 ≤ q0 < ∞ and q1 = ∞, then

‖u‖F

sθ/ν

q(θ),r

≤ c ‖u‖1−θ

Fs0/νq0,r0

‖u‖θ

Bs1/ν∞

for u ∈ Fs0/νq0,r0 ∩B

s1/ν∞ (E).

Proof (1) We apply the preceding lemma to ak := |ψk(D)u(x)|E and use Holder’sinequality. Then

‖u‖F

sθ/ν

q(θ),r

=∥∥(

2ksθψk(D)u)∥∥

Lq(θ)(`r)

≤ c∥∥∥∥∥(

2ks0ψk(D)u)∥∥1−θ

`∞

∥∥(2ks1ψk(D)u

)∥∥θ

`∞

∥∥∥q(θ)

≤ c ‖u‖1−θ

Fs0/νq0,∞

‖u‖θ

Fs1/νq1,∞

.

Claim (i) follows now from Theorem 5.2.3(i).(2) Similarly as in step (1),

‖u‖F

sθ/ν

q(θ),r

≤ c∥∥∥∥∥(


`∞

∥∥(2ks1ψk(D)u

)∥∥θ

`∞

∥∥∥q(θ)

≤ c∥∥∥∥∥(


`∞

∥∥∥q0/(1−θ)

∥∥∥∥∥(

2ks1ψk(D)u)∥∥θ

`∞

∥∥∥∞

≤ c ‖u‖1−θ

Fs0/νq0,∞

∥∥(2ks1ψk(D)u

)∥∥θ

`∞(L∞).

This proves (ii). ¥


Optimal Sobolev-Type Embeddings

First we consider Triebel–Lizorkin spaces.

5.6.3 Theorem Let s0, s1 ∈ R and 1 ≤ q0, q1 < ∞ satisfy s1 − |ω|/q1 ≥ s0 − |ω|/q0

and s1 ≥ s0. If 1 ≤ r0, r1 < ∞, then Fs1/νq1,r1 (E) → F

s0/νq0,r0 (E).

Proof Due to Theorem 5.2.3(iii) we can assume, by making s1 smaller if neces-sary, that s1 − |ω|/q1 = s0 − |ω|/q0 and s1 > s0. Thus q1 < q0 and we can defineθ ∈ (0, 1) by 1/q0 = (1 − θ)/q1.

Suppose σ0 6= σ1, 1 ≤ p0 < ∞, and 1/p := (1 − θ)/p0. Given ρ, ρ0 ∈ [1,∞],Lemma 5.6.2(ii) implies

‖u‖F

σθ/νp,ρ

≤ c ‖u‖1−θ

Fσ0/νp0,ρ0

‖u‖θ

Bσ1/ν∞

. (5.6.2)

We now set σ1 := s1 − |ω|/q1, σ0 := s1, and p0 := q1. Then p = q0 and

σθ = (1 − θ)s1 + θ(s1 − |ω|/q1) = s1 − θ |ω|/q1

= s1 − |ω|/q1 + (1 − θ) |ω|/q1 = s1 − |ω|/q1 + |ω|/q0 = s0 .

Consequently, letting ρ0 := r1 and ρ := r0 in (5.6.2),

‖u‖F

s0/νq0,r0

≤ c ‖u‖1−θ

Fs1/νq1,r1

‖u‖θ

B(s1−|ω|/q0)/ν∞

. (5.6.3)

We infer from Theorem 5.2.3(i) and (2.2.3) that

F s1/νq1,r1

(E) → Bs1/νq1,∞(E) → B(s1−|ω|/q1)/ν

∞ (E) .

The assertion follows now from (5.6.3). ¥

5.6.4 Corollary Let K be a corner in Rd and X ∈ Rd,K. Suppose s0, s1 ∈ R andq0, q1 ∈ [1,∞] satisfy s1 − |ω|/q1 ≥ s0 − |ω|/q0 and s1 ≥ s0. Then

Bs1/νq1

(X, E) → Bs0/νq0

(X, E) .

Proof Due to Theorem 2.8.3, we can assume that X = Rd. By Bs/νq → B

t/νq for

s > t, it suffices to consider the case s1 − |ω|/q1 = s0 − |ω|/q0.

If q0 < ∞, then the claim follows from the theorem and Bs/νq = F

s/νq,q . Oth-

erwise, Bs1/νq1 → B

s0/ν∞,q1 → B

s0/ν∞ by Theorem 2.2.2. ¥

Now we can prove the Sobolev embedding theorems alluded to above. Observethat there is no restriction whatsoever on E.


5.6.5 Theorem Let K be a corner in Rd and X ∈ Rd,K.(i) Suppose m0,m1 ∈ νN, 1 < p1 < p0 < ∞, and m1 − |ω|/p1 ≥ m0 − |ω|/p0.

Then Wm1/ν

p1 (X, E)d

→ Wm0/ν

p0 (X, E).(ii) Let −∞ < s0 < s1 < ∞, 1 < p0, p1 < ∞, and s1 − |ω|/p1 ≥ s0 − |ω|/p0.

Then Hs1/νp1 (X, E)

d→ H

s0/νp0 (X, E).

Proof Let X = Rd. Then we get from Theorems 5.5.2 and 5.6.3

Wm1/νp1

(E) → Fm1/νp1,∞ (E) → F

m0/νp0,1 (E) → Wm0/ν

p0(E)

andHs1/ν

p1(E) → F s1/ν

p1,∞(E) → Fs0/νp0,1 (E) → Hs0/ν

p0(E) .

If X = K, the assertion follows now by extension and restriction, due to Theorems1.3.1 and 4.1.1. The density is a consequence of the density of S(X, E) in thesespaces. ¥

Sharp Embeddings of Intersection Spaces

By means of Corollary 5.6.4 we can improve the embedding given in Theorem 2.7.3by admitting the limiting case s − |ω|/q = sθ − |ω|/q(θ). Furthermore, there is ananalogous result for Bessel potential spaces.

5.6.6 Theorem Let (E0, E1) be an interpolation couple, s, s0, s1 ∈ R with s0 6= s1,1 ≤ q, q0, q1 < ∞, and 0 < θ < 1. Suppose K is a corner in Rd, X ∈ Rd,K,

1/q(θ) ≥ 1/q ≥ 0 and sθ − |ω|/q(θ) ≥ s − |ω|/q .

(i) It holds

Bs0/νq0

(X, E0) ∩Bs1/νq1

(X, E1) → Bs/νq (X, Eθ,q(θ)) .

(ii) If E0 and E1 are ν-admissible and 1 < q, q0, q1 < ∞, then

Hs0/νq0

(X, E0) ∩Hs1/νq1

(X, E1) → Hs/νq (X, E[θ]) .

Proof (1) Due to

Bs0/νq0

(X, E0) ∩ Bs1/νq1

(X, E1) → Bsj/νqj

(X, Ej) , j = 0, 1 ,

interpolation gives

Bs0/νq0

(X, E0) ∩Bs1/νq1

(X, E1) → (Bs0/ν

q0(X, E0), Bs1/ν

q1(X, E1)

)θ,q(θ)

.

Hence, using Theorems 2.7.2(i) and 2.8.3,

Bs0/νq0

(X, E0) ∩Bs1/νq1

(X, E1) → Bsθ/νq(θ) (X, Eθ,q(θ)) .

Now assertion (i) follows from Corollary 5.6.4.


(2) Since

Hs0/νq0

(X, E0) ∩Hs1/νq1

(X, E1) → Hsj/νqj

(X, Ej) , j = 0, 1 ,

complex interpolation and Theorem 4.5.5 imply

Hs0/νq0

(X, E0) ∩ Hs1/νq1

(X, E1) → [Hs0/ν

q0(X, E0), Hs1/ν

q1(X, E1)

]θ

.= Hsθ/νq(θ) (X, E[θ]) .

Now the assertion follows from Theorem 5.6.5(ii). ¥

5.6.7 Remark Given the hypotheses of the preceding theorem,

Bs0/νq0

(X, E0) ∩Bs1/νq1

(X, E1) → Bsθ/νq(θ) (X, E[θ]) .

Proof It suffices to use part (ii) of Theorem 2.7.2 instead of (i) in the aboveproof. ¥

5.7 Gagliardo–Nirenberg Type Estimates

By means of the preceding multiplicative inequalities and various embedding the-orems for Besov, Triebel–Lizorkin, and Sobolev spaces we prove now far reach-ing generalizations of the well-known Gagliardo–Nirenberg inequalities ([Gag59],[Nir59]).


• K is a corner in Rd and X ∈ Rd,K.We also suppose

• −∞ < s0 ≤ s < s1 < ∞ , 1 ≤ p, p0, p1 ≤ ∞ (5.7.1)

satisfy

s − |ω|p

= (1 − θ)(s0 − |ω|

p0

)+ θ

(s1 − |ω|

p1

), (5.7.2)

wheres − s0

s1 − s0≤ θ ≤ 1 . (5.7.3)

Nonhomogeneous Inequalities

First we prove multiplicative inequalities involving complete norms of the spacesunder consideration.


5.7.1 Theorem Let (5.7.1)–(5.7.3) be satisfied. Then

‖u‖B

s/νp

≤ c ‖u‖1−θ

Bs0/νp0

‖u‖θ

Bs1/νp1

(5.7.4)

for u ∈ Bs1/νp1 ∩B

s0/νp0 (X, E).

Proof It suffices to consider X = Rd. Then we get the assertions for X = K bymeans of extension and restriction, due to Theorems 1.3.1 and 2.8.2.

(1) We define sθ and p(θ) by (VI.2.3.1), admitting now the endpoint valuesθ = 0 or θ = 1 also. Then (5.7.2) can be written as

s − |ω|p

= sθ − |ω|p(θ)

(5.7.5)

or

θ − s − s0

s1 − s0=

|ω|s1 − s0

( 1p(θ)

− 1p

).

From this we read off that (5.7.3) is equivalent to

1p≤ 1

p(θ). (5.7.6)

Note that θ = 0 implies s = s0 and, consequently, p = p0. Thus (5.7.4) is trivial inthis case. Hence we can assume that θ > 0.

(2) Suppose

minp0, p1 < ∞ , θ < 1 . (5.7.7)

Lemma 5.6.2(i) guarantees

‖u‖F

sθ/ν

p(θ),1≤ c ‖u‖1−θ

Fs0/νp0,∞

‖u‖θ

Fs1/νp1,∞

, maxp0, p1 < ∞ . (5.7.8)

Similarly, we get from part (ii) of that lemma

‖u‖F

sθ/ν

p(θ),1≤ c ‖u‖1−θ

Fs0/νp0,∞

‖u‖θ

Bs1/ν∞

, p0 < p1 = ∞ . (5.7.9)

It follows from Theorem 5.2.3(ii) and Corollary 5.6.4 that

Fsθ/νp(θ),1 → B

sθ/νp(θ) → Bs/ν

p , (5.7.10)

due to (5.7.5) and (5.7.6). Hence we get from (5.7.8)

‖u‖B

s/νp

≤ c ‖u‖1−θ

Fs0/νp0,∞

‖u‖θ

Fs1/νp1,∞

, maxp0, p1 < ∞ . (5.7.11)


Similarly, (5.7.9) yields

‖u‖B

s/νp

≤ c ‖u‖1−θ

Fs0/νp0,∞

‖u‖θ

Bs1/ν∞

, p0 < p1 = ∞ . (5.7.12)

Employing Theorem 5.2.3(ii) once more, we obtain from these two estimates

‖u‖B

s/νp

≤ c ‖u‖1−θ

Bs0/ν0p0

‖u‖θ

Bs1/ν1p1

(5.7.13)

if either p0, p1 < ∞ or p0 < p1 = ∞. Interchanging in this estimate (θ, p0, p1) with(1 − θ, p1, p0), we see that it applies also if p1 < p0 = ∞.

Now assume that p0 = p1 = ∞. Then the interpolation Theorem 2.7.1(i) im-plies

‖u‖B

sθ/νp

≤ c ‖u‖1−θ

Bs0/νp0

‖u‖θ

Bs1/νp1

.

Thus, using the second part of (5.7.10), it follows that (5.7.13) holds in this casealso.

Lastly, assume θ = 1. Then the validity of (5.7.13) is a consequence of (5.7.5),(5.7.6), and Corollary 5.6.4. This proves the theorem. ¥

Recall that Bt/νq = W

t/νq for 1 ≤ q < ∞ and B

t/ν∞ = BUCt/ν , provided t be-

longs to R+\νN. However, Bt/νq 6= W

t/νq and B

t/ν∞ 6= BUCt/ν for t ∈ νN. For ap-

plications, it is desirable to have estimates of the type given in the theorem if some(or all) of s, s0, s1 belong to νN. The next theorem shows that this is possible,provided some restrictions are imposed. For this we set, for s ≥ 0,

Ws/νp :=

W s/ν

p , if 1 ≤ p < ∞ ,

BUCs/ν , if p = ∞ .(5.7.14)

We write X Ã Y to indicate that X is replaced by Y.

5.7.2 Theorem Let (5.7.1)–(5.7.3) be satisfied and suppose θ < 1. Then Theo-rem 5.7.1 remains valid if any one of the following (independent) substitutions isapplied:

Bs/νp Ã Ws/ν

p if s ≥ 0 , (5.7.15)

and, for j = 0, 1,

Bsj/νpj

Ã Wsj/νpj

if sj ≥ 0 and 1 < pj ≤ ∞ if sj ∈ νN . (5.7.16)

Furthermore, θ = 1 can be admitted if either s, s1 ∈ νN and 1 < p, p1 < ∞, ors, s1 /∈ νN.


Proof As in the preceding proof, we can assume that X = Rd.

(1) Let (t, q) ∈ νN× (1,∞). Then Wt/ν

q → Ft/νq,∞ by Theorem 5.5.2(i). Thus,

if sj ∈ νN, we can replace either of the Fsj/νpj ,∞ in (5.7.11) by W

sj/νpj , and F

s0/νp0,∞ in

(5.7.12) by Ws0/ν0

p0 , provided p0, p1 ∈ (1,∞). Hence (5.7.13) holds in this case withB

s0/νp0 Ã W

s0/νp0 and/or B

s1/νp1 Ã W

s1/νp1 .

By (2.6.7), BUCt/ν → Bt/ν∞ . Thus (5.7.13) applies if

Bs0/ν∞ Ã BUCs0/ν and/or Bs1/ν

∞ Ã BUCs1/ν

if s0, s1 ∈ νN. This proves (5.7.16).(2) We obtain from (5.7.5), (5.7.6), and Theorem 5.6.3 that

Fsθ/νp(θ),1 → F

s/νp,1 .

From this and Fs/νp,1 → B

s/νp we see that B

s/νp Ã F

s/νp,1 is possible in (5.7.11) and

(5.7.12), hence in (5.7.13) as well.

Let s ∈ νN. Theorem 5.5.2(i) guarantees Fs/νp,1 → W

s/νp if 1 < p < ∞. More-

over, Fs/ν1,1 = B

s/ν1 → W

s/ν1 by Theorem 2.6.5. This proves (5.7.15).

(3) Assume s, s1 ∈ νN and 1 < p, p1 < ∞. Then Theorem 5.6.5(i) shows thatθ = 1 is admissible. ¥

5.7.3 Corollary Let (5.7.1)–(5.7.3) be satisfied with s0 ≥ 0. If 1 ≤ p, p0, p1 ≤ ∞,then

Ws1/νp1

∩Ws0/νp0

(K, E) →Ws/νp (K, E) (5.7.17)

and

‖u‖Ws/νp

≤ c ‖u‖1−θ

Ws0/νp0

‖u‖θ

Ws1/νp1

(5.7.18)

for u ∈ Ws1/νp1 ∩Ws0/ν

p0 (K, E), provided

(α) pj > 1 if sj ∈ νN;(β) θ < 1 unless either s, s1 ∈ νN and 1 < p, p1 < ∞, or s, s1 /∈ νN.

5.7.4 Remarks (a) The assertions of Theorem 5.7.1 remain valid, given any of thesubstitutions:

Bs/νp Ã Hs/ν

p , Bsj/νpj

Ã Hsj/νpj

,

provided p, pj ∈ (1,∞).

Proof This follows from the proof of Theorem 5.7.2 by using Theorem 5.5.2(ii). ¥


(b) If (5.7.2) is satisfied, then

s − s0

s1 − s0≤ θ ≤ 1 iff

1p(θ)

≥ 1p≥ 0 .

Proof See step (1) of the proof of Theorem 5.7.1. ¥

(c) The somewhat cumbersome formulation of Theorem 5.7.2 and its corollarycannot be avoided, since the restrictions on the integrability parameters are es-sential if the regularity indices are integer multiples of ν. Indeed, it has beenobserved by J. Bourgain, H. Brezis, and P. Mironescu [BoBM00, Remark D.1]that W 1

1 ∩ L∞(R) is not contained in W1/22 (R), although

(s, p) = (1/2, 2) , (s0, p0) = (0,∞) , (s1, p1) = (1, 1)

satisfy (5.7.2) for any θ ∈ [0, 1]. ¥

Let E1 → E → E0 and 0 < θ < 1. Then E is of class J(θ, E0, E1) if

‖x‖E ≤ c ‖x‖1−θE0

‖x‖θE1

, x ∈ E1 .

It is known (e.g., [BeL76, Theorem 3.5.2] or [Tri95, Lemma 1.10.1]) that

E is of class J(θ, E0, E1) iff Eθ,1 → E , (5.7.19)

where Eϑ,r := (E0, E1)ϑ,r for 0 < ϑ < 1 and 1 ≤ r ≤ ∞.

5.7.5 Remark Suppose 0 < θ < 1 and m0,m1,mθ ∈ νN with m0 < m1. Then itfollows from Corollary 5.7.2 that

Wmθ/νp(θ) (X, E) is of class J

(θ,Wm0/ν

p0(X, E),Wm1/ν

p1(X, E)

),

provided 1 < p0, p1 ≤ ∞. Thus

|||u|||mθ/ν,p(θ) ≤ c |||u|||1−θm0/ν,p0

|||u|||θm1/ν,p1

for u ∈ Wm1/νp1 (X, E). This shows, in particular, that the multiplicative inequali-

ties (5.7.4) cannot be obtained by interpolation, in general, even if p = p(θ). Indeed,in the case where p0 = p1 = ∞, BUCmθ (Rd) cannot be obtained by interpolationbetween BUCm0(Rd) and BUCm1(Rd). ¥

Homogeneous Estimates

The multiplicative inequalities of Theorems 5.7.1 and 5.7.2 can be considerablyimproved, inasmuch as the norms can be replaced by homogeneous seminorms.


This depends on a scaling argument which, in turn, rests on the following simpleobservation.

We use the seminorms [·]θ,q,r and [[·]]θ,q,r;i introduced in Subsection 3.5. Nowit is convenient to set

[·]s,p,r :=

‖·‖p , if s = 0 ,

[·]s,p,r , if s > 0 ,(5.7.20)

for 1 ≤ p, r ≤ ∞.

5.7.6 Lemma Suppose ω = ν1, 1 ≤ p ≤ ∞, and s ≥ 0. Then [σtu]s/ν,p,r equalsts−|ω|/p[u]s/ν,p,r for t > 0.

Proof If s = 0, then this is clear from Proposition VI.3.1.3. Thus assume s > 0.Note that t q y = tνy implies

4yσtu(x) = u(t q x + t q y) − u(t q x) = σt4tνyu(x)

for x, y ∈ X and t > 0. Hence, by induction, 4ky σt = σt 4k

tνy for k ∈ N.Let r < ∞. Then

[σtu]s/ν,p,r =(∫

Id

(|y|−s/ν ‖4[s/ν]+1y σtu‖p

)r dy

|y|d)1/r

= t−|ω|/p(∫

Id

(|y|−s/ν ‖4[s/ν]+1tνy u‖p

)r dy

|y|d)1/r

= ts−|ω|/p[u]s/ν,p,r .

If r = ∞, then we argue correspondingly. ¥

Suppose s > 0 and 1 ≤ p, r ≤ ∞. Then we endow Bs/νp,r (X, E) with the norm12

‖u‖B

s/νp,r

:=∑

i=1

[s/νi]−∑

j=0

(‖∇j

xiu‖p + [[∇j

xiu]]

s/νi−[s−νi]−,p,r;i

).

Due to Theorem 3.5.2 and the remark following it, this is possible.Let s ≤ 0 and 1 ≤ p, r ≤ ∞. We denote by m = m(s) the smallest element

of νN satisfying m + s > 0. Then Theorem 2.6.3 and Remark 2.6.4 guarantee thatwe can equip B

s/νp,r (X, E) with the norm

‖u‖B

s/νp,r

:= inf(‖u0‖B

(s+m)/νp,r

+∑

i=1

‖∇m/νixi

u‖B

(s+m)/νip,r

),

12[·]− is the largest integer strictly smaller than t ∈ R.


where the infimum is taken over all representations

u = u0 +∑

i=1

∇m/νixi

ui , u0, u1, . . . , u` ∈ B(s+m)/νip,r . (5.7.21)

Then we put

‖u‖ pB

s/νp,r

:=∑

i=1

[[∇[s/νi]−xi u]]s/νi−[s/νi]−,p,r;i , s > 0 (5.7.22)

and

‖u‖ pB

s/νp,r

:= inf(∑

i=1

[[∇m/νixi ui]]s+m−[s+m]−,p,r;i

), s ≤ 0 ,

where, again, the infimum is taken over all representations (5.7.21). Lastly,

‖·‖′B

s/νp,r

:= ‖·‖B

s/νp,r

− ‖·‖ pB

s/νp,r

, s ∈ R . (5.7.23)

If s ∈ νN, then Ws/νp is given the norm |||·|||s/ν,p introduced in (1.1.4). Then

‖·‖ pWs/ν

p,r:=

∑α ppppω=s

‖∂α q ‖p (5.7.24)

and‖·‖′Ws/ν

p:= |||·|||s/ν,p − ‖·‖ p

Ws/νp

if s ∈ νN.

The following lemma shows that the seminorms (5.7.22) and (5.7.24) are thehomogeneous part of the corresponding norms.

5.7.7 Lemma Suppose s ∈ R and 1 ≤ p, r ≤ ∞. Then

‖σtu‖ pB

s/νp,r

= ts−|ω|/p‖u‖ pB

s/νp,r

, t > 0 ,

and t−s+|ω|/p ‖σtu‖′B

s/νp,r

→ 0 as t → 0. If s ≥ 0, then

‖σtu‖ pWs/ν

p= ts−|ω|/p ‖u‖ p

Ws/νp

, t > 0 ,

and t−s+|ω|/p ‖σtu‖′Ws/νp

→ 0 as t →∞.

Proof This is a consequence of Proposition VI.3.1.3 and Lemma 5.7.6. ¥


Now we can prove the main result of this subsection, the homogeneous versionof estimate (5.7.4) and its variants.

5.7.8 Theorem Let (5.7.1)–(5.7.3) be satisfied. Then

‖u‖ pB

s/νp

≤ c ‖u‖1−θpB

s0/νp0

‖u‖θpB

s1/νp1

(5.7.25)

for u ∈ Bs1/νp1 ∩B

s0/νp0 (K, E). Furthermore, any one of the following substitutions

is possible:‖·‖ p

Bs/νp

Ã ‖·‖ pWs/ν

pif s ≥ 0 ,

and, for j = 0, 1,

‖·‖ pB

sj/νpj

Ã ‖·‖ pWsj/ν

pj

if sj ≥ 0 and 1 < pj ≤ ∞ ,

provided θ < 1 unless s, s1 ∈ νN and 1 < p, p1 < ∞ or s, s1 /∈ νN.

Proof We replace u in (5.7.4) by σtu. Then it follows from (5.7.5) and Lem-ma 5.7.7 that

ts−|ω|/p ‖u‖ pB

s/νp

≤ ts−|ω|/p(‖u‖1−θp

Bs0/νp0

‖u‖θpB

s1/νp1

+ c(t, u))

,

where c(t, u) → 0 as t →∞. Now we divide this inequality by ts−|ω|/p and lett →∞. This proves (5.7.25). The other claims follow similarly. ¥

We illustrate this theorem by some prototypical examples, which, in par-ticular, are related to results available in the literature (see Subsection 5.8 forreferences).

Isotropic Multiplicative Inequalities

All examples below follow from Theorem 5.7.8.First we consider isotropic E-valued spaces. More precisely, we assume ω = 1.

Recall definition (5.7.20).

5.7.9 Examples (a)(Classical estimates) Assume j, m ∈ N with j < m and p0, p1

belong to (1,∞]. Set

1π

:=j

d+ θ

( 1p1

− m

d

)+

1− θ

p0(5.7.26)

for j/m ≤ θ ≤ 1.(α) Assume 1/π ≥ 0. Then

‖∇ju‖π ≤ c ‖u‖1−θp0

‖∇mu‖θp1

, u ∈ Wmp1

∩W0p0

(X, E) ,

provided θ < 1 if p1 = ∞.


(β) Suppose 1/π < 0. Set k := [−d/π] and σ := −k − d/π ∈ [0, 1). Then

[∇j+ku]σ,∞ ≤ c ‖u‖1−θp0

‖∇mu‖θp1

, u ∈ Wmp1

∩W0p0

(X, E) ,

for (j + k + σ)/m ≤ θ < 1.

Proof Set s0 := m and s1 := m.(1) Letting s := j, it is immediate that (α) holds.(2) If 1/π < 0, we set s := k + j + σ and p := ∞. Then −1/π = (k + σ)/d

and s/m ≤ θ < 1. This and (5.7.26) imply that (5.7.2) and (5.7.3) are satisfied. ¥

(b)(Negative order estimates) Let s0 ≤ 0, m ∈qN, and p0, p1 ∈ [1,∞] with 1 < p1.

Assume

1p

:=1

p(θ)− sθ

d≥ 0 , − s0

m− s0≤ θ < 1 . (5.7.27)

Then

‖u‖p ≤ c ‖u‖1−θpB

s0p0

‖∇mu‖θp1

, u ∈ Wmp1

∩Bs0p0

(X, E) . (5.7.28)

In particular, if 0 < θ < 1, then

‖u‖p1/θ ≤ c ‖u‖1−θpB−θm/(1−θ)∞

‖∇mu‖θp1

. (5.7.29)

Proof Let s := 0 and s1 := m. It follows from (5.7.27) and Remark 5.7.4(b) that(5.7.2) and (5.7.3) apply. This proves (5.7.28). Then (5.7.29) is obtained by settingp0 := ∞ and p = p(θ) = p1/θ, so that sθ = 0 and hence s0 = −θm/(1− θ). ¥

5.7.10 Remark In (5.7.28), ‖u‖ pB

s0p0

can be replaced by [∇−[s0]−u1]s0−[s0]−,p0 , where

u = u0 + ∇−[s0]−u1 is any representation of u ∈ Bs0p0

(X, E) with u0, u1 belonging

to Bs0−[s0]−p0 (X, E).

Proof Since, see Remark 2.6.4,

‖u‖Bs0p0

≤ ‖u0‖ + [∇−[s0]−u1]s0−[s0]−,p0 (5.7.30)

for any such representation, we can replace ‖u‖Bs0p0

in (5.7.4) by the right sideof (5.7.30). Now the assertion follows by the scaling argument of the proof ofTheorem 5.7.8. ¥

Sobolev Inequality

Next we show that, for isotropic E-valued spaces, the widely known Sobolev in-equality is also valid. For this we recall a well-known generalization of Holder’sinequality. Observe that we admit here Lp spaces with 0 < p < ∞.


5.7.11 Lemma (Generalized Holder inequality) Suppose X is a σ-compact metriz-able space and µ a positive Radon measure on it. Let 0 < pj ≤ ∞ for 1 ≤ j ≤ k

and put 1/p :=∑k

j=1 1/pj. Then

‖u1 · · ·uk‖p ≤k∏

j=1

‖uj‖pj , uj ∈ Lpj (X,µ) ,

Proof We can assume that p < ∞. Otherwise, the assertion is obvious. Putqj := pj/p so that 1 =

∑1/qj . If k = 2, Holder’s inequality implies

‖u1u2‖pp ≤

∥∥ |u1|p |u2|p∥∥

1≤

∥∥ |u1|p∥∥

q1

∥∥ |u2|p∥∥

q2= ‖u1‖p

p1‖u2‖p

p2.

Now the general case follows by induction. ¥

5.7.12 Theorem (Sobolev inequality) Suppose ω = 1. Let 1 ≤ p < m with m ∈ N.Then

‖u‖q ≤ c ‖∇mu‖p , u ∈ Wmp (X, E) , (5.7.31)

where 1/q := 1/p− m/d.

Proof (1) If p > 1, then the claim follows from Example 5.7.9(a) by settings0 = 0 = j, s1 = m, p1 = p, and θ = 1 in (5.7.26).

(2) Suppose p = 1. By extension and restriction we can assume that X = Rd.Since D is dense in Wm

1 , it suffices to prove (5.7.31) for u ∈ D.

Using obvious notation,

|u(x)|E ≤∫ xj

−∞|∂ju|E dxj ≤

∫|∂ju|E dxj , x ∈ Rd , 1 ≤ j ≤ d , (5.7.32)

where we integrate over R if nothing else is indicated. Hence

|u(x)|d/(d−1)E ≤

( d∏

j=1

∫|∂ju|E dxj

)1/(d−1)

.

The generalized Holder inequality implies

∫|u|d/(d−1)

E dx1 ≤∫ (∫

|∂1u|E dx1d∏

j=2

∫|∂ju|E dxj

)1/(d−1)

dx1

≤∫

|∂1u|E dx1( d∏

j=2

∫

R2|∂ju|E dxj dx1

)1/(d−1)

.


By integrating this inequality successively with respect to dx2, dx3, . . . , dxd over Rd

and applying each time the generalized Holder inequality, we arrive at

‖u‖d/(d−1) ≤( d∏

j=1

‖∂ju‖1

)1/d

.

Now the elementary inequality

(ξ1 q · · · q ξd)1/d ≤ (ξ1 + · · · + ξd)/d , ξj ∈ R+ ,

which is a consequence of the concavity of the logarithm, implies

‖u‖d/(d−1) ≤1d

d∑

j=1

‖∂ju‖1 =1d

∫

Rd

d∑

j=1

|∂ju|E dx .

We define p∗ by 1/p∗ := 1/p− 1/d. Then ‖u‖1∗ ≤ ‖∇u‖1/d. Consequently,

‖∇u‖1∗ ≤ ‖∇2u‖1/d .

Hence, by step (1), ‖u‖(1∗)∗ ≤ c ‖∇2u‖1, where 1/(1∗)∗ = 1/1∗ − 1/d = 1 − 2/d.Now (5.7.31) follows by induction. ¥

5.7.13 Remark If d = 1, then W 11 (X, E) → C0(X, E) and

‖u‖∞ ≤ ‖∂u‖1 , u ∈ W 11 (X, E) .

Proof This is a consequence of (5.7.32) and the density of D in W 11 and in C0. ¥

Parabolic Estimates

Next we give a simple application of Theorem 5.7.8 for the case of parabolic weightsystems.

5.7.14 Examples We suppose that [m, ν] is a parabolic weight system and use thenotations of Examples 3.6.5.

(a)(Classical Estimates) Let 1 < p0, p1 ≤ ∞. Set

1q

:= θ( 1

p1− ν

m + ν

)+

1− θ

p0=

−θν

m + ν+

1p(θ)

.

(α) Assume 1/q ≥ 0 and 0 < θ < 1. Then

‖u‖q ≤ c ‖u‖1−θp0

(‖∇νu‖p1 + ‖∂tu‖p1

)θ, u ∈ W(ν,1)

p1∩W(0,0)

p0(Y× J,E) .

(β) Suppose 1/q < 0 and 0 < θ ≤ 1. Set

k :=[−(m + ν)/q

]and σ := −k − (m + ν)/q .

Then 0 ≤ k < ν and 0 ≤ σ < 1. Assume (k + σ)/ν ≤ θ < 1.


If σ > 0, then

[[∇ku]]σ,∞;t + [[u]]σ/ν,∞;x ≤ c ‖u‖1−θp0

(‖∇νu‖p1 + ‖∂tu‖p1

)θ.

If σ = 0 and k ≥ 1, then

[[∇k−1u]]1,∞;t + ‖u‖∞ ≤ c ‖u‖1−θp0

(‖∇νu‖p1 + ‖∂tu‖p1

)θ.

Proof In this situation, |ω| = m + ν. Put s0 := 0, s1 := ν. In case (α), set s := 0.In case (β), let s := k + σ and p := ∞. Then (5.7.1)–(5.7.3) are satisfied. Hencethe assertion follows from Theorem 5.7.8. ¥

(b)(Negative order estimates) Suppose s0 ≤ 0, s1 := ν, and p0, p1 ∈ [1,∞] withp1 < ∞. Also assume −s0/ν ≤ θ < 1 and

1p

:=1

p(θ)− sθ

m + ν≥ 0 .

Then‖u‖p ≤ c ‖u‖1−θp

B(s0,s0/ν)p0,∞

(‖∇νu‖p1 + ‖∂tu‖p1

)θ

for u ∈ W(ν,1)p1 ∩B

(s0,s0/ν)p0,∞ (Y× J,E). In particular,

‖u‖p1/θ ≤ c ‖u‖1−θpB

−θν1−θ

(1,1/ν)∞

(‖∇νu‖p1 + ‖∂tu‖p1

)θ

for 0 < θ < 1.

Proof Theorem 5.7.8 with s := 0. ¥

5.8 Notes

In the scalar-valued isotropic case, the basic references for Triebel–Lizorkin spacesare, of course, Triebel’s books, [Tri83] in particular. There many more results canbe found, as well as detailed historical references.

Quite a few of the proofs of [Tri83] can be adapted to our anisotropic vector-valued setting. Notably, our demonstrations of Lemmas 5.1.4 and 5.1.5 are modi-fications and adaptions of similar results given in Chapter I of [Tri83]. The proofof Lemma 5.1.7 uses arguments of E.M. Stein [Ste93, II. §2.1].

The fact that Triebel–Lizorkin spaces (and Besov spaces too) posses theFatou property has first been observed in the scalar-valued isotropic case byJ. Franke [Fra86]. Our proof follows H.-J. Schmeißer and W. Sickel [SS01], whoconsidered the isotropic vector-valued case (also see S. Dachkowski [Dac03]).

In the classical isotropic case, there are various Fourier multiplier theoremsfor Triebel–Lizorkin spaces in [Tri83] (see, in particular, Theorem 2.3.7 therein).


Theorem 5.5.2 is due to H.-J. Schmeißer and W. Sickel, as are the proofsof the Sobolev embedding theorems 5.6.3 and 5.6.5 –– in the isotropic vector-valued case (see [SS01] and [SS05]). The demonstrations of the basic Lemmas 5.6.1and 5.6.2 follow H. Brezis and P. Mironescu [BrM01]. That paper contains alsoinequality (5.7.18) in the particular (isotropic scalar) case, where either s0 ≥ 0,1 < p0, p1 < ∞, or s0 = 0, p0 = ∞, and, in either case, p = p(θ). In a more re-cent paper [BrM18] these authors find necessary and sufficient conditions on theparameters for (5.7.18) to be valid.

Theorem 5.6.6 seems to be new. Related results have been obtained byR. Denk, J. Saal, and J.Seiler [DSS08], R. Denk and M Kaip [DK13], and M. Mey-ries and M. Veraar [MeV14].

Example 5.7.9(a) is precisely the E-valued version of the Gagliardo–Nirenberginequality as it appears in L. Nirenberg’s paper [Nir59], except for some limitingcases. Most notably is our restriction p0, p1 > 1. For this reason we have given ––using Nirenberg’s method –– a direct proof of the Sobolev inequality in the casep = 1.

There are several extensions of the classical Gagliardo–Nirenberg inequalityto fractional order spaces. Most of them consider the case s0 = 0 and p = p(θ) anduse homogeneous Bessel potential spaces as the right endpoint space. A generalstudy in the framework of homogeneous Triebel–Lizorkin and Bessel potentialspaces has been carried out by H. Triebel [Tri14] for the case s0 ≥ 0. Inequalitiesvalid in that setting require, of course, that p0 and p1 belong to (1,∞). Theseminorm in the homogeneous Bessel potential space

qHs

p(Rd) is u 7→ ‖(−∆)s/2u‖p.This essentially restricts the applicability of such inequalities to Rd. In contrast,the fractional order inequalities of Subsection 5.7 are much more flexible and allowextensions to manifolds with boundary, for example. This is already witnessed bythe fact that our results apply to distributions defined on corners. (Observe that,although we use homogeneous seminorms, the functions belong to inhomogeneousspaces.)

It should also be mentioned that there is a number of variants of Gagliardo–Nirenberg inequalities involving BMO, Lorentz, Morrey, or Sobolev spaces withweights. See N.A. Dao, J.I. Dıaz, and Q.-H. Nguyen [DDN18], D.S. McCormick,J.C. Robinson, and J.L. Rodrigo [McRR13], H. Kozono and H. Wadade [KoW08],H. Wadade [Wad10], and the references therein, for example.

Gagliardo–Nirenberg type inequalities involving homogeneous Besov spacesof negative order have been studied, among others, by M. Ledoux [Led03], V.I.Kolyada and F.J. Perez Lazaro [KoPL14], H. Bahouri and A. Cohen [BaC11] usingentirely different techniques like semigroup and rearrangement methods.

Multiplicative inequalities of the type studied here are of great importancefor the theory of nonlinear partial differential equations. For example, they allowto reduce the problem of global existence in evolution equations to find a prioribounds in relatively weak norms. See, for instance, [Ama85]. That paper seems to


contain the first (non-optimal) extension of Gagliardo–Nirenberg inequalities tofractional order spaces.

Another important line of applications rests on the fact that such inequalitiescan be used to estimate seminorms of intermediate strength by arbitrarily smallcontributions of higher order seminorms. This is due to the following well-knownsimple observation:

Suppose x, y, z > 0 and 0 < θ < 1. Then there exists a constant c0 ≥ 1 such that

x ≤ c0y1−θzθ (5.8.1)

iff there exists a constant c1 ≥ 1 such that

x ≤ εz + c1ε−θ/(1−θ)y , ε > 0 . (5.8.2)

Proof (1) Let (5.8.1) be satisfied. Write

c0y1−θzθ = (εz)θ

(c1/(1−θ)0 ε−θ/(1−θ)y

)1−θ

and apply Young’s inequality ξ1−θηθ ≤ (1− θ)ξ + θη < ξ + η for ξ, η > 0.(2) Assume (5.8.2) applies. Then (5.8.1) is obtained by minimizing the func-

tion f(t) := tz + c1t−θ/(1−θ)y, t > 0. ¥

It is well-known (see [Tri95]) that

F sp,2(Rd) .= Hs

p(Rd) , s ∈ R , 1 < p < ∞ . (5.8.3)

Hence, since F sp,p = Bs

p,

Bs2(Rd) .= Hs

2(Rd) , s ∈ R .

The proof of (5.8.3) relies on the Paley–Littlewood theorem which is known tohold in the vector-valued case iff E can be given an equivalent Hilbert space norm(see Remark 5 in [SS05]). Thus

Bs2(Rd, E) .= Hs

2(Rd, E) , s ∈ R , (5.8.4)

holds iff E is a Hilbert space.


6 Point-Wise Multiplications

For many applications it is of great importance to know optimal or nearly optimalconditions on the function u0 that guarantee that the product u0u1 with a func-tion u1 belongs to the same function space as u1 does. In this section we studythis question –– in a vector-valued anisotropic setting –– assuming that u1 belongsto one of the spaces introduced earlier. In particular, we show that Besov–Holderspaces of appropriate order are universal point-wise multiplier spaces.

In the better part of this section we study spaces of continuous functions,classical Sobolev spaces, and positive order Besov spaces. In doing so we take ad-vantage of their characterizations by the norms and seminorms introduced earlier,in Section 3 in particular.

Our approach rests on optimal embedding theorems and the optimal behaviorof the derivative operators in Besov spaces. The latter facts are collected in thepreparatory first subsection. In Subsection 6.2 we present –– besides the trivialresults for spaces of bounded continuous functions –– optimal point-wise multipliertheorems for Sobolev spaces. By a simple duality argument, negative order spacesare covered as well.

The most important theorems of this section are contained in Subsections 6.3and 6.5. In the first one, point-wise multiplier assertions for positive order Besovspaces are established. In Subsection 6.5 it is shown that Besov–Holder spacesprovide multipliers for Bessel potential spaces.

Subsection 6.4 is concerned with multiplications in negative order Besovspaces. The main result guarantees that positive order Besov–Holder spaces actas multiplier spaces for negative order Besov spaces. This is true without anyassumption on the target spaces.

In the last subsection we derive, by a simple method, point-wise multiplierresults for space-dependent multiplication operators.

Throughout this section it is assumed that

• K is a corner in Rd .

• X ∈ Rd,K .

• 1 ≤ q < ∞, 1 ≤ r ≤ ∞ .

Recall that E is an arbitrary Banach space and assumption (VI.3.1.20) applies.

6.1 Preliminaries

In this subsection we collect preparatory embedding theorems and introduce somenotation.

VII.6 Point-Wise Multiplications 223

Continuity of Derivatives

The following continuity theorems are of independent interest. They form the basisfor many of the proofs of this section.

6.1.1 Theorem Suppose s > 0 and α q ω ≤ s.(i) If 13

1q≥ 1

p>

(1q− s − α q ω

|ω|)

+,

then ∂α ∈ L(B

s/νq,r (X, E), Lp(X, E)

). The equality sign can be admitted if ei-

ther s − α q ω > |ω|/q or r = 1.(ii) Assume K is closed and s − t ≥ α q ω. If

1q≥ 1

p≥

(1q− s − t− α q ω

|ω|)

+,

then

∂α ∈ L(Bs/ν

q,r (X, E), Bt/νp,r (X, E)

) ∩ L(Bs/ν

q (X, E), Bt/νp (X, E)

).

Proof We omit (X, E). It follows from Theorems 2.2.2 and 2.8.3 that

Bs/νq,r → Bσ/ν

p,r if1q≥ 1

p≥

(1q− s − σ

|ω|)

+(6.1.1)

for 0 ≤ σ ≤ s.(1) Using Theorem 2.6.5,

Bσ/νp,r → Wα ppppω/ν

p∂α

−→ Lp

if either σ > α q ω or r = 1. This and (6.1.1) imply (i).(2) If K is closed, then

∂α ∈ L(Bσ/ν

p,r , Bσ−α ppppωp,r

)

by Theorem 2.8.6. From this and (6.1.1) we get the first part of (ii) by settingσ = t + α q ω. For the second part we have to invoke Corollary 5.6.4. ¥

6.1.2 Corollary Suppose m ∈ νN. Then

∂α ∈ L(Wm/ν

q (X, E), Lp(X, E))

, α q ω ≤ m ,

provided1q≥ 1

p≥

(1q− m− α q ω

|ω|)

+

and either13x+ := maxx, 0 and x− := minx, 0 for x ∈ R.


(i) the second equality is strict.or(ii) m− α q ω > |ω|/q.

or(iii) p, q ∈ (1,∞) and α q ω ∈ νN.

Proof Use either Wm/ν

q → Bm/νq,∞ and the theorem, or Theorem 5.6.5. ¥

Point-Wise Products

We denote by L(E0, E1; E2) the Banach space of all maps β : E0 × E1 → E2 whichare bilinear and bounded, endowed with the norm

β 7→ |β| := sup |β(e0, e1)|E2 ; |e0|E0 ≤ 1, |e1|E1 ≤ 1

.

With β ∈ L(E0, E1; E2) we associate the continuous linear map

Bβ : E0 7→ L(E1, E2) , e0 7→ β(e0, ·) .

Then |Bβ | ≤ |β|. Conversely, given B ∈ L(E0,L(E1, E2)

), we put

βB : E0 × E1 7→ E2 , (e0, e1) 7→ (Be0)e1 .

It follows βB ∈ L(E0, E1; E2) and |βB| ≤ |B|. Note that βBβ= β and BβB

= B.Hence

L(E0, E1; E2) → L(E0,L(E1, E2)

), β 7→ Bβ (6.1.2)

is an isometric isomorphism by which we often identify these two Banach spaces.If β ∈ L(E0, E1; E2), then its point-wise extension, mβ , over X is the bilinear

map

BC(X, E0) × L1,loc(X, E1) → L1,loc(X, E2) , (u0, u1) 7→ mβ(u0, u1)

defined bymβ(u0, u1)(x) := β

(u0(x), u1(x)

), a.a. x ∈ X .

We often write β(u0, u1) for mβ without fearing confusion.Throughout the rest of this section

• β ∈ L(E0, E1; E2) ,

unless explicitly stated otherwise. Furthermore, we frequently simply write

• mβ : F0(X, E0) × F1(X, E1) → F2(X, E2)

formβ ∈ L(

F0(X, E0), F1(X, E1); F2(X, E2))

,

where Fj(X, Ej) are suitable subspaces of S ′(X, Ej).


6.2 Multiplications in Classical Spaces

First we prove point-wise multiplier theorems in spaces of bounded and continuousfunctions and Sobolev spaces.

Spaces of Bounded Continuous Functions

The following elementary point-wise multiplier theorem is basically a consequenceof Leibniz’ rule.

6.2.1 Theorem Suppose m ∈ νN.

(i) If K is closed and F ∈ BC,BUC, then

mβ ∈ L(Fm/ν(X, E0), Fm/ν(X, E1); Fm/ν(X, E2)

).

(ii) mβ ∈ L(BCm/ν(X, E0),W

m/νq (X, E1); W

m/νq (X, E2)

).

Proof (1) Let α ∈ Nd and uj ∈ C |α|(X, Ej) for j = 0, 1. Then, by Leibniz’ rule,

∂α(β(u0, u1)

)=

∑

γ≤α

(αγ

)β(∂α−γu0, ∂

γu1) . (6.2.1)

Moreover,‖β(u0, u1)‖r ≤ |β| ‖u0‖∞ ‖u1‖r .

From this we get

|||β(u0, u1)|||m/ν,r ≤ c(m) |β| |||u0|||m/ν,∞ |||u1|||m/ν,r . (6.2.2)

This implies (i) if F = BC.

(2) We infer from

β(u0, u1)(x) − β(u0, u1)(y)

= β(u0(x) − u0(y), u1(x)

)+ β

(u0(y), u1(x) − u1(y)

)

that β(u0, u1) ∈ BUC(X, E2) if uj ∈ BUC(X, Ej). Now (1) and the definition ofBUCm/ν imply (i) for F = BUC.

(3) Suppose u0 ∈ BC∞(X, E0) and u1 ∈ S(X, E1). It follows from (VI.1.1.2),(1.1.9), and (6.2.1) that

∂α(β(u0, u1)

)|∂jK = 0 , α ∈ Nd , j ∈ J∗

K,

if X = K. From this and (6.2.2) we deduce β(u0, u1) ∈ S(X, E2) and that (6.2.2)applies. Since S(X, E1) is dense in W

m/νq (X, E1), estimate (6.2.2) holds equally


well for u0 in BC∞(X, E0) and u1 in Wm/ν

q (X, E1). Now we apply Theorem 1.6.4to arrive at assertion (ii). ¥

6.2.2 Remarks (a) BCm/ν(X, E0) is a multiplier space for Wm/ν

q (X, E1), inde-pendently of q. Thus it is universal.

(b) The assumption that u0 belongs to BCm/ν(K, E) means that u0 does not needto vanish on ∂jK for j ∈ J∗ if K is not closed.

(c) Let K be closed. In agreement with Theorem 1.4.1, we set

Wm/ν∞ (X, E) :=

u ∈ S ′(X, E) ; ∂αu ∈ L∞(X, E) for α q ω ≤ m

.

It is a Banach space with the norm |||·|||m/ν,∞ which contains BCm/ν(X, E) as aclosed linear subspace.

Clearly, Theorem 6.2.1(ii) remains valid if K is closed and BCm/ν(X, E0) isreplaced by W

m/ν∞ (X, E0).

(d) In Theorem 6.2.1 the map β 7→ mβ is linear and continuous. This is also truein all other point-wise multiplier theorems of this section. ¥

Sobolev Spaces

The following theorem shows, in particular, that we can replace BCm/ν(X, E0) byW

m/νq0 (X, E0) if m > |ω|/q0.

6.2.3 Theorem Let K be a corner in Rd and X ∈ Rd,K. Suppose m0,m ∈ νNwith m ≤ m0 and 1 ≤ q0 < ∞ are such that m0 > |ω|/q0. Then

mβ : Wm0/νq0

(X, E0)× Wm/νq (X, E1) → Wm/ν

q (X, E2) , 1 ≤ q < ∞ ,

where q = q0 > 1 if m = m0.

Proof Assume

α0 + α1 = α , α q ω ≤ m . (6.2.3)

Then, by Holder’s inequality,

‖β(∂α0u0, ∂α1u1)‖q ≤ |β| ‖∂α0u0‖p0 ‖∂α1u1)‖p1 , (6.2.4)

with p0, p1 ∈ [1,∞] satisfying

1p0

+1p1

=1q

. (6.2.5)


We set m1 := m and q1 := q. Suppose we can choose pi satisfying (6.2.5) suchthat

1qi

≥ 1pi

≥( 1

qi− mi − αi

q ω|ω|

)+

, (6.2.6)

where the second inequality is strict unless either mi − αiq ω > |ω|/qi or αi

q ωi

belongs to νN and qi > 1. Then it follows from Corollary 6.1.2 that

‖β(∂α0u0, ∂α1u1)‖p ≤ c |β| |||u0|||m0/ν,q0

|||u1|||m1/ν,q1. (6.2.7)

If α0 = 0 and α q ω = m1, then we can choose p0 = ∞ and p1 = q1.Let either α0 > 0 and α q ω = m1, or α q ω < m1. Then α0

q ω ≤ m1 andα1

q ω < m1. Hence, if either α0q ω < m1 or m0 > m1,

1q0

− m0 − α0q ω

|ω| <1q0

and 1/q1 − (m1 − α1q ω)/|ω| < 1/q1. Consequently, by (6.2.3),

( 1q0

− m0 − α0q ω

|ω|)

+( 1

q1− m1 − α1

q ω|ω|

)<

1q0

− m0

|ω| +1q1

<1q1

<1q0

+1q1

.

This shows that we can fix pi satisfying (6.2.6) with a strict inequality sign in thesecond place such that (6.2.5) holds true.

Lastly, if α0q ω = m1 = m0, then the choice p0 = q0 = q1 and p1 = ∞ is pos-

sible.These considerations show that (6.2.7) is true whenever (6.2.3) is satisfied,

provided q0 = q if m0 = m. Now the assertion is a consequence of (6.2.1). ¥

6.2.4 Corollary Suppose m ∈ νN satisfies m > |ω|/q with 1 < q < ∞. If E is acontinuous multiplication algebra, then W

m/νq (X, E) is one as well.

A slight modification of the preceding proof leads to the following point-wisemultiplier theorem for Sobolev spaces of low regularity.

6.2.5 Theorem Let K be a corner in Rd and X ∈ Rd,K. Suppose m0, m1, and m2,belonging to νN, and q0, q1, q2 ∈ [1,∞) satisfy

mi ≤ |ω|/qi for i = 0, 1 , m2 ≤ m1 ≤ m0 , 1/q2 ≤ 1/q0 + 1/q1 ,

and1q2

− m2

|ω| >1q0

− m0

|ω| +1q1

− m1

|ω| .

Thenmβ : Wm0/ν

q0(X, E0)× Wm1/ν

q1(X, E1) → Wm2/ν

q2(X, E2) .


Proof Let α0 + α1 = α with α q ω ≤ m2. Then

( 1q0

− m0 − α0q ω

|ω|)

+( 1

q1− m1 − α1

q ω|ω|

)

=1q0

− m0

|ω| +1q1

− m1

|ω| +α q ω|ω| ≤ 1

q0− m0

|ω| +1q1

− m1

|ω| +m2

|ω| <1q2

.

This shows that we can choose pi such that (6.2.6) holds with strict inequalitiesand 1/p0 + 1/p1 = 1/q2. Hence (6.2.7) applies with p = q2. Now the claim followsonce more from (6.2.1). ¥

Spaces of Negative Order

Let E1 and E2 be reflexive, p1, p2 ∈ (1,∞), and denote by |||·|||−m/ν,pithe norm

of W−m/ν

pi (X, Ei). Assume

|||β(u0, u1)|||−m2/ν,p2≤ c |β| |||u0|||m0/ν,q0

|||u1|||−m1/ν,p1

for u0 ∈ Wm0/ν

q0 (X, E0) if 1 ≤ q0 < ∞, and u0 ∈ BCm0/ν(X, E0) for q0 = ∞, andfor u1 ∈ S(X, E1). Then, by the density of S(X, E1) in W

−m1/νq1 (X, E1), there exists

a unique mβ such that

mβ : Wm0/νq0

(X, E0) × W−m1/ν1p1

(X, E1) → W−m2/νp2

(X, E2) (6.2.8)

if 1 ≤ q0 < ∞, and

mβ : BCm0/ν(X, E0)× W−m1/ν1p1

(X, E1) → W−m2/νp2

(X, E2) (6.2.9)

if q0 = ∞, such that mβ(·, u1) = mβ(·, u1) for u1 ∈ S(X, E1). Thus it is feasible towrite again mβ for mβ .

The following theorems extend point-wise multiplication to Sobolev spacesof negative order.

6.2.6 Theorem Let K be a corner in Rd and X ∈ Rd,K. Suppose E1 and E2 arereflexive, p0, p ∈ (1,∞), and m0,m ∈ νN. Then

(i) mβ : BCm/ν(X, E0) × W−m/ν

p (X, E1) → W−m/ν

p (X, E2).

(ii) If m0 > |ω|/p0 and 0 ≤ m < m0, then

mβ : Wm0/νp0

(X, E0)× W−m/νp (X∗, E1) → W−m/ν

p (X∗, E2) .


Proof (1) We define a bilinear map

β′ : E0 × E′2 → E′

1 , (e0, e′2) 7→

⟨e′2, β(e0, ·)

⟩E2

.

Then

|β′(e0, e′2)|E′

1= sup

|e1|≤1

∣∣⟨β′(e0, e′2), e1

⟩E1

∣∣ = sup|e1|≤1

∣∣⟨e′2, β(e0, e1)⟩

E2

∣∣

≤ |β| |e0|E0 |e′2|E′2

.

This shows

β′ ∈ L(E0, E′2; E

′1) , |β′| ≤ |β| . (6.2.10)

(2) It follows from Theorem 6.2.1(ii) and cl(K∗) = K that

mβ′ ∈ L(BCm/ν(X, E0),W

m/νp′ (X∗, E′

2); Wm/ν

p′ (X∗, E′1)

)(6.2.11)

and ‖mβ′‖ ≤ c |β|. Assume u1 ∈ S(X, E1). Then the definition of β′ implies⟨mβ′(u0, u

′2), u1

⟩=

⟨u′2, mβ(u0, u1)

⟩

for u0 ∈ BCm/ν(X, E0) and u′2 ∈ Wm/ν

p′ (X∗, E′2). Hence we infer from (6.2.11),

Theorem 1.5.1, and definition (1.2.5)∣∣⟨u′2, mβ(u0, u1)

⟩∣∣ ≤∥∥mβ′(u0, u

′2)

∥∥W

m/ν

p′ (X∗,E′1)‖u1‖(W

m/ν

p′ (X∗,E′1))

′

≤ c |β| |||u0|||m/ν,∞ ‖u′2‖Wm/ν

p′ (X∗,E′2)|||u1|||−m/ν,p .

Consequently, using Theorem 1.5.1 once more, we see that (6.2.9) is satisfied withm1 = m2 = m and p1 = p2 = p. This proves the first claim.

(3) By Theorem 6.2.3,

mβ′ : Wm0/νp0

(X, E0) × Wm/ν

p′2(X, E′

2) → Wm/ν

p′1(X, E′

1)

and ‖mβ′‖ ≤ c |β|. Now an obvious modification of step (2) proves statement (ii). ¥

Clearly, ‘dualizing’ Theorem 6.2.5, we can obtain a multiplication theoremfor Sobolev spaces of negative order in which the multipliers possess low regularity.

6.3 Multiplications in Besov Spaces of Positive Order

First we prove a general multiplier theorem for positive order Besov spaces. It hasa number of important consequences which we derive afterwards.


6.3.1 Theorem Let K be a closed corner in Rd and X ∈ Rd,K. Suppose

0 < s2 ≤ s1 ≤ s0 , 1 ≤ p0, p1, p2 ≤ ∞ with 1/p2 ≤ 1/p0 + 1/p1 (6.3.1)

satisfy

sk 6= |ω|/pk , k = 0, 1 , (6.3.2)

and

1p2

− s2

|ω| ≥

( 1p0

− s0

|ω|)

++

( 1p1

− s1

|ω|)

+, if max

k=0,1

1pk

− sk

|ω|

> 0 ,

maxk=0,1

1pk

− sk

|ω|

otherwise ,

(6.3.3)

where

s2 < s1 if mink=0,1

1pk

− sk

|ω|

> 0 , (6.3.4)

and (6.3.3) is a strict inequality if at least one of the s0, s1, and s2 is an integer.Assume, moreover, that either

1 ≤ r0 ≤ r1 ≤ r2 ≤ ∞ , (6.3.5)

or (6.3.3) is a strict inequality and

1 ≤ r0 ≤ ∞ , 1 ≤ r1 ≤ r2 ≤ ∞ . (6.3.6)

Then

mβ : Bs0/νp0,r0

(X, E0) × Bs1/νp1,r1

(X, E1) → Bs2/νp2,r2

(X, E2) . (6.3.7)

Proof It suffices to prove the theorem for r0 = r1 = r2 = r ∈ [1,∞]. Then, ap-plying this result with r = r1 and using (2.2.1), we obtain (6.3.7) under the as-sumption (6.3.5).

(1) First we suppose

s0, s1, s2 /∈ N . (6.3.8)

Theorem 6.1.1(i) implies

‖∂α0u0‖π0 ≤ c ‖u0‖Bs0/νp0,r

if α0q ω < s0 and

1p0

≥ 1π0

>( 1

p0− s0 − α0

q ω|ω|

)+

, (6.3.9)


where the equality sign is permitted if |ω|/p0 < s0 − α0q ω. Similarly,

‖∂α1u1‖π1 ≤ c ‖u1‖Bs1/νp1,r

if α1q ω < s1 and

1p1

≥ 1π1

>( 1

p1− s1 − α1

q ω|ω|

)+

, (6.3.10)

the equality being admissible if |ω|/p1 < s1 − α1q ω. Thus, if these condition are

satisfied and α0 + α1 = α with α q ω < s2, then

‖β(∂α0u0, ∂α1u1)‖p2 ≤ c |β| ‖u0‖B

s0/νp0,r

‖u1‖Bs1/νp1,r

, (6.3.11)

provided

1/p2 = 1/π0 + 1/π1 . (6.3.12)

We observe that, for α q ω < s2,

1p0

− s0 − α0q ω

|ω| +1p1

− s1 − α1q ω

|ω| =1p0

− s0

|ω| +1p1

− s1

|ω| +α q ω|ω|

<1p0

− s0

|ω| +1p1

− s1

|ω| +s2

|ω| .

(6.3.13)

If the first alternative of (6.3.3) applies, then the last expression is majorized by

( 1p0

− s0

|ω|)

++

( 1p1

− s1

|ω|)

++

s2

|ω| ≤1p2

.

This shows that, in this case, we can choose π0 and π1 such that (6.3.9), (6.3.10),and (6.3.12) are satisfied.

If the second alternative of (6.3.3) is active, then the last expression of (6.3.13)is bounded from above by

maxk=0,1

1pk

− sk

|ω|

+s2

|ω| ≤1p2

.

Hence in this case we can also find π0 and π1 satisfying (6.3.9), (6.3.10), and(6.3.12).

Define mi ∈ νiN by mi < s2 < mi + νi for 1 ≤ i ≤ `. Then the above obser-vations and Leibniz’ rule (6.2.1) imply

∑

i=1

mi/νi∑

j=0

∥∥∇jxi

(β(u0, u1)

)∥∥p2

≤ c |β| ‖u0‖Bs0/νp0,r

‖u1‖Bs1/νp1,r

. (6.3.14)


(2) Note that

4yβ(u0, u1) = β(4yu0, u1(· + y)

)+ β(u0,4yu1)

for y ∈ Xd. Thus ‖4yβ(u0, u1)‖π2 ≤ |β| (‖4yu0‖π0 ‖u1‖π1 + ‖u0‖π0 ‖u1‖π1

), if

1π2

=1π0

+1π1

=1π0

+1π1

.

Consequently, given θ ∈ (0, 1),[β(u0, u1)

]θ,π2,r

≤ |β| ([u0]θ,π0,r ‖u1‖π1 + ‖u0‖π0 [u1]θ,π1,r

). (6.3.15)

(3) Fix i ∈ 1, . . . , ` and suppose α0 + α1 = α ∈ Ndi and α q ω = |α| νi = mi.By Theorem 6.1.1(ii),

[[∂α0xi

u0]](s2−mi)/νi,π0,r;i≤ c ‖u0‖B

s0/νp0,r

(recall Theorem 3.5.2), provided

1p0

≥ 1π0

≥( 1

p0− s0 − s2 + mi − |α0| νi

|ω|)

+. (6.3.16)

Similarly,[[∂α1

xiu1]](s2−mi)/νi,π1,r;i

≤ c ‖u1‖Bs1/νp1,r

,

if

1p1

≥ 1π1

≥( 1

p1− s1 − s2 + mi − |α1| νi

|ω|)

+. (6.3.17)

Hence, if we can find π0 and π1 satisfying (6.3.12), (6.3.16), and (6.3.10), as well asπ0 and π1 with 1/p2 = 1/π0 + 1/π1 such that (6.3.9) and (6.3.17) are valid, thenit follows from (6.3.15) that

[[β(∂α0

xiu0, ∂

α1xi

u1)]]

(s2−mi)/νi,p2,r;i≤ c |β| ‖u0‖B

s0/νp0,r

‖u1‖Bs1/νp1,r

. (6.3.18)

We take the sum of the two expressions in ( · · · )+ in (6.3.16) and (6.3.10) toget

1p0

− s0

|ω| +s2

|ω| −mi − |α0| νi

|ω| +1p1

− s1

|ω| +|α1| νi

|ω|=

1p0

− s0

|ω| +1p1

− s1

|ω| +s2

|ω| =: a .

Analogously, by adding the corresponding terms of (6.3.9) and (6.3.17), we arriveat

1p0

− s0 − |α0| νi

|ω| +1p1

− s1

|ω| +s2

|ω| −mi − |α1| νi

|ω| = a .


If the first alternative of (6.3.3) is in force, then

a ≤( 1

p0− s0

|ω|)

++

( 1p1

− s1

|ω|)

++

s2

|ω| ≤1p2

.

Otherwise,

a ≤ maxk=0,1

1pk

− sk

|ω|

+s2

|ω| ≤1p2

.

In each case we can thus choose π0, π1, and π0, π1, as desired. From this, (6.3.18)and (6.2.1) we obtain

[[∇mi/νixi

(β(u0, u1)

)]](s2−mi)/νi,p2,r;i

≤ c |β| ‖u0‖Bs0/νp0,r

‖u1‖Bs1/νp1,r

.

This is true for each i ∈ 1, . . . , `. Hence we see from it, (6.3.14), and Theo-rem 3.5.2 that the assertion is valid in this case.

(4) We now drop assumption (6.3.8). Since (6.3.3) is a strict inequality, wecan choose σj < sj for j = 0, 1, and σ2 > s2 such that (6.3.2), (6.3.3), and (6.3.4)hold with s0, s1, s2 replaced by σ0, σ1, σ2. Then, by what we have alreadyproved,

mβ : Bσ0/νp0,r0

(X, E0) × Bσ1/νp1,r1

(X, E1) → Bσ2/νp2,r2

(X, E2) .

Since Bsj/νpj ,rj → B

σj/νpj ,rj for j = 0, 1 and B

σ0/νp0,r0 → B

s0/νp0,r0 , the assertion follows.

(5) Lastly, suppose (6.3.3) is a strict inequality and (6.3.6) applies. Then wefix σ0 < s0 such that we can apply the foregoing results to get

mβ : Bσ0/νp0,1 (X, E0) × Bs1/ν

p1,r1(X, E1) → Bs2/ν

p2,r2(X, E2) .

Now we get the claim from Bs0/νp0,r0 → B

σ0/νp0,1 . The theorem is proved. ¥

6.3.2 Remarks Let (6.3.1) be satisfied.

(a) Assume s0 > |ω|/p0 and

s0 − |ω|/p0 ≥ s1 − |ω|/p1 , s1 6= |ω|/p1 . (6.3.19)

Then conditions (6.3.2) and (6.3.3) reduce to

s2 − |ω|/p2 ≤ s1 − |ω|/p1 . (6.3.20)

(b) Suppose s0 < |ω|/p0 and (6.3.19) is satisfied. Then (6.3.2) and (6.3.3) boildown to

s2 − |ω|/p2 ≤ s0 − |ω|/p0 + (s1 − |ω|/p1)− .

Proof This is obvious in both cases. ¥


These general results have major implications some of which we collect in thenext theorem.

6.3.3 Theorem Let K be a closed corner in Rd, X ∈ Rd,K, and 1 ≤ p0, p ≤ ∞.(i) If s0 > |ω|/p0 and 0 < s ≤ s0 are such that

s0 − |ω|/p0 ≥ s − |ω|/p , (6.3.21)

then

mβ : Bs0/νp0,r0

(X, E0)× Bs/νp,r (X, E1) → Bs/ν

p,r (X, E2) , 0 < s ≤ s0 ,

where r0, r ∈ [1,∞] satisfy r0 ≤ r, unless (6.3.21) is a strict inequality, ands0 6= |ω|/p if s = s0.

(ii) For 0 < s < s0,

mβ : Bs0/ν∞ (X, E0) × Bs/ν

p,r1(X, E1) → Bs/ν

p,r2(X, E2) , 1 ≤ r1 ≤ r2 ≤ ∞ .

If r1 = r2 = ∞, then this applies to s = s0 also.

(iii) mβ : bs0/ν∞ (X, E0)× b

s/ν∞ (X, E1) → b

s/ν∞ (X, E2), 0 < s ≤ s0.

(iv) Suppose s0 < |ω|/p and 0 < s1 ≤ s0. Then

mβ : Bs0/νp,r0

(X, E0)× Bs1/νp,r1

(X, E1) → Bs0+s1−|ω|/pp,r2

(X, E2)

with 1 ≤ r0 ≤ r1 ≤ r2 ≤ ∞.

Proof (1) Statement (i) follows from Remark 6.3.2(a) and Theorem 6.3.1, pro-vided s 6= |ω|/p.

Assume s = |ω|/p. Then s < s0. Thus we can choose 0 < t0 < s < t1 < s0

such that tj 6= |ω|/p. Then, given u0 ∈ Bs0/νp,r0 ,

β(u0, ·) ∈ L(Btj/ν

p,r , Btj/νp,r

), ‖β(u0, ·)‖ ≤ c |β| ‖u0‖B

s0/νp,r0

. (6.3.22)

Set θ := (s − t0)/(t1 − t0). Then we obtain from (6.3.22) and Theorems 2.7.1(i)and 2.8.3, by interpolating with the functor (·, ·)θ,r,

β(u0, ·) ∈ L(Bs/ν

p,r , Bs/νp,r

), ‖β(u0, ·)‖ ≤ c |β| ‖u0‖B

s0/νp,r0

.

This implies (i).(2) Claim (ii) is a special case of (i).(3) Claim (iii) is an easy consequence of (ii) (with r1 = r2 = ∞), defini-

tion (2.2.7), and Theorem 2.8.3.(4) The last assertion is clear from Remark 6.3.2(b) and Theorem 6.3.1. ¥


6.3.4 Corollary Suppose 1 ≤ p < ∞.

(i) Let s > |ω|/p0 > 0. Then

mβ : W s/νp0

(X, E0) × W t/νp (X, E1) → W t/ν

p (X, E2)

if either s > t and s − |ω|/p0 > t − |ω|/p or s = t and p0 = p.

(ii) mβ : BUCs/ν(X, E0) × Wt/ν

p (X, E1) → Wt/ν

p (X, E2), 0 ≤ t < s.

(iii) mβ : BUCs/ν(X, E0) × BUCt/ν(X, E1) → BUCt/ν(X, E2)andmβ : bucs/ν(X, E0) × buct/ν(X, E1) → buct/ν(X, E2)for 0 ≤ t ≤ s.

Proof Recall that Wr/ν

p = Br/νp and BUCr/ν = B

r/ν∞ for r ∈ R+\νN.

Suppose s, t /∈ νN. Then the claims follow from the theorem. If s, t ∈ νN, thenthey are contained in Theorems 6.2.1 and 6.2.3.

If s /∈ νN and t ∈ νN, or s ∈ νN and t /∈ νN, then we use –– in an obviousmanner –– the embeddings Fs/ν → Ft/ν for s > t and F ∈ Wq, BUC, buc. ¥

6.4 Multiplications in Besov Spaces of Negative Order

Now we consider multiplications in Besov spaces with one factor being a singulardistribution.

The Reflexive Case

First we restrict ourselves to the reflexive case and employ the duality argumentintroduced in Subsection 6.2.

6.4.1 Theorem Let K be a closed corner in Rd, X ∈ Rd,K, and E1, E2 reflexive.Assume

−s0 ≤ s2 ≤ s1 < 0 and p0 ∈ [1,∞], p1, p2 ∈ (1,∞) with 1/p2 ≤ 1/p0 + 1/p1

satisfy

s0 6= |ω|/p0 , s2 6= −|ω| (1 − 1/p2) . (6.4.1)

Also assume r0 ∈ [1,∞] and r1, r2 ∈ (1,∞) are such that

r0 ≤ r′2 ≤ r′1 . (6.4.2)

Thenmβ : Bs0/ν

p0,r0(X, E0) × Bs1/ν

p1,r1(X∗, E1) → Bs2/ν

p2,r2(X∗, E2) ,


provided

s0 + s1 ≥ |ω|( 1

p0+

1p2

− 1)

(6.4.3)

and one of the following conditions is satisfied:

(i) s0 > |ω|/p0 and s2 − |ω|/p2 ≤ s1 − |ω|/p1.

(ii) s0 < |ω|/p0 and

s0 + s1 − |ω|( 1

p0+

1p1

− 1)≥

(s2 + |ω|

(1− 1

p2

))+

.

where s2 < s1 if s2 + |ω| (1 − 1/p2) > 0.

If the second inequality in (i), resp. (ii), is strict, then the restriction r0 ≤ r′2 canbe omitted.

Proof We set σ0 := s0, σ1 := −s2, σ2 := −s1 and π0 := p0, π1 := p′2, π2 := p′1,as well as ρ0 := r0, ρ1 := r′2, ρ2 := r′1. Then σj and πj satisfy conditions (6.3.1)and (6.3.2). It is also verified that assumptions (i) and (ii) guarantee that (6.3.3)holds (with sj , pj , rj replaced by σj , πj , ρj). Thus, since ρ0 ≤ ρ1 ≤ ρ2, it followsfrom Theorem 6.3.1 that

mβ′ : Bs0/νp0,r0

(X, E0)× B−s2/νp′2,r′2

(X, E′2) → B

−s1/νp′1,r′1

(X, E′1) .

From this we obtain the assertion by Theorem 2.8.4 and the duality argumentemployed in the proof of Theorem 6.2.6. The last assertion is a consequence ofRemarks 6.3.2(a) and (b). ¥

6.4.2 Corollary Let p0 ∈ [1,∞) and p ∈ (1,∞). Assume 0 < s ≤ s0 with s < s0 ifs0 ∈ νN.

(i) If s0 > |ω|/p0, then

mβ : W s0/νp0

(X, E0) × B−s/νp (X∗, E1) → B−s/ν

p (X∗, E2) ,

provided

s0 − s ≥ |ω|( 1

p0+

1p− 1

)≥ 0 . (6.4.4)

(ii) mβ : BUCs0(X, E0)× B−s/νp (X∗, E1) → B

−s/νp (X∗, E2) if 0 < s < s0.

(iii) Suppose 0 < s0 < |ω|/p0 and (6.4.4) is satisfied. Then

mβ : W s0/νp0

(X, E0) × B−s/νp (X∗, E1) → B(s0−s−|ω|/p0)/ν

p (X∗, E2) .


Proof (1) If s0 /∈ νN and s 6= |ω|/p′, then the assertion is immediate by the theo-rem, due to the fact that the second inequality in (6.4.4) implies r0 = p0 ≤ p′ = r′2.The case where s = |ω|/p′ is then handled by interpolation. If s0 ∈ νN, we fix σwith s < σ < s0 and σ /∈ νN. Then we obtain the assertion from what has justbeen shown and W

s0/νp0 → B

σ/νp0 .

(2) Claims (ii) and (iii) follow analogously. ¥

The Non-Reflexive Case

It is the purpose of the following considerations to prove that BUCs0/ν is a point-wise multiplier space for negative order Besov spaces even if E1 and E2 are notreflexive. For this we need some preparation. We use the notations and conventionsof Section VI.3.

First we study the case X = Rd. In this situation we employ the followingstipulations: if F(Rd, E) is a linear subspace of D′(Rd, E), then we denote it sim-ply by F(E), and F := F(C). We fix a ν-quasinorm Q on Rd, set Ω := [Q < 1].Let ψ be Ω-adapted. Then

((Ωk)(ψk)

)is the ν-dyadic partition of unity in-

duced by (Ω, ψ). Thus, see Example VI.3.6.1, ψ0 := ψ, ψj := σ2−j ψ, whereψ := ψ − σ2ψ, Ω0 = [Q < 2], and Ωk = [2k−1 < Q < 2k+1] for k ≥ 1.

Since F−1ψk ∈ S, it follows (see (III.4.2.14) or Proposition 1.2.7 of the Ap-pendix) that

ψk(D)u = F−1ψk ∗ u ∈ OM (E) , u ∈ S ′(E) . (6.4.5)

Hence

Snu :=n∑

k=0

ψk(D)u ∈ OM (E) , n ∈ N , u ∈ S ′(E) . (6.4.6)

We get from (6.4.6) and Theorem 1.6.4 of the Appendix that

β(Snu0, Snu1) ∈ OM (E2) , uj ∈ S ′(Ej) , j = 0, 1 . (6.4.7)

Suppose s > −t > 0. It is the purpose of the following considerations to show that

π(u0, u1) := limn→∞

β(Snu0, Snu1)

exists in Bt/νq,r (E2) for (u0, u1) ∈ B

s/ν∞ (E0)× B

t/νq,r (E1), that π is a multiplication,

and π(u0, u1) = β(u0, u1) if u1 ∈ Bs0/νq,r (E1) → B

t/νq,r (E1) for some s0 > 0. In this

sense, π is the point-wise product in Bt/νq,r (E2) of (u0, u1) ∈ B

s/ν∞ (E0) × B

t/νq,r (E1).


For abbreviation, Sj := ψj(D). Then we set, for n ≥ 3,

πn1 (u0, u1) :=

n∑

j=3

β(Sju0, Sj−3u1) ,

πn2 (u0, u1) :=

n∑

j=0

2∑

i=−2

β(Sju0, Sj+iu1) − β(Sn−1u0, Sn+1u1)

− β(Snu0, (Sn+1 + Sn+2)u1

),

πn3 (u0, u1) :=

n∑

j=3

β(Sj−3u0, Sju1) ,

where Sj := 0 for j < 0. Then

πn :=n∑

j=0

β(Sju0, Sju1) = πn1 + πn

2 + πn3 . (6.4.8)

The importance of this decomposition stems from the following support properties.

6.4.3 Lemma Let (u0, u1) ∈ S ′(E0) × S ′(E1).(i) If j ≥ 3, then Fβ(Sju0, S

j−3u1) and Fβ(Sj−3u0, Sju1) are supported in[2j−2 < Q < 2j+2].

(ii) If j ∈ N, then supp(Fβ

(Sju0,

∑j+2k=j−2 Sku1

)) ⊂ [Q < 2j+4].

Proof (1) Using the notations of the Appendix, we set v0q v1 := β(v0, v1) for

(v0, v1) ∈ S ′(E0)×OM (E1). By replacing β by |β|−1 β, we can assume that q isa multiplication in the sense of the Appendix, that is, it has norm at most one.Hence the convolution theorem for vector-valued distributions (Theorem 1.9.10 ofthe Appendix) guarantees

(v0 ∗ q v1) = v0q v1 , (v0, v1) ∈ S ′(E0) × S(E1) , (6.4.9)

where ∗ q denotes the convolution induced by β.Theorem 1.6.4 of the Appendix and the continuity of

on S and S ′ imply

β(v0, v1)

= β(v

0, v

1) , (v0, v1) ∈ S ′(E0) × S(E1) .

Recall F−1 = (2π)−dF

(see (III.4.2.2)). From this and (6.4.9) we deduce

F−1(v0 ∗ q v1) = (2π)dF−1v0qF−1v1

and thus, setting wj := F−1vj ,

Fβ(w0, w1) = (2π)−dw0 ∗ q w1 , (w0, w1) ∈ S ′(E0)× S(E1) .


Now we get from Remark 1.9.6(f) of the Appendix that

supp(Fβ(w0, w1)

) ⊂ supp(w0) + supp(w1) , (6.4.10)

whenever w0 and w1 are compactly supported.(2) Suppose w0 ∈ OM (E0) and w0 is compactly supported. Let (w1,j) be a

sequence in S(E1) converging in S ′(E1) towards w1 such that supp(w1,j) ⊂ K forsome K b Rd. Then, by (6.4.10),

supp(Fβ(w0, w1,j)

) ⊂ supp(w0) + K , j ∈ N . (6.4.11)

It follows from Theorem 1.6.4 of the Appendix that β(w0, ·) is a continuous linearmap from S ′(E1) into S ′(E2). Thus Fβ(w0, ·) has this property also. From thisand (6.4.11) we deduce that

supp(Fβ(w0, w1)

) ⊂ supp(w0) + K . (6.4.12)

Assume u1 ∈ S ′(E1). Since Sk = ψk(D) ∈ L(S ′(E1))

by Lemma VI.3.6.3(ii),and S(E1) is dense in S ′(E1), we can find a sequence (u1,j) in S(E1) such thatw1,j := Sku1,j → Sku1 in S ′(E1). Note that

supp(w1,j) = supp(ψku1,j) ⊂ Ωk , j ∈ N .

Hence it follows from (6.4.12) that

supp(Fβ(w0, Sku1)

) ⊂ supp(w0) + Ωk , k ∈ N , u1 ∈ S ′(E1) , (6.4.13)

provided w0 belongs to OM (E0) and w0 is is compactly supported. Similarly, givenw1 ∈ OM (E1) with w1 being compactly supported,

supp(Fβ(Sju0, w1)

) ⊂ Ωj + supp(w1) (6.4.14)

for j ∈ N and u0 ∈ S ′(E0).

(3) Let m ∈ N. Then Smu =∑m

k=0 ψku for u ∈ S ′(E). Hence (see (VI.3.6.1)and (VI.3.6.3))

supp(Smu

) ⊂ 2m+1 q Ω = [Q < 2m+1] .

Consequently, if k ≥ 3, we obtain from (6.4.13)

supp(Fβ(Sk−3u0, Sku1)

) ⊂ [Q < 2k−2] + [2k−1 < Q < 2k+1] .

Thus |Q(ξ) − Q(η)| ≤ Q(ξ + η) ≤ Q(ξ) + Q(η) implies

supp(Fβ(Sk−3u0, Sku1)

) ⊂ [2k−1 − 2k−2 < Q < 2k+1 + 2k−2]

⊂ [2k−2 < Q < 2k+2]

for (u0, u1) ∈ S ′(E0)× S ′(E1), due to (6.4.7). This proves one half of (i). Thesecond half follows analogously.


(4) Since∑j+2

k=j−2 ψk equals σ2−j−2ψ − σ2−j+1ψ if j ≥ 3, respectively σ2−j−2ψ

if 0 ≤ j ≤ 2, we find that its support is contained in [2j−3 < Q < 2j+3] if j ≥ 3,and in [Q < 2j+3] otherwise. We deduce from

[2j−1 < Q < 2j+1] + [2j−3 < Q < 2j+3] ⊂ [Q < 2j+4]

that, in either case,

supp(Fβ(Sju0,

j+2∑

k=j−2

Sku1))⊂ [Q < 2j+4]

for (u0, u1) ∈ S ′(E0)× S ′(E1). This proves (ii). ¥

Henceforth, setting πn0 := πn, we define

πi(u0, u1) := limn→∞

πni (u0, u1) , i = 0, . . . , 3 ,

for those (u0, u1) ∈ S ′(E0) × S ′(E1) for which this limit exists in S ′(E2).

6.4.4 Lemma Assume s0 > −s1 > 0. Then π is a continuous bilinear map fromB

s0/ν∞ (E0) × B

s1/νq,r (E1) into B

s1/νq,r (E2). Its norm is bounded by c |β|.

Proof (1) It suffices to prove the assertion for r = 1. Indeed, then we applythis fact with si replaced by σi, where s0 > −σ0 > −s1 > −σ1 > 0. Thus, givenu0 ∈ B

s0/ν∞ (E0),

π(u0, ·) ∈ L(B

σi/νq,1 (E1), B

σi/νq,1 (E2)

), ‖π(u0, ·)‖ ≤ c |β| ‖u0‖B

s0/ν∞

for i = 0, 1. Hence interpolation with the real interpolation functor (·, ·)θ,r, whereθ = (s1 − σ0)/(σ1 − σ0), gives

π(u0, ·) ∈ L(Bs1/ν

q,r (E1), Bs1/νq,r (E2)

), ‖π(u0, ·)‖ ≤ c |β| ‖u0‖B

s0/ν∞

,

due to Theorem 2.7.1(i). This being true for all u0 ∈ Bs0/ν∞ (E0), the assertion

follows.(2) It suffices to prove the claim with π replaced by πi, 1 ≤ i ≤ 3. Clearly,

this is a consequence of (6.4.8).(3) We infer from supp(ψk) ⊂ Ωk and Lemma 6.4.3(i) that

ψk(D)β(Sju0, Sj−3u1) = 0 , |j − k| ≥ 3 .

Hence

ψk(D)πn1 (u0, u1) =

k+2∑

j=k−2

ψk(D)β(ψj(D)u0, S

j−3u1

), k ≤ n + 2 ,

and ψk(D)πn1 (u0, u1) = 0 for k > n + 2.


Let 3 ≤ m < n. Then we find

ψk(D)(πn1 − πm

1 )(u0, u1) =k+2∑

j=k−2

ψk(D)β(ψj(D)u0, S

j−3u1

)

if m− 1 ≤ k ≤ n + 2, and ψk(D)(πn1 − πm

1 )(u0, u1) = 0 otherwise. Thus we inferfrom (2.1.1), since s1 < 0 and s0 + s1 > 0, that

∞∑

k=0

2ks1‖ψk(D)(πn1 − πm

1 )(u0, u1)‖q

≤ c |β|n+2∑

k=m−1

2ks1

2∑

i=−2

‖ψk+i(D)u0‖∞k+i−3∑

j=0

‖ψj(D)u1‖q

≤ c |β|∑

k≥m−1

2∑

i=−2

2ks12−(k+i)s02−(k+i−3)s1 ‖u0‖Bs0/ν∞

‖u1‖Bs1/νq,1

≤ c |β|( ∑

k≥m−1

2ks1

)‖u0‖B

s0/ν∞

‖u1‖Bs1/νq,1

.

From this we deduce that (πn1 ) is a Cauchy sequence in B

s1/νq,1 (E2). Thus it con-

verges therein towards π1(u0, u1). Obvious modifications of the preceding estimatesyield

‖π1(u0, u1)‖Bs1/νq,1

= limn→∞

‖πn1 (u0, u1)‖B

s1/νq,1

≤ c |β| ‖u0‖Bs0/ν∞

‖u1‖Bs1/νq1,1

.

(4) Analogously to the above considerations we find, for τ > 0,∞∑

k=0

2ks1‖ψk(D)(πn3 − πm

3 )(u0, u1)‖q

≤ c |β|∑

k≥m−1

2ks1

2∑

i=−2

k+i−3∑

j=0

‖ψj(D)u0‖∞ ‖ψk+i(D)u1‖q

≤ c |β|( ∑

k≥m−1

2ks1 ‖ψk(D)u1‖q

) ∞∑

j=0

2jτ ‖ψk(D)u0‖∞

= c |β|∑

k≥m−1

(2ks1 ‖ψk(D)u1‖q

)‖u0‖B

τ/ν∞,1

.

Since the sum over k ≥ m− 1 converges to 0 for m →∞ if u1 ∈ Bs1/νq,1 (E1), we

see that π3(u0, u1) exists in Bs1/νq,1 (E2) and satisfies

‖π3(u0, u1)‖Bs1/νq,1

≤ c(τ) |β| ‖u0‖Bτ/ν∞,1

‖u1‖Bs1/νq,1

(6.4.15)

for (u0, u1) ∈ Bτ/ν∞,1(E0) × B

s1/νq,1 (E1).


Finally, the assertion for π3 is obtained by choosing τ such that s0 > τ > −s1

and recalling Bs0/ν∞ (E0) → B

τ/ν∞,1(E0).

(5) The definition of πn2 and Lemma 6.4.3(ii) imply that the support of Fπn

2

is contained in [Q < 2n+4]. Hence ψk(D)πn2 = 0 for k ≥ n + 5. Thus, given n > m,

ψk(D)(πn2 − πm

2 )(u0, u1)

= ψk(D)( n∑

j=m+1

2∑

i=−2

β(Sju0, Sj+iu1)

− β(Sn−1u0, Sn+1u1)− β(Snu0, (Sn+1 + Sn+2)u1

)

+ β(Sm−1u0, Sm+1u1) + β(Smu0, (Sm+1 + Sm+2)u1

))

vanishes if k ≥ n + 5. From this and (2.1.1) we infer that∞∑

k=0

2ks1 ‖ψk(D)(πn2 − πm

2 )(u0, u1)‖q

≤ c

n+4∑

k=0

2ks1

n∑

j=m

2∑

i=−2

‖β(Sju0, Sj+iu1)‖q

≤ c |β|( ∞∑

k=0

2ks1

) ∞∑

j=m

2∑

i=−2

2−(js0+(j+i)s1) ‖2js0ψj(D)u0‖∞ ‖2(j+i)s1ψj+i(D)u1‖q

≤ c |β| 2−m(s0+s1)∥∥(‖2js0ψj(D)u0‖∞

)∥∥`∞

∥∥(‖2js1ψj(D)u1‖q

)∥∥`1

= c |β| 2−m(s0+s1) ‖u0‖Bs0/ν∞

‖u1‖Bs1/νq,1

.

(6.4.16)

Since s0 + s1 > 0, we see that(πn

2 (u0, u1))

is a Cauchy sequence in Bs1/νq,1 (E2).

From this and estimate (6.4.16) we deduce that π2 is a continuous bilinear mapfrom B

s0/ν∞ (E0)× B

s1/νq,1 (E1) into B

s1/νq,1 (E2), whose norm is bounded by c |β|. This

proves the lemma. ¥

The next lemma shows that π is an extension of the point-wise multiplicationoperator mβ studied in the preceding subsection.

6.4.5 Lemma Suppose s0 > s > 0. Then π(u0, u1) = mβ(u0, u1) for u0 ∈ Bs0/ν∞ (E0)

and u1 ∈ Bs/νq,r (E1).

Proof (1) Assume t ∈ R and u ∈ Bt/νp,r (E), where 1 ≤ p ≤ ∞. We claim that

Snu → u in Bt/νp,r (E).

To see this, let n > m > 0. Then

ψk(D)(Sn − Sm)u =n∑

j=m+1

ψk(D)ψj(D)u =1∑

i=−1

ψk(D)ψk+i(D)u


for m ≤ k ≤ n + 1, and ψk(D)(Sn − Sm)u = 0 otherwise. Hence, using once more(2.1.1),

∑

k≥0

2kt ‖ψk(D)(Sn − Sm)u‖p ≤ c

∞∑

k=m−1

2kt ‖ψk(D)u‖p .

This shows that (Snu) is a Cauchy sequence in Bt/νp,1 (E). Therefore Snu converges

in Bt/νp,1 (E), consequently, due to B

t/νp,1 (E) → B

t/νp,r (E) → S ′(E), in B

t/νp,r (E) and

in S ′(E) towards some v. From Lemma VI.3.6.2 we know that Snu → u in S ′(E).Hence v = u. This proves the claim.

(2) Let (u0, u1) ∈ S ′(E0)× S ′(E1). It is obvious from (6.4.7) that

πn(u0, u1) = β(Snu0, Snu1) = mβ(Snu0, S

nu1) , n ∈ N . (6.4.17)

Suppose u0 ∈ Bs0/ν∞ (E0) and u1 ∈ B

s/νq,r (E1). Step (1) and Theorem 6.3.3(ii) imply

mβ(Snu0, Snu1) → mβ(u0, u1) in Bs/ν

q,r (E2) .

Hence, by Bs/νq,r (E2) → B

−s/νq,r (E2), in B

−s/νq,r (E2). Now we deduce from (6.4.17)

that π(u0, u1) = mβ(u0, u1). ¥

Let K be closed, 0 < s < s0, and r < ∞. We set, for u0 ∈ Bs0/ν∞ (K, E0) and

u1 ∈ B−s/νq,r (K, E1),

πK(u0, u1) := RKπ(EKu0, EKu1) .

It follows from Lemma 6.4.4 and Theorem 2.8.2 that πK is a continuous bilin-ear map from B

s0/ν∞ (K, E0) × B

−s/νq,r (K, E1) into B

−s/νq,r (K, E2), whose norm is

bounded by c |β|.Assume u1 ∈ S(K, E1). Then (EKu0, EKu1) ∈ B

s0/ν∞ (E0) × S(E1) by Theo-

rems 2.8.2 and VI.1.2.3. Hence, by Lemma 6.4.5,

πK(u0, u1) = RKmβ(EKu0, EKu1) = mβ,K(u0, u1) ,

where, for clarity, we write mβ,K for the mβ of Subsection 6.3 if X = K. Indeed,this is obvious by the point-wise definition of mβ,K and the fact that RK is thepoint-wise restriction. Since S(K, E1) is dense in B

−s/νq,r (K, E1), this implies that,

given u0 ∈ Bs0/ν∞ (K, E0),

πK(u0, ·) ∈ L(B−s/ν

q,r (K, E1), B−s/νq,r (K, E2)

)

is the unique continuous extension of mβ,K(u0, ·). Thus it is feasible to writeagain mβ,K for πK. Moreover, as before, we drop the index K and write simply mβ ,even if X = K. Thus, if K = K, r < ∞, and s0 > s > 0,

mβ ∈ L(Bs0/ν∞ (K, E0), B−s/ν

q,r (K, E1); B−s/νq,r (K, E2)

). (6.4.18)

From this we get, by obvious real interpolation with the functor (·, ·)θ,∞, that wecan admit r = ∞ also. Furthermore, mβ is universal, that is, independent of theparticular choice of the parameters s0, s, q, and r.


Now we can easily prove the following general point-wise multiplier result.

6.4.6 Theorem Suppose K is a closed corner in Rd, X ∈ Rd,K, and

1 ≤ q < ∞ , 1 ≤ r ≤ ∞ , s0 > s > −s0 .

Then

mβ : Bs0/ν∞ (X, E0) × Bs/ν

q,r (X, E1) → Bs/νq,r (X, E2) (6.4.19)

and

mβ : Bs0/ν∞ (X, E0)× bs/ν

q,∞(X, E1) → bs/νq,∞(X, E2) . (6.4.20)

Proof Due to Theorem 6.3.3 and (6.4.18) it suffices to prove (6.4.19) for s = 0and assertion (6.4.20). The first case is settled by interpolation with the func-tor (·, ·)θ,∞, and the second one by employing (6.4.19) and (·, ·)0θ,∞. ¥

6.5 Multiplications in Bessel Potential Spaces

Now it is not difficult to prove that Besov–Holder spaces are point-wise multipliersfor Bessel potential spaces as well.

6.5.1 Theorem Suppose K is a corner in Rd and X ∈ Rd,K. Let E1 and E2 beν-admissible and 1 s > −s0.

Proof (1) Suppose s0 > t > s > 0. We can assume that ε := t− s < ν. We fix min νN with m > s and set θ := s/m. It follows from Theorems 4.3.2 and 4.5.1 that

Hs/νp (X, Ej)

.=[Lp(X, Ej),Wm/ν

p (X, Ej)]θ

(6.5.1)

for j = 1, 2.Theorems 2.7.1(v) and 2.8.3 guarantee, due to

(1− θ)ε + θ(m + ε) = s + ε = t ,

that

bt/ν∞ (X, E0)

.=[bε/ν∞ (X, E0), b(m+ε)/ν

∞ (X, E0)]θ

. (6.5.2)

It is a consequence of b(j+ε)/ν∞ (X, E0) → BUCj/ν(X, E0) for j ∈ 0,m and The-

orem 6.2.1 that

mβ ∈ L(b(j+ε)/ν∞ (X, E),W j/ν

p (X, E1); W j/νp (X, E2)

)(6.5.3)


for j ∈ 0,m. Now we use properties of A.P. Calderon’s second complex interpo-lation functor [·, ·]θ, 0 < θ < 1. Namely, suppose (F0, F1) is an interpolation pairof Banach spaces. Then

[F0, F1]θ → [F0, F1]θ (6.5.4)

and

[F0, F1]θ = [F0, F1]θ if F0 is reflexive (6.5.5)

(see [Cal64, Sections 6–9] or [BeL76, Theorems 4.1.4 and 4.3.1]). Hence, by (6.5.1)and (6.5.5),

Hs/νp (X, Ej) =

[Lp(X, Ej), Wm

p (X, Ej)]θ

, j = 1, 2 . (6.5.6)

It follows from (6.5.2) and (6.5.4) that

bt/ν∞ (X, E0) → [

bε/ν∞ (X, E0), b(m+ε)/ν

∞ (X, E0)]θ =: X θ . (6.5.7)

Using (6.5.6) and Calderon’s multilinear interpolation theorem (cf. [Cal64, Sec-tion 10.1] or [BeL76, Theorems 4.4.1]), we obtain from (6.5.3) that

mβ ∈ L(X θ,Hs/νp (X, E1); Hs/ν

p (X, E0))

.

Thus (6.5.7) and Bs0/ν∞ → b

t/ν∞ imply

mβ ∈ L(Bs0/ν∞ (X, E0),Hs/ν

p (X, E1); Hs/νp (X, E0)

).

(2) Suppose −s0 < s < 0. Then the claim follows from (1) by duality, see theproof of Theorem 6.2.6, due to that theorem and Theorem 4.4.2.

(3) The case s = 0 is already contained in (6.5.3). ¥

6.6 Space-Dependent Bilinear Maps

For certain applications, in particular in situations involving Riemannian metricson manifolds, it is necessary to consider point-wise multiplications which dependon the underlying space variables also. Such situations are considered in the presentsubsection. More precisely, suppose

β ∈ BC(X,L(E0, E1; E2)

)(6.6.1)

and write mβ for the point-wise extension,

BC(X, E0) × L1,loc(X, E1) → L1,loc(X, E2) , (u0, u1) 7→ mβ(u0, u1) ,

now defined by

mβ(u0, u1)(x) := β(x)(u0(x), u1(x)

), a.a. x ∈ X .

Observe that the point-wise multiplier theorems proved so far are universal in thesense that they depend neither on the particular structure of β nor on the Banach


spaces E0, E1, and E2, except for occasional reflexivity assumptions. In otherwords, suppose Fj(X, Ej) → S ′(X, Ej) and

mβ : F0(X, E0)× F1(X, E1) → F2(X, E2) .

Then, given Banach spaces F0, F1, F2, and γ ∈ L(F0, F1; F2), it follows

mγ : F0(X, F0)× F1(X, F1) → F2(X, F2) .

To express this fact we use the somewhat unprecise but intuitive notation

F0 × F1 → F2 .

Based on this observation, we can employ a simple argument to carry over manyof the preceding results to space-dependent multiplications.

6.6.1 Lemma Suppose Fj = Fj(X, Ej) → S ′(X, Ej), j = 0, 1, 2, are such that

F0 × F0 → F0 and F0 × F1 → F2 . (6.6.2)

Thenmβ : L(

F0(X, E0), F1(X, E1); F2(X, E2))

.

Proof Consider the multiplication

β0 : L(E0, E1; E2) × E0 → L(E1, E2) , (β, e0) 7→ β(e0, ·)and let m0 be its point-wise extension. Then the first part of (6.6.2) implies

m0 ∈ L(F0

(X,L(E0, E1; E2)

), F0(X, E0); F0(X, E2)

). (6.6.3)

Next we introduce the multiplication

L(E1, E2) × E1 → E2 , (A, e) 7→ Ae

and denote its point-wise extension by m1. Then, by the second part of (6.6.2),

m1 ∈ L(F0

(X,L(E1, E0)

), F1(X, E1); F2(X, E2)

). (6.6.4)

Note that

mβ(u0, u1) = m1

(m0(β, u0), u1

), (u0, u1) ∈ EX0 × EX1 .

This proves the lemma. ¥

6.6.2 Examples Assume K is closed.

(a) Let one of the following conditions be satisfied:(α) m ∈ νN, F0 ∈ BCm/ν , BUCm/ν.(β) s0 > 0, F0 ∈ BUCs0/ν , bucs0/ν.(γ) s0 > |ω|/p0 > 0, F0 ∈ W s0/ν

p0 , Bs0/νp0,r .

Then F0 × F0 → F0.


Proof Theorem 6.2.1 implies (α). From (α) and Theorem 6.3.3 we get (β). Lastly,(γ) follows from Corollary 6.3.4(i). ¥

(b) Theorems 6.2.1, 6.2.3, 6.2.6, 6.3.3(i)–(iii), Corollary 6.3.4, Theorem 6.4.1(i)and parts (i) and (ii) of its corollary, and Theorems 6.4.6 and 6.5.1 apply to space-dependent multiplications as well.

Proof Clear by (a). ¥

6.7 Notes

A detailed study of point-wise multiplication in scalar-valued Besov (and Triebel–Lizorkin) spaces on Rd can be found in Th. Runst and W. Sickel [RS96] for isotropicspaces, and in J. Johnson [Joh95] for the anisotropic case. It follows from thosepublications that the multiplier theorems of this section are optimal, except for afew limit cases. The proofs given here in the case of positive order spaces and ––in the reflexive setting –– for negative order spaces are essentially the same as theones in our paper [Ama91]. There we formulated the results for general Banach-space-valued functions. However, the proofs were restricted to finite-dimensionalsettings. This is now rectified by the reasoning in this section. In fact, the theoremspresented here are even sharper than the ones in our earlier paper.

It should be noted that [Ama91] is the first paper in which the optimalconditions for the validity of the point-wise multiplier theorems for positive or-der Sobolev and Besov spaces are given. In earlier work (see [Han85], [Val85],[Val88], [Zol78]) additional restrictions had been imposed. The reader should con-sult [RS96] for references to earlier work as well as the book by V.G. Maz’ya andT.O. Shaposhnikova [MaS85].

In [Ama91] and [RS96] there are also considered multiplications with morethan two factors. It is clear that this could be done here also.

The proofs of Subsection 6.4 in the non-reflexive setting follow the work ofW. Sickel and J. Johnson inasmuch as we use paramultiplication (also see [Yam86]).For the proof of the support properties of Lemma 6.4.3 we have to employ the fullstrength of the theory of vector-valued distributions, namely the Schwartz kerneltheorem, presented in the Appendix. Given this lemma, our proof of Lemma 6.4.5is simpler and more direct than the demonstrations of the other authors since wedo not rely on convergence theorems of M. Yamazaki. In [Ama01] we have given a(somewhat sketchy) proof of Theorem 6.4.6 by following closely the arguments ofJ. Johnson [Joh95] and Th. Runst and W. Sickel [RS96].

M. Meyries and M. Veraar [MeV15] extend many of the results known forpoint-wise multipliers on scalar-valued Bessel potential spaces to the vector-valuedsetting. More precisely, they consider weighted Bessel potential spaces

Hsp,w(Rd, E) := J−sLp(Rd, w, E) ,


where w is a power weight acting on one variable only, and E is a UMD space.In this setting, they obtain the analogue of Theorem 6.5.1. By R-boundednesstechniques, ‘irregular’ multiplier spaces of the form Bs

q,∞(Rd, w,L(E1, E2)

)are

considered also.The idea of using multilinear interpolation in the proof of Theorem 6.5.1 is

taken from [MeV15].M. Kohne and J. Saal [KoS17] present a study of point-wise multiplication

in anisotropic vector-valued Sobolev, Besov, and Bessel potential spaces. Theyconsider the case of possibly more than two factors and positive order spaces.These authors base their proofs on the results of [Ama09]. Thus they have toassume that all occurring target spaces are ν-admissible.

We refer to the book [RS96] for rather detailed references to earlier work onpoint-wise multiplication in scalar Besov (and Triebel–Lizorkin) spaces.

VII.7 Compactness 249

7 Compactness

This section is concerned with compact embeddings of Sobolev, Besov, and Besselpotential spaces. Such results are fundamental for many investigations of qualita-tive properties of evolution and nonlinear partial differential equations.

The main result of the first subsection is an extension to the vector-valuedsetting of the sufficiency part of the Frechet–Kolmogorov theorem characterizingcompact subsets of Lebesgue spaces. This and the Arzela–Ascoli theorem providethe basis for compact embedding theorems involving Besov and Bessel potentialspaces. They are derived in Subsection 7.2.

In the rest of this section we consider vector-valued function spaces on com-pact intervals. In Subsection 7.3 it is shown how these spaces relate to the gen-eral theory established in the earlier sections. First we define E-valued Sobolev–Slobodeckii spaces on compact intervals by means of the classical seminorms. Thenwe prove a retraction-coretraction theorem. On its basis, we can transpose embed-ding and interpolation theorems, proved earlier for function spaces on corners, tothe present setting.

Vector-valued function spaces on intervals play a fundamental role in thetheory of evolution equations, which is seen, for example, by the theory devel-oped in Volume I. Here they are introduced to establish, in the last subsection,generalizations of compact embedding theorems of Aubin-Lions type.

Throughout this section,

• K is a closed corner in Rd.

• X ∈ Rd,K.• K is a compact subset of X with nonempty interior.

(7.0.1)

Given an LCS F(X, E) → D′(X, E), we set

FK(X, E) :=

u ∈ F(X, E) ; supp(u) ⊂ K

. (7.0.2)

7.1 Equicontinuity

In the scalar-valued isotropic case, the fundamental Rellich–Kondrachov compactembedding theorem is based on the Arzela–Ascoli and the Frechet–Kolmogorovtheorem. Motivated by this, we give in this subsection sufficient conditions guar-anteeing compactness in the spaces BUC(X, E) and Lq(X, E).

Compact Sets in BUC

Our first result is a simple application of the Arzela–Ascoli theorem. Recall thatb and −→ stand for compact embeddings.


7.1.1 Theorem Let (7.0.1) be satisfied, E1 −→ E0, and K ⊂ BUCK(X, E1). As-sume:

(i) K is bounded in BUC(X, E1).

(ii) K is equicontinuous in C(X, E0), that is, for each x ∈ X

limy→0y∈X

|u(x + y) − u(x)|E0 = 0 ,

uniformly with respect to u ∈ K.

Then K b BUC(X, E0).

Proof Since E1 −→ E0, it follows from (i) that K(x) :=

u(x) ; u ∈ K is for

each x ∈ K relatively compact in E0. Due to (ii) and Lemma 1.1.1, the Arzela–Ascoli theorem (e.g., [DuS57, Theorem IV.6.7]) implies the assertion. ¥

Compact Sets in Lq

The next theorem is a vector-valued variant of the sufficiency part of the Frechet–Kolmogorov theorem (e.g., [DuS57, Theorem IV.8.2]).

7.1.2 Theorem Let (7.0.1) be satisfied, E1 −→ E0, and 1 ≤ q < ∞. Suppose:

(i) K is bounded in Lq(X, E1).

(ii) K is equicontinuous in Lq(X, E0), that is,

limy→0y∈X

∫

X|u(x + y) − u(x)|qE0

dx = 0 ,

uniformly with respect to u ∈ K.

(iii) For each ε > 0,

limR→∞

∫

X∩[|x|≥R]

|u(x)|qE0dx = 0 ,

uniformly for u ∈ K.

Then K b Lq(X, E0).

Proof (1) We set Bt := x ∈ X ; |x| ≤ t , (Tyu)(x) := u(x + y), and

Mtu :=1

|Bt|∫

Bt

Tyu dy , t > 0 .

Then, by Holder’s inequality (if q > 1) and the Fubini–Tonelli theorem, we getfrom

(Mtu − u)(x) =1

|Bt|∫

Bt

(u(x + y)− u(x)

)dy , a.a. x ∈ X ,


the estimate

‖Mtu− u‖Lq(X,E0) ≤(∫

X

( 1|Bt|

∫

Bt

|u(x + y) − u(x)|E0 dy)q

dx)1/q

≤ 1|Bt|

(∫

X

∫

Bt

|u(x + y)− u(x)|qE0dy |Bt|q/q′ dx

)1/q

=( 1|Bt|

∫

Bt

∫

X|u(x + y)− u(x)|qE0

dx dy)1/q

≤ sup|y|≤ty∈X

‖Tyu− u‖Lq(X,E0) .

Let ε > 0. Then this estimate and (ii) imply the existence of τ > 0 such that

‖Mτu− u‖Lq(X,E0) ≤ ε/6 , u ∈ K . (7.1.1)

(2) Similarly as above,

|Mτu(x)|E1 ≤1

|Bτ |∫

Bτ

|u(x + y)|E1 dy ≤( 1|Bτ |

∫

Bτ

|u(x + y)|qE1dy

)1/q

≤ 1|Bτ |1/q

(∫

X|u(x + y)|qE1

dy)1/q

≤ 1|Bτ |1/q

‖u‖Lq(X,E1) .

By (i),

|Mτu(x)|E1 ≤ c , a.a. x ∈ X , u ∈ K . (7.1.2)

By the same arguments we find

|Mτu(x + y) − Mτu(x)|E0 ≤1

|Bτ |∫

Bτ

|u(x + y + z) − u(x + z)|E0 dz

≤( 1|Bτ |

∫

Bτ

|u(x + y + z) − u(x + z)|qE0dz

)1/q

≤ 1|Bτ |1/q

(∫

X|u(x + y) − u(x)|qE0

dx)1/q

.

Thus we infer from E1 → E0, (7.1.2), and (ii) that Mτu ; u ∈ K is equicon-tinuous in BUC(X, E0).

(3) Using the arguments of step (1) once more, we find

∫

X\BR

|Mτu|qE0dx ≤ 1

|Bτ |∫

Bτ

∫

X\BR

|u(x + y)|qE0dx dy ≤

∫

X\BR

|u(x)|qE0dx .


Hence it follows from (iii) that we can fix R > 0 such that, setting K := BR,

(∫

X\K

|Mτu|qE0dx

)1/q

≤ ε/6 , u ∈ K .

(4) Since K is compact, we deduce from step (2) and Theorem 7.1.1 thatMτu |K ; u ∈ K is relatively compact in BUC(K, E0), hence totally bounded.

Let B(v, r) be the open ball in BUC(K, E0) with center at v and radius r,and Bq(v, r) the one in Lq(K,E0). There are v0, . . . , vn ∈ Mτu ; u ∈ K suchthat

Mτu |K ; u ∈ K ⊂n⋃

j=0

B(vj |K, ε/3 |K|) .

Since

‖Mτu− vj‖Lq(K,E0) =(∫

K

|Mτu(x) − vj(x)|qE0dx

)1/q

≤ ε/3

for Mτu ∈ B(vj , ε/3 |K|), we see that Mτu |K ; u ∈ K ⊂ ⋃nj=0 Bq(vj , ε/3).

(5) Assume u ∈ Lq(X, E0). By step (4) we can find uj ∈ K such that

∫

K

|Mτu− Mτuj |qE0dx ≤ (ε/3)q . (7.1.3)

Note that (7.1.1) implies

‖u − uj‖Lq(X,E0) ≤ ‖u − Mτu‖Lq(X,E0) + ‖Mτu − Mτuj‖Lq(X,E0)

+ ‖Mτuj − uj‖Lq(X,E0)

≤ ε/3 + ‖Mτu − Mτuj‖Lq(X,E0) .

(7.1.4)

From (7.1.3) and step (3) we infer

‖Mτu − Mτuj‖Lq(X,E0)

≤ ‖Mτu − Mτuj‖Lq(K,E0) +(∫

X\K

|Mτu− Mτuj |qE0dx

)1/q

≤ ‖Mτu − Mτuj‖Lq(K,E0) +(∫

X\K

|Mτu|qE0dx

)1/q

+(∫

X\K

|Mτuj |qE0dx

)1/q

≤ 2ε/3 .

Thus it follows from (7.1.4) that ‖u− uj‖Lq(X,E0) ≤ ε. This shows that K is totallybounded in Lq(X, E0), hence relatively compact. ¥


7.2 Compact Embeddings

For preparation, we include some simple observations.

7.2.1 Lemma L1,loc,K(X, E) is a closed linear subspace of L1,loc(X, E).

Proof Let (uj) be a sequence in L1,loc,K(X, E) converging in L1,loc(X, E) to-wards u. Given ϕ ∈ D(X, E) with supp(ϕ) ⊂ X\K, it follows that

0 =∫

Xϕuj dx →

∫

Xϕudx .

Hence supp(u) ⊂ K. ¥

7.2.2 Corollary Let F and F1 be LCSs satisfying F1 → F → L1,loc(X, E). ThenFK is a closed linear subspace of F and F1,K → FK .

Let X, X0, and X1 be Banach spaces. Recall that K(X1, X0) is the set of allcompact T ∈ L(X1, X0). Below we shall repeatedly use the following well-knownand easy to prove fact:

Let T ∈ K(X1, X0), S0 ∈ L(X0, X), and S1 ∈ L(X, X1).Then S0T ∈ K(X1, X) and TS1 ∈ K(X, X0).

(7.2.1)

Compact Embeddings of Besov Spaces

First we consider general Besov spaces.

7.2.3 Theorem Let (7.0.1) be true. Assume E1 −→ E0, 1 ≤ q < ∞, 1 ≤ r ≤ ∞,and s0 < s1 with s1 > 0. Then

Bs1/νq,r,K(X, E1) −→ Bs0/ν

q,r (X, E0) (7.2.2)

and

Bs1/ν∞,K(X, E1) −→ Bs0/ν

∞ (X, E0) . (7.2.3)

Proof We use the notations of Subsection 3.5.(1) Let K be a bounded subset of B

s1/νq,r,K(X, E1). Then

K is bounded in Lq,K(X, E1). (7.2.4)

We fix s ∈ (0, s1) with s < νi for 1 ≤ i ≤ `. It follows from Theorems 2.2.2, 2.2.5,and 2.8.3 and Corollary 7.2.2 that K is a bounded subset of B

s/νq,∞(X, E0). Now


Proposition 3.5.4 implies that

Ki :=

xi 7→ u(xi; ·) ; u ∈ K (7.2.5)

is equicontinuous in Lq

(Xi, Lq(Xı, E0)

). Given x, y ∈ X,

u(x + y) − u(x) =∑

i=1

(u(x1, . . . , xi−1, xi + yi, . . . , x` + y`)

− u(x1, . . . , xi, xi+1 + yi+1, . . . , x` + y`))

.

From this we infer

‖u(x + y) − u(x)‖Lq(X,E0)

≤∑

i=1

(∫

X1

· · ·∫

X`

∣∣∣u(x1, . . . , xi−1, xi + yi, . . . , x` + y`)

− u(x1, . . . , xi, xi+1 + yi+1, . . . , x` + y`)∣∣∣q

E0

dx` · · · dx1

)1/q

≤∑

i=1

(∫

Xi

∥∥u(xi + yi; ·)− u(xi; ·)∥∥q

Lq(Xı,E0)dxi

)1/q

,

where the last inequality follows by the variable substitutions xj + yj → ξj fori + 1 ≤ j ≤ `. Thus (7.2.5) shows that K is equicontinuous in Lq(X, E0). Fromthis, (7.2.4), and the fact that condition (iii) of Theorem 7.1.2 is trivially satisfiedwe deduce by that theorem that K b Lq(X, E0). This proves

Bs1/νq,r,K(X, E1) −→ Lq(X, E0) .

(2) Suppose s0 > 0. Let K be a bounded sequence in Bs1/νq,r,K(X, E1). By (1)

there exists a subsequence (uj) which is a Cauchy sequence in Lq(X, E0). It followsfrom Theorems 2.7.1(i) and 2.8.3 and embedding (2.6.6) with m = 0 that

Bs0/νq,r (X, E0)

.=(Lq(X, E0), Bs1/ν

q,r (X, E0))s0/s1,r

.

This and Proposition I.2.2.1 give

‖uj − uk‖s0/ν,q,r ≤ c ‖uj − uk‖1−s0/s1q ‖uj − uk‖s0/s1

s1/ν,q,r

≤ c ‖uj − uk‖1−s0/s1q

for j, k ∈ N, where the norms belong to E0-valued spaces. The last inequalityholds since K is bounded in B

s1/νq,r (X, E0). Hence (uj) is a Cauchy sequence in


Bs0/νq,r (X, E0). This implies that K is sequentially compact, hence compact, in

Bs0/νq,r (X, E0). Consequently, (7.2.2) applies if 0 < s0 < s1.

(3) Suppose s0 ≤ 0. Choose s ∈ (0, s1). Then, by (2),

Bs1/νq,r,K(X, E1) −→ Bs/ν

q,r (X, E0) → Bs0/νq,r (X, E0) .

Now (7.2.2) follows from (7.2.1).(4) The proof of (7.2.3) is essentially the same, except that we have to apply

Theorem 7.1.1. ¥

Compact Embeddings of Holder, Sobolev–Slobodeckii, and Bessel Potential Spaces

On the basis of the preceding results it is not difficult to prove compact embeddingtheorems for the spaces listed in this subtitle. The reader should recall the defini-tions of the Sobolev–Slobodeckii, Holder, and little Holder scales in Subsection 3.6.

7.2.4 Theorem Let (7.0.1) be satisfied. Suppose E1 −→ E0 and 0 ≤ s0 < s1. Then

BUCs1/νK (X, E1) −→ BUCs0/ν(X, E0)

andbuc

s1/νK (X, E1) −→ bucs0/ν(X, E0) .

Proof We fix t0, t1 /∈ νN with s0 < t0 < t1 < s1. Then

bucs1/νK (X, E1) → BUC

s1/νK (X, E1) → B

t1/ν∞,K(X, E1)

−→ Bt0/ν∞ (X, E0) → bucs0/ν(X, E0) → BUCs0/ν(X, E0) .

Thus the assertion follows from (7.2.1). ¥

As an application of these results we can easily prove the following far-reaching generalizations of the classical Rellich–Kondrachov embedding theorem.

7.2.5 Theorem Let assumption (7.0.1) be satisfied and suppose E1 −→ E0.(i) Assume q0, q1 ∈ [1,∞), 0 ≤ s0 < s1, and s1 − |ω|/q1 > s0 − |ω|/q0. Then

Ws1/ν

q1,K (X, E1) −→ W s0/νq0

(X, E0) and Hs1/νq1,K (X, E1) −→ Hs0/ν

q0(X, E0) .

(ii) If 1 ≤ q < ∞ and s − |ω|/q > t ≥ 0, then

Ws/ν

q,K (X, E1) −→ buct/ν(X, E0) and Hs/νq,K (X, E1) −→ buct/ν(X, E0) .


Proof (1) We fix s, t, t0 with s0 < s < t < t1 < s1 and t− |ω|/q1 ≥ s − |ω|/q0.Then we get from Theorems 2.2.2, 2.6.5, and 2.8.3, Corollary 7.2.2 and Theo-rem 7.2.3

Ws1/ν

q1,K (X, E1) → Bt1/νq1,K(X, E1) −→ Bt/ν

q1(X, E0)

→ Bs/νq0,q1

(X, E0) → W s0/νq0

(X, E0) .

This implies the first part of (i).(2) Let 0 ≤ t < s < t0 < t1 < s1 be such that t1 − |ω|/q > t0. Then, similarly

as in the preceding step,

Ws1/ν

q,K (X, E1) → Bt1/νq,K (X, E1) → Bt0/ν

∞ (X, E1)

−→ Bs/ν∞ (X, E0) → bucs0/ν(X, E0) ,

where the last embedding is implied by (2.2.10). Hence the first part of (ii) isproved.

(3) The assertions about the Bessel potential spaces are now obvious conse-quences of Theorem 4.1.3(i), taking Theorems 2.2.4(i) and 2.8.3 into account. ¥

7.3 Function Spaces on Intervals

It is the purpose of this subsection to introduce the basic spaces of distributionson compact intervals and to put them in relation to general results obtained inthe preceding sections. It is no loss of generality to assume throughout

• 0 < T < ∞ , J = [0, T ] .

Classical Spaces

Since continuous functions on compact metric spaces are bounded and uniformlycontinuous,

BUCk(J,E) = Ck(J,E) , k ∈ N ,

andD(J,E) = BUC∞(J,E) = C∞(J,E) .

E-valued distributions on J are the elements of

D′(J,E) := L(D(J), E)

.

As usual, we identify u ∈ L1(J,E) with the regular distribution

ϕ 7→∫

J

uϕdx = 〈u, ϕ〉J , ϕ ∈ D(J) . (7.3.1)

Then

C(J,E) → Lq(J,E)d

→ Lr(J,E)d

→ L1(J,E) → D′(J,E) (7.3.2)

for 1 ≤ r < q ≤ ∞, where the first embedding is dense as well if q < ∞.


The Sobolev space W kq (J,E), 1 ≤ q < ∞, k ∈ N, is defined to be the comple-

tion of Ck(J,E) in Lq(J,E) with respect to the norm

u 7→ ‖u‖k,q = ‖u‖k,q,J :=k∑

j=0

‖∂ju‖Lq(J,E) ,

and ‖·‖0,q = ‖·‖q = ‖·‖q,J .Given 0 < θ < 1 and 1 ≤ q ≤ ∞, we introduce the seminorm

[u]θ,q,J :=(∫

J2

( |u(x) − u(y)|E|x− y|θ

)q d(x, y)|x− y|

)1/q

if q < ∞, and

[u]θ,∞,J := supx,y∈Jx 6=y

|u(x) − u(y)|E|x− y|θ

if q = ∞. Then the Slobodeckii space W sq (J,E) of order s ∈ R+\N (over Lq(J,E))

is defined by

W sq (J,E) :=

u ∈ W [s]

q (J,E) ; [∂[s]u]s−[s],q,J < ∞, 1 ≤ q < ∞ ,

endowed with the norm

u 7→ ‖u‖s,q,J := ‖u‖[s],q,J + [∂[s]u]s−[s],q,J .

As usual, if s ∈ R+\N, then

Cs(J,E) :=

u ∈ C [s](J,E) ; [∂[s]u]s−[s],∞,J < ∞,

equipped with the norm

u 7→ ‖u‖s,∞,J := ‖u‖[s],∞,J + [∂[s]u]s−[s],∞,J ,

is the Holder space of order s on J .

Besov Spaces

Let 0 < θ < 1, 1 ≤ q < ∞, and 1 ≤ r ≤ ∞. Then we introduce Besov spaces on Jas follows. We set

[u]θ,q,r,J :=(∫ T/2

0

(∫ T/2

0

( |4τu(t)|Eτθ

)q

dt +∫ T

T/2

( |4−τu(t)|Eτθ

)q

dt

)r/qdτ

τ

)1/r

if r < ∞. If r = ∞, then the integral with respect to dτ/τ is to be replacedby sup0<τ<T/2. Similarly,

[u]θ,∞,∞,J := sup0<τ<T/2

(sup

0<t<T/2

|4τu(t)|Eτθ

+ supT/2<t≤T

|4−τu(t)|Eτθ

).


The little Holder space cs(J,E), s ∈ R+\N, consists of all u ∈ Cs(J,E) satis-fying

limτ→∞

sup|x−y|≤τ

|∂[s]u(x) − ∂[s]u(y)|E|x− y|s−[s]

= 0 .

As in Subsection 3.6, the little Holder space scale[cs(J,E) ; s ≥ 0

]is defined by

ck(J,E) := Ck(J,E) for k ∈ N.Given s ∈ R+\N, 1 ≤ q < ∞, and 1 ≤ r ≤ ∞,

Bsq,r(J,E) :=

u ∈ W [s]

q (J,E) ; [∂[s]u]s−[s],q,r,J < ∞, (7.3.3)

endowed with the norm

u 7→ ‖u‖s,q,r,J := [u][s],q,J + [∂[s]u]s−[s],q,r,J , (7.3.4)

is a Besov space of order s on J . Similarly,

Bs∞,∞(J,E) :=

u ∈ C [s](J,E) ; [∂[s]u]s−[s],∞,∞,J < ∞

is given the norm

u 7→ ‖u‖s,∞,∞,J := ‖u‖[s],∞,J + [∂[s]u]s−[s],∞,∞,J .

7.3.1 Proposition Suppose s ∈ R+\N. Then(i) Bs

q,q(J,E) .= W sq (J,E) for 1 ≤ q < ∞.

(ii) Bs∞,∞(J,E) .= Cs(J,E).

Proof It suffices to consider the case s = θ ∈ (0, 1).(1) Assume 1 ≤ q < ∞. The Fubini–Tonelli theorem and the substitution

x + τ = y give∫ T/2

0

∫ T/2

0

( |4τu(x)|τθ

)q

dxdτ

τ

=∫ T/2

0

∫ x+T/2

x

( |u(y) − u(x)||y − x|θ

)q dy

|y − x| dx ≤ [u]qθ,q,J .

(7.3.5)

Similarly, using x− τ = y,∫ T/2

0

∫ T

T/2

( |4−τu(x)|τθ

)q

dxdτ

τ

=∫ T

T/2

∫ x

x−T/2

( |u(y) − u(x)||y − x|θ

)q dy

|y − x| dx ≤ [u]qθ,q,J .

(7.3.6)

This proves [·]θ,q,q,J ≤ c [·]θ,q,J .


(2) We write J2 = J × J =⋃6

i=1 Mi, where

M1 :=

(x, y) ; 0 ≤ x ≤ T/2, x ≤ y ≤ x + T/2

,

M3 :=

(x, y) ; 0 ≤ x ≤ T/2, 0 ≤ y ≤ x

,

M5 :=

(x, y) ; 0 ≤ x ≤ T/2, x + T/2 ≤ y ≤ T

,

M2 :=

(x, y) ; T/2 ≤ x ≤ T, x − T/2 ≤ y ≤ x

,

M4 :=

(x, y) ; T/2 ≤ x ≤ T, x ≤ y ≤ T

,

M6 :=

(x, y) ; T/2 ≤ x ≤ T, 0 ≤ y ≤ x− T/2

.

Note that Mi ∩ Mj is a 2-dimensional Lebesgue null set if i 6= j.

Consider the map

ϕ : J2 → J2 , (x, y) 7→ (y, x) .

Then ϕ(M3) ⊂ M1 and ϕ(M4) ⊂ M2. Thus, setting

f(x, y) := |u(x) − u(y)|q |x − y|−1−θq , z := (x, y) ∈ J2 , x 6= y ,

it follows from ϕ∗f = f ϕ = f and | det ∂ϕ| = 1 that

∫

M3

f dz ≤∫

M1

f dz ,

∫

M4

f dz ≤∫

M2

f dz . (7.3.7)

If (x, y) ∈ M5 ∪ M6, then |x− y| ≥ T/2. Hence

∫

M4∪M5

f dz ≤ c

∫

J2|u|q dz . (7.3.8)

Lastly, we read off (7.3.5) and (7.3.6) that

∫

M1∪M2

f dz = [u]qθ,q,q,J . (7.3.9)

Now we get from (7.3.7)–(7.3.9)

[u]qθ,q,J =6∑

i=1

∫

Mi

f d(x, y) ≤ c([u]qθ,q,q,J + ‖u‖q

q,J

).

From this and step (1) we obtain assertion(i).

(3) The proof of the second claim is similar. ¥


A Retraction-Coretraction Theorem

The next theorem relates these function spaces on J to the corresponding oneson R+ and allows the transposition of the result proved in earlier sections to thepresent setting. It implies, in particular, that all the spaces introduced above arecomplete.

7.3.2 Theorem There exists an r-c pair (RJ , RcJ) for

(S ′(R+, E2),D′(J,E))

, (7.3.10)

which is universal with respect to E. It restricts to an r-c pair for(X (R+, E2),Y(J,E)

)

where:

(i) (X ,Y) = (S,D) .

(ii) (X ,Y) ∈ (BUCk, Ck), (W k

q ,W kq )

, k ∈ N, 1 ≤ q < ∞ .

(iii) (X ,Y) = (Bsq,r, B

sq,r), s ∈ R+\N, 1 ≤ q < ∞, 1 ≤ r ≤ ∞ .

(iv) (X ,Y) ∈ (BUCs, Cs), (bucs, cs)

, s ∈ R+\N .

Proof (1) We set I0 := R+ and I1 := (−∞, T ] and consider the smooth diffeo-morphism κ : I1 → R+, x 7→ −(x− T ). We define an open covering U0, U1 of Jby

U0 := [0, 5T/8) , U1 := (3T/8, T ] . (7.3.11)

Then we fix πj ∈ D(Uj , [0, 1]

)such that π0 | [0, T/2] = 1 and π1 | [T/2, T ] = 1 and

set πj := πj

/√π2

0 + π21 . Thus

πi ∈ D(Uj , [0, 1]

), π2

0 + π21 = 1 . (7.3.12)

For u ∈ D(J,E) we set14

Rcu = RcJu :=

(π0u, κ∗(π1u)

)

and, given v = (v0, v1) ∈ S(R+, E2),

Rv = RJv := π0v0 + π1κ∗v1 .

It is obvious that

Rc ∈ L(D(J,E),S(R+, E2)) ∩ L(D(J , E),S((0,∞),R2)

)(7.3.13)

14As usual, κ∗ denotes the pull-back and κ∗ = (κ∗)−1 the push-forward by κ.


and

R ∈ L(S(R+, E2),D(J,E)) ∩ L(S((0,∞), E2),D(J , E)

). (7.3.14)

For u ∈ S ′(I1, E) we define κ∗u ∈ S ′(R+, E) by

〈κ∗u, ϕ〉R+ := 〈u, κ∗ϕ〉I1 , ϕ ∈ S((0,∞), E

). (7.3.15)

Similarly,

〈κ∗v, ψ〉I1 := 〈v, κ∗ψ〉R+ , v ∈ S ′(R+, E) , ψ ∈ D(I1, E) . (7.3.16)

Then

κ∗ ∈ Lis(S ′(I1, E),S ′(R+, E)

), (κ∗)−1 = κ∗ . (7.3.17)

It follows

Rc ∈ L(D′(J,E),S ′(R+, E2))

, R ∈ L(S ′(R+, E2),D′(J,E))

. (7.3.18)

It is obvious from (7.3.12) and (7.3.17) that

RRcu = π0(π0u) + π1κ∗(κ∗(π1u)

)= (π2

0 + π21)u = u

for u ∈ D′(J,E). This proves (7.3.10). Now (i) follows from this, (7.3.13), and(7.3.14).

(2) Since ∂ κ∗ = −κ∗ ∂, Leibniz’ rule gives for k ∈ N

‖Rcu‖k,q ≤ c ‖u‖k,q,J , u ∈ Ck(J,E) , 1 ≤ q ≤ ∞ ,

and‖Rv‖k,q,J ≤ c ‖v‖k,q , , 1 ≤ q ≤ ∞ ,

where v ∈ S(R+, E2) if q < ∞, and v ∈ BUC∞(R+, E2) otherwise. This and adensity argument prove (ii).

(3) We write |·| for |·|E . Suppose 0 < s < 1. Let u ∈ Bsq,r(J,E) and r < ∞.

Then

4τ (π0u) = π04τu + u(· + τ)4τπ0 (7.3.19)

implies∫ ∞

0

(∫ ∞

0

( |4τ (π0u)(x)|τ s

)q

dx)r/q dτ

τ

≤ c

∫ ∞

0

(∫ ∞

0

( |(π04τu)(x)|τ s

)q

dx)r/q dτ

τ+ c

(∫

J

|u|q dx)r/q

,


due to π0 ∈ D(R+) ⊂ Bs∞(R+). Since supp(π0) ⊂ [0, 5T/8], the first term on the

right can be estimated by

c

∫ 3T/8

0

(∫ 5T/8

0

(4τu(x)|τ s

)q

dx)r/q dτ

τ

+ c

∫ T

3T/8

(∫ 5T/8

0

(4τ u(x)|τ s

)q

dx)r/q dτ

τ,

(7.3.20)

where u : R+ → E is the trivial extension of u. Here the second term is boundedfrom above by c ‖u‖r

q,J . Moreover, the substitution y = x + τ yields

∫ 5T/8

T/2

( |4τu(x)|τ s

)q

dx =∫ τ+5T/8

τ+T/2

( |4−τu(y)|τ s

)q

dy .

Using this, the first term of (7.3.20) can be majorized by

c

∫ T/2

0

(∫ T/2

0

( |4τu(x)|τ s

)q

dx)r/q dτ

τ

+ c

∫ 3T/8

0

(∫ τ+5T/8

τ+T/2

( |4−τu(y)|τ s

)q

dy)r/q dτ

τ,

where, in turn, the last summand is majorized by

c

∫ T/2

0

(∫ T

T/2

( |4−τu(x)|τ s

)q

dx)r/q dτ

τ.

By collecting these estimates we find∫ ∞

0

(∫ ∞

0

( |4τ (π0u)(x)|τ s

)q

dx)r/q dτ

τ≤ c

([u]s,q,r,J + ‖u‖q

)r. (7.3.21)

Note that x = κ−1(y) = T − y implies

4τ

(κ∗(π1u)

)(x) = (π14−τu)(T − y) +

(4−τπ1(y))u(T − y) .

We deduce from (7.3.11) and (7.3.12) that supp(y 7→ π1(T − y)

) ⊂ [0, 5T/8]. Fromthis and the smoothness of π1 we get

∫ ∞

0

(∫ ∞

0

( |4τ (κ∗(π1u))(x)|τ s

)q

dx)r/q dτ

τ

≤ c

∫ ∞

0

(∫ T

−∞

( |(π14−τu)(z)|τ s

)q

dz)r/q dτ

τ+ c ‖u‖r

q .


The first term on the right can be majorized by

c

∫ 3T/8

0

(∫ T

3T/8

( |4−τu(x)|τ s

)q

dx)r/q dτ

τ

+ c

∫ T

3T/8

(∫ T

3T/8

( |4−τ u(x)|τ s

)q

dx)r/q dτ

τ,

(7.3.22)

where u denotes the extension of u by zero over (−∞, 0). The second term isestimated from above by c ‖u‖r

q,J . Similarly as above, we find, setting y = x− τ ,

∫ T/2

3T/8

( |4−τu(x)|τ s

)q

dx =∫ −τ+T/2

−τ+3T/8

( |4τu(y)|τ s

)q

dy .

Using this, we obtain for the first term in (7.3.22) the upper bound

c

∫ T/2

0

(∫ T

T/2

( |4−τu(x)|τ s

)q

dx)r/q dτ

τ+ c

∫ T/2

0

(∫ T/2

0

( |4τu(x)|τ s

)q

dx)r/q dτ

τ.

Putting these estimates together, we arrive at∫ ∞

0

(∫ ∞

0

( |4τ (κ∗(π1u))(x)|τ s

)q

dx)r/q dτ

τ≤ c

([u]s,q,r,J + ‖u‖q

)r.

From this and (7.3.21) we see

Rc ∈ L(Bs

q,r(J,E), Bsq,r(R+, E2)

)(7.3.23)

if 0 < s < 1 and r < ∞. Obvious modifications of these arguments show that(7.3.23) holds for r = ∞ as well.

(4) Suppose 0 < s < 1 and v = (v0, v1) ∈ Bsq,r(R+, E2). Then

[RJv]s,q,r,J ≤ [π0v0]s,q,r,J + [π1κ∗1v1]s,q,r,J .

By means of (7.3.19) we find easily

[π0v0]s,q,r,J ≤ c([v0]s,q,r,R+ + ‖v0‖s,q,r,R+

)= c ‖v0‖Bs

q,r(R+,E) .

Similarly,[π1κ

∗1v1]s,q,r,J ≤ c ‖v1‖Bs

q,r(R+,E) .

This provesR ∈ L(

Bsq,r(R+, E2), Bs

q,r(J,E))

.

From this and (7.3.23) we get (iii) if 0 < s < 1.

(5) Let k < s < k + 1 for some k ∈qN. Then Leibniz’ rule and appropriate

modifications of the arguments of steps (3) and (4) establish assertion (iii) in thiscase also.


(6) Obvious modifications of the previous steps show that (RJ , RcJ) is an r-c

pair for(Bs∞(R+, E2), Bs

∞,∞(J,E)). This and Proposition 7.3.1(ii) imply the first

part of (iv). The validity of the second part is then clear. ¥

In the next corollary we use notation (7.0.2) and the r-e pair (RR+ , ER+)introduced in Section VI.1.

7.3.3 Corollary Suppose (X ,Y) equals (S ′,D′) or any of the pairs (i)–(iv). Then(i) Rc

J ∈ L(Y(J,E),XJ(R+, E2)).

(ii) (RJ RR+ , ER+ RcJ ) is an r-c pair for

(X (R, E2),Y(J,E)).

Proof The above proof shows that supp(RcJu) ⊂ [0, 5T/8] for u ∈ Y(J,E). This

gives (i). Assertion (ii) is obvious by the properties of (RR+ , ER+). ¥

Interpolations and Embeddings

Similarly as in Subsections 2.8 and 4.1 we define, in general, Besov spaces on J by

Bsq,r(J,E) := RJBs

q,r(R+, E2) , 1 ≤ q, r ≤ ∞ , (7.3.24)

and Bessel potential spaces on J by

Hsq (J,E) := RJHs

q (R+, E2) , 1 ≤ q < ∞ ,

for s ∈ R. This is justified since Theorem 7.3.2 guarantees that the ‘new definition’(7.3.24) gives for s ∈ R+\N the same spaces as definitions (7.3.3) and (7.3.4),except for equivalent norms.

7.3.4 Theorem The Besov and Bessel potential spaces on J possess the sameinterpolation and embedding properties as the corresponding spaces on R.

Proof This follows from Proposition I.2.3.2, Lemma 2.8.1, and Corollary 7.3.3. ¥

It is a consequence of these theorems that

Bsq,r(J,E) .=

(W k

q (J,E), W k+1q (J,E)

)s−k,r

(7.3.25)

for 1 ≤ q < ∞ and 1 ≤ r ≤ ∞, and

Cs(J,E) .=(Ck(J,E), Ck+1(J,E)

)s−k,∞ (7.3.26)

for k < s < k + 1 with k ∈ N. Similarly, if E is a UMD space,

Hsp(J,E) .=

[Hk

p (J,E),Hk+1p (J,E)

]s−k

, 1 < p < ∞ , (7.3.27)

and

Hkp (J,E) .= W k

p (J,E) , 1 < p < ∞ , (7.3.28)

for k < s < k + 1 and k ∈ N.


The embedding (7.3.2) implies a similar result for Sobolev–Slobodeckii andBessel potential spaces on J .

7.3.5 Theorem Suppose s ≥ 0 and 1 ≤ p0 < p1 < ∞. Then

Cs(J,E) → W sp1

(J,E) → W sp0

(J,E) .

If E is a UMD space and p0 > 1, then

Cs(J,E) → Hsp1

(J,E) → Hsp0

(J,E) .

Proof Assume k ∈ N. If s = k, then the assertion follows from (7.3.2). From thiswe get the claim for k < s < k + 1 by interpolation, due to (7.3.25)–(7.3.28). ¥

The Rellich–Kondrachov Theorem

As a consequence of Theorem 7.3.2 and its corollary and the compactness resultsof Subsection 7.2, it is easy to prove, in particular, a vector-valued version of theRellich–Kondrachov theorem on compact intervals.

7.3.6 Theorem Suppose E1 −→ E0.(i) If 1 ≤ q0, q1 < ∞, 0 ≤ s0 < s1, and s1 − 1/q1 > s0 − 1/q0, then

W s1q1

(J,E1) −→ W s0q0

(J,E0) and Hs1q1

(J,E1) −→ Hs0q0

(J,E0) .

(ii) Suppose 1 ≤ q < ∞ and s − 1/q > s0 ≥ 0. Then

W sq (J,E1) −→ cs0(J,E0) and Hs

q (J,E1) −→ cs0(J,E0) .

(iii) If 0 ≤ s0 < s1, then

Cs1(J,E1) −→ Cs0(J,E0) and cs1(J,E1) −→ cs0(J,E0) .

Proof Let the assumptions of (i) be satisfied. It follows from Theorem 7.3.2, itscorollary, E2

1 −→ E20 , and Theorem 7.2.5(i) that the following diagram

W s1q1 (J, E1) W s1

q1,J(R+, E21)

W s0q0 (J, E0) W s0

q0 (R+, E20)

RcJ

RJ

-

¾?

¤¡

?

¤¡¤¡

is commutative. Hence, by (7.2.1), the left vertical embedding is compact too. Thisproves the first part of (i).


The proofs of the other assertions follow analogously by also invoking Theo-rems 7.2.5(ii) and 7.2.4. ¥

Of course, J is a very simple instance of a compact Riemannian manifoldwith boundary. Spaces of functions and distributions on Riemannian manifolds aretreated in detail in Volume III. There the reader can find further results, dualitytheorems for example. For this reason we refrain here from presenting additionalproperties of function spaces on compact intervals.

7.4 Aubin–Lions Type Theorems

In this subsection we prove a rather sharp compact embedding theorem for theintersection of two Sobolev–Slobodeckii spaces.

The General Result

As before, we assume again that 0 < T < ∞ and put J := [0, T ]. Recall defini-tion (5.7.19).

7.4.1 Theorem Let (E0, E1) be an interpolation couple, s0, s1 ∈ R+ with s0 6= s1,q, q0, q1 ∈ [1,∞), and 0 < θ < 1. Suppose E1 −→ E0 and E is of class J(θ, E0, E1).

(i) If 0 ≤ s < sθ and s − 1/q < sθ − 1/q(θ), then

W s1q1

(J,E1) ∩W s0q0

(J,E0) −→ W sq (J,E) .

(ii) Assume 0 ≤ s < sθ − 1/q(θ). Then

W s1q1

(J,E1) ∩W s0q0

(J,E0) −→ cs(J,E) .

Proof (1) For 0 < ϑ < 1 and 1 ≤ r ≤ ∞ we put Eϑ,r := (E0, E1)ϑ,r. It followsfrom (I.2.7.2) (also see [BeL76, Corollary 3.8.2]) that

E1 −→ Eϑ,r −→ Eθ,1 , θ < ϑ < 1 .

From this, (7.2.1), and (5.7.19) we get

Eϑ,r −→ E , θ < ϑ < 1 , 1 ≤ r ≤ ∞ . (7.4.1)

(2) Let the hypotheses of (i) be satisfied. We fix ϑ ∈ (θ, 1), σj < sj withσj /∈ Z, and t > s with t /∈ N such that s < sϑ and

s − 1/q < t − 1/q < σϑ − 1/q(ϑ) . (7.4.2)

Then

W s1q1

(J,E1) ∩W s0q0

(J,E0) → Bσ1q1

(J,E1) ∩Bσ0q0

(J,E0) . (7.4.3)


Hence we obtain from (7.4.2) and Theorems 2.7.3 and 7.3.4

W s1q1

(J,E1) ∩W s0q0

(J,E0) → Btq(J,Eϑ,1) = W t

q (J,Eϑ,1) .

Since W tq (J,Eϑ,1) −→ W s

q (J,E) by (7.4.2) and Theorem 7.3.6(i), assertion (i) fol-lows.

(3) Assume 0 ≤ s < sθ − 1/q(θ). Fix σ0, σ1, t ∈ R+\N with σj < sj and selectϑ ∈ (θ, 1) such that s < t < σϑ − 1/q(ϑ). Then (7.4.3) applies. Hence we obtainfrom Theorem 2.7.3

W s1q1

(J,E1) ∩W s0q0

(J,E0) → bt∞(J,Eϑ,1) .

Since bt∞ = ct, we now get assertion (ii) from (7.4.1) and Theorem 7.3.6(iii). ¥

Limit Cases

In the limit case where the strict inequality signs in Theorem 7.4.1(i) are replacedby equality signs, we cannot expect compact embeddings anymore. However, un-der suitable additional restrictions, there is still a continuous embedding result.Recall that, given an interpolation couple (E0, E1), we set Eθ,r := (E0, E1)θ,r andE[θ] := [E0, E1]θ for 1 ≤ r ≤ ∞ and 0 < θ < 1.

7.4.2 Theorem Let (E0, E1) be an interpolation couple. Assume 0 ≤ s0 < s1 < ∞,q0, q1 ∈ [1,∞), and 0 < θ < 1.

(i) If s0, s1, sθ ∈ R+\N, then

W s0q0

(J,E0) ∩W s1q1

(J,E1) → W sθ

q(θ)(J,Eθ,q(θ)) ∩W sθ

q(θ)(J,E[θ]) .

(ii) Suppose s0, s1, sθ ∈ N, q0, q1 > 1, and E0 and E1 are UMD spaces. Then

W s0q0

(J,E0) ∩ W s1q1

(J,E1) → W sθ

q(θ)(J,E[θ]) .

Proof The first assertion follows from Theorem 5.6.6(i) and Remark 5.6.7, Propo-sition 7.3.1(i), and Theorem 7.3.4.

The second claim is a consequence of Theorem 5.6.6(ii), assertion (7.3.28),and Theorem 7.3.4. ¥

Applications

Now we consider some particular instances of Theorem 7.4.1 which are of impor-tance in concrete applications.


7.4.3 Theorem Suppose 0 < θ < 1 and

E1 −→ E0 , Eθ,1 → E → E0 . (7.4.4)

Assume

(α) K is bounded in Lq1(J,E1);

(β) 1 ≤ q0 ≤ q1 < ∞;

(γ) ∂u ; u ∈ K is bounded in Lq0(J,E0).

Then K is relatively compact in:

(i) W sq (J,E) if s − 1/q < 1 − θ − 1/q(θ),

(ii) cs(J,E) if 0 ≤ s < 1− θ − 1/q(θ).

Proof It follows from (α) and (β) that K is bounded in Lq0(J,E0). From thisand (γ) we get that it is bounded in W 1

q0(J,E0). Thus K is bounded in the intersec-

tion space Lq1(J,E1) ∩W 1q0

(J,E0). Since s0 = 1 and s1 = 0 results in sθ = 1 − θ,the assertion follows from Theorem 7.4.1. ¥

In our next application of Theorem 7.4.1 we weaken assumption (γ) by re-placing it with an equicontinuity condition on K.

7.4.4 Theorem Suppose 0 < θ < 1 and

E1 −→ E0 , Eθ,1 → E → E0 .

Assume

(α) K is bounded in Lq1(J,E1);

(β) 0 < σ < 1, 1 ≤ q0 ≤ q1 < ∞;

(γ) K is σ-equicontinuous in Lq0(J,E0), that is,

(∫ T−h

0

|u(x + h)− u(x)|q0E0

dx)1/q0 ≤ c hσ (7.4.5)

for 0 < h < T and u ∈ K.

Then K is relatively compact in:

(i) W sq (J,E) if 0 ≤ s < (1− θ)σ and s − 1/q < (1 − θ)σ − 1/q(θ),

(ii) cs(J,E) if 0 ≤ s < (1 − θ)σ − 1/q(θ).

Proof (1) Let 1 ≤ p < ∞ and 0 < ϑ < 1. Set

Iϑ,p(u) := sup0<τ<T

(∫ T−τ

0

( |4τu(x)|τϑ

)p

dx)1/p


for u ∈ Lp(J,E). Since∫ T

T/2

|4−τu(x)|p dx =∫ T−τ

T/2−τ

|4τu(y)|p dy ,

it is obvious that [·]ϑ,p,∞,J ≤ 21/pIϑ,p. The estimate Iϑ,p ≤ [·]ϑ,p,∞,J is equallyclear. Hence

[·]ϑ,p,∞,J ∼ Iϑ,p . (7.4.6)

(2) Due to Lq1(J,E1) → Lq0(J,E0), we infer from (α), (γ), Proposition 3.5.4,and (7.4.6) that K is bounded in Bσ

q0,∞(I, E0). We fix 0 < s0 < σ such that

s < (1 − θ)s0 and s − 1/q < (1 − θ)s0 − 1/q(θ)

if the conditions of (i) are satisfied, and s < (1− θ)s0 − 1/q(θ) if case (ii) is beingconsidered. Then Bσ

q0,∞(J,E0) → W s0q0

(J,E0) and

K is bounded in Lq1(J,E1) ∩ W s0q0

(J,E0).

Hence we get the assertion from Theorem 7.4.1 with s1 := 0. ¥

These compactness theorems are very useful in the theory of nonlinear evo-lution equations and are employed by numerous authors. In typical applications,(uk) is a sequence of approximate solutions to a nonlinear equation. If it is possi-ble to bound this sequence in Lp1(J,E1) and if one can bound the sequence (∂uk)in Lp0(J,E0), then the classical Aubin–Lions lemma gives sufficient conditionswhich allow to extract a subsequence converging in Lp(J,E). If it is then possibleto pass to the limit in the approximating equations, whose solutions are the uk,and the limiting equation coincides with the original one, then the existence of asolution to the original problem has been established. It is the advantage of theabove theorems over the classical Aubin–Lions lemma that they guarantee moreregularity than just membership in Lp(J,E). This often facilitates the passage tothe limit in nonlinear problems, or makes it even possible.

7.5 Notes

The proof of Theorem 7.1.2 is an adaption of the arguments of K. Yosida [Yos65,Theorem X.1].

Subsection 7.2 is of preparatory nature. The compact embedding theoremsproved there will be exploited in Volume III to derive compact embedding theoremson compact Riemannian manifolds.

The retraction-coretraction Theorem 7.3.2 is prototypical for the much morecomplex and sophisticated r-c theorems on uniformly regular Riemannian mani-folds proved in the next volume.


We can trace back the Aubin–Lions compactness lemma to J.-P. Aubin[Aub63] and J.-L. Lions [Lio69]. It was extended by J.A. Dubinskii [Dub65]. Infact, that author proved a slightly more general result by replacing the Lp1(J,E1)norm by a more general functional. A detailed study of compact sets in Banach-space-valued Lebesgue spaces on compact intervals has been undertaken by J. Si-mon [Sim87]. In particular, he removed unnecessary assumptions in the Aubin–Lions lemma and showed that the boundedness of the derivatives can be replacedby an equicontinuity assumption. In fact, he obtains the compactness theorems ofSubsection 7.4, but only for s = 0 (see [Sim87, Corollaries 8 and 9]) The limit caseresults of Theorem 7.4.2 are new.

So-called ‘nonlinear’ Aubin–Lions lemmas, that is, variants and extensionsof Dubinskii’s result, have been given by E. Maitre [Mai03] and, more recently,by J.W. Barrett and E. Suli [BaS12], X. Chen and J.-G. Liu [CL12], M. Dreherand A. Jungel [DrJ12], X. Chen, A. Jungel, J.-G. Liu [CJL14] and A. Moussa[Mou16]. However, in all those papers compactness is established only in Lq(J,E)or C(J,E).

Theorem 7.4.1 –– in fact, a multidimensional version thereof, in which J isreplaced by a bounded domain in Rd with a smooth boundary –– appears firstin [Ama00a]. In Volume III we give an extension to compact Riemannian manifolds.

VII.8 Parameter-Dependent Spaces 271

8 Parameter-Dependent Spaces

For establishing resolvent and related estimates in the next volume, it will some-times be convenient to deal with parameter-dependent Besov and Bessel potentialspaces. Such spaces are introduced in this section and their basic properties arederived. Particular attention is paid to the parameter-dependence of the norms ofvarious mappings, like embedding and Fourier multiplier operators, acting betweenparameter-dependent spaces.

We assume throughout this section:

• K is a corner in Rd .

• X ∈ Rd,K .

• s ∈ R, 1 ≤ q, r ≤ ∞, 1 < p < ∞ .

Recall that assumption (VI.3.1.20) is in force.

8.1 Sobolev Spaces and Bounded Continuous Functions

Recall the notations introduced in Section VI.3, in Subsection VI.3.3 in particular.For u ∈ S ′(X, E) we set

T sq,[η]u := [η]−s+|ω|/qσ[η]u , η ∈

qH .

SinceK is invariant under the dilation t 7→ t q x, it follows from Proposition VI.3.1.3and definition (VI.1.2.1) that

T sq,[η] ∈ Laut

(S ′(X, E))

, (T sq,[η])

−1 = T sq,1/[η] . (8.1.1)

Furthermore, T sq,[η] is well-defined on each linear subspace of S ′(X, E) which is

invariant under t 7→ σt.

In the next lemma we collect the most important properties of these linearoperators.The reader should bear definition (VI.3.4.3) in mind.

8.1.1 Lemma Assume η ∈q

H.

(i) T sq,[η] commutes with (RK, EK).

(ii) ∂α T sq,1/[η] = T s−α ppppω

q,1/[η] ∂α, α ∈ Nd.

(iii) ‖T sq,1/[η]u‖q = [η]s ‖u‖q, u ∈ Lq(X, E).

(iv) T 0q,1/[η] Js

η = Js T sq,1/[η].


Proof (1) It is clear that T sq,[η] commutes with ε`

i defined in (VI.1.1.6). Hence it

commutes with E`k and R`

k (see (VI.1.1.9) and (VI.1.1.14)). Now claim (i) is obviousfrom the definition of (RK, EK). Assertions (ii) and (iii) follow from PropositionVI.3.1.3(i) and (ii) and the invariance of K under t 7→ σt.

(2) Note that σ[η]Λsη = [η]sΛs

1. From this and Proposition VI.3.1.3(iii) wededuce for u ∈ S ′(Rd, E)

T 0q,1/[η](J

sηu) = [η]−|ω|/qσ1/[η]F−1(Λs

ηu)

= [η]|ω| (1−1/q)F−1(σ[η](Λs

ηu))

= [η]s+|ω| (1−1/q)F−1(Λs

1σ[η](Fu))

= [η]s−|ω|/qF−1(Λs

1F(σ1/[η]u))

= JsT sq,1/[η]u .

From this we get (iv). ¥

It follows from (ii) and (iii) that

‖∂α(T sq,1/[η]u)‖q = [η]s−α ppppω ‖∂αu‖q .

Consequently, given m ∈ νN,

|||Tmq,1/[η]u|||m/ν,q =

∑

α ppppω≤m

[η]m−α ppppω ‖∂αu‖q =: |||u|||m/ν,q;η (8.1.2)

for η ∈q

H. Now we define parameter-dependent Sobolev spaces by

Wm/νq;η (X, E) :=

(Wm/ν

q (X, E), |||·|||m/ν,q;η

), q < ∞ ,

and parameter-dependent spaces of bounded continuous functions by

BCm/νη (X, E) :=

(BCm/ν(X, E), |||·|||m/ν,∞;η

)

for m ∈ νN and η ∈q

H. Thus

Wm/νq;η (X, E) .= Wm/ν

q (X, E) , q < ∞ , (8.1.3)

and

BCm/νη (X, E) .= BCm/ν(X, E) (8.1.4)

for η ∈q

H. Also note that it follows from (1.2.3), (1.2.1), (1.1.6), and (8.1.1) that(cf. Remarks VI.2.2.1)

Wm/νq;η (X, E) = Tm

q,[η]Wm/ν

q (X, E) , q < ∞ , (8.1.5)

and

BCm/νη (X, E) = Tm

∞,[η]BCm/ν(X, E) (8.1.6)

for η ∈q

H.


We see from (8.1.4) that

BUCm/νη (X, E) :=

(BUCm/ν(X, E), |||·|||m/ν,∞;η

)

is a closed linear subspace of BCm/νη (X, E). Furthermore, since BUC(X, E) is

invariant under t 7→ σt,

BUCm/νη (X, E) = Tm

∞,[η]BUCm/ν(X, E)

for η ∈q

H.

Suppose Xη = Xη(X, E0) and Yη = Yη(X, E1) are parameter-dependent Ba-nach spaces. Then we write

A ∈ L(Xη,Yη) η-uniformly ,

if the norm of A can be bounded independently of η ∈q

H. Similarly, if X is a non-empty set and fη, gη : X → R+ for η ∈

qH, then fη and gη are uniformly equivalent,

in symbols fη ∼η

gη, if there exists a constant c ≥ 1 such that

gη/c ≤ fη ≤ cgη , η ∈q

H .

Furthermore, Xη.=ηYη means that Xη and Yη are equal except for uniformly equiv-

alent norms.

8.1.2 Lemma Suppose m ∈ νN. Then (RK, EK) is an η-uniform universal r-e pairfor

(i)(Wm/ν

q;η (Rd, E),Wm/νq;η (K, E)

), q < ∞ ,

(ii)(BCm/ν

η (Rd, E), BCm/νη (K, E)

),

(iii)(BUCm/ν

η (Rd, E), BUCm/νη (K, E)

).

Proof Set Fq = W if q < ∞ and let F∞ ∈ BC, BUC. Then we get from (8.1.1),Lemma 8.1.1(i), and Theorem 1.3.1

‖RKu‖F

m/νq;η (K,E)

= ‖Tmq,1/[η]RKu‖

Fm/νq (K,E)

= ‖RKTmq,1/[η]u‖F

m/νq (K,E)

≤ c ‖Tmq,1/[η]u‖F

m/νq (Rd,E)

= c ‖u‖F

m/νq;η (Rd,E)

,

where c is the norm of RK in L(F

m/νq (Rd, E), Fm/ν

q (K, E)). Analogously,

‖EKu‖F

m/νq;η (Rd,E)

≤ c ‖u‖F

m/νq;η (K,E)

,

η-uniformly. ¥


8.2 Besov and Bessel Potential Spaces

Motivated by (8.1.5) and (8.1.6), we define parameter-dependent Besov spaces by

Bs/νq,r;η(X, E) := T s

q,[η]Bs/νq,r (X, E) , η ∈

qH . (8.2.1)

Similarly, we introduce for 1 ≤ q < ∞ parameter-dependent Bessel potential spacesby

Hs/νq;η (X, E) := T s

q,[η]Hs/νq (X, E) , η ∈

qH . (8.2.2)

Note that Lemma 8.1.1(iii) implies

H0/νq;η (X, E) = Lq(X, E) , η ∈

qH . (8.2.3)

The following properties are almost obvious.

8.2.1 Proposition

(i) Parameter-dependent Besov and Bessel potential spaces enjoy η-uniformlythe same interpolation properties as the spaces B

s/νq,r (X, E) and H

s/νq (X, E).

(ii) Suppose F ∈ B, B, b, H. Then (RK, EK) is an η-uniform universal r-e pairfor

(F

s/νq,r;η(Rd, E), Fs/ν

q,r;η(K, E)).

(iii) (RK, EK) is an η-uniform universal r-e pair for(H

s/νq;η (Rd, E),Hs/ν

q;η (K, E))

if 1 ≤ q < ∞.

Proof (1) Assertion (i) is immediate from the fact that T sq,[η] is an isometric

isomorphism from the parameter-free onto the corresponding parameter-dependentspaces.

(2) Claims (ii) and (iii) follow by the arguments of the proof of Lemma 8.1.2and from Theorems 2.8.2 and 4.1.1, respectively. ¥

In the case where X = Rd, parameter-dependent Besov and Bessel potentialspaces can also be characterized by means of the parameter-dependent Besselkernels introduced in (VI.3.4.3). This will be useful for establishing embeddingproperties of parameter-dependent spaces in the next subsection.

8.2.2 Lemma

(i) Suppose F ∈ B, B, b. Then

Fs/νq,r;η(Rd, E) .=

ηJ−s

η F0/νq,r;η(Rd, E) .

(ii) If q < ∞, thenHs/ν

q;η (Rd, E) .=η

J−sη Lq(Rd, E) .


Proof Let F ∈ B, B, b. Then we deduce from Theorem 2.4.1 and part (iv) ofLemma 8.1.1 that

‖u‖F

s/νq,r;η

= ‖T sq,1/[η]u‖F

s/νq,r

.= ‖JsT sq,1/[η]u‖F

0/νq,r

= ‖T 0q,1/[η]J

sηu‖

F0/νq,r

= ‖Jsηu‖

F0/νq,r;η

= ‖u‖J−s

η F0/νq,r;η

for η ∈q

H. This proves (i). Assertion (ii) follows similarly from definition (4.1.1)and (8.2.3). ¥

The next proposition shows that we can characterize parameter-dependentspaces of negative order η-uniformly by duality.

8.2.3 Proposition With respect to 〈·, ·〉,(Bs/ν

q,r;η(X, E))′ .=

ηB−s/νq′,r′;η(X∗, E′)

if E is either reflexive or has a separable dual, and(Hs/ν

p;η (X, E))′ .=

ηH

−s/νp′;η (X∗, E′)

if E is ν-admissible.

Proof For u ∈ S(X∗, E) and v ∈ S(X, E′) we find by a change of variables

〈T−sq′,1/[η]v, T s

q,1/[η]u〉 =∫

X

⟨[η]−|ω|/q′σ1/[η]v, [η]−|ω|/qσ1/[η]u

⟩E

dx

=∫

Xσ1/[η]

(〈v, u〉E)[η]−|ω| dx = 〈v, u〉 .

From this the assertion follows by density, due to Theorems 2.3.1, 2.8.4, and4.4.2. ¥

Since the parameter-dependent spaces differ only by equivalent norms fromtheir parameter-free counterparts, they posses the same embedding properties asthe latter. However, these embeddings do not hold η-uniformly, in general.

8.2.4 Theorem Let s0, s1 ∈ R with s0 < s1, 1 ≤ r0, r1 ≤ ∞, and F ∈ B, B, b.Then

‖u‖F

s0/νq,r0;η(X,E)

≤ c [η]s0−s1 ‖u‖F

s1/νq,r1;η(X,E)

, η ∈q

H . (8.2.4)

If q < ∞, then

‖u‖H

s0/νq;η (K,E)

≤ c [η]s0−s1 ‖u‖H

s1/νq;η (K,E)

, η ∈q

H . (8.2.5)


Assume q0, q1 ∈ [1,∞] satisfy

s1 − |ω|/q1 = s0 − |ω|/q0 , (8.2.6)

then

Fs1/νq1,r;η(X, E) → Fs0/ν

q0,r;η(X, E) η-uniformly , (8.2.7)

and, provided 1 < q0, q1 < ∞,

Hs1/νq1;η (X, E) → Hs0/ν

q0;η (X, E) η-uniformly . (8.2.8)

Proof Observe that

T s0q,1/[η] = [η]s0−s1T s1

q,1/[η] , η ∈q

H .

Hence we get from Theorems 2.2.2, 2.2.4, and 2.8.3, and from definition (8.2.1)

‖u‖F

s0/νq,r0;η

= ‖T s0q,1/[η]u‖F

s0/νq,r

≤ c ‖T s0q,1/[η]u‖F

s1/νq,r1

= c [η]s0−s1 ‖T s1q,1/[η]u‖F

s1/νq,r1

= c [η]s0−s1 ‖u‖F

s1/νq,r1;η

.

This proves (8.2.4). Estimate (8.2.5) follows analogously from (8.2.2) and Corol-lary 4.1.2.

If (8.2.6) is satisfied, then

T s0q0,1/[η] = [η]s0−|ω|/q0σ1/[η] = T s1

q1,1/[η] .

Thus (8.2.7) is a consequence of (2.2.3) and Theorem 2.8.3. Assertion (8.2.8) isimplied by Theorem 5.6.5(ii). ¥

Derivatives are compatible with parameter-dependent spaces in the followingsense.

8.2.5 Theorem Suppose K is closed and α ∈ Nd.(i) If F ∈ B, B, b, then

∂α ∈ L(F(s+α ppppω)/ν

q,r;η (X, E), Fs/νq,r;η(X, E)

)η-uniformly .

(ii) If E is ν-admissible and 1 < p < ∞, then

∂α ∈ L(H(s+α ppppω)/ν

p;η (X, E), Hs/νp;η (X, E)

)η-uniformly .

Proof Lemma 8.1.1(ii) and Theorems 2.6.1, 2.8.6, and 4.3.1(i). ¥



We consider the decomposition X = X1 × · · · × X` associated with the weight sys-tem [`, d, ν] and use the notation introduced in the beginning of Subsection 3.5.Then we extend Theorems 3.6.2 and 4.6.1 to the parameter-dependent setting.

8.3.1 Proposition Suppose s > 0 and let M be a subset of L := 1, . . . , `.(i) Assume K is closed. Then

Bs/νq;η (X, E) .=

η

⋂

µ∈M

Lq

(Xµ, Bs/νµ

q;η (Xµ, E)) ∩

⋂

λ∈L\MBs/νλ

q;η

(Xλ, Lq(Xλ, E)

)

if q < ∞, and

Bs/ν∞;η(X, E) .=

η

⋂

µ∈M

B(Xµ, Bs/νµ∞;η (Xµ, E)

) ∩⋂

λ∈L\MBs/νλ∞;η

(Xλ, BUC(Xλ, E)

).

(ii) If E is ν-admissible , then

Hs/νp;η (X, E) .=

η

⋂

µ∈M

Lp

(Xµ,Hs/νµ

p;η (Xµ, E)) ∩

⋂

λ∈L\MHs/νλ

p;η

(Xλ, Lp(Xλ, E)

).

Proof For 1 ≤ i ≤ ` we denote by T sq,[η](i), resp. T s

q,[η](ı), the operator T sq,[η] acting

on Xi, resp. Xı. Then

T sq,[η] = T s

q,[η](i) T 0q,[η](ı) = T 0

q,[η](ı) T sq,[η](i) .

Now the assertions follow from (8.1.5) and (the analogue of) (8.1.6) with m = 0,definitions (8.2.1) and (8.2.2), and Theorems 3.6.2 and 4.6.1, resp. ¥

Similarly as the Sobolev spaces and spaces of bounded continuous functions,Besov–Slobodeckii and Besov–Holder spaces can also be characterized by explicitparameter-dependent norms. For this we prepare the following lemmas, using thenotations of Subsections 3.4 and 3.5.

8.3.2 Lemma Suppose ω = ν1 and s > m with m ∈ νN. Then

[∇m/νT sq,1/[η]u](s−m)/ν,q = [∇m/νu](s−m)/ν,q .

Proof Lemma 5.7.6 implies

[T sq,1/[η]u](s−m)/ν,q = [u](s−m)/ν,q .

Since∇m/ν T s

q,1/[η] = T s−mq,1/[η] ∇m/ν

by Lemma 8.1.1(ii), the assertion follows. ¥


8.3.3 Theorem Let K be closed and s > 0. Set

‖u‖∗s/ν,q;η := [η]s ‖u‖q +∑

i=1

[[u]]s/νi,q;i

and, if mi ∈ νiN with mi < s < mi + νi for 1 ≤ i ≤ `,

|||u|||∗s/ν,q;η :=∑

i=1

(mi/νi∑

j=0

[η]s−jνi ‖∇jxi

u‖q + [[∇mi/νixi u]](s−mi)/νi,q;i

).

Then ‖·‖∗s/ν,q;η and |||·|||∗s/ν,q;η are η-uniformly equivalent norms for Bs/νq;η (X, E).

Proof It follows from Theorem 3.4.1 and definition (8.2.1) that

‖u‖B

s/νq;η (X,E)

= ‖T sq,1/[η]u‖B

s/νq (X,E)

.= ‖T sq,1/[η]u‖∗s/ν,q

.= |||T sq,1/[η]u|||∗s/ν,q .

Now the assertion is implied by Lemmas 8.1.1(ii) and 8.3.2. ¥

8.4 Fourier Multipliers

By Remark VI.3.4.7(b), the parameter-dependent multiplier spaces Mη(Rd, E) areobtained by endowing M(Rd, E) with the norm15

a 7→ ‖a‖Mη:= max

α ppppω≤k(ν)‖Λα ppppω

η ∂αa‖∞ , η ∈q

H .

The following theorem shows that these spaces are well-adapted to parameter-dependent Besov and Bessel potential spaces.

8.4.1 Theorem

(i) Let a ∈ F ∈ B, B, b. Suppose aη ∈Mη

(Rd,L(E1, E0)

)for η ∈

qH. Then

aη(D) ∈ L(Fs/ν

q,r;η(Rd, E1), Fs/νq,r;η(Rd, E0)

)

and‖aη(D)‖ ≤ c ‖aη‖Mη , η ∈

qH .

(ii) Assume E is ν-admissible. If aη ∈Mη(Rd), then

aη(D) ∈ L(Hs/ν

p;η (Rd, E))

and‖aη(D)‖ ≤ c ‖aη‖Mη , η ∈

qH .

15For the results of this subsection, k(ν) can be replaced by 2 |ω|.


Proof First we note that Proposition VI.3.1.3(iii) implies

σ1/[η]

(aη(D)u

)= σ1/[η]F−1aηFu = [η]|ω|F−1σ[η](aηu)

= F−1(σ[η]aη)F(σ1/[η]u)= (σ[η]aη)(D)σ1/[η]u ,

whenever u is appropriately smooth. Thus

T sq,1/[η] aη(D) = (σ[η]aη)(D) T s

q,1/[η] , η ∈q

H .

From Remark VI.3.4.7(b) we know that ‖σ[η]aη‖M = ‖aη‖Mη . Hence, given theassumptions of (i), we deduce from Theorem 2.4.2

‖aη(D)u‖F

s/νq,r;η(Rd,E0)

= ‖T sq,1/[η]aη(D)u‖

Fs/νq,r (Rd,E0)

= ‖(σ[η]aη)(D)T sq,1/[η]u‖F

s/νq,r (Rd,E0)

≤ c ‖σ[η]aη‖M ‖T sq,1/[η]u‖F

s/νq,r (Rd,E1)

= c ‖aη‖Mη ‖u‖Fs/νq,r;η(Rd,E1)

for η ∈q

H. This proves (i). By invoking Theorem 4.2.5 we get (ii) analogously. ¥

Particularly important situations occur if we consider positively homogeneoussymbols.

8.4.2 Theorem Suppose Rez ≥ 0.(i) Let F ∈ B, B, b. If a ∈ H−z

(Z,L(E1, E0)

), then

aη(D) ∈ L(Fs/ν

q,r;η(Rd, E1), Fs/νq,r;η(Rd, E0)

)

and

‖aη(D)‖ ≤ c [η]−Rez ‖a‖H−z, η ∈

qH . (8.4.1)

(ii) Assume E is ν-admissible and a ∈ H−z(Z). Then

aη(D) ∈ L(Hs/ν

p;η (Rd, E))

, η ∈q

H ,

and estimate (8.4.1) applies also.

Proof It follows from Lemma VI.3.3.2(ii) that

Λα ppppωη (ξ) |∂αaη(ξ)| ≤ Λ−Rez

η (ξ) ‖a‖H−z ≤ [η]−Rez ‖a‖H−z , ξ ∈ Rd ,

for α q ω ≤ k(ν) and η ∈q

H. Consequently,

‖aη‖Mη ≤ c [η]−Rez ‖a‖H−z , η ∈q

H .

Now the assertions are implied by Theorem 8.4.1. ¥


8.4.3 Example Suppose Rez ≥ 0.

(i) If F ∈ B, B, b, then J−zη ∈ L(

Fs/νq,r;η(Rd, E)

)and

‖J−zη ‖L(F

s/νq,r;η)

≤ c [η]−Rez , η ∈q

H .

(ii) Suppose E is ν-admissible. Then J−zη ∈ L(

Hs/νp;η (Rd, E)

)and

‖J−zη ‖L(H

s/νp;η (Rd,E))

≤ c [η]−Rez , η ∈q

H .

Proof Example VI.3.3.9 and the preceding theorem. ¥

8.5 Notes

The observation that parameter-dependent L2 Sobolev spaces are very useful inthe study of resolvent constructions for elliptic and parabolic boundary value prob-lems is due to M.S. Agranovich and M.I. Vishik [AgV64]. Scalar-valued isotropicand parabolic anisotropic parameter-dependent L2 Bessel potential, hence (see(5.8.4)) Sobolev–Slobodeckii spaces have been extensively used by G. Grubb in nu-merous articles on a parameter-dependent Boutet de Monvel theory for pseudodif-ferential boundary value problems. This work and a functional calculus for suchproblems is well documented in [Gru96]. The parameter-dependent calculus is ex-tended by G. Grubb and N.J. Kokholm in [GruK93] to isotropic Bessel potentialand Besov spaces in the Lp setting for 1 < p < ∞.

The considerations of this section simplify and amplify corresponding resultsof [Ama09].

Chapter VIII

Traces and Boundary Operators

In the theory of elliptic and parabolic boundary value problems it is of fundamentalimportance to understand well the behavior of regular distributions on lower-dimensional manifolds. Such ‘trace theorems’ form the heart of this chapter whichconsists of two sections. The first one is concerned with various properties of(higher order) traces. The results developed there form the basis for the secondsection. In it we study boundary operators on half-spaces, herewith laying thefundament for the theory of boundary value problems for elliptic and parabolicdifferential equations.


281


282 VIII Traces and Boundary Operators

1 Traces

In connection with boundary value problems, trace operators, that is, restrictionsto lower-dimensional subspaces of Rd, play a fundamental role. In this section weinvestigate the behavior of trace operators of arbitrary order on the function spacesstudied in the preceding sections.

After preliminary considerations in the first subsection, the basic trace re-traction theorem is proved in Subsection 1.2. In Subsection 1.3 we turn to traceson closed half-spaces. The results developed there are the fundament for the char-acterization of spaces with vanishing traces, given in Subsection 1.4. In the nextsubsection we characterize spaces of functions with vanishing boundary data by the‘speed’ with which they approach zero. The last subsection is concerned with tracetheorems on corners. In particular, we prove an r-e theorem for general corners.

It should be observed that, except when duality arguments are involved, thereis no restriction on the nature of the Banach space E.

The reader is advised to recall the definitions and notations of weight sys-tems [`, d, ν] introduced at the beginning of Subsection VI.3.1. In particular,ν = LCM(ν).

Throughout this section it is always assumed that

• d = (1, dq), ν = (ν1, ν

q) .

• 1 ≤ q, r ≤ ∞ .(1.0.1)

We set νq:= LCM(ν

q) and write x = (y, x

q) for the general point of the (1, d

q)-

clustering Rd = R× Rd2 × · · · × Rd` . Also, 0 × Rd−1 is identified with Rd−1, ifconvenient. Moreover, we use –– without further ado –– the natural identificationu(y)(x

q) := u(x) for x = (y, x

q) ∈ R× Rd−1 = Rd and any function u from Rd into

some nonempty set.These notations are well-defined if d ≥ 2 and ` ≥ 2. In order to include also

the case d = 1 in what follows, we employ the following conventions :

• R0 := 0, and if F denotes any vector space ofE-valued distributions, then F(R0, E) := E.

• If d = 1, then dqand ν

qhave to be unconsidered and

obvious adaptions have to be employed.

(1.0.2)

For example, 0 × R0 is to be identified with R0 = 0.

1.1 Trace Operators

Suppose m ∈ N. Then the trace operator of order m (from Rd onto 0 × Rd−1) isdefined by

trm : Cm(R,D′(Rd−1, E)

) → D′(Rd−1, E) , u 7→ ∂mu(0) .

VIII.1 Traces 283

It is obviously a continuous linear map, and tr := tr0 is the trace operator. Notethat trm = tr ∂m.

Assume X is a LCS such that X → D′(Rd−1, E). Then Cm(R,X ) injectscontinuously into Cm

(R,D′(Rd−1, E)

). It is obvious that

trm ∈ L(Cm(R,X ),X )

. (1.1.1)

1.1.1 Lemma Suppose X ∈ C(Rd−1, E), Lq(Rd−1, E)

. Then1

C(R,X ) → S ′(Rd, E) → D′(Rd, E) . (1.1.2)

Proof (1) First we prove that

C(R,X ) → L1,loc(Rd, E) . (1.1.3)

Since C(R, C(Rd−1, E)

)= C(Rd, E), this is clear for X = C(Rd−1, E).

Let X := Lq(Rd−1, E) with q < ∞. It follows from C(R,X ) → L1,loc(R,X )and Fubini’s theorem that (1.1.3) applies. Thus let X := L∞(Rd−1, E). (Recall thatL∞(R,X ) 6= L∞(Rd, E); see Remark X.6.23(a) in [AmE08].) Assume f =

∑vjχIj

is a λ1-simple2 X -valued function. (We refer to Section X.1 in [AmE08] for detailson Banach-space-valued measurable functions.) Clearly, vjχIj is λd-measurable.Hence f has this property also. Let u ∈ C(R,X ) be λ1-measurable. There existsa sequence (uj) of λ1-simple X -valued functions converging λ1-a.e. towards u.Then (uj) is a sequence of λd-measurable E-valued functions converging λd-a.e.towards u. Thus u, too, is λd-measurable. Now it is clear that (1.1.3) holds in thiscase also.

(2) Suppose u ∈ C(R,X ) and ϕ⊗ ψ ∈ D(R) ⊗D(Rd−1) with supp(ϕ) con-tained in the compact subset K of R. Since, by step (1), u is a regular distribution,there exists k ≥ 2 such that

|〈u, ϕ ⊗ ψ〉Rd |E ≤∫

Rd

|(ϕ⊗ ψ)u|E dx

≤∫

R

∫

Rd−1|u(y, x

q)|E |ϕ(y)ψ(x

q)| dx

qdy

≤∫

R|ϕ(y)| ‖u(y, ·)‖q ‖ψ‖q′ dy

≤ c ‖u‖C(K,X ) q2,0(ϕ) qk,0(ψ) ≤ c ‖u‖C(K,X ) qk,0(ϕ⊗ ψ) .

(1.1.4)

By Theorem 1.8.1 of the Appendix we know that D(R)⊗D(Rd−1) is dense inS(Rd). From this and (1.1.4) we get the embedding C(R,X ) → S ′(Rd, E). Thelast inclusion of the statement is clear. ¥

1Recall that the spaces of continuous functions carry the compact-open topology. In the caseof locally convex target spaces, this is the topology of uniform convergence on compact sets.

2λn denotes the n-dimensional Lebesgue measure.


Let (ψk) be a ν-dyadic partition of unity on Rd subordinate to the ν-dyadicopen covering (Ωk) of Rd, induced by some bounded, symmetric, ν-starshaped,and open 0-neighborhood Ω (cf. Subsection VI.3.6). Then

ψk(D)u = F−1(ψku) ∈ C(Rd, E) , u ∈ S ′(Rd, E) . (1.1.5)

We infer from Lemma VI.3.6.2 and (1.1.2) that

u =∞∑

k=0

ψk(D)u in S ′(Rd, E) , u ∈ C(Rd, E) . (1.1.6)

Using (1.1.1), we see

n∑

k=0

tr(ψk(D)u

) ∈ C(Rd−1, E) → S ′(Rd−1, E) , u ∈ N . (1.1.7)

These considerations suggest the following Fourier representation of the trace op-erator.

1.1.2 Lemma Suppose

u ∈ C(R, Lq(Rd−1, E)

)and

∞∑

k=0

ψk(D)u exists in C(R, Lq(Rd−1, E)

).

Then

tru =∞∑

k=0

tr(ψk(D)u

)in Lq(Rd−1, E) . (1.1.8)

Proof Assumption and (1.1.6) imply u =∑∞

k=0 ψk(D)u in C(R, Lq(Rd−1, E)

).

Hence, by (1.1.1),

tru = tr limn→∞

n∑

k=0

ψk(D)u = limn→∞

trn∑

k=0

ψk(D)u

= limn→∞

n∑

k=0

tr(ψk(D)u

)=

∞∑

k=0

tr(ψk(D)u

)

in Lq(Rd−1, E). ¥

Now we can guarantee the existence of the trace operator on Besov andTriebel–Lizorkin spaces of sufficiently high order.

VIII.1 Traces 285

1.1.3 Theorem Suppose s > ν1/q. Then tr ∈ L(B

s/νq,r (Rd, E), Lq(Rd−1, E)

)and

tru =∞∑

k=0

tr(ψk(D)u

)in Lq(Rd−1, E) . (1.1.9)

Proof (1) We fix t with ν1/q < t < s. Then we infer from (VII.2.2.2) and Theo-rem VII.3.5.1 that

Bs/νq,r (Rd, E) → Bt/ν

q (Rd, E) → Bt/ν1q

(R, Lq(Rd−1, E)

).

By means of (VII.2.2.2), (VII.2.2.3), and (VII.2.6.7) we get for s0 := t− ν1/q > 0

Bt/ν1q

(R, Lq(Rd−1, E)

)→ B

s0/ν1∞,1

(R, Lq(Rd−1, E)

)→ B

0/ν1∞,1

(R, Lq(Rd−1, E)

)

→ BUC(R, Lq(Rd−1, E)

).

Thus

Bs/νq,r (Rd, E) → C

(R, Lq(Rd−1, E)

). (1.1.10)

(2) Let 0 < m < n. Then ψk(D)∑n

j=m ψj(D)u = 0 if k < m − 1 or k > n + 1.Moreover, the sum

∑nj=m ψk(D)ψj(D)u has, for each k ∈ N, at most 3 nonzero

terms. Thus it follows from (VII.2.1.1) that

∑

k

2tk∥∥∥ψk(D)

n∑

j=m

ψj(D)u∥∥∥

q≤ c

∞∑

k=m−1

2tk ‖ψk(D)u‖q .

From this, (VII.2.1.18), and Bs/νq,r → B

t/νq,1 we infer that

(∑nk=0 ψk(D)u

)is for

u ∈ Bs/νq,r a Cauchy sequence in B

t/νq,1 . Hence

∑∞k=0 ψk(D)u exists in B

t/νq,1 . Since

Bt/νq,1 → S ′, it follows from Lemma VI.3.6.2 that

∑∞k=0 ψk(D)u = u in B

t/νq,1 . Thus

(1.1.10) (with s and r replaced by t and 1, resp.) shows that∑∞

k=0 ψk(D)u existsin C

(R, Lq(Rd, E)

). Now the assertion follows from (1.1.10) and Lemma 1.1.2. ¥

1.1.4 Corollary Suppose 1 ≤ p < ∞ and s > ν1/p. Then

tr ∈ L(F s/ν

p,r (Rd, E), Lp(Rd−1, E))

and the representation (1.1.9) holds in Lp(Rd−1, E) for u ∈ Fs/νp,r (Rd, E).

Proof This follows from Fs/νp,r (Rd, E) → B

s/νp,∞(Rd, E) and the theorem. ¥


1.2 The Retraction Theorem

Now we improve Theorem 1.1.3 and its corollary considerably by determining theprecise range of the trace operator. In fact, we prove a much stronger retractionproperty.

1.2.1 Theorem Let (1.0.1) be true. Suppose

B ∈ B, b , m ∈ N , s > ν1(m + 1/q) .

Then trm is a universal retraction

from Bs/νq,r (Rd, E) onto B(s−ν1(m+1/q))/ν

ppppq,r (Rd−1, E)

and, if 1 < q < ∞,

from F s/νq,r (Rd, E) onto B(s−ν1(m+1/q))/ν

ppppq (Rd−1, E) .

There exists a universal3 map, trcm, sending

B(t−ν1(m+1/q))/νpppp

q,r (Rd−1, E) into Bt/νq,r (Rd, E)

and, if 1 < q < ∞,

B(t−ν1(m+1/q))/νpppp

q (Rd−1, E) into F t/νq,r (Rd, E)

for t ∈ R, such that trcm is a coretraction for trm satisfying trj trc

m = 0 for0 ≤ j < m.

This theorem is a consequence of the following series of propositions andlemmas. To simplify the presentation we introduce some abbreviations.

We write Bs/νq,r := Bs/ν

q,r (Rd, E) and Bs/νpppp

q,r := Bs/νpppp

q,r (Rd−1, E) for s ∈ R. Similarconventions apply to Triebel–Lizorkin spaces. The general point of R, resp. Rd−1,is denoted by x∗, resp. x

q. Furthermore, the superscript ∗ , resp. q , at the symbol

for a function indicates that the latter is defined on R, resp. Rd−1. In particular,‖·‖∗q := ‖·‖Lq(R,E) and ‖·‖ q

q := ‖·‖Lq(Rd−1,E).We make use of the fact that, by Lemmas VII.2.1.1 and VII.5.2.1, Besov and

Triebel–Lizorkin spaces are independent of the particular choice of Ω and ψ, exceptfor equivalent norms. Thus we can consider pairs (Ω, ψ) which are particularly welladapted to our purposes. For this we rely on Examples VI.3.2.1 and VI.3.6.1. Moreprecisely, we employ the non-reduced version [d,1, ω] of [`, d, ν] so that ω = ν. Wealso set

N := Nω , N∗(ξ∗) := N(ξ∗, 0) , Nq(ξ

q) := N(0, ξ

q) .

Note that N∗ and Nqare quasinorms on R and Rd−1, respectively.

3This means that it has a representation which is independent of s, p, q, r, and E.

VIII.1 Traces 287

1.2.2 Lemma Let m ∈qN and set4

Q(ξ) :=( d∑

j=1

ξ2mω/ωj

j

)1/2mω

, ξ ∈ Rd . (1.2.1)

Then Q is a quasinorm on Rd, and [Q ≤ R] is for each R > 0 a compact convex0-neighborhood in Rd.

Proof It suffices to prove the convexity assertion if d ≥ 2. We put a := 2mω/ω1

and ai := 2mω/ωi for 2 ≤ i ≤ d. Then we show that the function f , defined by

f(ξq) :=

(R2mω −

d∑

i=2

ξaii

)1/a

for ξq ∈ (R+)d−1 with f(ξ

q) > 0, is concave (from below).

Let j, k ∈ 2, . . . , k. We compute

∂kf(ξq) = −ak

af1−a(ξ

q)ξak−1

k

and

∂j∂kf(ξq) = −f1−a(ξ

q)[a − 1f(ξ q)

aj

a

ak

aξ

aj−1j ξak−1

k +ak(ak − 1)

aξak−2k δjk

].

Hence, given λ2, . . . , λd ∈ R,

∑

j,k

∂j∂kf(ξq)λjλk

= −f1−a(ξq)[a − 1f(ξ q)

(∑

j

aj

aλjξ

aj−1j

)2

+∑

k

ak(ak − 1)a

λ2kξak−2

k

]≤ 0 .

This proves that [Q ≤ R] ∩ [ ξ1 > 0, ξi ≥ 0, 2 ≤ i ≤ d ] is convex. Now the asser-tion follows from the symmetry of [Q ≤ R] under reflection on each one of thecoordinate hyperplanes. ¥

1.2.3 Corollary Let R > 0. Then [N ≤ R], [N∗ ≤ R], and [Nq ≤ R] are compact

convex 0-neighborhoods in Rd, R, and Rd−1, respectively.

Proof It is enough to observe that N, N∗, and Nqare of the form (1.2.1), where

m = 1, m = 1, and m = ω/ωq, respectively, with ω

q:= LCM(ω2, . . . , ωd). ¥

4Observe that ξj = ξj since ` = d.


1.2.4 Proposition Suppose B ∈ B, B, b and s > ν1/q. Then

tr ∈ L(Bs/νq,r ,B(s−ν1/q)/ν

ppppq,r ) .

Proof (1) We set Ω := [N < 1] and fix an Ω-adapted ψ ∈ D(Rd). We also fixη0 ∈ D(

[N < 4])

with η0 | [N ≤ 2] = 1 and put ηk := σ2−kη0 for k ∈ N. Then ηk issupported in [N < 2k+2] and equals 1 on [N < 2k+1].

Suppose

v ∈ C ∩ Lq , supp(v) ⊂ [N < 2k+1] . (1.2.2)

By the convolution theorem (cf. (1.9.26) and Remark 1.9.6(b) of the Appendix),v = F−1v = F−1(ηkv) = v ∗ F−1ηk. Hence

v(0, xq) =

∫

Rd−1

∫

Rv(−x∗, x

q− xq)F−1ηk(x∗, x

q) dx∗ dx

q.

Holder’s inequality gives

|v(0, xq)|E ≤

∫

Rd−1

(∫

R|v(−x∗, x

q− xq)|qE dx∗

)1/q

‖F−1ηk(·, x q)‖∗q′ dx

q,

where(∫R · · ·

)1/q has to be replaced by supR if q = ∞. From this and Young’sinequality it follows

‖v(0, ·)‖ qq ≤ ak,q ‖v‖q (1.2.3)

with

ak,q :=∫

Rd−1‖F−1ηk(·, x q

)‖∗q′ dxq.

Proposition VI.3.1.3(iii) implies F−1ηk = 2|ω| kσ2kF−1η0. Thus, using also part (ii)of that proposition and Fubini’s theorem,

ak,q = 2|ω| k∫

Rd−1‖(σ2kF−1η0)(·, x q

)‖∗q′ dxq= 2|ω| k−ω1k/q′−|ω pppp| ka0,q

= 2ω1k/qa0,q

for k ∈ N. Hence (1.2.3) implies

‖v(0, ·)‖ qq ≤ c 2ν1k/q ‖v‖q , k ∈ N , (1.2.4)

whenever v satisfies (1.2.2).

VIII.1 Traces 289

Let u ∈ Bs/νq,r . It follows from (VI.3.6.3), (1.1.5), and (VII.2.1.18) (also see

Example VI.3.6.1(a)) that v := ψk(D)u satisfies (1.2.2). Therefore we get from(1.2.4)

2k(s−ν1/q)∥∥tr

(ψk(D)u

)∥∥ qq≤ c 2ks ‖ψk(D)u‖q , k ∈ N .

Consequently,

∥∥(2k(s−ν1/q) ‖tr(ψk(D)u)‖ q

q

)∥∥`r≤ c ‖u‖

Bs/νq,r

(1.2.5)

for u ∈ Bs/νq,r .

(2) By Corollary 1.2.3, Kk := [N ≤ 2k+1] is a compact convex 0-neighborhoodin Rd. We denote by Hk its support function, that is,

Hk(ξ) := supη∈Kk

〈ξ, η〉 , ξ ∈ Rd .

Suppose u ∈ Bs/νq,r . Then the distribution ψku has its support in Kk. Thus

the Paley–Wiener–Schwartz theorem (cf. [Hor83, Theorem 7.3.1] or [Pet83, The-orem 8.8 in Chapter II], for example, and note that the proofs carry over to theE-valued setting) guarantees that vk := ψk(D)u is an E-valued entire analyticfunction on Cd := Rd + iRd, such that, for some n ∈ N and c ≥ 1,

|vk(ζ)|E ≤ c (1 + |ζ|)neHk(Imζ) , ζ ∈ Cd .

Writing ζ = (ζ∗, ζq) ∈ C× Cd−1, we get

|vk(0, ζq)|E ≤ c (1 + |ζ q|)neHk(0,Imζ

pppp) , ζ

q ∈ Cd−1 . (1.2.6)

We set Hqk(ξ

q) := Hk(0, ξ

q) and observe that H

qk is the support function of the

set Kk ∩(0 × Rd−1

), that is, of [N

q ≤ 2k+1]. By Corollary 1.2.3, the latter isa compact convex 0-neighborhood of Rd−1. Now we apply the sufficiency partof the Paley–Wiener–Schwartz theorem to deduce from (1.2.6) that there ex-ists a distribution wk on Rd−1 which is supported in [N

q ≤ 2k+1] and satisfies5

F−1d−1wk = vk(0, ·) = tr

(ψk(D)u

).

(3) We put Ωq:= [N

q< 1] and fix an Ω

q-adapted ψ

q ∈ D(Rd−1). Then wewrite

((Ω

qk)(ψ

qk)

)for the ω

q-dyadic open covering of Rd−1 induced by (Ω

q, ψ

q). By

Example VI.3.6.1(a) we know that supp(ψqj) ⊂ [2j−1 < N

q< 2j+1] for j ≥ 1. From

this and step (2) we get

ψqj(D

q)tr

(ψk(D)u

)= F−1

d−1(ψqjwk) = 0 , k + 1 < j − 1 . (1.2.7)

5Fn is the Fourier transform on Rn.


Hence we infer from Lemma VII.2.1.1, Theorem 1.1.3, (VII.2.1.1), and (1.2.7),setting t := s − ν1/q,

‖tru‖B

t/ν ppppq,r

≤ c∥∥(

2tj ‖ψ qj(D

q)tru‖ q

q

)∥∥`r

= c∥∥∥(2tj

∥∥ψqj(D

q)

∞∑

k=0

tr(ψk(D)u

)∥∥ qq

)∥∥∥`r

≤ c∥∥∥(2tj

∑

k≥j−2

∥∥ψqj(D

q)tr

(ψk(D)u

)∥∥ qq

)∥∥∥`r

≤ c∥∥∥(2tj

∞∑

k=0

∥∥tr(ψk+j−2(D)u

)∥∥ qq

)∥∥∥`r

= c∥∥∥( ∞∑

k=0

2(2−k)t2(k+j−2)t∥∥tr

(ψk+j−2(D)u

)∥∥ qq

)∥∥∥`r

≤ c

∞∑

k=0

2(2−k)t∥∥∥(2(k+j−2)t

∥∥tr(ψk+j−2(D)u

)∥∥ qq

)∥∥∥`r

.

Since t > 0, we obtain from this and (1.2.5)

‖tru‖B

(s−ν1/q)/ν ppppq,r

≤ c(t) ‖u‖B

s/νq,r

, u ∈ Bs/νq,r .

This proves the assertion for B = B.(4) For k, m ∈ N and u ∈ S(Rd, E) we set

pk,m(u) := supy∈R

supz∈Rd−1

maxβ∈Nd−1

|β|≤m

〈z〉k ‖∂(0,β)u‖E .

Then pk,m is a continuous seminorm on S(Rd, E).Let u = v ⊗ w ∈ S(R) ⊗ S(Rd−1, E). Then tru = v(0)w ∈ S(Rd−1, E) and

qk,m(tru) = |v(0)| qk,m(w) ≤ ‖v‖∞ qk,m(w) = pk,m(u) .

From this and the density of S(R)⊗ S(Rd−1, E) in S(Rd, E) (see Corollary 1.8.2of the Appendix) we get

tr ∈ L(S(Rd, E),S(Rd−1, E))

. (1.2.8)

Using the validity of the claim for B = B, it follows from (1.2.8) and defini-tion (VII.2.2.6) that the assertion applies to B = B.

(5) Suppose t > s. Then, due to step (3),

tr ∈ L(Ba/ν

q,r , B(a−ν1/q)/νpppp

q,r

), a ∈ s, t . (1.2.9)

By (VII.2.2.10), Bt/νq,r is dense in b

s/νq,r . Since b

(s−ν1/q)/νpppp

q,r is a closed linear subspaceof B

(s−ν1/q)/νpppp

q,r , we get the assertion for B = b from (1.2.9). ¥

VIII.1 Traces 291

1.2.5 Proposition Let 1 ≤ p < ∞ and s > ν1/p. Then

tr ∈ L(F s/νp,∞, B(s−ν1/p)/ν

ppppp ) .

Proof We use the notation of the preceding proof as well as definition (VII.5.1.4)of Tk.

(1) Suppose u ∈ Fs/νp,∞. By Corollaries VII.2.1.2 and 1.1.4,

tru =∞∑

k=0

ψk(D)u(0, ·) =∞∑

k=0

(ψk(D)

(χk(D)u

))(0, ·) (1.2.10)

in Lp(Rd−1, E).

We set uk := χk(D)u. Since (0, xq)− y = x − (

(x∗, 0) + y)

for y ∈ Rd, we get∣∣uk

((0, x

q) − y

)∣∣E≤ Λ2 |ω|

1

(2k q ((x∗, 0) + y

))Tkuk(x) , x ∈ Rd .

Hence, using (VII.5.1.6),∣∣uk

((0, x

q) − y

)∣∣E≤ cΛ2 |ω|

1

(2k q (x∗, 0)

)Λ2 |ω|

1 (2k q y)Tkuk(x) (1.2.11)

for x, y ∈ Rd and k ∈ N. Moreover, see Proposition VI.3.1.3,

(F−1ψk)σ2kΛ2 |ω|1 = 2k |ω|σ2k

(F−1ψΛ2 |ω|1

)

and F−1ψ ∈ S(Rd) imply ‖(F−1ψk)Λ2 |ω|1 (2k q )‖1 = c for k ∈

qN. From this and

(1.2.11) we infer

|ψk(D)uk(0, xq)|E =

∣∣∣∫

Rd

(F−1ψk)(y)uk

((0, x

q) − y

)dy

∣∣∣E

≤ cΛ2 |ω|1

(2k q (x∗, 0)

)Tkuk(x)

(1.2.12)

for x ∈ Rd and k ∈ N. Let

I∗k := [2−k−1 < N∗ ≤ 2−k] = [2−ν1(k+1) < |x∗| ≤ 2−ν1k] .

ThenΛ2 |ω|

1

(2k q (x∗, 0)

)=

(1 + (2kN∗(x∗))2ν1

)2 |ω|/2ν1 ≤ c

for x∗ ∈ I∗k and k ∈ N. Hence (1.2.12) implies

‖ψk(D)uk(0, ·)‖ qp ≤ c ‖Tkuk(x∗, ·)‖ q

p , x∗ ∈ I∗k , k ∈ N .

By integrating this inequality, raised to the power p, over I∗k and using |I∗k | ∼ 2−ν1k,we obtain

(‖ψk(D)uk(0, ·)‖ qp

)p ≤ c 2kν1

∫

I∗k

(‖Tkuk(x∗, ·)‖ qp

)pdx∗ , k ∈ N . (1.2.13)


Thus, recalling (VII.2.1.1),

∥∥ψqj(D

q)(ψk(D)uk(0, ·))

∥∥ qp≤ c ‖ψk(D)uk(0, ·)‖ q

p

≤ c 2kν1/p(∫

I∗k


)pdx∗

)1/p

for j, k ∈ N.(2) From (1.2.10) and (1.2.13) we find, due to (1.2.7),

‖ψ qj(D

q)tru‖ q

p ≤∞∑

k=j−2

∥∥ψqj(D

q)(ψk(D)uk(0, ·))

∥∥ qp

≤ c

∞∑

k=j−2

2kν1/p(∫

I∗k


)pdx∗

)1/p

for j ≥ 2. Thus

2(s−ν1/p)j ‖ψ qj(D

q)tru‖ q

p ≤ c∑

k≥j−2

2(s−ν1/p)(j−k)(∫

I∗k

(2sk ‖Tkuk(x∗, ·)‖ q

p

)pdx∗

)1/p

for j ≥ 2. We fix ε ∈ (0, s − ν1/p) and, using |I∗k | ≤ |I∗j−2| for k ≥ j − 2, estimatethe right side by

c( ∑

k≥j−2

2(s−ν1/p−ε)(j−k)p′)1/p′( ∑

k≥j−2

2ε(j−k)p

∫

I∗k

(2sk ‖Tkuk(x∗, ·)‖ q

p

)pdx∗

)1/p

≤ c(∫

I∗j−2

∫

Rd−1

(sup

k2sk |Tkuk(x∗, x

q)|E

)pdx

qdx∗

)1/p

.

Finally, we obtain

‖tru‖B

(s−ν1/p)/ν ppppp

≤ c( ∞∑

j=0

(2(s−ν1/p)j ‖ψ q

j(D)tru‖ qp

)p)1/p

≤ c( ∞∑

j=0

∫

I∗j

∫

Rd−1

∥∥(2skTkuk(x∗, x

q))∥∥p

`∞dx

qdx∗

)1/p

≤ c(∫

Rd

∥∥(2skTkuk(x)

)∥∥p

`∞dx

)1/p

= c ‖(2skTkuk)‖Lp(`∞) .

(3) Since supp(uk) = supp(χku) ⊂ 4 q Ωk, it follows from Lemma VII.5.1.4that

‖(2skTkuk)‖Lp(`∞) ≤ c∥∥(

M |2skχk(D)u|1/2E

)2∥∥Lp(`∞)

≤ c∥∥M |2skχk(D)u|1/2

E

∥∥2

L2p(`∞).

VIII.1 Traces 293

Now we apply Theorem VII.5.1.3 to infer that the last term is estimated fromabove by

c∥∥(

2skχk(D)u)‖Lp(`∞) ≤ c

∥∥(2skψk(D)u

)∥∥Lp(`∞)

= c ‖u‖F

s/νp,∞

.


The following considerations lead to the construction of a coretraction for trm

with the asserted properties, provided B ∈ B, b.

1.2.6 Lemma Suppose a ∈ D(R) and b ∈ D(Rd−1). Then

F−11 a⊗ b(D

q)v = F−1(a⊗ b Fd−1v)

for v ∈ S ′(Rd−1, E).

Proof Let ϕ := ϕ1 ⊗ ϕq ∈ S(R) ⊗ S(Rd−1). Then Fϕ = F1ϕ1 ⊗Fd−1ϕ

qby Fu-

bini’s theorem. Hence, see Theorem 1.8.4 in the Appendix,⟨F(F−1

1 a ⊗ b(Dq)v

), ϕ

⟩Rd =

⟨F−11 a ⊗ b(D

q)v,Fϕ

⟩Rd

=⟨F−1

1 a ⊗ b(Dq)v,F1ϕ1 ⊗Fd−1ϕ

q⟩Rd

= 〈F−11 a,F1ϕ1〉S(R)

⟨b(D

q)v,Fd−1ϕ

q⟩Rd−1

= 〈a, ϕ1〉S(R) 〈bFd−1v, ϕq⟩Rd−1

= 〈a ⊗ bFd−1v, ϕ1 ⊗ ϕq〉Rd = 〈a ⊗ bFd−1v, ϕ〉Rd .

Thus, by the density of S(R)⊗ S(Rd−1) in S(Rd) (see Corollary 1.8.2 in the Ap-pendix),

F(F−11 a ⊗ b(D

q)v

)= a ⊗ b Fd−1v .

This implies the claim. ¥

Now we fix η0 ∈ D(−1, 1) and η ∈ D(1, 2) satisfying

F−11 η0(0) = F−1

1 η(0) = 1 . (1.2.14)

For m ∈ N we put (recall (VI.3.1.6))

ηm0 := (−D)mη0/m! , ηm

j := (−D)mσν12−j η/m! , j ≥ 1 .

Given v ∈ S(Rd−1, E), we set

Kmv :=∞∑

j=0

2−jν1F−11 ηm

j ⊗ (ψ

qj(D

q)v

), (1.2.15)

whenever this series converges in S ′(Rd, E).


1.2.7 Lemma Suppose s ∈ R and B ∈ B, b. Set σ := s − ν1(m + 1/q) and letv ∈ Bσ/ν

ppppq,r . Then Kmv exists in B

s/νq,r if r < ∞, and in B

t/νq,1 , for any t < s, if

r = ∞. In either case, Kmv ∈ Bs/νq,r and Km ∈ L(Bσ/ν

ppppq,r ,Bs/ν

q,r ).

Proof (1) We put

Ω∗ := [N∗ < 1] , Ωq:= [N

q< 1] , Ω := Ω∗ × Ω

q.

We fix an Ω∗-adapted ψ∗ ∈ D(R), an Ωq-adapted ψ

q ∈ D(Rd−1), set ψ := ψ∗ ⊗ ψq,

and suppose v ∈ Bσ/ν

ppppq,r . Then we put

v := Fd−1v , vj := F−11 ηm

j ⊗ (ψ

qj(D

q)v

), uj := 2−jν1vj (1.2.16)

for j ∈ N.

By Lemma 1.2.6,

ψk(D)vj = F−1((ψ∗

k ⊗ ψqk)(ηm

j ⊗ ψqj v)

)= F−1

((ψ∗

kηmj ) ⊗ (ψ

qkψ

qj v)

).

Note that, for j ≥ 1,

supp(ηmj ) ⊂ (2jν1 , 2(j+1)ν1) ⊂ Ω∗

j , Ωqi ∩ Ω

qk = ∅ , |i− k| ≥ 2 .

Consequently,

ψk(D)vj = 0 , |j − k| ≥ 2 , j, k ∈ N . (1.2.17)

(2) Proposition VI.3.1.3(i)–(iii) imply

‖F−11 ηm

j ‖∗q = 2j(1−m−1/q)ν1 ‖F−11 ηm‖∗q , j ≥ 1 . (1.2.18)

Let Snv :=∑n

j=0 uj for n ∈ N. Then we infer from (1.2.16)–(1.2.18) and (VII.2.1.1)that

2ks ‖ψk(D)Snv‖q ≤ c

1∑

i=−1

2(k+i)s ‖uk+i‖q

≤ c

1∑

i=−1

2(k+i)σ ‖ψ qk+i(D

q)v‖ q

q

(1.2.19)

for 0 ≤ k ≤ n + 1 (letting ‖ψ q−1 := 0), and

2ks ‖ψk(D)Snv‖q = 0 , k ≥ n + 2 . (1.2.20)

VIII.1 Traces 295

(3) Assume r < ∞. It follows from (1.2.17), (1.2.19), and (1.2.20) that, given0 < ` < n,

‖Snv − S`v‖Bs/νq,r

≤ c∥∥(

2ks ‖ψk(D)(Snv − S`v)‖q

)∥∥`r

≤ c( n+2∑

j=`−1

(2σj ‖ψ q

j(Dq)v‖ q

q

)r)1/r

.(1.2.21)

Since∥∥(

2σj ‖ψ qj(D

q)v‖ q

q

)∥∥`r

< ∞, the right side converges to 0 as ` →∞. This

shows that (Snv) is a Cauchy sequence in Bs/νq,r . Hence Sv := Kmv =

∑∞j=0 uj

exists in Bs/νq,r .

(4) Let r = ∞. Fix t < s and set τ := t− ν1(m + 1/q) < σ. Then v ∈ Bτ/ν

ppppq,1 .

Thus Sv exists in Bt/νq,1 by step (3). From

2kt ‖ψk(D)(Snv − S`v)‖q ≤ c ‖Snv − S`v||Bt/νq,1

and (1.2.21) we infer that(ψk(D)Snv

)n∈N is, for each k ∈ N, a Cauchy sequence

in Lq. Hence, given k ∈ N, wk := limn→∞ ψk(D)Snv exists in Lq, hence in S ′. FromB

t/νq,1 → S ′ we infer that Snv → Sv in S ′. Furthermore, ψk(D) ∈ L(S ′) implies

ψk(D)Snv → ψk(D)Sv in S ′. Thus wk = ψk(D)Sv and(ψk(D)Snv

)n≥1

convergesin Lq towards ψk(D)Sv.

(5) Now let 1 ≤ r ≤ ∞. Due to (3) and (4), we can take the limit as n →∞in (1.2.19) to deduce

2ks ‖ψk(D)Sv‖q ≤ c

1∑

i=−1

2(k+i)σ ‖ψ qk+i(D

q)v‖ q

q , k ∈ N .

These estimates imply∥∥(

2ks ‖ψk(D)Sv‖q

)∥∥`r≤ c

∥∥(2σk ‖ψ q

k(Dq)v‖ q

q

)∥∥`r≤ c ‖v‖

Bσ/ν ppppq,r

,

where c is independent of v. Hence Sv ∈ Bs/νq,r , the map v 7→ Sv is linear, and

S ∈ L(Bσ/νpppp

q,r , Bs/νq,r ). Since S = Km, this proves the assertion if B = B.

(6) We consider the case B = b. Due to Lemma VII.2.2.3, we can assumer = ∞. Let v ∈ b

σ/νpppp

q,∞ and ε > 0. By Theorem VII.2.2.4 there is a w ∈ B(1+σ)/ν

ppppq,1

satisfying‖v − w‖ q

Bσ/ν ppppq,∞

< ε/‖Km‖ ,

where ‖Km‖ is the norm in L(Bσ/νpppp

q,∞ , Bs/νq,∞). Thus

‖Kmv − Kmw‖B

s/νq,∞

< ε .


Since Kmw ∈ B(1+s)/νq,1 by step (5), we see that Kmv belongs to the closure of

B(1+s)/νq,1 in B

s/νq,∞. Hence Kmv ∈ b

s/νq,∞ and Km maps b

σ/νpppp

q,∞ continuously into bs/νq,∞.


1.2.8 Lemma Suppose 1 < q < ∞, m ∈ N, and σ := s − ν1(m + 1/q). Then Km

belongs to L(Bσ/νpppp

q , Fs/νq,1 ).

Proof By the preceding lemma we know that the series (1.2.15) exists in Bs/νq ,

hence in S ′.Let v ∈ B

σ/νpppp

q . It follows from (1.2.17) that

ψk(D)Kmv =1∑

i=−1

2−(k+i)ν1ψk(D)(F−1

1 ηmk+i ⊗ (ψ

qk+i(D

q)v)

)

=1∑

i=−1

2−(k+i)ν1F−11 (ψ∗

kηmk+i) ⊗ ψ

qk(D

q)ψ

qk+i(D

q)v .

(1.2.22)

Observe that

|F−11 (ψ∗

kηmk+i)| ≤ c12(k+i−m)ν1(1 + 2(k+i)ν1 |·|∗)−2 , k ≥ 0 ,

where, for k ≥ 1,

max−1≤i≤1

∥∥F−11

((σν1

2i ψ∗)(−D)mζ)〈·〉2∥∥∞ ≤ c

with ζ ∈ η0, η (recall the definition of an Ω∗-adapted ψ∗).From this and (1.2.22) we infer, recalling (VII.2.1.1), that

‖Kmv‖q

Fs/νq,1

=∥∥∥

∞∑

k=0

2ks |ψk(D)Kmv|E∥∥∥

q

q

≤ c

∫

R

( ∞∑

k=0

2k(s−mν1)(1 + 2kν1 |·|∗)−2 ‖ψ qk(D

q)v‖ q

q

)q

dx∗

≤ c

∫

R

( ∞∑

k=0

2kν1/q(1 + 2kν1 |·|∗)−2ak

)q

dx∗ ,

where ak := 2kσ ‖ψ qk(D

q)v‖ q

q.

We set I := [ |·|∗ > 1] and Ij := [2−(j+1)ν1 < |·|∗ < 2−jν1 ]. Then we decom-pose the last integral to obtain

∫

R( · · · )q dx∗ =

(∫

I

+∞∑

j=0

∫

Ij

)( · · · )q dx∗ =: A + B .

VIII.1 Traces 297

We estimate

|A| ≤ c

∫

I

( ∞∑

k=0

2−kν1(2−1/q)ak

)q dx∗

|x∗|2q≤ c

∞∑

k=0

(2−kν1(2−1/q)ak

)q

≤ c

∞∑

k=0

aqk .

As for B,

|B| ≤ c

∞∑

j=0

2−jν1

(( j∑

k=0

2kν1/qak

)q

+( ∞∑

k=j+1

2kν1/q 2−2(k−j)ν1ak

)q)

=: B1 + B2 .

By multiplying each term of the inner sum of B1 by 2−(j−k)ν1/2q 2(j−k)ν1/2q wefind, by q ≥ 1,

B1 ≤ c

∞∑

j=0

2−jν1

j∑

k=0

2(j−k)ν1/22kν1aqk = c

∞∑

k=0

∞∑

j=k

2−jν1/2 2−kν1/2aqk

= c

∞∑

k=0

2−kν1/2aqk

∞∑

j=k

2−jν1/2 ≤ c

∞∑

k=0

aqk .

Finally, by Holder’s inequality,

B2 = c

∞∑

j=0

2−jν1

( ∞∑

k=j+1

2kν1/q 2−2(k−j)ν1ak

)q

≤ c

∞∑

j=0

2−jν1

∞∑

k=j

2kν1−(k−j)ν1qaqk

= c

∞∑

k=0

aqk

k∑

j=0

2−(k−j)ν1(q−1) ≤ c

∞∑

k=0

aqk .

Thus the estimates for A, B1, and B2 imply

‖Kmv‖F

s/νq,1

≤ c ‖v‖B

σ/ν ppppq

, v ∈ Bσ/νpppp

q .


Proof of Theorem 1.2.1 (1) Let s > ν1(m + 1/q). It follows from Theorem VII.2.6.1and Proposition 1.2.4 that

trm = tr ∂m1 ∈ L(Bs/ν

q,r ,B(s−ν1(m+1/q))/νpppp

q,r ) .

Similarly, Theorem VII.5.5.1(i), Proposition 1.2.5, and Fs/νq,r → F

s/νq,∞ imply

trm ∈ L(F s/νq,r , B(s−ν1(m+1/p))/ν

ppppq ) , 1 ≤ q < ∞ .

(2) First we observe (cf. (III.4.2.2) and (III.4.2.4)) that

F−11

((−D)mw

)= (2π)−1F

1

((−D)mw

)= (2π)−1xmF

1w = xmF−11 w


for w ∈ S(R). Hence, recalling Proposition VI.3.1.3(iii),

F−11 ηm

j =2ν1jxm

m!σν1

2j F−11 ζ , j ∈ N ,

where ζ := η if j ≥ 1, and ζ := η0 for j = 0. From (1.2.14) and trk = tr ∂k1 it thus

follows

trk

(2−jν1F−1

1 ηmj ⊗ (ψ

qj(D

q)v)

)= δk,mψ

qj(D

q)v , 0 ≤ k ≤ m . (1.2.23)

We suppose v ∈ Bσ/νpppp

q,r and s > t > ν1(m + 1/q). Then we know from Lemma 1.2.7that the series in (1.2.15) converges in Bt/ν

q,1 . From this, step (1), and (1.2.23) weinfer

trk(Kmv) =∞∑

j=0

trk

(2−jν1F−1

1 ηmj ⊗ (ψ

qj(D

q)v)

)= δk,m

∞∑

j=0

ψqj(D

q)v = δk,mv

for 0 ≤ k ≤ m, where the last equality is a consequence of Lemma VI.3.6.2. Thisand the last part of Lemma 1.2.7 show that trc

m := Km is a continuous right inverseon B

(s−ν1(m+1/q))/νpppp

q,r for trm if s > ν1(m + 1/q). Thus the theorem is proved in thecase of Besov spaces. Lemma 1.2.8 and the embedding F

s/νp,1 → F

s/νp,r guarantee

that Km maps B(s−ν1(m+1/p))/ν

ppppp continuously into F

s/νp,r . Thus the assertion holds

for Triebel–Lizorkin spaces also. ¥

By means of the ‘sandwich theorem’ of Subsection VII.5.5 we can now showthat the r-c pair (trm, trc

m) of Theorem 1.2.1 is an r-c pair for Sobolev and Besselpotential spaces too. Recall the definition of Slobodeckii spaces (VII.3.6.3).

1.2.9 Theorem Suppose 1 ν1(m + 1/p), then (trm, trcm) is an r-c pair for

(W k/ν

p (Rd, E), B(k−ν1(m+1/p))/νpppp

p (Rd−1, E))

. (1.2.24)

(ii) Let s > ν1(m + 1/p). Then (trm, trcm) is an r-c pair for

(Hs/ν

p (Rd, E), B(s−ν1(m+1/p))/νpppp

p (Rd−1, E))

.

Furthermore,

trcm ∈ L(

B(t−ν1(m+1/p))/νpppp

p (Rd−1, E), Ht/νp (Rd, E)

)

for t ∈ R.

Proof This is immediate from Theorems VII.5.5.2 and 1.2.1. ¥

VIII.1 Traces 299

1.2.10 Corollary Let k ∈ νN satisfy k > ν1(m + 1/p) and suppose that eitherk − ν1(m + 1/p) /∈ ν

qN or E is νq-admissible. Then (1.2.24) can be replaced by

(W k/ν

p (Rd, E),W (k−ν1(m+1/p))/νpppp

p (Rd−1, E))

.

Proof If t := k − ν1(m + 1/p) /∈ νqN, then B

t/νpppp

p.= W

t/νpppp

p by (VII.3.6.3). Other-wise, this equivalence is implied by Theorem VII.4.3.2. ¥

1.3 Traces on Half-Spaces

We consider first the most important spaces, namely the scales of Bessel potential,Sobolev–Slobodeckii, Holder, and little Holder spaces. For the sake of a uniformnotation, we leave out Besov spaces B

s/νq,r with q 6= r and Triebel–Lizorkin spaces.

Afterwards, general Besov spaces are easily included by interpolation.

We write Ft/νq , t ∈ R, to denote either one of the spaces

• Bt/νq , 1 ≤ q ≤ ∞ ,

• bt/νq , 1 ≤ q ≤ ∞ ,

• W t/νq , 1 < q < ∞ , t ∈ νN ,

• Ht/νq , 1 < q < ∞ ,

(1.3.1)

where we recall thatbt/νq = Bt/ν

q , 1 ≤ q < ∞ .

We denote byH = Hd := R+ × Rd−1

the closed right half-space in Rd, that is, H is the standard closed 1-corner in Rd. Fur-thermore, we identify ∂H naturally with Rd−1, if convenient, and n = n1 = n∂His the inner (unit) normal (for H) on ∂H, defined by n(x

q) = e1 = (1, 0, . . . , 0) ∈ Rd

for xq ∈ ∂H.Suppose j ∈ N and s > s0 > ν1(j + 1/q). Set

• X := Lq if q < ∞ and X := BUC if q = ∞ . (1.3.2)

It follows from Theorems VII.2.6.5, VII.2.8.3, and VII.3.6.2 that

Bs0/νq (H, E) → Bs0/ν1

q

(R+,X (∂H, E)

)→ Cj

(R+,X (∂H, E)

). (1.3.3)

Furthermore, by the definition of bs/νq and Theorems VII.2.2.2, VII.2.6.5, VII.2.8.3,

VII.4.1.3, Corollary VII.4.1.2,

Fs/νq (H, E) → Bs0/ν

q (H, E) .


From this we deduce that the trace operator of order j on ∂H, γj = γj∂H = ∂j

n (inthe direction of the interior normal) is defined for u ∈ F

s/νq (H, E) by

γju := ∂ju|x1=0 = ∂ju(0, ·) ∈ X (∂H, E) .

We set

γ = γ∂H := γ0 = ∂0n (1.3.4)

and call it trace operator on ∂H. Note

γj = γ ∂j1 , (1.3.5)

and γj maps Fs/νq (H, E) continuously into X (∂H, E). The following theorem ex-

hibits its precise image.For abbreviation, we set

t(j, q) := t− ν1(j + 1/q)

and

∂Ft/νpppp

q :=

Bt/ν

pppp∞ , if q = ∞ and Ft/ν

∞ = Bt/ν∞ ,

bt/νpppp

q in all other cases ,(1.3.6)

for t ∈ R. Moreover, ω = (ω1, ωq) = (ν1, ω

q).

Using these notations, we can prove the following fundamental boundary re-

traction theorem for half-spaces.

1.3.1 Theorem Suppose j ∈ N and s > ν1(j + 1/q). Let Fs/νq satisfy (1.3.1). Then

γj is a universal retraction from

Fs/νq (H, E) onto ∂Fs(j,q)/ν

ppppq (∂H, E) .

There exists a universal map

(γj)c ∈ L(∂Ft(j,q)/ν

ppppq (∂H, E), Ft/ν

q (H, E))

(1.3.7)

for t ∈ R. It is a coretraction for γj if t > ν1(j + 1/q) and satisfies γi (γj)c = 0for 0 ≤ i < j.

It commutes with ‘tangential derivatives’, that is, given t ∈ R and β ∈ Nd−1,the diagram

(γj)c

(γj)c

∂Ft(j,q)/ν

ppppq (∂H, E) F

t/νq (H, E)

∂F(t(j,q)−β ppppω pppp

)/νpppp

q (∂H, E) F(t−β ppppω pppp

)/νq (H, E)

∂βx pppp ∂β

x pppp

-

-? ?

(1.3.8)

is commutative, provided E is ν-admissible if 6 F = H.6Recall that then W

t/νp

.= H

t/νp for t ∈ νN by Theorem VII.4.3.2.

VIII.1 Traces 301

Proof We know from Theorems VII.1.3.1, VII.2.8.2 and VII.4.1.1 that (RH, EH)is a universal r-e pair for

(F

s/νq (Rd, E), Fs/ν

q (H, E)). From this and Theorems 1.2.1

and 1.2.9 we deduce that

trj EH ∈ L(Fs/ν

q (H, E), ∂Fs(j,q)/νpppp

q (∂H, E))

.

Suppose u ∈ Fs/νq (H, E). Then v := EHu ∈ F

s/νq (Rd, E). Since the derivative of a

continuously differentiable function equals its right derivative, we find

trj EHu = trjv = ∂j1v|x1=0 = ∂j

1(RHv)|x1=0

= ∂j1(RHEHu)|x1=0 = ∂j

1u|x1=0 = γju ∈ X (∂H, E) .(1.3.9)

Thusγj ∈ L(

Fs/νq (H, E), ∂Fs(j,q)/ν

ppppq (∂H, E)

).

We define (γj)c := RH trcj . Theorems 1.2.1 and 1.2.9 imply (1.3.7). If 0 ≤ i ≤ j,

then it follows from (1.3.9) and Theorems 1.2.1 and 1.2.9 that

γi (γj)cg = ∂i1(RH trc

jg)|x1=0 = tri(trcjg) = δijg

for g ∈ ∂Fs(j,q)/ν

ppppq (∂H, E).

It is clear that ∂βx pppp commutes with RH. Hence it suffices to show that ∂β

x pppp com-mutes with (trj)c = Kj . It follows from Lemma 1.2.6 that

∂βx pppp(2−iν1F−1

1 ηji ⊗ ψ

qi(D

q)v

)= 2−iν1F−1

1 ηji ⊗ ∂β

x ppppψ qi(D

q)v .

Since ∂βx pppp = (iD

q)β commutes with ψ

qi(D

q), assertion (1.3.8) follows from (1.2.15)

and Lemmas 1.2.7 and 1.2.8, by also using Theorems VII.2.8.6 and VII.4.3.1(i). ¥

Now it is not difficult to deduce from this the following general boundary

retraction theorem, the main result of this subsection.

1.3.2 Theorem Suppose k ∈ N and s > ν1(k + 1/q). Let Fs/νq satisfy (1.3.1). Then

~γk := (γ0, . . . , γk) = (∂0n, . . . , ∂k

n)

is a universal retraction from

Fs/νq (H, E) onto ∂Fs(~k,q)/ν

ppppq (∂H, E) :=

k∏

j=0

Fs(j,q)/νpppp

q (∂H, E) . (1.3.10)

It possesses a universal coretraction (~γk)c.

If β ∈ Nd−1 and s > ν1(k + 1/q) + β q ω q, then (~γk)c commutes with ∂β

x pppp , pro-vided E is ν-admissible if F = H.


Proof For abbreviation, we set

X s := Fs/νq (H, E) , Ys

j := ∂Fs(j,q)/νpppp

q (∂H, E) . (1.3.11)

Then ~γk ∈ L(X s,∏k

j=0 Ysj

)by the preceding theorem.

Suppose (v0, . . . , vk) ∈ ∏ki=0 Ys

i . Set u0 := (γ0)cv0 ∈ X s. Let 1 ≤ j ≤ k andassume u0, . . . , uj−1 are already defined. Put

uj := uj−1 + (γj)c(vj − γjuj−1) = pjuj−1 + (γj)cvj ,

where pj := 1 − (γj)cγj ∈ L(X s). Then

uk = pk · · · p1(γ0)cv0 + pk · · · p2(γ1)cv1 + · · · + (γk)cvk . (1.3.12)

Note that γjpj = 0 and γipj = γi for 0 ≤ i < j. From this we infer γjuk = vj for0 ≤ j ≤ k. Hence, setting (~γk)c(v0, . . . , vk) := uk, we see that (~γk)c is a universalcoretraction for ~γk.

Let s > ν1(k + 1/q) + β q ω qso that s > ν1(j + 1/q) + β q ω q

for 0 ≤ j ≤ k.Since ∂β

x pppp ∈ L(X s,X s−β ppppω pppp) and γj commutes with ∂β

x pppp , we get from Theorem 1.3.1that

∂βx ppppγj = γj∂β

x pppp ∈ L(X s,Ys−β ppppω pppp) .

Hence(γj)cγj∂β

x pppp ∈ L(X s,X s−β ppppω pppp) , 0 ≤ j ≤ k .

This and the commutativity of (γj)c and ∂βx pppp imply

∂βx pppppj = pj∂

βx pppp ∈ L(X s,X s−β ppppω pppp

) , 0 ≤ j ≤ k .

Now the second part of the assertion is a consequence of the representation (1.3.12)of (~γk)c. ¥

Parameter-Dependence

The following technical considerations elucidate the behavior of(~γk, (~γk)c

)on

parameter-dependent spaces. For this we make use of (VII.8.1.5) and definitions(VII.8.2.1) and (VII.8.2.2).

We denote by σqt = σν

ppppt the action of the multiplicative group

((0,∞), q) as-

sociated with the dilation t 7→ t qνpppp ξ q

on ∂H (see (VI.3.1.6)).

1.3.3 Theorem Suppose k ∈ N and s > ν1(k + 1/q). Then ~γk is an η-uniform uni-

versal retraction from Fs/νq;η (H, E) onto ∂F

s(~k,q)/νpppp

q;η (∂H, E). Let (~γk)c be a universalcoretraction for ~γk for the case η = 1 and set

(~γk)cη(v0, . . . , vk) := σ[η]

((~γk)cσ

q1/[η](v

0, [η]−ν1v1, . . . , [η]−kν1vk))

. (1.3.13)

VIII.1 Traces 303

Then(~γk)c

η ∈ L(∂Fs(~k,q)/ν

ppppq;η (∂H, E), Fs/ν

q;η (H, E))

η-uniformly ,

and it is a universal coretraction for ~γk on Fs/νq;η (H, E).

Proof Setq

T sq,[η] := [η]−s+|ω pppp|/qσ

q[η] and use the notation (1.3.11) also for the

parameter-dependent case.

(1) Let 0 ≤ j ≤ k. We define a surjection γjη ∈ L(X s

η ,Ysj;η) by the commuta-

tivity of the diagram

T sq,1/[η]

∼=

Ts(j,q)

q,[η]

∼=

X sη X s

Ysj;η Ys

j

γjη γj

-

¾? ?

(1.3.14)

Then, for v ∈ X sη ,

γjηv = [η]−s(j,q)+|ω pppp|/qσ

q[η]

(γj([η]s−|ω|/qσ[1/η]v)

)

= [η]ν1jσq[η]

(γj(σ[1/η]v)

)= γjv ,

(1.3.15)

due to γj(σ[1/η]v) = [η]−jν1σq1/[η]γ

jv. From this and the fact that the horizontalarrows represent isometric isomorphisms we see that ~γk is an η-uniform continuoussurjection from X s

η onto∏k

j=0 Ysj;η.

(2) We define (~γk)cη by the commutativity of the diagram

diagq

Ts(j,q)

q,1/[η]

∼=

T sq,[η]

∼=

kj=0 Ys

j;ηkj=0 Ys

j

X sη X s

(~γk)cη (~γk)c

-

¾? ?

Given (v0, . . . , vk) ∈ ∏kj=0 Ys

j;η, it follows that (~γk)cη is given by the right side

of (1.3.13). Now the assertion follows from the result of step (1), (1.3.14), and(1.3.15). ¥

General Besov spaces

Finally, we indicate the easy extension of the findings of this section to generalBesov spaces. For this we can be rather brief.


1.3.4 Theorem Assume 1 ≤ q, r ≤ ∞. Let Fs/νq,r ∈ Bs/ν

q,r , bs/νq,r and set

∂Ft/νpppp

q,r :=

Bt/ν

ppppq,r , if Ft/ν

q,r = Bt/νq,r ,

bt/νpppp

q,r , if Ft/νq,r = bt/ν

q,r .

Then all theorems of this subsection remain valid if (Fs/νq , ∂F

t/νpppp

q ) is replaced by(Fs/ν

q,r , ∂Ft/ν

ppppq,r ) with the appropriate choices of t.

Proof Fix s1 > s > s0 > ν1(j + 1/q). Then

Fs1/νq → Fs/ν

q,r → Fs0/νq , ∂Fs1(j,q)/ν

ppppq → ∂Fs(j,q)/ν

ppppq,r → ∂Fs0(j,q)/ν

ppppq

by the embedding theorems of Section VII.2. From this, Theorem 1.3.1, and the in-terpolation Theorems VII.2.7.1 and VII.2.8.3 we get the analogue of Theorem 1.3.1in the present setting. Now the counterparts of Theorems 1.3.2 and 1.3.4 follow byusing the parameter-dependent version of Theorem 1.3.1 in the respective proofs. ¥

1.4 Spaces of Vanishing Traces

On the basis of the preceding results we can now give alternative characterizationsof (most of the) Sobolev–Slobodeckii, Bessel potential, and Holder spaces on H.

Recalling convention (1.3.1), we put for k ∈ N and s > ν1(k + 1/q)

Fs/νq,~γk

(H, E) :=

u ∈ Fs/νq (H, E) ; ~γku = 0

. (1.4.1)

Furthermore, we define for q < ∞:

Fs/νq (H, E) is the closure of D(H, E) in Fs/ν

q (H, E) . (1.4.2)

We investigate the relations between these spaces and Fs/νq (H, E). First we con-

centrate on the Sobolev–Slobodeckii space scale.

Sobolev–Slobodeckii Spaces

Henceforth, unless stated otherwise,

• s ∈ R+ and 1 ≤ q < ∞ with q > 1 if s ∈ νN . (1.4.3)

Recall that the Slobodeckii spaces are defined by Ws/ν

q := Bs/νq = b

s/νq if s /∈ νN.

1.4.1 Theorem The following dense embeddings prevail:

D(H, E)d

→ S(H, E)d

→ W s/νq (H, E)

d→ W s/ν

q (H, E) .

Proof Theorem VII.3.7.7 guarantees that Ws/ν

q (H, E) → Ws/ν

q (H, E), provideds /∈ νN. If s ∈ νN, then we get this from definition (VII.1.2.2).

VIII.1 Traces 305

It is a consequence of (VII.2.2.8) that S(Rd, E) is dense in Bs/νq (Rd, E).

Hence Lemma VI.1.1.2 and Theorems VI.1.2.3 and 2.8.3 imply that S(H, E) isdense in B

s/νq (H, E). Thus, recalling definition (VII.1.2.2) once more, we see that

the middle dense embedding prevails. The left one follows from Lemma VI.1.1.3(ii).Now the statement on the right is implied by definition (1.4.2). ¥

1.4.2 Theorem Let (1.4.3) be satisfied.(i) If k ∈ N and

ν1(k + 1/q) < s < ν1(k + 1 + 1/q) , (1.4.4)

thenW

s/ν

q,~γk(H, E) = W s/νq (H, E) .

(ii) If 0 < s < ν1/q, then

W s/νq (H, E) = W s/ν

q (H, E) .

Proof We omit E.(1) Since W

s/ν

q,~γk(H) is the kernel of the continuous linear map ~γk on Ws/ν

q (H),

it is a closed linear subspace of Ws/ν

q (H). Hence D(H) ⊂ Ws/ν

q,~γk(H) implies

W s/νq (H) → W

s/ν

q,~γk(H) .

Consequently, the assertion follows, provided we show that

D(H)d⊂ W

s/ν

q,~γk(H) . (1.4.5)

Similarly, in order to prove assertion (ii), we have to verify that

D(H)d⊂ W s/ν

q (H) , 0 < s < ν1/q . (1.4.6)

(2) Let s ∈ R+\ν1(N+ 1/q), u ∈ Ws/ν

q (H), and ε > 0. Since S(H) is densein this space, we can find v ∈ S(H) satisfying

‖u − v‖W

s/νq (H)

< ε . (1.4.7)

By Theorem 1.3.2 we can choose a coretraction (~γk)c for ~γk such that

(~γk)c ~γk ∈ L(W t/ν

q (H))

, t > ν1(k + 1/q) . (1.4.8)

From this, (VII.2.2.1), (VII.2.6.7), and Theorem VII.2.8.3 it follows that

w := v − (~γk)c~γkv ∈ C∞(H) .


Hence, if s > ν1(k + 1/q), then

w ∈ C∞ ∩ W s/νq (H) , ~γkw = 0 .

Furthermore, (1.4.8) and ~γku = 0 imply, due to (1.4.7),

‖u− w‖W

s/νq (H)

≤ ‖u − v‖W

s/νq (H)

+ ‖(~γk)c~γk(u − v)‖W

s/νq (H)

≤ c ε ,

where c is independent of ε > 0. This shows that we can –– and do –– assume that

u ∈ C∞ ∩ W s/νq (H) where ~γku = 0 if s > ν1(k + 1/q) . (1.4.9)

(3) We fix ϕ ∈ D(Rd, [0, 1]

)satisfying ϕ(x) = 1 for x ∈ H with |x| ≤ 1, and

set ϕt(x) := ϕ(tx) for t > 0. Then ϕt(x) = 1 for x ∈ H with |x| ≤ 1/t and

∂αϕt = t|α|(∂αϕ)t , α ∈ Nd . (1.4.10)

Hence ϕt ∈ D(H) and

‖ϕt‖BCn/ν∞

≤ c(n) , 0 < t ≤ 1 , n ∈ νN . (1.4.11)

Consequently (see Theorem VII.6.2.1(i)),

ϕtu ∈ D(H) . (1.4.12)

Let m ∈ νN. Then

∂α(ϕtu) = ϕt∂αu +

∑

0<β≤α

(αβ

)(∂βϕt)∂α−βu , α q ω ≤ m .

From this, (1.4.10), and (1.4.11) we obtain

‖∂α(u − ϕtu)‖q ≤ ‖(1 − ϕt)∂αu‖q + ct |||u|||m/ν,q , 0 < t ≤ 1 , α q ω ≤ m .

Since‖(1− ϕt)∂αu‖q

q ≤∫

|x|≥1/t

|∂αu|q dx → 0 as t → 0 ,

we see thatlimt→0

ϕtu = u in Wm/νq (H) , m ∈ νN .

Thus, if s /∈ νN, we obtain by interpolation that

limt→0

ϕtu = u in W s/νq (H) .

Hence, in either case, it follows from (1.4.12) that we can assume that u ∈ D(H).Furthermore, we see from

γj(ϕtu) =j∑

i=0

(ji

)∂j−iϕt(0)γiu , 0 ≤ j ≤ m ,

VIII.1 Traces 307

that we can assume

u ∈ D(H) with ~γku = 0 if s > ν1(k + 1/q) . (1.4.13)

(4) We write Y := R and choose ψ ∈ D(Y +, [0, 1]

)such that ψ(y) = 0 for

0 ≤ y ≤ 1/2, and ψ(y) = 1 for y ≥ 1. Then, using the notation introduced above,ψ1/tu ∈ D(H) for t > 0.

Let X := Lq(∂H) and Y ∈ X ,Ws/ν

ppppq (∂H)

. Set v(y) := u(y, ·). Then

‖(1 − ψ1/t)v‖Lq(Y +,Y) ≤(∫ t

0

‖v(y)‖qY dy

)1/q

≤ ct1/q , t > 0 . (1.4.14)

Note that

∆h

((1 − ψ1/t)v

)(y) = −∆hψ1/t(y)v(y + h) + (1 − ψ1/t)∆hv(y) . (1.4.15)

Moreover,∆hψ(y) = 0 for y ≥ 1 , |∆hψ(y)| ≤ c minh, 1 ,

and ∆hψ1/t = (∆h/tψ)1/t for h, t > 0. Let σ ∈ (0, 1/q). Then it follows that∫ ∞

0

h−σq

∫

Y +

∥∥(∆hψ1/t)(y)v(y + h)∥∥q

X dydh

h

= t−σq

∫ ∞

0

(h/t)−σq

∫

Y +|∆h/tψ(y/t)|q ‖v(y + h)‖q

X dydh

h

≤ ct−σq

∫ ∞

0

τ−σq−1 minτ, 1q dτ

∫ t

0

‖v(y + tτ)‖qX dy ≤ ct1−σq

for t > 0. Similarly, using |∆hv(y)| ≤ c minh, 1 for y ∈ Y +,∫ ∞

0

h−σq

∫

Y +

∥∥(1 − ψ1/t)(y)∆hv(y)∥∥q

X dydh

h

≤ c

∫ ∞

0

h−σq−1 minh, 1q dh

∫ t

0

dy ≤ ct

for t > 0. From these estimates and (1.4.15) we obtain[(1 − ψ1/t)v

]Xσ,q

≤ ct−σ+1/q , 0 < t ≤ 1 . (1.4.16)

Thus, using (1.4.14) (with Y = X ), it follows that

limt→0

ψ1/tv = v in W s/ν1q

(Y +, Lq(∂H)

)if 0 < s < ν1/q .

Since (1.4.14) implies also

ψ1/tv → v in Lq

(Y +,W s/ν

ppppq (∂H)

), (1.4.17)

it is a consequence of Corollary VII.3.6.3 that (1.4.6) is true. This proves (ii).


(5) Now we suppose that s > ν1(k + 1/q). Hence ~γku = 0. Thus Taylor’s for-mula implies for 0 ≤ j ≤ k

∂jv(y) =m∑

i=k−j+1

1i!

∂i+jv(0)yi + Rm(∂jv)(y)ym

for m ≥ k − j + 1, where

Rm(w)(y) :=∫ 1

0

(1 − t)m−1

(m + 1)!(∂mw(ty) − ∂mw(0)

)dt .

Since y ≤ c on supp(v), it thus follows that

‖∂jv(y)‖X ≤ c yk−j+1 , 0 ≤ j ≤ k + 1 . (1.4.18)

Note that

∂iψ1/t = t−i(∂iψ)1/t (1.4.19)

implies |∂iψ1/t| ≤ ct−i. Moreover, ∂iψ1/t(y) = 0 for y ≥ t > 0. From this and

∂j((1− ψ1/t)v

)= (1 − ψ1/t)∂jv −

j∑

i=1

(ji

)∂iψ1/t∂

j−iv

we infer

∥∥∂j((1 − ψ1/t)v

)∥∥q≤ c

(‖(1 − ψ1/t)yk−j+1‖q +

j∑

i=1

t−i ‖yk−j+i+1‖Lq(0,t)

)

for 0 ≤ j ≤ k + 1. The first norm on the right is majorized by(∫ t

0

y(k−j+1)q dy)1/q

≤ ctk−j+1+1/q , t > 0 , 0 ≤ j ≤ k + 1 .

The sum too can be majorized by ctk−j+1+1/q. Hence∥∥∂j

((1 − ψ1/t)v

)∥∥Lq(Y +,X )

≤ ctk−j+1+1/q , t > 0 , 0 ≤ j ≤ k + 1 ,

and thus

‖(1 − ψ1/t)v‖W k+1q (Y +,X ) ≤ ct1/q , 0 < t ≤ 1 . (1.4.20)

(6) Leibniz’ rule implies[∂k+1

((1 − ψ1/t)v

)]Xσ,q

≤ [(1 − ψ1/t)∂k+1v

]Xσ,q

+ c

k+1∑

j=1

[(∂jψ1/t)∂k+1−jv

]Xσ,q

.(1.4.21)

VIII.1 Traces 309

By replacing v in step (4) by ∂k+1v we see that

[(1 − ψ1/t)∂k+1v

]Xσ,q

≤ ct−σ+1/q , 0 < t ≤ 1 , (1.4.22)

where 0 < σ < 1/q. Using (1.4.18) and (1.4.19) we get

|∂jψ1/t(y)∂k+1−jv(y)| ≤ ct−j |(∂jψ)1/tyj |X .

Hence

‖(∂jψ1/t)∂k+1−jv‖q ≤ ct−j(∫ t

0

yjq dy)1/q

= ct1/q (1.4.23)

for 0 < t ≤ 1. Similarly,∥∥∂

((∂jψ1/t)∂k+1−jv

)∥∥q≤ t−j−1 ‖(∂j+1ψ)1/ty

j‖q + t−j ‖(∂jψ)1/tyj−1‖q

≤ cy−1+1/q

(1.4.24)

for 0 < t ≤ 1. Thus t1/q ≤ t−1+1/q for 0 < t ≤ 1 and (1.4.23), (1.4.24) imply

‖(∂jψ1/t)∂k+1−jv‖W `p (Y +,X ) ≤ ct−`+1/q , 0 < t ≤ 1 , ` = 0, 1 .

From this and the interpolation inequality of Proposition I.2.2.1 it follows[(∂jψ1/t)∂k+1−jv

]Xσ,q

≤∥∥(∂jψ1/t)∂k+1−jv

∥∥W σ

q (Y +,X )

≤ ct(1−σ)/q+σ(−1+1/q) = ct−σ+1/q

for 0 < t ≤ 1 and 0 < j ≤ k + 1. By combining this with (1.4.22), we get from(1.4.21) that

[∂k+1

((1− ψ1/t)v

)]Xσ,q

≤ ct−σ+1/q , 0 < t ≤ 1 .

Thus, taking (1.4.20) into consideration, we see that, given 0 < σ < 1/q,

ψ1/tv → v in W k+1+σq (Y +,X ) as t → 0 .

From this and (1.4.17) we obtain, due to Corollary VII.3.6.3, that

ψ1/tv → v in W ν1(k+1+σ)/νq (H) as t → 0 , (1.4.25)

provided 0 < σ < 1/q.Let s satisfy (1.4.4). Choose σ with 0 < σ < 1/q such that s/ν1 < k + 1 + σ.

Then it follows from Wν1(k+1+σ)/ν

q (H) → Ws/ν

q (H) and (1.4.25) that ψ1/tv → v

in Ws/ν

q (H). This proves (1.4.5), hence (i). ¥


1.5 Weighted Spaces

There is an alternative characterization of Sobolev–Slobodeckii spaces Ws/ν

p (H, E)for 1 < p < ∞ and s /∈ ν1(N+ 1/p) by the order with which their elements vanishon ∂H. This relation is expounded in the present subsection. Its proof relies onwell-known Hardy inequalities which are proved here as well.

Weighted Lebesgue Spaces

For 1 ≤ q < ∞ and ρ ∈ R we define a weighted Lq space on H (with respect to thepower weight yρ) by

Lq,ρ(H, E) := Lq(H, yρdx; E) ,

where we recall that x ∈ H is written (y, xq) with y ∈ R+ and x

q ∈ ∂H.

1.5.1 Lemma

(i) D(H, E) is dense in Lq,ρ(H, E).

(ii) Suppose 1 < p < ∞ and set ρ′ := ρ/(p − 1). If E is reflexive, then Lp,ρ(H, E)is reflexive and

(Lp,ρ(H, E)

)′ = Lp′,−ρ′(H, E′) (1.5.1)

with respect to the Lp(H, E)-duality pairing 〈·, ·〉.

Proof (1) Suppose u ∈ Lq,ρ(H, E). We choose ψ ∈ D(R+, [0, 1]

)with ψ(y) = 0

for 0 ≤ y ≤ 1/2 and ψ(y) = 1 for y ≥ 1. Then ψ1/t(y) := ψ(y/t) = 0 for y ≤ t/2and ψ1/t(y) = 1 for y ≥ t. It follows that ψ1/tu belongs to Lq,ρ(H, E), vanishes for0 ≤ y ≤ t/2, and

‖(1 − ψ1/t)u‖Lq,ρ(H,E) ≤(∫ t

0

‖u(y, ·)‖qLq(∂H,E)y

ρ dy)1/q

→ 0

as t → 0. Hence we can assume that u is supported in [δ,∞] × Rd−1 for some δ > 0.

Denote by χr the characteristic function of [ |x| ≤ r]. Then χru ∈ Lp,ρ(H, E)and

‖χru − u‖Lq,ρ(H,E) → 0 as r →∞ .

Thus we can assume that u is compactly supported in H. Now assertion (i) followsby a standard mollification procedure (cf. Subsection III.4.2).

(2) Suppose d = 1 so that H = R+. We write Lp,ρ(E) for Lp,ρ(H, E), etc.Note that it equals y−ρ/pLp(E), the image space of Lp(E) in Lp,loc(E) under thepoint-wise multiplication

mp,ρ := (u 7→ yρ/pu) ∈ L(Lp(E), L1,loc(E)

)

VIII.1 Traces 311

(recall (VI.2.2.1) and Remark VI.2.2.1(a)). Hence

mp,ρ ∈ Lis(Lp(E), Lp,ρ(E)

), (mp,ρ)−1 = mp,−ρ , (1.5.2)

and this isomorphism is isometric.Observe that ρ′/p′ = ρ/p, so that

〈v, mp,ρu〉 = 〈mp′,ρ′v, u〉 , u ∈ D(E) , v ∈ D(E′) . (1.5.3)

Thus (i), (1.5.2), and(Lp(E)

)′ = Lp′(E′) imply

|〈mp′,ρ′v, u〉| = |〈v, (mp,−ρ)−1u〉| ≤ ‖v‖Lp′ (E′) ‖(mp,−ρ)−1u‖Lp(E)

= ‖v‖Lp′ (E′) ‖u‖Lp,ρ(E)

for u ∈ Lp,ρ(E) and v ∈ Lp′(E′). From this we get

‖mp′,ρ′v‖(Lp,ρ(E))′ ≤ ‖v‖Lp′ (E′) , v ∈ Lp′(E′) ,

that is,

mp′,ρ′ ∈ L(Lp′(E′), (Lp,ρ(E))′

). (1.5.4)

Let u ∈ D(E) and v ∈ D(E′). Then mp,−ρu ∈ D(E). Hence, by (1.5.2) and (1.5.4),

|〈v, u〉| =∣∣∣∫ ∞

0

⟨v(y), u(y)

⟩E

dy∣∣∣ =

∣∣∣∫ ∞

0

⟨mp′,−ρ′v(y), mp,ρu(y)

⟩E

dy∣∣∣

= |〈mp′,−ρ′v, mp,ρu〉| ≤ ‖mp′,−ρ′v‖Lp′ (E′) ‖mp,ρu‖Lp(E)

= ‖v‖Lp′,−ρ′ (E′) ‖u‖Lp,ρ(E) .

Thus, by (i),(v 7→ 〈v, ·〉) ∈ L(

Lp′,−ρ′(E′), (Lp,ρ(E))′)

. (1.5.5)

Conversely, suppose f ∈ (Lp,ρ(E)

)′. Then

〈f, u〉Lp,ρ(E) = 〈f, mp,ρmp,−ρu〉Lp,ρ(E) =⟨(mp,ρ)′f, mp,−ρu

⟩Lp(E)

for u ∈ D(E), where (mp,ρ)′ ∈ L((Lp,ρ(E))′, Lp′(E′)

)is the dual of (1.5.2). Hence,

using (1.5.3) and the density of D(E′) in Lp′(E′), we obtain from (1.5.2) that

〈f, u〉Lp,ρ(E) = 〈v, u〉 , u ∈ D(E) ,

with v := mp′,−ρ′((mp,ρ)′f

) ∈ Lp′,−ρ′(E′), due to (1.5.2). From this and (i) we seethat f is represented by 〈v, ·〉 with a suitable v ∈ Lp′,−ρ′(E′). This shows that


(1.5.5) is surjective. Hence it is an isometric isomorphism, which proves (1.5.1).The reflexivity assertion follows from the reflexivity of Lp(E).

(3) Now we assume d > 1. Then, by Fubini’s theorem,

Lp,ρ(H, E) = Lp,ρ

( qR+, Lp(∂H, E)

).

Hence, by step (2) and Theorem VII.1.2.1,

(Lp,ρ(H, E)

)′ = Lp′,−ρ′( qR+, Lp′(∂H, E′)

)= Lp′,−ρ′(H, E′) .

This proves (1.5.1) as well as the reflexivity assertion. ¥

Hardy Inequalities

For completeness, we include proofs of the following well-known Hardy inequalities.Here Lp,ρ = Lp,ρ(

qR+, E).

1.5.2 Theorem Set

Sf(x) :=1x

∫ x

0

f(t) dt , Tf(x) :=1x

∫ ∞

x

f(t) dt .

Letc(q, ρ) := q/|1 + ρ− q| , 1 ≤ q < ∞ , ρ ∈ R\q − 1 .

Then

‖Sf‖Lq,ρ ≤ c(q, ρ) ‖f‖Lq,ρ if 1 + ρ < q (1.5.6)

and

‖Tf‖Lq,ρ≤ c(q, ρ) ‖f‖Lq,ρ

if 1 + ρ > q > 1 (1.5.7)

for f ∈ Lq,ρ(qR+, E). These estimates are optimal.

Proof Obvious changes of variables and Minkowski’s inequality for integrals give

‖Sf‖Lq,ρ=

(∫ ∞

0

∣∣∣ 1x

∫ x

0

f(t) dt∣∣∣q

Exρ dx

)1/q

=∥∥∥

∫ 1

0

f(xτ)xρ/q dτ∥∥∥

Lq(R+,dx;E)≤

∫ 1

0

‖f(xτ)xρ/q‖Lq(R+,dx;E) dτ

≤ ‖f(y)yρ/q‖Lq(R+,dy;E)

∫ 1

0

τ−(ρ+1)/q dτ = c(q, ρ) ‖f‖Lq,ρ ,

provided ρ + 1 < q. This proves (1.5.6). Inequality (1.5.7) is obtained analogously.

VIII.1 Traces 313

Set E := C and fy := χ[y, y+1] for y ≥ 0. Then∫ x

0fy(t) dt = 1 for x ≥ y + 1.

Consequently, if q ≤ ρ + 1,

‖Sfy‖qLq,ρ

≥∫ ∞

y+1

xρ−q dx →∞ as y →∞ .

Similarly, if ρ + 1 ≤ q,

‖Tfy‖qLq,ρ

≥∫ y

0

xρ−q dx →∞ as y → 0 .

This implies the asserted optimality. ¥

1.5.3 Corollary Suppose 1 < p < ∞ and ρ 6= p − 1. Then∥∥∥u

x

∥∥∥Lp,ρ

≤ c(p, ρ) ‖∂u‖Lp,ρ

for all u ∈ Lp,loc(qR+, E) with ∂u ∈ Lp,ρ(

qR+, E).

Proof Set u(x) :=∫ x

0f(t) dt if ρ < p − 1, respectively u(x) :=

∫ ∞x

f(t) dt other-wise. ¥

Analogous arguments yield the following dual Hardy inequalities.

1.5.4 Theorem Put

S′f(x) :=∫ ∞

x

f(t)t

dt , T ′f(x) :=∫ x

0

f(t)t

dt , (1.5.8)

andc′(p, ρ) := p/|1 + ρ| , 1 −1 (1.5.9)

and

‖T ′f‖Lp,ρ≤ c′(p, ρ) ‖f‖Lp,ρ

if ρ < −1 (1.5.10)

for f ∈ Lp,ρ(qR+, E).

Proof We proceed as in the proof of the preceding theorem:

‖S′f‖Lp,ρ =(∫ ∞

0

∣∣∣∫ ∞

x

f(t)t

dt∣∣∣p

Exρ dx

)1/p

=∥∥∥

∫ ∞

1

f(τx)xρ/p

τdτ

∥∥∥Lp(R+,dx;E)

≤∫ ∞

1

‖f(τx)xρ/p‖Lp(R+,dx;E)dτ

τ

= ‖f(y)yρ/p‖Lp(R+,dy;E)

∫ ∞

1

τ−1−(ρ+1)/p dτ = c′(p, ρ) ‖f‖Lp,ρ ,


provided ρ + 1 > 0. This gives (1.5.9). The demonstration of (1.5.10) follows thesame line of arguments. ¥

Observe that

〈v, Su〉 = 〈S′v, u〉 , 〈v, Tu〉 = 〈T ′v, u〉for u ∈ D(

qR+, E) and v ∈ D(

qR+, E′), which explains the name ‘dual Hardy in-

equalities’. This observation can be used –– in conjunction with Lemma 1.5.1 ––to give an alternative proof of Theorem 1.5.4, provided we assume that E is re-flexive.

Weighted Space Characterizations of Sobolev–Slobodeckii Spaces

Now we can characterize the elements of Ws/ν

p (H, E) by means of the order withwhich they vanish on ∂H. First we consider the case where s is small.

1.5.5 Theorem Let 1 0. We omit E.(1) We set

X := Lp(∂H) , Y := W s/νpppp

p (∂H) .

Then Theorem 1.4.2(ii) and Corollary VII.3.6.3 imply

W s/νp (H) = W s/ν

p (H) = W s/ν1p (R+,X ) ∩ Lp(R+,Y) .

Thus suppose that we show that

W s/ν1p (R+,X ) → Lp,−sp/ν1(

qR+,X ) . (1.5.11)

Then it follows that

W s/νp (H) → Lp,−sp/ν1(

qR+,X ) ∩ Lp(

qR+,Y)

→ Lp,−sp/ν1(qR+,X ) = Lp,−sp/ν1(H) .

Hence the assertion is a consequence of the density of D(H) in Ws/ν

p (H) andin Lp,−sp/ν1(H).

(2) For u ∈ D(qR+,X ) we set

v(y) :=1y

∫ y

0

(u(y) − u(t)

)dt , w(y) :=

∫ ∞

y

v(t)t

dt , y > 0 .

Then u = v − w. Indeed, v(y) → 0 and w(y) → 0 as y →∞. Hence the claim fol-lows from ∂u = ∂v − ∂w, which is immediate.

VIII.1 Traces 315

We put σ := s/ν1. Then (1.5.11) will follow from the density of D(qR+,X ) in

W σp (R+,X ), provided we show that

‖v‖Lp,−σp(R+,X ) ≤ c ‖u‖W σp (R+,X ) (1.5.12)

and

‖w‖Lp,−σp(

pR+,X )

≤ c ‖u‖W σp (R+,X ) . (1.5.13)

Holder’s inequality, applied with 1 q (u(x)− u(t)), yields

|v(y)|pX ≤ 1y

∫ y

0

|u(y) − u(t)|pX dt .

Hence∫ ∞

0

y−σp |v(y)|pX dy ≤∫ ∞

0

y−σp−1

∫ y

0

|u(y)− u(t)|pX dt dy

=∫ ∞

0

∫ ∞

t

y−σp−1 |u(y)− u(t)|pX dy dt

=∫ ∞

0

∫ ∞

0

(t + τ)−σp−1 |u(t + τ)− u(t)|pX dτ dt

≤∫ ∞

0

τ−σp ‖∆τu‖pLp(R+,X )

dτ

τ= [u]pσ,p

≤ ‖u‖pW σ

p (R+,X ) ≤ c ‖u‖p

W σp (H)

.

(1.5.14)

This implies (1.5.12).Inequality (1.5.13) is a consequence of (1.5.12) and

‖w‖Lp,−σp(

pR+,X )

≤ c ‖v‖Lp,−σp(

pR+,X )

.

The latter follows from (1.5.9) since s < ν1/p implies −σp > −1. This proves(1.5.11), hence the theorem. ¥

Next we extend tis result to the case where s > ν1/p.

1.5.6 Theorem Suppose 1 < p < ∞ and s ∈ R+\ν1(N+ 1/p). Then

W s/νp (H, E) .= W s/ν

p (H, E) ∩ Lp,−sp/ν1(H, E) . (1.5.15)

Thus u ∈ Ws/ν

p (H, E) iff u ∈ Ws/ν

p (H, E) and ys/ν1u ∈ Lp(H, E).

Proof We omit E and use the notations of the preceding proof. Due to Theo-rem 1.5.5, we can assume that

ν1(k + 1/p) < s < ν1(k + 1 + 1/p) (1.5.16)

for some k ∈ N.


(1) Assume s 6= ν1(k + 1). It follows from Theorem VII.3.4.1(i) that (1.5.14)still holds.

We set

w1(y) :=∫ y

0

v(t)t

dt , y > 0 .

By u(0) = v(0) = w1(0) for u ∈ D(qR+,X ) and ∂u = ∂v + ∂w1, we get u = v + w1.

From (1.5.10) we obtain

‖w1‖Lp,−σp(pR+,X )

≤ c ‖v‖Lp,−σp(

pR+,X )

,

since s > ν1/p implies ρ := −σp < −1. From this we obtain, similarly as in step (2)of the proof of the preceding theorem, that

W s/νp (H) → Lp,−sp/ν1(H) , (1.5.17)

due to the density of D(qR+,X ) in W σ

p (R+,X ).

(2) Suppose s = ν1(k + 1). For u ∈ D(qR+,X ) set

w(t) :=1t

∫ t

0

u(τ) dτ , t > 0 .

Repeated partial integration gives

w(t) =(−1)k

tk!

∫ t

0

τk∂ku(τ) dτ =(−1)ktk

tk!

∫ 1

0

sk∂ku(st) ds .

Hence, given ρ ∈ R,

(∫ ∞

0

tρ |w(t)|pX dt)1/p

= c(∫ ∞

0

∣∣∣∫ 1

0

tk+ρ/psk∂ku(st) ds∣∣∣p

Xdt

)1/p

≤ c

∫ 1

0

‖tk+ρ/psk∂ku(st)‖Lp(R+,dt;X ) ds

= c

∫ 1

0

sk(∫ ∞

0

tkp+ρ |∂ku(st)|pX dt)1/p

ds

= c

∫ ∞

0

s−(ρ+1)/p ds ‖yp+ρ/p∂ku‖Lp(R+,dy;X ) ,

where the last equality results from the substitution y = st. Now we set ρ := −kpto obtain

(∫ ∞

0

t−kp |w(t)|pX dt)1/p

≤ c ‖∂ku‖Lp(R+,X ) (1.5.18)

with c :=(∫ 1

0sk−1/p ds

)1/p< ∞.

VIII.1 Traces 317

Observe that

u(t) =∫ t

0

∂u(τ) dτ = t1t

∫ t

0

∂u(τ) dτ .

By replacing u in the preceding considerations by ∂u, we obtain from (1.5.18)(∫ ∞

0

t−(k+1)p |u(t)|pX dt)1/p

≤ c ‖∂k+1u‖Lp(R+,X ) . (1.5.19)

Definition (VII.1.2.2) of the norm of W(k+1)/ν

p (H) implies

‖∂k+1u‖Lp(R+,X ) ≤ |||u|||(k+1)/ν,p .

From this, (1.5.19), and the density of D(H) in W(k+1)/ν

p (H) we infer that (1.5.17)holds for s = ν1(k + 1) also.

(3) Assume v ∈ S(H), but u /∈ Ws/ν

p (H). Then ~γkv 6= 0 by Theorem 1.4.2.Let j be the least integer for which γjv 6= 0. Since S(H) → Cj(R+,X ), we de-duce from Taylor’s theorem that v(y, ·) = yj

(γjv + r(y, ·)) with |r(y, ·)|X → 0 as

y → 0. Hence we can choose κ, ε > 0 such that |v(y, ·)|X ≥ κyj for 0 ≤ y ≤ ε. Con-sequently,

∫ ∞

0

y−sp/ν1 |v(y, ·)|pX dy ≥ κ

∫ ε

0

y−(s−ν1j)p/ν1 dy = ∞ ,

since (1.5.16) implies (s − ν1j)p/ν1 > 1. Thus v /∈ Lp,−sp/ν1(H). From this and thedensity of S(H) in W

s/νp (H) we infer that u ∈ W

s/νp (H) ∩ Lp,−sp/ν1(H) belongs to

Ws/ν

p (H). Now we get from the validity of (1.5.17) for each admissible s that

W s/νp (H) → W s/ν

p (H) ∩ Lp,−sp/ν1(H) ⊂ W s/νp (H) ,

thanks to the fact that Ws/ν

p (H) is a closed linear subspace of Ws/ν

p (H). Thisproves the theorem. ¥

1.6 Further Characterizations of Spaces with Vanishing Traces

With the aid of Theorem 1.5.6 we can establish a further property of Ws/ν

p (H, E),which turns out to have important consequences.

1.6.1 Theorem Suppose 1 0.(1) We write (R, E) for the r-e pair (RH, EH), so that E is the trivial extension

operator (VI.1.1.12). Recall that Ws/ν

p = Bs/νp if s /∈ νN.


Suppose we show that

E ∈ L(W s/ν

p (H),W s/νp (Rd)

). (1.6.1)

Theorems VII.1.3.1 and VII.2.8.2 imply R ∈ L(W

s/νp (Rd),W s/ν

p (H)). Hence we

get from (1.6.1) and u ∈ REu that

W s/νp (H) → W s/ν

p (H) .

Then the assertion follows from Theorem 1.4.1. Thus we have to establish thevalidity of (1.6.1).

(2) Let s ∈ νN. It is obvious that

‖∂αEu‖Lp(Rd) = ‖E∂αu‖Lp(Rd) = ‖∂αu‖Lp(H)

for a q ω ≤ s/ν and u ∈ D(H). From this and the density of D(H) in Ws/ν

p (H, E)property (1.6.1) follows in this case.

(3) Assume that s /∈ νN. Let X ∈ Rd,H, X1 ∈ R,R+. Then X = X1 × ∂Hand ∂H = Rd−1. Corollary VII.3.6.3 guarantees that

W s/νp (X) .= W s/ν1

p

(X1, Lp(H)

) ∩ Lp(X, Bs/νpppp

p (∂H))

. (1.6.2)

For abbreviation, we set X := Lp(∂H) and Y := Bs/ν

ppppp (∂H). We also write

u := Eu. Then, obviously,

‖u‖Lp(R,Z) = ‖u‖Lp(R+,Z) , u ∈ D(H) , Z ∈ X ,Y . (1.6.3)

From this, (1.6.2), and the denisty of D(H) in Ws/ν

p (H) we infer that it suffices toshow that

‖u‖W

s/ν1p (R,X )

≤ c ‖u‖W

s/νp (R+,X )

, u ∈ D(H) . (1.6.4)

(4) First we establish an auxiliary estimate. Suppose r > 0 and u ∈ D(H).Then, given 0 < ϑ < ∞,

Ir(u) :=∫ ∞

0

∫ 0

−rh

|u(x + rh)|pX dxdh

h1+ϑp=

∫ ∞

0

∫ rh

0

|u(y)|pX dydh

h1+ϑp

=∫ ∞

0

∫ ∞

y/r

dh

h1+ϑp|u(y)|pX dy = ϑp rϑp

∫ ∞

0

|y−ϑu(y)|pX dy .

Thus Theorem 1.5.6 guarantees

Ir(u) ≤ c(r, ϑ) ‖u‖pW ϑ

p (R+,X ), u ∈ D(H) , (1.6.5)

provided ϑ /∈ N+ 1/p.

VIII.1 Traces 319

(5) Suppose ν1(k + 1/p) < s < ν1(k + 1 + 1/p) for some k ∈ N, and s /∈ ν1

qN.

Set ϑ := s/ν1. By Theorem VII.3.4.1(i),

v 7→ ‖v‖Lp(X1,X ) + [v]ϑ,p

is a norm for Wϑp (X1,X ), where

[v]pϑ,p :=∫ ∞

0

∫

X1

‖4k+1h v‖p

Xhϑp

dh

h.

Hence we infer from (1.6.3) that

‖u‖W ϑp (R,X ) ≤ c

(‖u‖Lp(R+,X ) + [u]ϑ,p

), u ∈ D(H) . (1.6.6)

Note that

4k+1h v(x) =

k+1∑

j=0

(−1)k+1−j(

k + 1j

)v(x + jh) (1.6.7)

(e.g., [AmE05, p. 124]). Moreover,

[u]pϑ,p = [u]pϑ,p + I ,

where

I :=∫ ∞

0

∫ 0

−∞

‖4k+1h u‖p

Xhϑp

dxdh

h.


[u]pϑ,p ≤ c([u]pϑ,p +

k+1∑

j=0

Ij(u))

, u ∈ D(H) .

By applying (1.6.5) to each element of the last sum, we see that

[u]pϑ,p ≤ c([u]pϑ,p + ‖u‖p

W ϑp (R+,X )

) ≤ c ‖u‖pW ϑ

p (R+,X ), u ∈ D(H) .

Now we get (1.6.4) from (1.6.6). The theorem is proved. ¥

Because of their importance, we collect the properties of the spaces Ws/ν

p inthe following theorem.

1.6.2 Theorem Suppose 1 < p < ∞.(i) If k ∈ N and ν1(k + 1/p) < s < ν1(k + 1 + 1/p), then


p (H, E) .= Ws/ν

p,~γk(H, E) .

(ii) If 0 < s < ν1/p, then


p (H, E) .= W s/νp (H, E) .

Proof Theorems 1.4.2 and 1.6.1. ¥


General Besov Spaces

By interpolation we can now extend the preceding results to general Besov spaces.For this we make use of the J method of real interpolation, which we recall briefly.

1.6.3 Remark Let E0 and E1 be Banach spaces with E1 → E0, 0 < θ < 1, and1 ≤ q ≤ ∞. For t ∈ R we set

J(t, e) := max‖e‖E0 , t ‖e‖E1 , e ∈ E1 .

Then e ∈ (E0, E1)θ,q iff there exists f ∈ C((0,∞), E1

)satisfying

‖u‖Jθ,q := ‖t−θJ(t, e)‖Lq((0,∞),dt/t) < ∞ , e =

∫ ∞

0

f(t) dt/t , (1.6.8)

where the integral converges in E0, and ‖·‖Jθ,q is a norm for (E0, E1)θ,q (e.g.,

Chapter 3 of [BeL76] or Section 1.6 in [Tri95]). ¥

1.6.4 Theorem Suppose 1 < p < ∞ and 1 ≤ r ≤ ∞.(i) If k ∈ N and ν1(k + 1/p) < s < ν1(k + 1 + 1/p), then

Bs/νp,r (H, E) = B

s/ν

p,r,~γk(H, E) .

(ii) If 0 < s < ν1/p, then

Bs/νp,r (H, E) .= Bs/ν

p,r (H, E) .

Proof We use the following abbreviations;

X t := W t/νp (H, E) , X t

ρ := Bt/νp,ρ (H, E) , Yt := bt/ν

ppppp (∂H, E) ,

as well asX t

0 := W t/νp (H, E) , X t

ρ,0 := Bt/νp,ρ (H, E)

for t ∈ R and 1 ≤ ρ ≤ ∞. Note that X t .= X tp .

(1) Assume

ν1(k + 1/p) < s0 < s < s1 < ν1(k + 1 + 1/p) .

Set θ := (s − s0)/(s1 − s0) so that sθ = (1 − θ)s0 + θs1 = s. Put B := ~γk. It fol-lows from X si

B → X si and Theorem 1.6.2(i) that

(X s00 ,X s1

0 )θ,r.= (X s0

B ,X s1B )θ,r → (X s0 ,X s1)θ,r . (1.6.9)

Theorems VII.2.7.1(i), VII.2.7.4, and VII.2.8.3 imply

(X s00 ,X s1

0 )θ,r.= X s

r,0 , (X s0 ,X s1)θ,r.= X s

r . (1.6.10)

VIII.1 Traces 321

Hence

X sr,0 → X s

r . (1.6.11)

(2) Supposeu ∈ (X s0

B ,X s1B )θ,r

.= X s0,r .

Fix f such that (1.6.8) is satisfied with q = r, e = u, and (E0, E1) = (X s0B ,X s1

B ).It is obvious that

B∫ 1/ε

ε

f(t) dt/t =∫ 1/ε

ε

Bf(t) dt/t = 0 , 0 < ε < 1 ,

since f(t) ∈ X s1B for t > 0. This implies Bu = 0. Thus we get from (1.6.11) that

X sr,0 → X s

r,B . (1.6.12)

(3) Assume v ∈ X sr,B, that is, v ∈ X s

r.= (X s0 ,X s1)θ,r and Bv = 0. There ex-

ists g such that (1.6.8) applies with q = r, e = v, f = g, and (E0, E1) = (X s0 ,X s1).We know from Theorem 1.3.2 that B is a retraction from X si onto Ysi(~k,p) andthat it possesses a coretraction Bc which is universal with respect to i = 0, 1. HenceP := 1X si − BcB is a projection from X si onto X si

B (cf. Subsection I.2.3). Thus,similarly as above,

v = Pv = P

∫ ∞

0

g(t) dt/t =∫ ∞

0

Pg(t) dt/t ,

where Pg ∈ C((0,∞),X s1

B)

and the integral converges in X s0B . Consequently, by

Remark 1.6.3,v ∈ (X s0

B ,X s1B )θ,r = X s

r,0 .

The last equivalence is due to (1.6.9) and (1.6.10). Hence we get X sr,0

.= X sr,B from

(1.6.12). This proves assertion (i).(4) Let 0 < s0 < s < s1 < ν1/p. Defining θ as above, we find from Theorems

1.6.2(ii), VII.2.7.1(i), VII.2.7.4, and VII.2.8.3 that

X sr = (X s0 ,X s1)θ,r

.= (X s00 ,X s1

0 )θ,r.= X s

0,r .

Thus assertion (ii) is also true. ¥

1.6.5 Corollary Suppose r < ∞ and s > 0 with s /∈ ν1(N+ 1/p). Then

Bs/νp,r (H, E) .= Bs/ν

p,r (H, E) .

Proof We know from Theorem 1.6.2 that X s1 = X s10 , that is, D(H) is dense in X s

0 .Since r < ∞ and Xr,0

.= (X s00 ,X s1

0 )θ,r, interpolation theory guarantees that X s10 is


dense in Xr,0 (see (I.2.5.2)). Hence D(H) is dense in X sr,0. Since X s

r,0.= X s

r,B, it isa closed linear subspace of X s

r . Now the assertion is clear. ¥

Bessel Potential Spaces

In order to prove partially analogous results for Bessel potential spaces we preparea simple technical density theorem.

1.6.6 Lemma Let Xj and Yj be Banach spaces for j = 0, 1 such that X1 → X0

and Y1 → Y0. Suppose (r, rc) is an r-c pair for (Xj , Yj), j = 0, 1. Set

Xj,r := x ∈ Xj ; rx = 0 = ker(r |Xj) .

If X1d

→ X0, then X1,rd

→ X0,r.

Proof Clearly, X1,r → X0,r → X0. Let x ∈ X0,r. Since X1 is dense in X0, thereexists a sequence (xj) in X1 with xj → x in X0. Set yj := (1 − rcr)xj . Thenyj ∈ X1,r and yj − x = (1 − rcr)(xj − x) → 0 in X0. Hence X1,r is dense in X0,r. ¥

By means of this lemma we can prove an embedding theorem for Besselpotential and Besov spaces with vanishing traces.

1.6.7 Theorem Suppose k ∈ N, 1 < p < ∞, and

ν1(k + 1/p) < s < ν1(k + 1 + 1/p) .

Then

Bs/ν

p,1,~γk(H, E)d

→ Hs/ν

p,~γk(H, E) → Bs/ν

p,∞,~γk(H, E) .

Proof Theorem VII.4.1.3(i) implies

Bs/νp,1 (H, E)

d→ Hs/ν

p (H, E) → Bs/νp,∞(H, E) . (1.6.13)

Hence the claim follows from Theorem 1.3.2 and the (proof of the) precedinglemma, setting (r, rc) :=

(~γk, (~γk)c

). ¥

It should be remarked that the proof of this theorem uses in an essential waythe fact that

(~γk, (~γk)c

)is simultaneously a universal r-c pair for Besov, Bessel

potential, and Sobolev spaces.

Now it is easy to derive an analogue of Theorem 1.6.1 for Bessel potentialspaces.

VIII.1 Traces 323

1.6.8 Theorem Suppose 1 < p < ∞.(i) If k ∈ N and

ν1(k + 1/p) < s < ν1(k + 1 + 1/p) , (1.6.14)

then

Hs/ν

p,~γk(H, E) .= Hs/νp (H, E) . (1.6.15)

(ii) If 0 ≤ s < ν1/p, then

Hs/νp (H, E) = Hs/ν

p (H, E) .

Proof We omit E.(1) Let (1.6.14) be satisfied. Then we deduce from Theorem 1.6.4 and its

corollary and from Theorem 1.6.7 that

D(H)d

→ Bs/νp,1 (H) .= B

s/ν

p,1,~γk(H)d

→ Hs/ν

p,~γk(H) .

Thus D(H) is dense in Hs/ν

p,~γk(H), which is a closed linear subspace of Hs/νp (H).

From this and the definition of Hs/νp (H) we get (1.6.15).

(2) Let 0 ≤ s < ν1/p. The claim is obvious if s = 0. If s > 0, then we find,similarly as above,

D(H)d

→ Bs/νp,1 (H) .= B

s/νp,1 (H)

d→ Hs/ν

p (H) ,

where we also made use of (1.6.13). Now the second claim is clear. ¥

Holder Spaces

If s is suitably restricted, then the following theorem gives a related characteriza-tion of the Holder spaces BUC

s/ν~γk (H, E) and buc

s/ν~γk (H, E).

1.6.9 Theorem Suppose ν1k < s < ν1(k + 1) for some k ∈ N. Then

BUCs/ν~γk (H, E) = BUCs/ν(H, E)

andbuc

s/ν~γk (H, E) = bucs/ν(H, E) .

Proof We use the notations of Subsection VI.1.1 and omit E.(1) Suppose u ∈ BUCs/ν(H). There exists v ∈ BUCs/ν(Rd) with u = RHv.

Writing x = (y, xq) ∈ H with x

q ∈ ∂H,

u(x) = v(x)−∫ ∞

0

h(t)v(−ty, xq) dt .


From this and Lemma VI.1.1.1 it follows that ~γku = 0. Hence

BUCs/ν(H) → BUCs/ν

~γk (H)

and, similarly, bucs/ν(H) → bucs/ν

~γk (H).

(2) Let Y be a Banach space and 0 < ϑ < 1. Suppose w ∈ BUCϑ(R+,Y)satisfies w(0) = 0. Denote by w its trivial extension. Then

w ∈ BUC(R,Y) , ‖w‖∞ = ‖w‖∞ ,

and|w(y) − w(z)|Y

|y − z|ϑ ≤ |w(0) − w(z)|Y|0− z|ϑ , y < 0 < z < ∞ .

This shows that w ∈ BUCϑ(R,Y) and ‖w‖ϑ,∞ ≤ ‖w‖ϑ,∞.

(3) Assume u ∈ BUCs/ν

~γk (H) and set v := EHu. It follows from step (2) and

Theorem VII.3.5.2 that v ∈ BUCs/ν(Rd). Hence u = RHv ∈ BUCs/ν(H). Conse-quently, BUC

s/ν

~γk (H) ⊂ BUCs/ν(H). Together with step (1), this proves the firstassertion.

(4) Letu ∈ buc

s/ν

γk (H) → bucs/ν(H) .

Choose t ∈ (s, ν1(k + 1)

). There exists a sequence (uj) in BUCt/ν(H) converging in

BUCs/ν(H) towards u. Since ~γku = 0, step (2) of the proof of Theorem 1.4.2 showsthat we can find a sequence (vj) in BUC

t/ν

γk (H) which converges in BUCs/ν(H)towards u. By steps (1) and (3) we know that

BUCt/ν

~γk (H) = BUCt/ν(H) → BUCs/ν(H) ,

where we also employed BUCr/ν = Br/ν∞ for r ∈ s, t and Theorems VII.2.2.2 and

VII.2.8.3. Since BUCs/ν(H) = BUCs/ν

~γk (H) is closed in BUCs/ν(H), we obtain from

vj → u in BUCs/ν(H) that u ∈ bucs(H). This shows that bucs~γk(H) ⊂ bucs/ν(H).

Due to step (1), the theorem is proved. ¥

1.7 Representation Theorems for Spaces of Negative Order

Assuming that E is reflexive, we can now derive from the preceding results usefulrepresentations for spaces of negative order.

Spaces of Mildly Negative Order

First we consider the case where s > ν1(−1 + 1/p), a particularly simple situation.

VIII.1 Traces 325

1.7.1 Theorem Let E be reflexive, 1 < p < ∞, and 1 < r < ∞. Suppose that

ν1(−1 + 1/p) < s < ν1/p .

ThenBs/ν

p,r (H, E) .= Bs/νp,r (H, E) .= Bs/ν

p,r (H, E) .

Proof By part (ii) of Theorem 1.6.4 and its corollary we can assume that

ν1(−1 + 1/p) < s ≤ 0 .

(1) Suppose s < 0. Then Theorem VII.2.8.4 gives

Bs/νp,r (H, E) .=

(B−s/νp′,r′ (H, E′)

)′.

Thus we get from 0 < −s < ν1/p and Theorem 1.6.4(ii) that

Bs/νp,r (H, E) .=

(B−s/νp′,r′ (H, E′)

)′ .= Bs/νp,r (H, E) ,

where the last equivalence follows by applying Theorem VII.2.8.4 once more.(2) Suppose ν1(−1 + 1/p) < s0 < 0 < s1 < ν1/p and set θ := −s0/(s1 − s0).

Then we deduce from step (1) and Theorems VII.2.7.1(i) and VII.2.8.3 that

B0/νp,r (H, E) .=

(Bs0/ν

p,r (H, E), Bs1/νp,r (H, E)

)θ,r

.=(Bs0/ν

p,r (H, E), Bs1/νp,r (H, E)

)θ,r

.= B0/νp,r (H, E) .

This proves the first asserted equivalence.(3) We get from Theorems VII.2.2.4(i) and VII.2.8.3 that

Bt/νp (H, E) .= Bt/ν

p,r (H, E)d

→ Bs/νp,r (H, E)

for ν1(−1 + 1/p) < s ≤ 0 < t < ν1/p. From this we obtain

Bs/νp,r (H, E) = Bs/ν

p,r (H, E)

for ν1(−1 + 1/p) < s ≤ 0. The theorem is proved. ¥

For the sake of a unified notation we set

W−s/νp (H, E) := B−s/ν

p (H, E) (1.7.1)

for 1 0 with s /∈ νN. Observe that

W 0/νp (H, E) = Lp(H, E) 6= B0/ν

p (H, E) . (1.7.2)

1.7.2 Corollary Assume E is reflexive and 1 < p < ∞. Then


p (H, E) .= W s/νp (H, E)

for ν1(−1 + 1/p) < s < ν1/p.

Proof Observe (1.7.1) and (1.7.2). ¥


By means of the weighted space characterization of Sobolev–Slobodeckiispaces derived in Subsection 1.5 we can obtain a different representation of Slo-bodeckii spaces of negative order.

1.7.3 Theorem Let E be reflexive, 1 < p < ∞, and 0 < s < ν1/p′. Then

W−s/νp (H, E) .= Lp,sp/ν1(H, E) .

Proof By Theorems VII.2.8.4, 1.6.2(ii), and 1.5.5,

W−s/ν1p (H, E) .=

(W

s/ν1p′ (H, E′)

)′ =(Lp′,−sp′/ν1(H, E′)

)′.= Lp,sp/ν1(H, E) ,

where the last equivalence results from Lemma 1.5.1(ii). ¥

Spaces of Strongly Negative Order

Suppose 1 < p < ∞ and E is reflexive. Then we know that, given s ≥ 0,

W−s/νp (H, E) .=

(W

s/νp′ (H, E′)

)′ (1.7.3)

and

W−s/νp (H, E) .=

(W

s/νp′ (H, E′)

)′ (1.7.4)

with respect to 〈·, ·〉 = 〈·, ·〉H. By its definition, if s /∈ νN, then W−s/ν

p (H, E) isthe image of W

−s/νp (Rd, E) := B

−s/νp (Rd, E) by the point-wise restriction oper-

ator RH, thus by the operator of restriction in the sense of distributions (Theo-rem VI.1.2.3(ii)). This space is relatively simple. On the other hand, W

−s/νp (H, E)

is the image of the rather more complicated restriction operator RH. Hence it isdesirable to have alternative descriptions of W

−s/νp (H, E) which are more practical

to work with.If ν1(−1 + 1/p) < −s < 0, then Corollary 1.7.2 and Theorem 1.7.3 yield easy

answers to this problem. If −s < ν1(−1 + 1/p), then the solution is more compli-cated, since then W

−s/νp (H, E) contains distributions supported on ∂H. This is

made precise in the next theorem. Extending definition (1.3.10), we set

W s(~k,p)/νpppp

p (∂H, E) :=k∏

j=0

W s(j,p)/νpppp

p (∂H, E)

for s ∈ R and k ∈ N, recalling that s(j, p) = s − ν1(j + 1/p). Similarly,

W−s(~k,p)/νpppp

p (∂H, E) :=k∏

j=0

W−s(j,p)/νpppp

p (∂H, E) .

VIII.1 Traces 327

1.7.4 Theorem Suppose E is reflexive, 1 < p < ∞, k ∈ N, and

ν1(k + 1/p) < s < ν1(k + 1 + 1/p) .

ThenW−s/ν

p (H, E) ∼= W−s/νp (H, E) × W−s(~k,p)/ν

ppppp (∂H, E) .

An isomorphism is given by the dual of the map(1− (~γk)c~γk, ~γk

)

: Ws/ν

p′ (H, E′) → Ws/ν

p′ (H, E′) × Ws(~k,p′))/ν

ppppp′ (∂H, E′) .

(1.7.5)

Proof It is a consequence of Theorems 1.3.2 and 1.4.2 and of Lemma I.2.3.1 that(1.7.5) is a toplinear isomorphism. Thus the assertion follows from (1.7.3), (1.7.4)and Theorems VII.1.5.1, VII.2.8.4, and 1.6.1, taking (VII.1.2.5) into account. ¥

1.7.5 Corollary Let k and s be as in the theorem. Then f ∈ W−s/ν

p (H, E) iff there

exist v ∈ W−s/ν

p (H, E) and h ∈ W−s(~k,p)/ν

ppppp (∂H, E) such that

〈f, u〉 =⟨v,

(1− (~γk)c~γk

)u⟩H + 〈h,~γku〉∂H

for all u ∈ Ws/ν

p′ (H, E′).

1.7.6 Remark It is clear that, using Theorem 1.3.4 and Theorem 1.6.4 and itscorollary, we obtain an analogous result for general Besov spaces. ¥

Duality of Sums and Intersections

Next we use the weighted space characterization of Ws/ν

p to derive a represen-tation off W

−s/νp . Its proof is based on the following duality theorem for sums

and intersections of Banach spaces, which seems to be folklore and is often rathervaguely stated (e.g., Theorem 2.7.1 in [BeL76]). Thus, for the reader’s convenience,we include its precise formulation and a proof.

Let E1, . . . , En be Banach spaces satisfying Ei → Y for some LCS Y. Thesum space, ΣEi, of E1, . . . , En is defined by

ΣEi :=n∑

i=1

Ei := y ∈ Y ; ∃ xi ∈ Ei with y = x1 + · · · + xn

and is equipped with the norm

y 7→ ‖y‖ΣEi := inf n∑

i=1

‖xi‖Ei ; y = x1 + · · · + xn

.


1.7.7 Proposition Let X be an LCS such that X d→ Ei for 1 ≤ i ≤ n. Then:

(i) E′i → X ′, so that ΣE′

i is well-defined and

ΣE′i = (∩Ei)′


〈x′, x〉∩Ei := 〈x′1, x〉E1 + · · · + 〈x′n, x〉En , (x′, x) ∈ ΣE′i × ∩Ei

and any x′i ∈ E′i with x′1 + · · · + x′n = x′.

(ii) If E1, . . . , En are reflexive, then ∩Ei and ΣEi are so too.

Proof (1) Set F∞ :=∏n

i=1 Ei, endowed with the `∞ norm. Denote by M theclosed subspace of all (x1, . . . , xn) satisfying x1 = · · · = xn. Then

f : M → Y , (y, y, . . . , y) 7→ y

is an injective continuous linear map whose image equals⋂n

i=1 Ei. Hence f is anisometric isomorphism from M onto ∩Ei.

(2) Put F1 :=∏n

i=1 Ei, equipped with the `1 norm. Then

add : F1 → Y , (x1, . . . , xn) 7→n∑

i=1

xi

is a continuous linear map and ΣEi = add F1. Hence ΣEi is a Banach space byRemark VI.2.2.1(a).

(3) Let the hypotheses of (i) be satisfied. Set F ]1 :=

∏ni=1 E′

i, endowed withthe `1 norm. Theorem VI.2.1.1 guarantees that F ]

1 = (F∞)′ by means of the dualitypairing

〈y], y〉F∞ = 〈y]1, y1〉E1 + · · · + 〈y]

n, yn〉En, (y], y) ∈ F ]

1 × F1.

Due to step (1),

∩Eif←−∼= M

i→ F∞

where i(M) = M is closed in F∞. Thus, by duality,

F ]1

i′−→ M ′ f ′←−∼= (∩Ei)′.

From im = m for m ∈ M and i′(y′)(m) = y′(im) = y′(m) it follows that i′(y′) isthe restriction y′ |M to M for y′ ∈ F ]

1 .

VIII.1 Traces 329

Since i is injective and has closed range, i′ is injective,

im(i′) = ker(i)⊥ = M ′ , ker(i′) = im(i)⊥ = M⊥.

(4) Suppose x′ ∈ (∩Ei

)′ and set m′ = f ′(x′) ∈ M ′. Then

〈x′, x〉∩Ei=

⟨f ′−1(m′), x

⟩∩Ei

=⟨m′, f−1(x)

⟩M

= 〈y]1, x〉E1 + · · · + 〈y]

n, x〉En

(1.7.6)

for x ∈ ∩Ei and y] = (y]1, . . . , y

]n) ∈ F ]

1 with y] |M = m′ by step (3). Note thatE′

i → X ′ implies〈e′i, x〉Ei

= 〈e′i, x〉X , x ∈ X .

Hence it follows from X ⊂ ∩Ei and (1.7.6) that

〈x′, x〉∩Ei= 〈y]

1 + · · · + y]n, x〉X , x ∈ X .

Thus, if y] ∈ F ]1 also satisfies y] |M = m′, we find

y]1 + · · · + y]

n = y]1 + · · · + y]

n.

(5) Since f−1 is an isometry from ∩Ei onto M , we obtain from (1.7.6), moreprecisely from

〈x′, x〉∩Ei =⟨m′, f−1(x)

⟩M

,

that‖m′‖M ′ = ‖x′‖(∩Ei)′ .

The Hahn-Banach theorem guarantees the existence of m] ∈ F ]1 with m] |M = m′

and ‖m]‖F ]1

= ‖m′‖M ′ . Set y′ := m]1 + · · · + m]

n. Then y′ ∈ ΣE′i and

‖y′‖ΣE′i≤ ‖m]‖F ]

1= ‖m′‖M ′ = ‖x′‖(∩Ei)′ .

Hence we see from step (4) that

g : (∩Ei)′ → ΣE′i , x′ 7→ y′

is a well-defined bounded linear map of norm at most 1.(6) Suppose y′ ∈ ΣE′

i and y] ∈ F ]1 satisfy y′ = y]

1 + · · · + y]n. Then

h(y′)(x) := 〈y]1, x〉E1 + · · · + 〈y]

n, x〉En , x ∈ ∩Ei,

definesh(y′) ∈ (∩Ei)′ , ‖h(y′)‖(∩Ei)′ ≤ ‖y]‖F ]

1.

This being true for every such y], it follows that h maps ΣE′i onto (∩Ei)′ and has

norm at most 1. Combining this with the result of step (5) proves claim (i).


(7) Let E1, . . . , En be reflexive. Then M , being a closed linear subspace ofthe reflexive Banach space F∞, is reflexive. Hence its isomorphic image ∩Ei is alsoreflexive. Now (i) implies the reflexivity of ΣEi. ¥

Weighted Space Representations

We can now prove the desired representation theorem, provided the target spaceis reflexive.

1.7.8 Theorem Suppose E is reflexive, 1 < p < ∞, and s ∈ R+\ν1(N+ 1/p′).Then

W−s/νp (H, E) .= W−s/ν

p (H, E) + Lp,sp/ν1(H, E) .

Proof Definition (VII.1.2.5), Theorem VII.1.5.1, and the duality result of Theo-rem VII.2.8.4 imply

W−s/νp (H, E) .=

(W

s/νp′ (H, E′)

)′

with respect to 〈·, ·〉. Hence we infer from (1.5.15), Theorem 1.6.1, and Proposi-tion 1.7.7 that

W−s/νp (H, E) .=

(W

s/νp′ (H, E′)

)′ .=(W

s/νp′ (H, E′)

)′ + (Lp′,−sp′/ν1(H, E′)

)′.

Now the claim follows by the above arguments and Lemma 1.5.1. ¥

It is clear that the intersection space characterizations of Corollary VII.3.6.3and Proposition 1.7.7 can be used to obtain a further representation theoremfor W

−s/νp (H, E). We leave this to the reader.

1.8 Traces for Corners

Based on the preceding results for half-spaces and using the properties of the r-epairs introduced in Subsection VI.1.1, we now derive trace theorems for corners.


• K is a corner in Rd of type J with card(J) ≥ 1.

Recall that the type J of K is the set of indices j ∈ 1, . . . , d with ∂jK ⊂ K.Without loss of generality we can assume that

K = R+ ×K q,

where K qis a corner in Rd−1 of type J \1. Hence ∂1K = K q

(= 0 ×K q).

Unless explicit restrictions are mentioned,

• Conventions (1.3.1) and (1.3.6) are in force.

For the sake of a simpler notation we omit E in the proofs of this subsection.

VIII.1 Traces 331

Traces on a Single Face

Let s > ν1(m + 1/q) for some m ∈ N. The trace operator of order m for ∂1K isdefined by

γm1 u = ∂m

n1u := ∂m

1 u|x1=0 ∈ X (K q, E) , u ∈ Fs/ν

q (K, E) ,

(recall (1.3.2) and (1.3.3)). Thus γm1 = γm if K = H.

The following result extends Theorem 1.3.1 to this situation.

1.8.1 Theorem Suppose s > ν1(m + 1/q) with m ∈ N. Then ~γm1 := (γ0

1 , γ11 , . . . , γm

1 )is a universal retraction from F

s/νq (K, E) onto

∂Fs(~m,q)/νpppp

q (∂1K, E) :=m∏

j=0

∂Fs(j,q)/νpppp

q (∂1K, E) .

It possesses a universal coretraction, (~γm)c, which commutes with ∂2, . . . , ∂d.

Proof (1) In accordance with the definitions in Subsection VI.1.1, we can apply(RK pppp , EK pppp) to functions defined on K by letting these maps operate on the variablesx

q ∈ K q. Then, using Theorems VII.1.3.1, VII.2.8.2, and VII.4.1.1,

EK pppp ∈ L(Fs/ν

q (K), Fs/νq (H)

).

Hence, by Theorem 1.3.1,

γjEK pppp ∈ L(Fs/ν

q (K), ∂Fs(j,q)/νpppp

q (∂H))

.

Consequently,

RK ppppγjEK pppp ∈ L(Fs/ν

q (K), ∂Fs(j,q)/νpppp

q (∂1K))

. (1.8.1)

Since γj operates on the first variable of x = (y, xq) ∈ K, it commutes with RK pppp .

ThusRK ppppγjEK ppppu = ∂j

1u(0, ·) = γj1u , u ∈ Fs/ν

q (K) ,

that is, γj1 is represented by (1.8.1).

(2) We set (γj1)

c := RK pppp(γj)cEK pppp . By Theorem 1.3.1,

(γj1)

c ∈ L(∂Fs(j,q)/ν

ppppq (K q

), Fs/νq (K)

),

andγi1(γ

j1)

cv = RK ppppγi(γj)cEK ppppv = δijRK ppppEK ppppv = δijv

for v ∈ ∂1Fs(j,q)/ν

pppp(∂1K). Now the assertion follows by the arguments used in the

proof of Theorem 1.3.2. ¥


1.8.2 Remark It is obvious from Theorem 1.3.4 that the same assertions apply togeneral Besov spaces. ¥

Vanishing Traces on Corners

Now we extend Theorem 1.6.2 and Corollary 1.7.2 to corners, using obvious adap-tions of the notation employed there.

1.8.3 Theorem Suppose either 1 < q < ∞ and F = W , or q = ∞ and F ∈ B, b.(i) Let m ∈ N and

ν1(m + 1/q) < s < ν1(m + 1 + 1/q) . (1.8.2)

ThenF

s/νq,~γm

1(K, E) = Fs/ν

q

((0,∞) ×K q

, E)

.

(ii) Assume either 0 < s < ν1/q, or

E is reflexive and ν1(−1 + 1/q) < s < ν1/q .

ThenW s/ν

q (K, E) = W s/νq

((0,∞) ×K q

, E)

.

(iii) If q < ∞, then, in either case,

W s/νq

((0,∞) ×K q

, E)

is the closure of D((0,∞) ×K q

, E)

in W s/νq (K, E) .

Proof (1) Let (1.8.2) be satisfied. Suppose u ∈ Fs/νq,~γm

1(K). Then v := EK ppppu belongs

to Fs/νq,~γm

1(H). Hence v ∈ F

s/νq (H) by Theorem 1.6.2(i). Thus

u = RK ppppv ∈ Fs/νq

((0,∞) ×K q)

.

An analogous argument gives the converse inclusion. This proves (i).(2) Assertion (ii) follows similarly from Theorem 1.6.2(i) and Corollary 1.7.2.

(3) Let q < ∞ and u ∈ Ws/ν

q

((0,∞) ×K q)

. The theorems used in step (1)guarantee that v = EK ppppu ∈ W

s/νq (H) can be approximated arbitrarily closely in

Ws/ν

q (H) by elements w ∈ D(H). The continuity of RK pppp , which is established inTheorem VI.1.2.3, implies that RK ppppw ∈ D(

(0,∞) ×K q). From this we obtain as-

sertion (iii). ¥

Faces of Higher Codimensions

Now we write

ωqi := (ω1, . . . , ωi, . . . , ωd) , si(j, q) := s − ωi(j + 1/q)

for 1 ≤ i ≤ d. (Recall the definitions of Subsection VI.3.1.)

VIII.1 Traces 333

Suppose j ∈ J and s > ωj(mj + 1/q) with mj ∈ N. Then, given 0 ≤ k ≤ mj ,

γkj u := ∂k

nju := ∂k

j u|xj=0 , u ∈ Fs/νq (K, E) ,

is the trace operator of order k for ∂jK. Using obvious notation, we deduce fromTheorem 1.8.1 by a permutation of coordinates that ~γ

mj

j is a retraction fromF

s/νq (K, E) onto ∂F

sj(~mj ,q)/ωppppj

q (∂jK, E) and that it possesses a universal coretrac-tion (~γmj

j )c which commutes with ∂k for k 6= j.Generalizing the concept of (d − 1)-dimensional faces, that is, codim(1)-faces,

we introduce faces of higher codimension. Set ` := card(J). Suppose 1 ≤ r ≤ ` andput

Jr6= :=

(j) := (j1, . . . , jr) ∈ Jr ; ji 6= jk for i 6= k

so that J16= = J . Then, given (j) ∈ Jr

6=,

∂(j)K = ∂r(j)K :=

x ∈ K ; xi = 0, i ∈ j1, . . . , jr

is a codim(r)-face of K. It is (naturally identified with) a corner in Rd−r of typeJ

∖(j)

. Note that card(Jr

6=) =(

`r

). In particular, ∂`

(j1,...,j`)K is either an open

corner in Rd−`, or it equals Rd−`. Also observe that, writing (j) = (j1, q) ∈ Jr

6=with

q ∈ Jr−16= ,

∂r(j)K = ∂j1(∂

r−1( pppp) K) = ∂r−1

( pppp) ∂j1K . (1.8.3)

Hence

∂r(j)K = ∂j1∂j2 · · · ∂jrK = ∂jσ(1) · · · ∂jσ(r)K , (1.8.4)

where σ is any permutation of 1, . . . , r. We also write ∂j1,...,jrK for ∂r

(j)K.

Of particular interest are corners of codim(2). In this case we put

ωq qij := (ω1, . . . , ωi, . . . , ωj , . . . , ωd) , 1 ≤ i < j ≤ d .

Furthermore, given mi, mj ∈ N,

sij(mi,mj , q) := s − ωi(mi + 1/q) − ωj(mj + 1/q) .

Using this notation, we get the following simple, but important, properties of traceoperators on faces.

1.8.4 Lemma Suppose (i, j) ∈ J2 with i < j. Let mi, mj ∈ N satisfy

s > ωi(mi + 1/q) + ωj(mj + 1/q) .

Then

∂mini

∂mjnj ∈ L(

Fs/νq (K, E), ∂F

sij(mi,mj ,q)/ωpppp ppppij

q (∂ijK, E))

(1.8.5)

and ∂mini

∂mjnj = ∂

mjnj ∂mi

ni.


Proof Claim (1.8.5) is a consequence of Theorem 1.8.1 and the preceding consid-erations. The commutativity assertion is clear if the partial derivatives ∂mi

i ∂mj

j u

and ∂mj

j ∂mii u are continuous. This is the case if q = ∞. Otherwise, it is true if

u ∈ S(K, E). Since this space is dense in Fs/νq (K, E) with q < ∞, the assertion

follows for q < ∞ also, due to (1.8.4). ¥

It is clear from (1.8.3) that this lemma can be iterated to obtain analogousresults for more than two trace operators.

Compatibility Conditions

Let I be a subset of J with card(I) ≥ 2, and I26= := I2 ∩ J2

6=. We set

dIK :=⋃

i∈I

∂iK

and say that dK := dJK is the essential (set) boundary of K. Note that dK is asubset of bdry(K) = K\K, the topological boundary, which is proper unless K isclosed.

Supposemi ∈ N and s > ωi(mi + 1/q) , i ∈ I .

We set~mI := (mi)i∈I and ~γ ~mI

I := (~γmii )i∈I .

This is the trace operator of order ~mI for dIK, resp. the essential trace operator ifI = J . In particular, γdK := (γj)j∈J is the essential trace operator.

It follows from Theorem 1.8.1 that

~γ ~mI

I ∈ L(F

s/νq (K, E),

∏i∈I ∂F

si(~mi/q)ωppppi

q (∂iK, E))

.

In general, this map is not surjective. Indeed, given u ∈ Fs/νq (K, E), Lemma 1.8.4

implies that there may exist compatibility relations between ~γmii u and ~γ

mj

j u for(i, j) ∈ I2

6=. Namely, suppose

(i, j) ∈ I26= and 0 ≤ µρ ≤ mρ, with µρ ∈ N for ρ ∈ i, j,

satisfy ωi(µi + 1/q) + ωj(µj + 1/q) < s.(1.8.6)

Set gµρρ := γ

µρρ u. Then Lemma 1.8.4 implies

γµi

i gµj

j = γµj

j gµi

i . (1.8.7)

We define the trace space for ~γ ~mI

I , denoted by

∂Fs(~mI ,q)/ωpppp

q (dIK, E) , (1.8.8)

VIII.1 Traces 335

to be the closed linear subspace of∏

i∈I

∂Fsi(~mi,q)/ω

ppppi

q (∂iK, E)

consisting of all (gi)i∈I with gi = (g0i , . . . , gmi

i ) satisfying the compatibility condi-tions (1.8.7) whenever the relations (1.8.6) apply. Clearly,

~γ ~mI

I ∈ L(Fs/ν

q (K, E), ∂Fs(~mI ,q)/ωpppp

q (dIK, E))

. (1.8.9)

If I = J , then ∂Fs(~mJ ,q)/ωppppis called essential boundary trace space of order ~mJ .

The Retraction Theorem for Corners

Now we can prove the main result of this subsection, a retraction theorem forcorners. It applies, in particular, to the essential boundary.

1.8.5 Theorem Suppose that

either 1 < q < ∞ and F = W or q = ∞ and F ∈ B, b .

Recall that ∂F = B if F = B, and ∂F = b in all other cases.

Let s > 0 and mi ∈ N satisfy

ωi(mi + 1/q) < s < ωi(mi + 1 + 1/q) , i ∈ I . (1.8.10)

Also assume that, given (i, j) ∈ I26=,

s 6= ωi(µi + 1/q) + ωj(µj + 1/q) (1.8.11)

for µρ ∈ N with µρ ≤ mρ, ρ ∈ i, j. Then ~γ ~mI

I is a retraction

from Fs/νq (K, E) onto ∂Fs(~mI ,q)/ω

ppppq (dIK, E) .

It has a coretraction which is universal subject to (1.8.10) and (1.8.11).

Proof Due to (1.8.9), it suffices to construct a universal coretraction.

We can (and do) assume that I = 0, . . . , k so that K = (R+)k × L, whereL is a corner of type (J \I) in Rd−k. Then we proceed by induction on k.

(1) Suppose k = 2. Let (g1, g2) ∈ ∂Fs(~mI ,q)/ω

ppppq (dIK) be given. Thus

gi ∈ ∂Fsi(~mi,q)/ω

ppppi

q (∂iK) , i = 1, 2 ,

and (g1, g2) satisfies the compatibility conditions (1.8.6) and (1.8.7).


By applying Theorem 1.8.1, we set

u1 := (~γm11 )cg1 ∈ Fs/ν

q (K) . (1.8.12)

Thenh2 := g2 − ~γm2

2 u1 ∈ ∂Fs2(~m2,q)/ω

pppp2

q (∂2K) ,

where ∂2K = R+ × 0 × L = R+ × L. Hence

hi2 = gi

2 − γi2u1 ∈ ∂F

s2(i,q)/ωpppp2

q (R+ × L)

for 0 ≤ i ≤ m2. Thus

γj1h

i2 = γj

1gi2 − γj

1γi2(~γ

m11 )cg1

= γj1g

i2 − γi

2γj1(~γ

m11 )cg1 = γj

1gi2 − γi

2gj1 = 0

(1.8.13)

for 0 ≤ j ≤ m1 with

ω1(j + 1/q) < s2(i, q) = s − ω2(i + 1/q) .

Here we used Lemma 1.8.4 and the compatibility conditions (1.8.6). It follows fromassumption (1.8.11) that s2(i, q) /∈ ω1(N+ 1/q). Hence (1.8.13) and Theorem 1.8.3imply that hi

2 ∈ ∂Fs2(i,q)/ω

pppp2

q

((0,∞) × L)

for 0 ≤ i ≤ m2, that is,

h2 ∈ ∂Fs2(~m2,q)/ω

pppp2

q

((0,∞) × L)

.

It is a consequence of (1.8.10) and Theorem 1.8.3 that Fs/νq

((0,∞) × R+ × L)

isa closed linear subspace of F

s/νq (K). In fact,

Fs/νq

((0,∞) × R+ × L)

= Fs/ν

q,~γm11

(K) . (1.8.14)

Hence, setting

~Bm22 := ~γm2

2

∣∣Fs/νq

((0,∞) × R+ × L)

, (1.8.15)

we deduce from Theorem 1.8.1 that

~Bm22 ∈ L

(Fs/ν

q

((0,∞) × R+ × L)

, ∂Fs2(~m2,q)/ω

pppp2

q

((0,∞) × L))

and that this map is a retraction possessing a universal coretraction ( ~Bm22 )c.

Now we put

u2 := ( ~Bm22 )ch2 ∈ Fs/ν

q

((0,∞)× R+ × L)

.

Thenu := u1 + u2 ∈ Fs/ν

q (K) .

VIII.1 Traces 337

Moreover,~γm1

1 u = ~γm11 u1 + ~γm1

1 u2 = g1

by (1.8.12) and (1.8.14). Furthermore,

~γm22 u = ~γm2

2 u1 + ~γm22 u2 = ~γm2

2 u1 + ~γm22 ( ~Bm2

2 )c(g2 − ~γm22 u1) = g2 ,

due to (1.8.15). The map

(g1, g2) 7→ u = (~γm11 )cg1 + ( ~Bm2

2 )c(g2 − ~γm2

2 (~γm11 )cg1

)

is continuous and linear from ∂F(~mI ,q)/ω

ppppq (dIK) into F

s/νq (K), and it is a coretrac-

tion for ~γ ~mdIK. It is obviously universal. This proves the theorem if k = 2.

(2) Suppose k = 3. The following arguments are analogous to the ones of thepreceding step. Thus we can be brief.

We set~Bm33 := ~γm3

3

∣∣Fs/νq

((0,∞)2 × R+ × L)

.

It is a retraction from

Fs/νq

((0,∞)2 × R+ × L)

=

u ∈ Fs/νq (K) ; ~γmi

i u = 0, i = 1, 2

onto∂F

s3(~m3,q)/ωpppp3

q

((0,∞)2 × L)

.

Let u1 and u2 be as in step (1). Set

h3 := g3 − ~γm33 (u1 + u2) ∈ ∂F

s3(~m3,q)/ωpppp3

q (∂3K) .

The compatibility conditions imply, similarly as in step (1),

h3 ∈ ∂Fs3(~m3,q)/ω

pppp3

q

((0,∞)2 × L)

.

Denoting by ( ~Bm33 )c a universal coretraction for ~Bm3

3 , we put

u3 := ( ~Bm33 )ch3 ∈ Fs/ν

q

((0,∞)2 × R+ × L)

.

Hence ~γmii u3 = 0 for i = 1, 2.

We define u := u1 + u2 + u3 ∈ Fs/νq (K). Then

~γmii u = ~γmi

i (u1 + u2) = gi , i = 1, 2 .

Moreover,~γm3

3 u = ~γm33 (u1 + u2) + ~γm3

3 ( ~Bm33 )h3

= ~γm33 (u1 + u2) + h3 = g3 .

This proves the theorem if k = 3. The general case is now clear. ¥


1.8.6 Example (Parabolic weight systems) Let m ∈qN and consider the closed

2-corner K = Hm × R+ in Rm+1, so that

∂1K = 0 × Rm−1 × R+ = Rm−1 × R+

and∂m+1K = Hm × 0 = Hm .

Thus dK = ∂1K ∪ ∂m+1K. We denote by x = (y, xq) with x

q ∈ Rm−1 the generalpoint of Hm and by (x, t) the one of K.

Suppose ν ∈ N with ν ≥ 2 and ω := (1m, ν) so that [m, ν] =[2, (m, 1), (1, ν)

]is a reduced parabolic weight system on Rm+1 (cf. Example VI.3.1.1). Note that

ωq1 = (1m−1, ν) and ω

qm+1 = 1m .

In the following, we explicate the compatibility conditions (1.8.7) for smallvalues of s.

(a)(Sobolev–Slobodeckii spaces) Let 1 ν/p with s, s/ν /∈ N+ 1/p .

Setm1 := [s − 1/p] , m2 := [s/ν − 1] .

Assumegi1 ∈ W (s−i−1/p)/(1,ν)

p (Rm−1 × R+, E) , 0 ≤ i ≤ m1 ,

andgj2 ∈ W s−j−1/p

p (Hm, E) , 0 ≤ j ≤ m2 .

Suppose

CC(s, p, ν) ∂iygj

2(0, ·) = ∂jt gj

1(·, 0) ifν + 1

s − i − jν< p .

Then the essential boundary trace space of order (m1,m2) is the closed linearsubspace of

m1∏

i=0

W (s−i−1/p)/(1,ν)p (Rm−1 × R+, E)×

m2∏

j=0

W s−j−1/pp (Hm, E) ,

consisting of all (g1, g2) satisfying the compatibility conditions CC(s, p, ν).Let ν/p < s < ν(1 + 1/p). Then m2 = 0 and CC(s, p, ν) is given by

g02(0, ·) = g0

1(·, 0) ifν + 1

s< p <

ν + 1s − 1

,

VIII.1 Traces 339

by

g02(0, ·) = g0

1(·, 0) , ∂yg02(0, ·) = g1

1(·, 0) ifν + 1s − 1

< p <ν + 1s − 2

,

and in general by

∂iyg0

2(0, ·) = gi1(·, 0) , 0 ≤ i ≤ k ≤ m1 if

ν + 1s − k

< p <ν + 1

s − k − 1,

where (ν + 1)/(s − k − 1) := ∞ if k + 1 ≥ s.

(b)(Holder spaces) Suppose s ∈ R+\νN. Set m1 := [s] and m2 := [s/ν − 1]. Let

gi1 ∈ buc(s−i)/(1,ν)(Rm−1 × R+, E) , gj

2 ∈ bucs−j(∂H, E)

for 0 ≤ i ≤ m1, 0 ≤ j ≤ m2. Assume

CC(s,∞, ν) ∂iygj

2(0, ·) = ∂jt gi

1(·, 0) if 0 ≤ i + νj < s .

Then the essential boundary trace space of order (m1,m2) is the closed linearsubspace of

m1∏

i=0

buc(s−i)/(1,ν)(Rm−1 × R+, E) ×m2∏

j=0

bucs−j(K, E) ,

which is made up by all (g1, g2) satisfying the compatibility conditions CC(s,∞, ν).Thus, if 0 < s < ν, conditions CC(s,∞, ν) read

∂i1g

02(0, ·) = gi

1(·, 0) if 0 ≤ i ≤ m1 .

Observe that, in contrast to (a), where there are no compatibility conditions if0 ≤ s < ν/p, conditions CC(s,∞, ν) are never void. ¥

1.9 Notes

First we make some comments on our notation and its relation to common usage.For this we consider the scalar isotropic case and half-spaces.

In our setup it is natural to consider Sobolev, Besov, and Bessel potentialspaces on the closed half-space H. In contrast, everywhere else in the literaturethese spaces are studied on open subsets of Rd, thus notably on H. To elucidatethis we set X := H. Given m ∈ N and 1 ≤ q < ∞, the standard definition of theSobolev space Wm

q (X) is the following: The function u ∈ Lq(X) belongs to Wmq (X)

iff the distributional derivatives ∂αu for |α| ≤ m too belong to Lq(X). It followsfrom Theorem VII.1.4.1(ii) that Wm

q (H) = Wmq (X), that is,

Wmq (H) is the classical Sobolev space of Lq functions on H.

The standard way to obtain Besov spaces Bsq,r(X) and Bessel potential spaces

Hsq (X) is to define them as the image spaces of Bs

q,r(Rd) and Hsq (Rd), respectively,


under the restriction to X in the sense of distributions (e.g., [Tri83]). Due toTheorem VI.1.2.3(ii), this is precisely what we did in definitions (VII.2.8.1) and(VII.4.1.6). Thus

the Besov spaces Bsq,r(H) and Bs

q,r(H), as well asthe Bessel potential spaces Hs

q (H) and Hsq (H),

coincide with the respective standard spaces on H.

In the classical isotropic scalar case, the trace theorems of Subsections 1.1and 1.2 are, of course, well-known. We refer to [Tri83] for those results and forextensive historical remarks. H. Triebel, as well as most of the other authors withmore recent contributions, consider also the case where 0 < q, r < 1, which is ofno interest in our work.

More recently, W. Farkas, J. Johnsen, and W. Sickel [FaJS00] presented theextension of the above mentioned results to anisotropic scalar Besov and Triebel–Lizorkin spaces and included, in particular, a discussion of the ‘border-line cases’(s = ν1/q, for example).

In the vector-valued setting, trace theorems for isotropic Besov, Triebel–Lizorkin, Sobolev, and Bessel potential spaces have been given by H.-J. Schmeißerand W. Sickel [SS01] and B. Scharf, H.-J. Schmeißer, and W. Sickel [SSS12]. Inparticular, these authors prove the isotropic version of Theorems 1.2.1 and 1.2.9(with m = 0).

Our proof of Theorem 1.2.1, in the case of Besov spaces, is an adaption of thearguments found in [FaJS00] and [SSS12], as it has been sketched in [Ama09]. Inparticular, the use of the Paley–Wiener–Schwartz theorem in the proof of Proposi-tion 1.2.4 is a modification of the corresponding argument in [FaJS00]. In the proofsof Proposition 1.2.5 and Lemma 1.2.8 we follow essentially the demonstration of[Tri83, Theorem 2.7.2]. This allows us to avoid the use of atomic decompositionswhich is a basic tool in the work of Schmeißer and Sickel, and Scharf, Schmeißer,and Sickel.

It is an important feature of our results that our trace theorems are universal.This means, in particular, that one and the same coretraction for the trace operator(namely Km) applies to Besov, Bessel potential, Sobolev, and Triebel–Lizorkinspaces, without any restriction on the image space E and for all indices.

Our proof of Theorem 1.4.2 builds on some ideas of Theorem 2.9.3 in [Tri95].It is known that its assertion does not hold if s ∈ ν1(N+ 1/p) (e.g., J.-L. Lionsand E. Magenes [LM61]).

The proof of the Hardy inequalities follows A. Kufner, L. Maligranda, andL.-E. Persson [KuMP07]. Theorems 1.5.5 and 1.5.6 generalize considerably resultsof [LM61], who consider the (isotropic scalar) case with s ∈ (0, 1)\1/p. Our prooffor s /∈ ν1N uses arguments of [LM73], which go back to P. Grisvard [Gri63] (alsosee [Gri72]), who in turn, builds on the work of V.P. Il′in [Il′i59].

VIII.1 Traces 341

In the isotropic scalar case and for s = 1, Theorem 1.5.6 coincides with aresult of J. Kadlec and A. Kufner [KaK66]. If s is an integer bigger than 1,then our theorem sharpens the corresponding one of [KaK66]. It should be notedthat Kadlec and Kufner considered weighted Sobolev spaces as well (also see Sec-tion 2.9.2 and Chapter 3 of [Tri95]).

The fundamental Theorem 1.6.1 relates our spaces Ws/ν

p (H, E) to the ‘clas-sical’ spaces W

s/νp (H, E). Its proof is based on the fact that

EH ∈ L(W s/ν

p (H, E),W s/νp (Rd, E)

)(1.9.1)

for s /∈ ν1(N+ 1/p). In turn, (1.9.1) is derived from Theorem 1.5.6 using ideasof [LM61]. We do not have an analogy of (1.9.1) for Bessel potential spaces.

If we knew that

EH ∈ L(Hs/ν

p (H, E),Hs/νp (Rd, E)

), s ∈ R+\ν1(N+ 1/p) , (1.9.2)

then step (1) of the proof of Theorem 1.6.1 together with

D(H, E)d

→ S(H, E)d

→ Hs/νp (H, E)

d→ Hs/ν

p (H, E) (1.9.3)

would imply

Hs/νp (H, E) .= Hs/ν

p (H, E) . (1.9.4)

Note that (1.9.3) follows from Theorem VII.4.1.3(i).Observe that, due to Theorem 1.4.2, assertion (1.9.1) implies that the char-

acteristic function χH of H in Rd is a point-wise multiplier of Ws/ν

p (Rd, E) for0 < s < ν1/p, that is,

(u 7→ χHu) ∈ L(W s/ν

p (Rd, E))

, 0 < s < ν1/p .

This follows from the observation that (u 7→ χHu) = (u 7→ EHRHu) and the conti-nuity properties of RH.

It has been shown by E. Shamir [Sha62] and R.S. Strichartz [Str67] that

(u 7→ χHu) ∈ L(Hs

p(Rd, E))

, −1 + 1/p < s < 1/p , (1.9.5)

if E = C. Recently, M. Meyries and M. Veraar [MeV15, Theorem 1.1] proved that(1.9.5) applies if E is a UMD space. Their demonstration is based on paraproductand R-boundedness techniques. In the light of this result it is reasonable to expectthat

(u 7→ χHu) ∈ L(Hs/ν

p (Rd, E))

, −1 + 1/p < s < 1/p , (1.9.6)

if E is ν-admissible. From (1.9.6) and Theorem 1.6.8 it would follow by meansof Theorem VII.4.3.1(ii) that (1.9.2) is true. Hence we would get the validity of


(1.9.4) if E is ν-admissible. We did not attempt to prove (1.9.6). Hence we leaveit, and thus (1.9.4), as an open problem.

The idea to employ the J method for the proof of Theorem 1.6.4 comes fromD. Guidetti’s paper [Gui91] on interpolation with boundary conditions.

Due to the general properties of the r-e pair (R, E), Theorems 1.8.1 and 1.8.3are easy extensions of the corresponding trace theorems for closed half-spaces.Their importance lies in the fact that they apply to arbitrary corners having atleast one closed face. This is essential for our proof of the general retraction The-orem 1.8.5.

The content of this section corrects, generalizes and simplifies considerablyearlier results of the author given in [Ama09]. Besides of this work, the only studyrelated to Theorem 1.8.5, known to the author, is due to P. Grisvard [Gri66,Theoreme 4.2]. (Also see [Gri67] and J.-L. Lions and E. Magenes [LM73, Theo-rems IV.2.1 and IV.2.3] for the case p = 2). However, Grisvard –– and Lions andMagenes also –– do not impose the restrictions (1.8.11). In these exceptional casesthe trace spaces have to be replaced by spaces smaller than the spaces (1.8.8).They are characterized by non-local norms. For simplicity, we do not enter suchinvestigations.

VIII.2 Boundary Operators 343

2 Boundary Operators

Preparing for the study of boundary value problems, we investigate in this sectiondifferential operators on the boundary of a closed half-space. Our main interestconcerns retraction theorems which allow to reduce non-homogeneous boundaryvalue problems to homogeneous ones. In the last subsection we prove an importantinterpolation theorem for spaces with boundary conditions.

In view of the envisaged applications and for the sake of a simple presentationwe assume throughout that

• 1 < q ≤ ∞ .

• F = W if q < ∞ ,

F ∈ B, b if q = ∞ .

(2.0.1)

We recall that

∂F = B if F = B and ∂F = b in all other cases .

Moreover, s(j, q) = s − ν1(j + 1/q).

2.1 Boundary Operators on Half-Spaces

Let F be a Banach space and m ∈ N. We consider boundary differential operatorsof order m on ∂H of the form

B =∑

α ppppω≤mν1

bαγ∂α ,

where the bα are L(E, F )-valued functions on ∂H. Then, by definition, B is of type

(s,m, F, q) if

s > ν1(m + 1/q) and bα ∈ bσ/νpppp

∞(∂H,L(E,F )

), α q ω ≤ mν1,

where σ ≥ s(m, q) with a strict inequality sign if q < ∞.(2.1.1)

This somewhat clumsy definition is introduced in order to impose (almost) minimalregularity assumptions on the coefficients. This is of importance in connection withquasilinear differential equations.

We write x = (y, xq) ∈ H with x

q ∈ ∂H = Rd−1 and α = (j, β) ∈ Nd whereβ belongs to Nd−1. Then γ∂α = ∂β

x pppp∂jn. Hence

B =m∑

j=0

bj∂jn , (2.1.2)

wherebj = bj(·,∇x pppp) :=

∑

β ppppω pppp≤(m−j)ν1

b(j,β)∂βx pppp

is a ‘tangential’ differential operator.


2.1.1 Lemma Suppose B is of type (s,m, F, q). Then

B ∈ L(Fs/ν

q (H, E), ∂Fs(m,q)/νpppp

q (∂H, F ))

and‖B‖ ≤ c

∑

α ppppω≤mν1

‖bα‖σ/ν pppp,∞ .

Proof The trace theorem, that is, Theorem 1.3.1, implies

∂jn ∈ L(

Fs/νq (H, E), ∂Fs(j,q)/ν

ppppq (∂H, E)

). (2.1.3)

It follows from Theorem VII.2.6.1 that

∂βx pppp ∈ L(

∂Fs(j,q)/νpppp

q (∂H, E), ∂F(s(j,q)−β ppppω pppp)/ν

ppppq (∂H, E)

)

for β q ω q ≤ (m− j)ν1. Hence we get from ∂Ft/ν

ppppq → ∂F

τ/νpppp

q for t ≥ τ that

∂βx pppp ∈ L(

∂Fs(j,q)/νpppp

q (∂H, E), ∂Fs(m,q)/νpppp

q (∂H, E))

(2.1.4)

for β q ω q ≤ (m− j)ν1. By Corollary VII.6.3.4, bt/ν

pppp∞ → B

t/νpppp

∞ , and (2.1.1),

(v 7→ aαv) ∈ L(∂Fs(m,q)/ν

ppppq (∂H, E), ∂Fs(m,q)/ν

ppppq (∂H, F )

). (2.1.5)

Now we infer from (2.1.3)–(2.1.5) that

bj∂jn ∈ L(

∂Fs/νq (H, E), ∂Fs(m,q)/ν

ppppq (∂H, F )

).

This implies the claim. ¥

2.1.2 Remark It should be noted that this lemma holds also for the general Besovspaces B

s/νq,r and b

s/νq,r as well as for Bessel potential spaces. ¥

Normal Boundary Operators

Observe that the top-order coefficient of (2.1.2),

bm = bm(·,∇x pppp) ∈ bσ/νpppp

∞(∂H,L(E,F )

),

is independent of ∇x pppp . The operator B is called normal if bm(xq) ∈ L(E, F ) is a

retraction for xq ∈ ∂H and there exists a coretraction bc

m(xq) such that

bcm ∈ bσ/ν

pppp∞

(∂H,L(F,E)

). (2.1.6)

The following lemma implies a sufficient condition for B to be normal. It isparticularly useful in the important case where E and F are finite-dimensional.


For this we introduce the following definition: Suppose E and F are Hilbert spacesand X is a nonempty set. Then b : X → L(E, F ) is uniformly surjective on X ifthere exists a constant β > 0 such that

|b∗(x)f |E ≥ β |f |F , f ∈ F , x ∈ X ,

that is, if the (point-wise defined) adjoint map

b∗ : X → L(F, E) , x 7→ b(x)∗

is uniformly injective on X. This is equivalent to

bb∗(x) ≥ β2 , x ∈ X , (2.1.7)

which means that the (point-wise defined) self-adjoint operator bb∗ is uniformlypositive definite.

We denote by

bρ/ν

pppp∞,surj

(∂H,L(E,F )

), ρ > 0 , (2.1.8)

the set of all bρ/ν

pppp∞

(∂H,L(E, F )

)which are uniformly surjective on ∂H. It is clear

from |b(x q)|L(E,F ) = |b∗(x q

)|L(F,E) that (2.1.8) is open in bρ/ν

pppp∞

(∂H,L(E,F )

).

2.1.3 Lemma Let E and F be Hilbert spaces, ρ > 0, and

b0 ∈ bρ/ν

pppp∞,surj

(∂H,L(E, F )

). (2.1.9)

Then there exist a neighborhood U of b0 in bρ/ν

pppp∞

(∂H,L(E, F )

)and, for each b ∈ U ,

a right inverse7 bc belonging to bρ/ν

pppp∞

(∂H,L(F, E)

)such that the map

(b 7→ bc) : U 7→ bρ/νpppp

∞(∂H,L(F,E)

)(2.1.10)

is analytic.

Proof (1) Suppose a ∈ L(E, F ) satisfies

(aa∗f |f)F = |a∗f |2E ≥ α2 |f |2F , f ∈ F .

Then aa∗ ∈ Laut(F ) and |(aa∗)−1|L(F ) ≤ α−2. Thus

ac := a∗(aa∗)−1 ∈ L(F, E) , aac = 1F ,

and

|ac|L(F,E) ≤ α−2 |a∗|L(F,E) = α−2 |a|L(E,F ) . (2.1.11)

Since Laut(F ) is open in L(F ) and the inversion map b 7→ b−1 : Laut(F ) → L(F )is continuously differentiable, it is analytic (cf. [HillP57] and recall that we deal

7That is, bc(xq) is for each x

q ∈ ∂H a right inverse for b(xq).


with complex Banach spaces). Since ∂a∗ = (∂a)∗, it is clear that a 7→ a∗ is contin-uously differentiable. From this we easily infer that a 7→ ac is continuously differ-entiable, hence analytic, on the open subset

a ∈ L(E,F ) ; ∃ α = α(a) > 0 with aa∗ ≥ α

(2.1.12)

of L(E, F ).(2) Let x, ξ ∈ ∂H. Then

4ξ(uv)(x) = 4ξu(x)v(x + ξ) + u(x)4ξv(x)

implies

42ξ(uv)(x) = 42

ξu(x)v(x + 2ξ) + 24ξu(x)4ξv(x + ξ) + u(x)42ξv(x)

whenever u and v are functions on ∂H with values in some algebra. Similarly, weget from 4ξ

(u−1(x)u(x)

)= 0

4ξu−1(x) = −u−1(x)4ξu(x)u−1(x + ξ)

and, consequently,

42ξu

−1(x) = u−1(x)4ξu(x)u−1(x + ξ)4ξu(x + ξ)u−1(x + 2ξ)

− u−1(x)42ξu(x)u−1(x + 2ξ)

+ u−1(x)4ξu(x)u−1(x + ξ)4ξu(x + ξ)u−1(x + 2ξ)

if u maps into the group of invertible elements.

(3) Let b ∈ Bρ/ν

pppp∞

(∂H,L(E, F )

)satisfy (2.1.7). Set a := bb∗ and α := β2. Also

put bc := b∗a−1. Then it follows from step (2) and the fact that

b(x) 7→ b∗(x) =(b(x)

)∗

are isometric maps for x ∈ ∂H that

|4ξbc(x)|L(F,E) ≤ c(α−1, ‖b‖∞) |4ξb(x)|L(E,F ) , x, ξ ∈ ∂H . (2.1.13)

This and (2.1.11) imply, in particular,

bc ∈ BUC(∂H,L(F, E)

), ‖bc‖∞ ≤ c(α−1, ‖b‖∞) . (2.1.14)

Furthermore, if 0 < θ < 1, we get (recall (VII.3.4.2))

[bc]θ,∞;j ≤ c(α−1, ‖b‖∞) [b]θ,∞;j , 2 ≤ j ≤ d . (2.1.15)

Similarly,

‖42ξb

c‖∞ ≤ c(α−1, ‖b‖∞) (‖42ξb‖∞ + ‖4ξb‖2

∞) (2.1.16)


so that

[bc]1,∞;j ≤ c(α−1, ‖b‖∞)([b]1,∞;j + [b]21/2,∞;j

), 2 ≤ j ≤ d . (2.1.17)

(4) Suppose 0 < ρ/ωj ≤ 1 and set X := BUC(Rd−2,L(E,F )

). Then we ob-

tain from (2.1.14)–(2.1.17) that(xj 7→ bc(xj ; ·)) ∈ Bρ/ωj∞ (R,X ) (2.1.18)

and the norm of this map is bounded by a constant depending on the norm of(xj 7→ b(xj ; ·)) ∈ Bρ/ωj∞ (R,X ) , (2.1.19)

It is clear that (2.1.13) implies

[bc]δθ,∞;j ≤ c(α−1, ‖b‖∞) [b]δθ,∞;j , 2 ≤ j ≤ d , δ > 0 .

Similarly, we get from (2.1.17)

[bc]δ1,∞;j ≤ c(α−1, ‖b‖∞)([b]δ1,∞;j +

([b]δ1/2,∞;j

)2), 2 ≤ j ≤ d , δ > 0 .

Since b1∞(R,X ) → b

1/2∞ (R,X ), it follows from Theorem VII.3.7.1 that [bc]δ1,∞;j → 0

as δ → 0 if b ∈ b∞1 (R,X ). This shows that (2.1.18) holds for bρ/ωj∞ too.

(5) Now we assume k ∈qN and

(xj 7→ b(xj ; ·)) ∈ BUCk(R,X ). Then it is a

consequence of Leibniz’ rule and Lemma VI.3.3.7 that(xj 7→ bc(xj ; ·)) ∈ BUCk(R,X ) .

From this, step (4), and Theorems VII.3.6.2 and VII.3.7.1 it follows that

b ∈ bρ/ν

pppp∞,surj

(∂H,L(E, F )

)=⇒ bc ∈ bρ/ν

pppp∞

(∂H,L(F, E)

)

for ρ > 0.(6) Given b0 satisfying (2.1.9), we see from (2.1.12) and steps (3) and (4) that

there exists an open neighborhood U of b0 in bρ/ν

pppp∞

(∂H,L(E,F )

)such that bc is

well-defined for b in U and bc belongs to bρ/ν

pppp∞

(∂H,L(F,E)

). Furthermore, the map

(2.1.10) is continuously differentiable, hence analytic. This proves the lemma. ¥

2.2 Systems of Boundary Operators

In this subsection we consider systems of boundary operators of different order.We suppose

• 0 ≤ m0 < m1 · · · < mk are integers for some k ∈ N.

• F0, . . . , Fk are Banach spaces.• Bmi is a normal boundary operator of type (s, mi, Fi, q).


Then m := (m0, . . . , mk) is said to be an order sequence (of length k + 1), andF := (F0, . . . , Fk) is a sequence of boundary range spaces. Moreover,

B := (Bm0 , . . . ,Bmk)

is a normal boundary operator of type (s, m, F , q) on ∂H. Note that this implies

s > ν1(mk + 1/q) .

If Fi = F for 0 ≤ i ≤ k, then we simply write F instead of F .For abbreviation,

∂Fs(m,q)/νpppp

q (∂H, F ) :=k∏

i=0

∂Fs(mi,q)/νpppp

q (∂H, Fi) .

The Boundary Operator Retraction Theorem

2.2.1 Theorem Let F satisfy (2.0.1). Suppose B is a normal boundary operator oftype (s, m, F , q). Then B is a retraction from

Fs/νq (H, E) onto ∂Fs(m,q)/ν

ppppq (∂H, F ) .

There exists a coretraction Bc satisfying

∂jn Bc = 0 , 0 ≤ j < mk , j /∈ m0, . . . , mk−1 .

It is universal with respect to s and q, subject to condition (2.1.1).

Proof Lemma 2.1.1 implies that B is a continuous linear map. We write

Bmi =mi∑

j=0

bi,j∂jn , 0 ≤ i ≤ k ,

and fix a coretraction bci,mi

satisfying (2.1.6) for bi,mi . Then, recall part (iii) ofCorollary VII.6.3.4,

Cmi := −mi−1∑

j=0

bci,mi

bi,j∂jn , C0 := 0 ,

is a continuous linear map from Fs/νq (H, E) into ∂F

s(mi,q)/νpppp

q (∂H, E).We introduce

C := (C0, . . . , Cmk) ∈ L(Fs/ν

q (H, E),mk∏

j=0

∂Fs(j,q)/νpppp

q (∂H, E))

by setting Cj := 0 for 0 ≤ j < mk with j /∈ m0, . . . , mk−1.


Suppose g = (gm0 , . . . , gmk) ∈ ∂Fs(m,q)/ν

ppppq (∂H, F ). Define

h = (h0, . . . , hmk) ∈mk∏

j=0

∂Fs(j,q)/νpppp

q (∂H, E)

by hmi := bci,mi

gmi for 0 ≤ i ≤ k, and hj := 0 otherwise.

We deduce from Theorem 1.3.2 that there exists a universal coretraction γcj

for ∂jn satisfying

∂`n γc

j = δ`j id , j 6= ` . (2.2.1)

Let u0 := γc0h

0 ∈ Fs/νq (H, E) and u−1 := 0. Suppose u0, u1, . . . , uj−1 have already

been defined, where 1 ≤ j ≤ mk. Set

uj := uj−1 + γcj (h

j + Cjuj−1 − ∂jnuj−1) . (2.2.2)

This results in u0, u1, . . . , umk∈ F

s/νq (H, E). It follows from (2.2.1) that

∂jnuj = hj + Cjuj−1 , 0 ≤ j ≤ mk , (2.2.3)

and

∂inuj = ∂i

nuj−1 , 0 ≤ i < j ≤ mk . (2.2.4)

Suppose 0 ≤ ` ≤ j ≤ mk. Then we infer from (2.2.3) and (2.2.4) that

∂`nuj = ∂`

nuj−1 = · · · = ∂`nu` = h` + Cù`−1 . (2.2.5)

If, moreover, ` = mi ∈ m0, . . . , mk, then (2.2.4) implies

Cù`−1 = −`−1∑

j=0

bci,mi

bi,j∂jnu`−1

= −`−1∑

j=0

bci,mi

bi,j∂jnu` = Cù` = · · · = Cùj .

Consequently,

∂min uj = hmi + Cmiuj , mi ≤ j ≤ mk , 1 ≤ i ≤ k .

By multiplying these equations from the left by bi,mi , we obtain

Bmiuj = gmi , mi ≤ j ≤ mk . (2.2.6)


We define

Bmi,c ∈ L( i∏

j=0

∂Fs(mj ,q)/νpppp

q (∂H, Fj), Fs/νq (H, E)

)

by Bmi,c(gm0 , . . . , gmi) := umi for 0 ≤ i ≤ k. Then, for 0 ≤ α ≤ i ≤ j ≤ k,

BmαBmi,c(gm0 , . . . , gmi) = Bmαumi = Bmαumj

= BmαBmj ,c(gm0 , . . . , gmj ) = gmα ,

due to (2.2.6). Finally, we set Bc := Bmk,c. Then Bc is a coretraction for B. If` /∈ m0, . . . , mk, then it follows from h` = 0, C` = 0, and (2.2.5) that

∂`nBmk,c(gm0 , . . . , gmk) = ∂`

numk= 0 .

The asserted universality with respect to s, q, E, and F is clear from these argu-ments. This proves the theorem. ¥

2.2.2 Remark An analogous result applies to the general Besov spaces Bs/νq,r

and bs/νq,r as well as to Bessel potential spaces. ¥

Embeddings with Boundary Conditions

On the basis of the preceding results we can easily extend Theorem 1.6.7 to generalboundary conditions.

2.2.3 Theorem Let 1 < p < ∞ and suppose B is a normal boundary operator oftype (s, m, F , p).

(i) Assume ν1(mk + 1/p) < σ ≤ s. Then

Bσ/νp,1,B(H, E)

d→ H

σ/νp,B (H, E) → B

σ/νp,∞,B(H, E) .

(i) If ν1(mk + 1/p) < s0 < σ < s1 ≤ s, then

Bs1/νp,B (H, E)

d→ B

σ/νp,1,B(H, E) and B

σ/νp,∞,B(H, E)

d→ B

s0/νp,B (H, E) .

Proof (1) Assertion (i) follows from(1.6.13), the preceding remark, and Lemma1.6.6, setting (r, rc) := (B,Bc).

(2) We infer from Theorems VII.2.2.2, VII.2.2.4, and VII.2.8.3 that

Bs1/νp (H, E)

d→ B

σ/νp,1 (H, E) and Bσ/ν

p,∞(H, E)d

→ Bs0/νp (H, E) .

From these embeddings we get claim (ii) by again applying Lemma 1.6.6 with(r, rc) = (B,Bc). ¥


It is an immediate consequence of this theorem that

Bs1/νp,B (H, E)

d→ H

σ/νp,B (H, E)

d→ B

s0/νp,B (H, E) ,

provided ν1(mk + 1/p) < s0 < σ < s1 ≤ s. Our next theorem generalizes these em-beddings to cases in which s0, σ, and s1 may be further apart.

2.2.4 Theorem Let B be a normal boundary operator of type (s, m, F , p), where1 < p < ∞. Suppose 0 < s0 < σ < s1 < s with

s0, s1, σ /∈ ν1(mi + 1/p) ; 0 ≤ i ≤ k

.

ThenB

s1/νp,B (H, E)

d→ H

σ/νp,B (H, E)

d→ B

s0/νp,B (H, E) .

Proof We omit (H, E) and let F ∈ B,H.(1) It follows from Theorems VII.2.2.2, VII.2.8.3, and VII.4.1.3(i) that

Bs1/νp

d→ Hσ/ν

p

d→ Bs0/ν

p . (2.2.7)

Since Ft/νp,B = F

t/νp if t < ν1(m0 + 1/p), the assertion is clear if s1 < ν1(m0 + 1/p).

(2) Suppose σ > ν1(m0 + 1/p). Let `, resp. `1, be the largest integer ≤ ksuch that ν1(m` + 1/p) < σ, resp. ν1(m`1 + 1/p) < s1. Set n := (m0, . . . , m`) andn1 := (m0, . . . ,m`1) as well as G := (F0, . . . , F`) and G1 := (F0, . . . , F`1). ThenR := (Bm0 , . . . ,Bm`), resp. R1 := (Bm0 , . . . ,Bm`1 ), is a normal boundary opera-tor of type (s, n, G, p), resp. (s, n1, G1, p). Thus Theorem 2.2.1 (also recall Re-mark 2.2.2) guarantees that R1 is a retraction

from Fs1/νp onto ∂Fs1(n1,p)/ν

ppppp (∂H, G1)

possessing a universal coretraction Rc1.

Assumeg1 := (g0, . . . , g`1) ∈ ∂Fs1(n1,p)/ν

ppppp (∂H, G1) .

Then

g := (g0, . . . , g`) ∈ ∂Fs1(n,p)/νpppp

p (∂H, G) → ∂Fσ(n,p)/νpppp

p (∂H, G) . (2.2.8)

Moreover,

BmiRc1(g

0, . . . , g`1) = gmi , 0 ≤ i ≤ ` . (2.2.9)

Set

Rc(g0, . . . , g`) := Rc1(g

0, . . . , g`, 0, . . . , 0) . (2.2.10)


Then the construction of Rc1 in the proof of Theorem 2.2.1 shows that

Rc ∈ L(∂Fσ(n,p)/ν

ppppp (∂H, G), Fσ/ν

p

). (2.2.11)

It follows from the definition of R, R1, and (2.2.8)–(2.2.11) that (R,Rc) is auniversal r-c pair for

(F

σ/νp , ∂F

σ(n,p)/νpppp

p (∂H, G)). Also note that

Fs1/νp,B = F

s1/νp,R1

and Fσ/νp,B = F

σ/νp,R . (2.2.12)

(3) Let σ > ν1(m0 + 1/p). Suppose u ∈ Hσ/νp,B → H

σ/νp . We infer from (2.2.7)

that there exists a sequence (uj) in Bs1/νp converging in H

σ/νp towards u. Set

vj := (1 −Rc1R1)uj , g1,j := R1uj , gj := Ruj .

Then, given 0 ≤ i ≤ `, it is a consequence of (2.2.9), (2.2.10), the definition of(R,Rc), and (2.2.12) that

Bmivj = Bmiuj − BmiRc1g1,j = Bmiuj − BmiRcgj

= Bmi(1 −RcR)uj = 0

for j ∈ N. From this and Bs1/νp → H

σ/νp we deduce that vj ∈ H

σ/νp,B . Hence

vj − u = (1 −RcR)(uj − u) → 0 in Hσ/νp .

This proves

Bs1/νp,B

d→ H

σ/νp,B .

(4) Suppose s0 > ν1(m0 + 1/p). Then we obtain

Hσ/νp,B

d→ B

s0/νp,B

by interchanging the roles of B and H in step (3).(5) Assume σ > ν1(m0 + 1/p) > s0. Let i ≤ k be the largest integer such that

ν1(mi + 1/p) < σ and set ` := mi. Then

Hσ/ν

p,~γ` → Hσ/νp,B → Bs0/ν

p . (2.2.13)

We know from Theorem 1.6.8(i) that Hσ/ν

p,~γ`

.= Hσ/νp , and Theorem 1.6.2(ii) guaran-

tees that Bs0/νp = W

s0/νp

.= Ws0/ν

p . Hence D(H, E) is dense in Hσ/ν

p,~γ` and in Bs0/νp .

Thus, since Hσ/ν

p,~γ` → Bs0/νp by (2.2.13), we see that H

σ/ν

p,~γ`

d→ B

s0/νp . Now we de-

duce from (2.2.13) that Hσ/νp,B

d→ B

s0/νp .

(6) Finally, let s1 > ν1(m0 + 1/p) > σ. By replacing (Hσ/νp,B , B

s0/νp ) in the

preceding step by (Bs1/νp,B ,H

σ/νp ), we see that

Bs1/νp,B

d→ Hσ/ν

p

d→ Bs0/ν

p ,

where the last dense embedding is obtained from (2.2.7). The theorem is proved. ¥


2.3 Transmission Operators

We denote by S the hyperplane 0 × Rd−1 in Rd, oriented by the positive normal

nS = n := e1 = (1, 0, . . . , 0). If it is convenient and clear from the context, thenwe identify S naturally with Rd−1. By ρS we mean the reflection on S,

ρSx := (−x1, x2, . . . , xd) , x ∈ Rd . (2.3.1)

This defines an operation of the two element group ±1 on D′(Rd, E) by settingρSu := u ρS for u ∈ L1,loc(Rd, E), and

〈ρSu, ϕ〉Rd := 〈u, ρSϕ〉Rd , u ∈ D′(Rd, E) , ϕ ∈ D(Rd) .

The closed negative half-space H− of Rd is defined by H− := ρSH. Then H+ := Hand H− are two adjacent closed half-spaces of Rd whose common boundary is theinterface S = ∂H+ = ∂H− between H+ and H−.

It is clear that the entire theory of function spaces on H, the trace theorems inparticular, developed so far, can equally well be established by replacing H by H−.Then

ρS ∈ Lis(S ′(H−, E),S ′(H+, E)

), ρ−1

S = ρS .

Furthermore, this operator restricts to an

idempotent isometric isomorphismρS : G(H−, E) → G(H+, E) ,

if G is any one of the Banach subspaces of S ′introduced in the preceding sections.

(2.3.2)

Unless stated otherwise, we suppose

• s > ν1(m + 1/q) for some m ∈ N.

• F is a Banach space.

Setting X := Lq(Rd−1, E), we know that8 Fs/νq (H±, E) → Cm(±R+,X ).

Suppose v± ∈ Cm(±R+,X ). Then

v±(0) = limt→0+

v±(±t) = limt→0±

v±(t)

is the trace of v+, resp. v−, from the right, resp. left, at 0 = ∂R+ = ∂(−R+).We set

γ±u± = limy→0±

u(y, ·) , u± ∈ Fs/νq (H±, E) .

Hence γ+u+ = γu+ for u+ ∈ Fs/νq (H+, E). Note that

γ−u− = limy→0−

u−(y, ·) = limz→0+

u−(−z, ·) = limz→0+

(ρSu−)(z, ·) .

8Here and in similar situations, everywhere either the upper or the lower sign has to be chosen.


Consequently, γ− = γ ρS . On the basis of this observation we define the trace

operator of order m on H− byγm− := γm ρS .

Let X be open in Rd and F a Banach space. Given x ∈ X and e ∈ Rd,

∂v

∂e(x) := lim

τ→0+

v(x + τe)− v(x)τ

∈ F

is the directional derivative of v ∈ C1(X,F ) at x in the direction of e.

Suppose m ≥ 1, u− ∈ Fs/νq (H−, E), and x = (y, x

q) ∈ H−. Then

∂1u−(x) = limh→0

u−(y + h, xq) − u−(x)

h= lim

τ→0+

u−(y + τ, xq) − u−(x)

τ

= limτ→0+

u−(x + τn) − u−(x)τ

=∂u−∂n

(x)

by the continuity of ∂1u−(·, x q) (for a.a. x

q ∈ ∂H−). Hence

∂1u−(0, ·) = limy→0−

∂1u−(y, ·) =∂u−∂n

∣∣∣y=0

=: ∂nu− . (2.3.3)

By definition,γ1−u− = γ1(ρSu−) = lim

t→0+∂1

(ρSu−(t, ·))

= − limt→0+

∂1u−(−t, ·) = −∂1u−(0, ·) .

From this and (2.3.3) it follows that

γ1−u− = −∂u−

∂n

∣∣∣x1=0

= −∂nu− , (2.3.4)

that is, the trace operator of order 1 on H− is the negative of the derivative on ∂H−with respect to the exterior normal n of H−.

In general, we see from

γm− u− = γmρSu− = ∂m

1 (ρSu−)|x1=0

= (−1)m∂m1 u−|x1=0 = (−1)m∂m

n u−

that

γm− = (−1)m∂m

n . (2.3.5)

Let F satisfy (2.0.1). For abbreviation, we set

Fs/νq (H+,H−; E) := Fs/ν

q (H+, E) ⊕ Fs/νq (H−, E)


and denote its general point by u = u+ ⊕ u−. We suppose that

Cm± :=

m∑

j=0

c±j γj

are boundary operators of type (s, m, F, q) on ∂H. Setting

Cmu := Cm+ u+ − Cm

− ρSu− , u = u+ ⊕ u− ∈ Fs/νq (H+,H−; F ) ,

we infer from Lemma 2.1.1 and (2.3.2) that

Cm ∈ L(Fs/ν

q (H+,H−; E), ∂Fs(m,q)/νpppp

q (S, F ))

.

We say that Cm is a transmission operator across S (in the direction of n). It is oforder m if Cm

+ and Cm− are of order m.

2.3.1 Examples (a) (m = 0) Here C0± = c±γ and C0u = c+γ+u+ − c−γ−u−. In

particular, if F = E and c+ = c− = 1E , then

C0u = [[u]] := γ+u+ − γ−u− = limt→0+

(u+(t, ·)− u−(−t, ·)) ,

the jump of u across S (in the direction of n).

(b) (m = 1) Now

C1u = c+1 γ1

+u+ − c−1 γ1−u− + c+

0 γ+u+ − c−0 γ−u− .

Suppose F = E, c±1 = ±1E and c±0 = 0. Then

C1u = ∂nu+ − ∂nu−

is the jump, [[∂nu]], of the ‘normal derivative’ of u across S. ¥

To Cm we associate a boundary operator, B(Cm), on ∂H, acting on E2-valuedfunctions, as follows. We define an isomorphism

T ∈ Lis(Fs/ν

q (H+,H−; E), Fs/νq (H, E2)

)(2.3.6)

by setting T (u+ ⊕ u−) := (u+, ρSu−). Then

B(Cm) := Cm T−1 . (2.3.7)

HenceB(Cm) ∈ L(

Fs/νq (H, E2), ∂Fs(m,q)/ν

ppppq (∂H, F )

).

Given9

u = (u1, u2) =[u1

u2

]∈ Fs/ν

q (H, E2) =(Fs/ν

q (H, E))2

,

9Whenever convenient, we use standard matrix notation.


we infer from (2.3.6) that

B(Cm)u = Cm+ u1 − Cm

− u2 =m∑

j=0

(c+j ∂j

nu1 − c−j ∂jnu2)

=m∑

j=0

[c+j ,−c−j ] ∂j

n

[u1

u2

]=

m∑

j=0

bj(Cm)∂jnu ,

where bj(Cm) := [c+j ,−c−j ].

2.3.2 Lemma B(Cm) is a boundary operator of type (s, m, F, q) on10 ∂H. It isnormal if Cm

+ and Cm− are normal.

Proof The first claim is obvious.Suppose Cm

+ and Cm− are normal. Thus, given x

q ∈ S, there exist continu-ous linear right inverses bc

m,1(xq) for c+

m(xq) and bc

m,2(xq) for −c−m(x

q). We de-

fine bcm(x

q) ∈ L(F,E2) by bc

m(xq)f :=

(bcm,1(x

q)f, bc

m,2(xq)f

)/2 for f ∈ F . Then

bm(xq)bc

m(xq)f = f . This implies the second claim. ¥

Henceforth, Cm is said to be a normal transmission operator (across S) of type(s,m, F, q) if B(Cm) is a normal boundary operator on ∂H.

Now we can prove a retraction theorem which corresponds to Theorem 2.2.1.For this we suppose

• m = (m0, . . . , mk) is an order sequence.• F := (F0, . . . , Fk) is a sequence of Banach spaces.• s > ν1(mk + 1/q).• Cmi is a normal transmission operator of type (s,mi, Fi, q).

(2.3.8)

Then C := (Cm0 , . . . , Cmk) is said to be a normal transmission operator across S) of

type (s, m, F , q).

2.3.3 Theorem Let F satisfy (2.0.1). Suppose C is a normal transmission op-erator of type (s, m, F , q). Then C is a retraction from F

s/νq (H+,H−; E) onto

∂Fs(m,q)/ν

ppppq (S, F ).

Proof The preceding lemma guarantees that B(C) :=(B(Cm0), . . . ,B(Cmk)

)is a

boundary operator of type (s, m, F , q) on ∂H acting on E2-valued functions. Byassumption, it is normal. Hence, by Theorem 2.2.1, there exists a continuous rightinverse Bc(C) for it. By (2.3.6), T−1 Bc(C) is then a continuous right inversefor C. ¥

10Of course, S is identified with ∂H.


Patching Together Half-Spaces

We know that (R+, E+) := (RH, EH) and (R+, E+) := (RH, EH) are r-e pairs for(F


q (H, E))

and(F


q (H, E)), respectively. By replac-

ing H = H+ by H− in the definition of R1, E1, R1, and E1 in Subsection VI.1.1,we obtain r-e pairs (R−, E−) and (R−, E−) for

(Fs/ν

q (Rd, E), Fs/νq (H−, E)

)and

(Fs/ν

q (Rd, E), Fs/νq (H−, E)

),

possessing properties completely analogous to the ones of (R+, E+) and (R+, E+),respectively. These facts will now be used to characterize u ∈ F

s/νq (Rd, E) by means

of its restrictions R±u to H±.Generalizing the notation introduced in Examples 2.3.1, we define the jump,

[[∂mn u]], of the m-th order normal derivative across S by

[[∂mn u]] := ∂m

n u+ − ∂mn u− , u = u+ ⊕ u− ∈ Fs/ν

q (H+,H−; E) ,

provided s > ν1(m + 1/q). Then, given s > 0, we denote by

Fs/νq (H+,H−; E)

the set of all u = u+ ⊕ u− in Fs/νq (H+,H−; E) satisfying

[[∂jnu]] = 0 , j ∈ N , ν1(j + 1/q) < s . (2.3.9)

Clearly, it is a closed linear subspace of Fs/νq (H+,H−; E), and

Fs/νq (H+,H−; E) = Fs/ν

q (H+,H−; E) , s < ν1/q , (2.3.10)

since condition (2.3.9) is then void.

2.3.4 Theorem Let F satisfy (2.0.1). Suppose s > 0 and s /∈ ν1(N+ 1/q). Then

Fs/νq (Rd, E) → Fs/ν

q (H+,H−; E) , u 7→ R+u ⊕ R−u (2.3.11)

is a toplinear isomorphism.

Proof (1) It follows from the continuity properties of R± that u 7→ R+u⊕ R−u

is a continuous linear map from Fs/νq (Rd, E) into F

s/νq (H+,H−; E).

Suppose m ∈ N with

ν1(m + 1/q) < s < ν1(m + 1 + 1/q) . (2.3.12)

Recall Fs/νq (Rd, E) → Cm(R,X ), where X := Lq(Rd−1, E). Clearly, [[∂j

nv]] = 0 for0 ≤ j ≤ m if v ∈ Cm(R,X ). This implies, together with (2.3.9), that (2.3.11) iswell-defined and continuous.


(2) Let (2.3.12) be satisfied. For v := v+ ⊕ v− in Fs/νq (H+,H−; E) we put

w := v+ − R+E−v−. Then w ∈ Fs/νq (H+, E). Since R+ is the operator of point-

wise restriction,

∂jnw = ∂j

nv+ − ∂jnE−v− , 0 ≤ j ≤ m .

From E−v− ∈ Fs/νq (Rd, E) we infer

∂jnE−v− = ∂j

nv− .

Hence ∂jnw = 0 for 0 ≤ j ≤ m, that is, w ∈ F

s/νq (H+, E) by Theorems 1.6.2(i) and

1.6.9. Therefore u := E−v− + E+w belongs to Fs/νq (Rd, E). Note that

R+u = R+E−v− + R+E+w = v+

and R−u = R−E−v− = v−. Consequently, the map v 7→ u is a continuous rightinverse for (2.3.11). Since the latter is clearly injective, the assertion is proved inthis case.

(3) Assume 0 < s < ν1/q. Then Ws/ν

q (H±, E) .= Ws/ν

q (H±, E), due to The-orem 1.6.2(ii). Thus, given v = v+ ⊕ v− ∈ F

s/νq (H+,H−; E),

u := E+v+ + E−v− ∈ Fs/νq (Rd, E)

and R±u = v±. This proves that v 7→ u is a continuous right inverse for the injec-tive map (2.3.11). ¥

2.3.5 Corollary Suppose

ν1(m + 1/q) < s < ν1(m + 1 + 1/q) .

Then

u ∈ Fs/νq (Rd, E) ⇐⇒ R±u ∈ Fs/ν

q (H±, E) and [[∂jnu]] = 0 with 0 ≤ j ≤ m .

If 0 < s < ν1/q, then u ∈ Ws/ν

q (Rd, E) iff R±u ∈ Ws/ν

q (H±, E).

2.4 Interpolation With Boundary Conditions

In this subsection we introduce spaces with vanishing normal boundary conditionswhich generalize the spaces with vanishing traces studied in Subsection 1.4. Thenwe investigate the behavior of these spaces under real and complex interpolation.

Preliminaries

For the reader’s convenience, we review some facts of interpolation theory of whichwe make use in the proof of the main theorem. We set S := [0 < Rez < 1] ⊂ C and


Sx := [Rez = x] for 0 ≤ x ≤ 1. Let E0 and E1 be Banach spaces with E1 → E0.Recall from Example I.2.4.2 that F(E0, E1) is the Banach space of bounded andcontinuous functions from S into E0 + E1

.= E0, which are holomorphic in S andsatisfy f |Sj ∈ C0(Sj , Ej) for j = 0, 1, endowed with the norm

f 7→ ‖f‖F := maxsupt∈R ‖f(it)‖E0 , supt∈R ‖f(1 + it)‖E1

.

Then [E0, E1]θ is, for 0 < θ < 1, the image space of the evaluation map

F(E0, E1) → E0 , f 7→ f(θ) .

Assume f ∈ F(E0, E1) and 0 < ϑ < 1. Set gt(z) := f(z + it) for z ∈ S and t ∈ R.Then gt ∈ F(E0, E1) and gt(ϑ) := f(ϑ + it). Hence, cf. Remark VI.2.2.1,

‖f(ϑ + it)‖[E0,E1]ϑ ≤ c ‖f‖F , t ∈ R . (2.4.1)

Below we need the following extension of the reiteration theorem (I.2.8.4) tothe case where E1 is not necessarily dense in E0. Recall that [E0, E1][j] := Ej forj = 0, 1.

2.4.1 Lemma Suppose E1 → E0, θ0 ∈ [0, 1) and θ1, η ∈ (0, 1). Then[[E0, E1]θ0 , [E0, E1]θ1

]η

= [E0, E1](1−η)θ0+ηθ1 .

Proof Let X0 and X1 be Banach spaces with X1 → X0. Denote by X0 the closureof X1 in X0. Then

[X0, X1]ϑ = [X0, X1]ϑ , 0 < ϑ < 1 (2.4.2)

(e.g., [BeL76, Theorem 4.2.2(a)] or [Tri95, Theorem 1.9.3(c)]). Hence[[E0, E1]θ0 , [E0, E1]θ1

]η

=[[E0, E1]θ0 , [E0, E1]θ1

]η

and E1 is dense in [E0, E1]θ0 and in [E0, E1]θ1 . Thus it follows from Theorem 4.6.1in [BeL76] that

[[E0, E1]θ0 , [E0, E1]

]η

= [E0, E1](1−η)θ0+ηθ1 .

From this we get the assertion by applying (2.4.2) once more. ¥

For a proof of the following three lines theorem we refer to [DuS57, Theo-rem VI.10.3].

2.4.2 Lemma Let X be a Banach space. Suppose ϕ : S → X is bounded and ϕ |Sis holomorphic. Then

supy∈R

‖ϕ(θ + iy)‖ ≤ supy∈R

‖ϕ(iy)‖1−θ ‖ supy∈R

‖ϕ(1 + iy)‖θ

for 0 < θ < 1.


The Main Theorem

We assume throughout that

(i) 1 ≤ q ≤ ∞ .

(ii) m = (m0, . . . , mk) is an order sequence .

(iii) F = (F0, . . . , Fk) is a sequence of boundary range spaces .

(iv) B = (Bm0 , . . . ,Bmk) is a normal boundary operatorof type (s, m, F , q) .

(v) 0 < θ < 1 .

(vi) s0 < s1 ≤ s, where s1 < s if q < ∞, satisfys0, s1, sθ /∈

ν1(mi + 1/q) ; 0 ≤ i ≤ k

.

(2.4.3)

Recall that this implies that s > ν1(mk + 1/q). Also keep in mind that

sθ = (1 − θ)s0 + θs1 .

Generalizing definition (1.4.1), we set for t ≤ s and Ft/νq satisfying (1.3.1),

Ft/νq,B(H, E) :=

u ∈ Ft/ν

q (H, E) ; Bmiu = 0 if ν1(mi + 1/q) < t

.

Hence

Ft/νq,B(H, E) = Ft/ν

q (H, E) , t < ν1(m0 + 1/q) . (2.4.4)

Now we can prove the main theorem of this subsection.

2.4.3 Theorem

(i) Assume q < ∞. If either s0 > 0, or E is reflexive and s0 > ν1(−1 + 1/q),then

[Bs0/ν

q (H, E), Bs1/νq,B (H, E)

]θ

.=(Bs0/ν

q (H, E), Bs1/νq,B (H, E)

)θ,q

.= Bsθ/νq,B (H, E) .

(ii) Let q = ∞ and s0 > 0. Then[bs0/ν∞ (H, E), bs1/ν

∞,B (H, E)]θ

.=(bs0/ν∞ (H, E), bs1/ν

∞,B (H, E))0

θ,∞.= b

sθ/ν∞,B (H, E) .

Proof For abbreviation, with F ∈ B, b,X t := Ft/ν

q (H, E) , Yt(j) := ∂Ft(j,p)/νpppp

q (∂H, Fj)

for t ∈ R and j ∈ N. First we consider complex interpolation.


(1) It follows from (2.4.3)(vi) and the continuity of B that X s1B is a closed

linear subspace of X s1 . Hence X s1B → X s1 implies

[X s0 ,X s1B ]θ → [X s0 ,X s1 ]θ

.= X sθ , (2.4.5)

where the second part is implied by Theorems VII.2.7.1(iii) and (v) and VII.2.8.3.(2) Suppose u ∈ [X s0 ,X s1

B ]θ. Assume sθ > ν1(m0 + 1/q) and let i be thelargest integer ≤ k such that ν1(mi + 1/q) < sθ.

There exists f ∈ F(X s0 ,X s1B ) with f(θ) = u. We deduce from step (1) and

(2.4.1) that the restriction of f to [θ ≤ Rez ≤ 1] is a bounded continuous functionwith values in X sθ , which is holomorphic on [θ ≤ Rez < 1]. Thus Bmj f is for0 ≤ j ≤ i a bounded continuous function from [θ ≤ Rez ≤ 1] into Ysθ(mj), whichis holomorphic on [θ ≤ Rez < 1] and vanishes on S1. Hence, by Lemma 2.4.2,Bmj f vanishes identically. Consequently,

Bmj u = Bmj f(θ) = 0 , 0 ≤ j ≤ i .

From this and (2.4.5) we get

[X s0 ,X s1B ]θ → X sθ

B , (2.4.6)

provided sθ > ν1(m0 + 1/q). If sθ < ν1(m0 + 1/q), then X sθ

B = X sθ and (2.4.6)follows from (2.4.5).

(3) Let bi,mi be the top-order coefficient of Bmi for 0 ≤ i ≤ k. We fix a core-traction bc

i,mifor bi,mi

satisfying (2.1.6). Assume Bmiu = 0. Then

Bmiu := bci,mi

Bu = 0 .

Conversely, if Bmiu = 0, then Bmiu = bi,mi Bmiu = 0. Hence

ker(Bmi) = ker(Bmi) , 0 ≤ i ≤ k .

We write

Bmi =mi∑

j=0

bi,j∂jn , bi,j := bc

i,mibi,j .

In particular,

πmi:= bi,mi

∈ bσ/νpppp

∞(∂H,L(E)

)(2.4.7)

and πmi bi,j = bi,j for 0 ≤ j ≤ mi. Thus πmi(xq) ∈ L(E) is a projection for x

q ∈ ∂H,and

πmi Bmi = Bmi (2.4.8)

for mi ∈ M := m0, . . . , mk.


(4) Assume

s1 > ν1(mk + 1/q) , s1 /∈ ν1(N+ 1/q) . (2.4.9)

Let ` be the largest integer satisfying ν1(` + 1/q) < s1 and set ` := (0, 1, . . . , `).Then we define a normal boundary operator of type (s1, `, E, q),

C = (C0, . . . , C`) : X s1 → Ys1(`) :=∏

j=0

Ys1(j) ,

by

Cj :=

∂j

n , if j /∈ M ,

(1 − πj)∂jn ⊕ Bj , if j ∈ M .

Of course, now Ys(j) is a space of E-valued functions. Note that

πjCj = πjBj = Bj , j ∈ M . (2.4.10)

Suppose Cv = 0. Let 0 ≤ r ≤ `. Then

Crv = ∂rnv = 0 , r /∈ M . (2.4.11)

If r = mi ∈ M , then, by (2.4.7), (2.4.8), and (2.4.10),

Crv = (1 − πr)∂rnv ⊕ πrBrv = ∂r

nv +r−1∑

j=0

bi,j∂jnv = 0 .

From this and (2.4.11) we deduce successively

Cmiv = ∂min v = 0 , i = 0, 1, . . . , k .

This and (2.4.11) imply that Cv = 0 iff ~γ`v = 0. Hence

X s1C = X s1

~γ`

.= X s1(H, E) , (2.4.12)

where the equivalence is a consequence of (2.4.9), the choice of `, and Theorems1.6.4 and 1.6.9.

Note that Theorem 2.2.1 (and its proof) implies the existence of a coretrac-tion Cc for C which is universal with respect to s1 in

(ν1(` + 1/q), s

].

(5) Suppose u ∈ X sθ

B = X sθ

B . Let n be the largest integer such that ν1(n + 1/q)is strictly smaller than sθ. Define

gj :=

Cju , if j /∈ M, j ≤ n ,

(1 − πj)Cju , if j ∈ M, j ≤ n .


Thengj ∈ Ysθ(j) .= [Ys0(j),Ys1(j)]θ

by Lemma 2.1.1 and Theorems VII.2.7.1(iii) and (v) and VII.4.5.1. Hence thereexists fj ∈ F(Ys0(j),Ys1(j)) such that fj(θ) = gj for j ≤ n. We put

fj :=

fj , if j /∈ M ,

(1 − πj)fj , if j ∈ M .

Note that fj has the same properties as fj . Let F := Cc(f0, . . . , fn, 0, . . . , 0). Clear-ly, F ∈ F(X s0 ,X s1) and

v := F (θ) = Cc(g0, . . . , gn, 0, . . . , 0) .

Observe that F |S1 ∈ C0(S1,X s1) satisfies

BjF1 = πjBjF1 = πjCjCc(f0, . . . , fn, 0, . . . , 0) |S1 = 0 , j ∈ M ,

due to (2.4.10) and πjfj = πj(1 − πj)fj = 0 for j ∈ M . Thus F ∈ F(X s0 ,X s1

B )and, consequently,

v ∈ [X s0 ,X s1B ]θ . (2.4.13)

If j ∈ M with j ≤ n, then

Cj(u− v) = Cju − CjCc(g0, . . . , gn, 0, . . . , 0) = Cju − gj = 0 ,

since πjCju = Bju = 0 implies Cju = (1 − πi)Cju = gj . If j /∈ M , then Cju = gj .Hence u− v ∈ X sθ

C.= X sθ (H, E) (cf. step (4)). We know from Theorems VII.2.7.1

and VII.2.8.3 that

X sθ (H, E) .=[X s0(H, E),X s1(H, E)

]θ

.

It follows from Theorem VII.3.7.7 that

X s0(H, E) → X s0 . (2.4.14)

From this and the obvious embedding X s1(H, E) → X s1B we obtain

X sθ (H, E) → [X s0 ,X s1B ]θ .

Hence u = v + (u− v) ∈ [X s0 ,X s1B ]θ, due to (2.4.13). This and (2.4.6) show that

X sθ

B.= [X s0 ,X s1

B ]θ, provided (2.4.9) is satisfied and sθ > ν1(m0 + 1/p).Let sθ < ν1(m0 + 1/q). Then X sθ

B = X sθ and it is clear from the above thatX sθ = X sθ

B.= [X s0 ,X s1

B ]θ in this case too, provided (2.4.9) is satisfied.


(6) Assume

s1 > ν1(mk + 1/q) (2.4.15)

and there exists r ∈ N with s1 = ν1(r + 1/q). Fix t ∈ (s1, s] with t /∈ ν1(N+ 1/q)and set ϑ := (s1 − s0)/(t− s0). Then, by the preceding step, [X s0 ,X t

B]ϑ.= X s1

B .Hence Lemma 2.4.1 implies

[X s0 ,X s1B ]θ

.=[X s0 , [X s0 ,X t

B]ϑ]θ

.= [X s0 ,X tB]θϑ

.= X sθ

B ,

using step (5) once more. This proves that the assertions concerning the complexinterpolation functor are valid, provided (2.4.15) applies.

(7) Now suppose s1 < ν1(mk + 1/q). Fix t ∈ (ν1(mk + 1/q), s

]and define ϑ

as in the preceding step. Then the arguments used there show that

[X s0 ,X s1B ]θ

.= X sθ

B .

Hence the assertions of the theorem involving the complex interpolation functorhave been proved.

(8) Let ν1(mk + 1/q) < t0 < t1 ≤ s. Write (·, ·)θ := (·, ·)θ,q if q < ∞, so that

Ft/νq = B

t/νq , and (·, ·)θ := (·, ·)0θ,∞ if F

t/ν∞ = b

t/ν∞ . Then we know from the interpo-

lation results of Section VII.2 that (X t0 ,X t1)θ.= X tθ .

Assume u ∈ (X t0 ,X t1B )θ → (X t0 ,X t1)θ

.= X tθ . Choose f such that (1.6.8) issatisfied with e = u and (E0, E1) := (X t0

B ,X t1B ). Then Bf(t) = 0 implies11

Bu = B∫ ∞

0

f(t) dt/t =∫ ∞

0

Bf(t) dt/t = 0 ,

that is, u ∈ X tθ

B . (Recall that (E0, E1)0θ,∞ is the closure of E1 in (E0, E1)θ,∞.)

(9) From Theorem 2.2.1 we know that B is a retraction from X t onto Yt(m)

for ν1(mk + 1/q) < t ≤ s . It possesses a coretraction Bc which is universal withrespect to these t. Hence

P := 1X t − BcBis a projection from (X t0 ,X t1) onto (X t0

B ,X t1B ) (cf. Subsection I.2.3).

Suppose v ∈ X tθ satisfies Bv = 0. Choose g such that (1.6.8) holds with e = v,f = g, and (E0, E1) = (X t0 ,X t1). As above, since Bv = 0,

v = Pv =∫ ∞

0

Pg(t) dt/t

in X t0 , where Pg ∈ C((0,∞),X t1

). Thus, see Remark 1.6.3, v ∈ (X t0

B ,X t1B )θ. By

combining this with the result of step (8), we obtain

(X t0B ,X t1

B )θ.= X tθ

B . (2.4.16)

11Cf. the proof of Theorem 1.6.4.


(10) Choose 0 < ε < minθ, 1− θ such that

sθ±ε ∈(ν1(mi + 1/q), ν1(mi+1 + 1/q)

)

if 0 ≤ i ≤ k − 1 and sθ belongs to this interval. If sθ < ν1(m0 + 1/q), assume thats0 < sθ±ε < ν1(m0 + 1/q). Similarly,

ν1(mk + 1/q) < sθ±ε < s1 if sθ > ν1(mk + 1/q) .

Then, by the validity of the theorem for the complex interpolation functor, we ob-tain [X s0 ,X s1

B ]θ±ε.= X sθ±ε

B . Thus the reiteration theorem for the real interpolationfunctor (e.g., [BeL76, Theorem 3.5.3] or [Tri95, Theorem 1.10.2]) implies

(X s0 ,X s1B )θ

.=([X s0 ,X s1

B ]θ−ε, [X s0B ,X s1

B ]θ+ε

)1/2

= (X sθ−ε

B ,X sθ+ε

B )1/2

.= X sθ

B .

The last equivalence follows by applying the result of step (9) with mk replacedby mi and with the operator (Bm0 , . . . ,Bmi), provided s1 > ν1(m0 + 1/q). Other-wise, (X s0 ,X s1

B )θ = (X s0 ,X s1)θ.= X sθ = X sθ

B by (2.4.4). The theorem is proved. ¥

Generalizations

By means of the reiteration theorems it is now not difficult to prove importantextensions and complements of the main theorem. We start with complex inter-polation results.

2.4.4 Theorem Let (2.4.3) apply.(i) Assume q < ∞. If either s0 > 0 or E is reflexive and s0 > −ν1(1 − 1/q), then

[B

s0/νq,B (H, E), Bs1/ν

q,B (H, E)]θ


(ii) Suppose q = ∞ and s0 > 0. Then[bs0/ν∞,B (H, E), bs1/ν

∞,B (H, E)]θ

.= bsθ/ν∞,B (H, E) .

Proof Define X t as in the preceding proof. Fix σ0 < s0 with σ0 > 0 if s0 > 0, andσ0 > −ν1(1− 1/q) otherwise. Then, setting θ0 := (s0 − σ0)/(s1 − σ0), it followsfrom Theorem 2.4.3 that X s0

B.= [X σ0 ,X s1

B ]θ0 . Consequently,

[X s0B ,X s1

B ]θ.=

[[X σ0 ,X s1

B ]θ0 ,X s1B

]θ

.= [X σ0 ,X s1B ](1−θ)θ0+θ (2.4.17)

by Lemma 2.4.1. Note that σ0 +((1 − θ)θ0 + θ

)(s1 − σ0) = sθ. Hence, by Theorem

2.4.3, the third term of (2.4.17) equals X sθ

B . Thus [X s0B ,X s1

B ]θ.= X sθ

B . This provesthe theorem. ¥

Next we turn to real interpolation and consider general Besov spaces.


2.4.5 Theorem Let (2.4.3) be satisfied with q < ∞. Assume that r, r0, r1 ∈ [1,∞]and

either s0 > 0 or E is reflexive and s0 > −ν1(1 − 1/q) .

Then (B

s0/νq,r0,B(H, E), Bs1/ν

q,r1,B(H, E))θ,r

.= Bsθ/νq,r,B(H, E)

and (H

s0/νq,B (H, E),Hs1/ν

q,B (H, E))θ,r

.= Bsθ/νq,r,B(H, E) .

Proof For 1 ≤ ρ ≤ ∞ we set X tρ := B

t/νq,ρ (H, E).

(1) Theorems VII.2.7.1(i) and VII.2.8.3 imply (X s0r0

,X s1r1

)θ,r.= X sθ

r . Usingthis, the arguments of steps (8)–(10) of the proof of Theorem 2.4.3 show that

(X s0r0

,X s1r1,B)θ,r

.= X sθ

r,B .

Now we follow the proof of the preceding theorem, with [·, ·]θ replaced by (·, ·)θ,r,to obtain the first assertion.

(2) It is an obvious consequence of Theorems VII.4.1.3(i) that

Bt/νq,1,B → H

t/νq,B → B

t/νq,∞,B . (2.4.18)

From this and (1), the second assertion is immediate. ¥

2.4.6 Corollary

(B

s0/νq,r0,B(H, E), Hs1/ν

q,B (H, E))θ,r

.=(H

s0/νq,B (H, E), Bs1/ν

q,r1,B(H, E))θ,r

.= Bsθ/νq,r,B(H, E) .

Proof Using (2.4.18), we find

(Bs0/νq,r0,B, B

s1/νq,1,B)θ,r → (Bs0/ν

q,r0,B,Hs1/νq,B )θ,r → (Bs0/ν

q,r0,B, Bs1/νq,∞,B)θ,r .

Now the theorem implies

(Bs0/νq,r,B,H

s1/νq,B )θ,r

.= Bsθ/νq,r,B .

This proves the first half of the assertion. The second half follows analogously. ¥

We illustrate these results by an example which is motivated by boundaryvalue problems for elliptic and parabolic equations.


2.4.7 Example Let (2.4.3) be satisfied with 1 < q < ∞ and s0 > −ν1(1 − 1/q).Suppose E is ν-admissible. Then

(W s0/ν

q (H, E),W s1/νq,B (H, E)

)θ,q

.=(W

s0/νq,B (H, E),W s1/ν

q,B (H, E))θ,q


Recall thatW

sθ/νq,B (H, E) = B

sθ/νq,B (H, E) if sθ /∈ ν1N .

Proof Since Wsj/ν

q.= H

sj/νq if sj ∈ ν1N, the assertions are immediate from the

results above. ¥

Complex Interpolation of Bessel Potential Spaces

Real interpolation of Bessel potential spaces with vanishing boundary conditionsis covered by Theorem 2.4.5. Now we turn to the complex interpolation of thesespaces. Here we restrict ourselves to isotropic spaces.

2.4.8 Theorem Suppose (2.4.3) applies with 1 < q < ∞ and s0 > −1 + 1/q. If E isa UMD space, then

[Hs0

q (H, E),Hs1q,B(H, E)

]θ

.=[Hs0

q,B(H, E), Hs1q,B(H, E)

]θ

.= Hs0q,B(H, E) .

Proof (1) As explained in the notes to the preceding section, Theorem 1.1 in[MeV15] implies that (1.9.4) is true. From this and Theorem 1.6.8 we deduce that,given ` ∈ N and ` + 1/q < t < ` + 1 + 1/q, it follows that

Htq,~γ`(H, E) = Ht

q(H, E) . (2.4.19)

(2) SetX t := Ht

q(H, E) , Yt(j) := bt−j−1/qq (∂H, Fj) .

Then (2.4.19) guarantees that (2.4.12) and X sθ

C.= X sθ (H, E) are valid. Hence steps

(1)–(7) of the proof of Theorem 2.4.3 apply verbatim, provided we invoke Theo-rem VII.4.5.1 instead of the interpolation theorems for Besov spaces. ¥

2.4.9 Remark Suppose we knew that (1.9.2) applies. Then (1.9.4) and Theo-rem 1.6.8 would imply that, given ` ∈ N and ν1(` + 1/q) < t < ν1(` + 1 + 1/q),

Ht/ν

q,~γ`(H, E) = Ht/νq (H, E) . (2.4.20)

Then the arguments of step (2) of the preceding proof, with the obvious defini-tions of X t and Yt(j), would show that the analogue of Theorem 2.4.8 holds foranisotropic Bessel potential spaces. As mentioned in the notes to Section 1, weconjecture that (2.4.20) is true if E is ν-admissible. ¥


2.5 Notes

The first interpolation theorem for spaces with vanishing boundary conditions isdue to P. Grisvard [Gri67]. He considers the isotropic scalar case E = C and provesTheorem 2.4.8 for p = 2, s0 = 0, and s1 = r ∈

qN. In [Gri69] the author extends

his theorem to p 6= 2. More precisely, he proves that

(Lp,Wr

p,B)θ,p = (Lp, Brp,B)θ,p = Bθr

p,B .

Note that W r2 = Br

2 = Hr2 and (E0, E1)θ,2

.= [E0, E1]θ if E0 and E1 are Hilbert

spaces with E1d

→ E0 (e.g., [LM73, Theorem I.15.1]). An amplification of thesetheorems to general Besov spaces has been achieved by D. Guidetti [Gui91].

If E = Cn, then Theorem 2.4.8, with s0 = 0 and s1 ∈qN, is due to R. See-

ley [See72].None of these authors assumes that sθ /∈ mi + 1/q ; 0 ≤ i ≤ k . If sθ is such

an exceptional value, then[Hs0

q (H, E), Hs1q,B(H, E)

]θ

is not a closed linear subspaceof Hsθ

q (H, E), but carries a stronger topology. Furthermore, in each of the papersmentioned, H is replaced by a smooth compact (sub-)manifold Ω (of Rd) withboundary. However, as is well-known, the crucial step is the proof for the half-space. From this one passes to Ω by standard localization and partition of unityprocedures (see the next volume).

Steps (1)–(4) of the proof of Theorem 2.4.3 follow essentially R. Seeley. Asalready mentioned in the notes to Section 1, the use of the J-method to derive thereal interpolation results goes back to D. Guidetti.

Recently, N. Lindemulder, M. Meyries, and M. Veraar [LMV18] presented aproof for the interpolation result

[Hs0

q (R+, E), Hs1q (R+, E)

]θ

.= Hsθq (R+, E)

if s0 > −1 + 1/p and s0, s1, sθ /∈ N+ 1/q, where E is a UMD space (recall Theo-rem 1.6.8). They also consider weighted spaces which are of no concern to us.

AppendixVector-Valued Distributions

The theory of vector-valued distributions has been almost completely developedby L. Schwartz around the middle of the last century. His results are very generalsince he considered locally convex target spaces. For many purposes –– in particularfor use in this book –– it suffices to consider distributions with values in Banachspaces. In this case the general theory simplifies, and we develop the relevant partsin this more restricted setting in this appendix.

It is subdivided into two sections. The first, rather long one, is centeredaround kernel theorems and their main concrete realizations, namely tensor prod-ucts and convolutions. The short second section is devoted to a proof of the Rieszrepresentation theorem in the vector-valued setting.

Throughout this appendix we employ the conventions collected in ‘Notationsand Conventions’ of Volume I. In particular,

K stands now either for the real or for the complex number field.

Corners do not appear in what follows.

© Springer Nature Switzerland AG 2019H. Amann, Linear and Quasilinear Parabolic Problems, Monographsin Mathematics 106, https://doi.org/10.1007/978-3-030-11763-4

369

370 Appendix

1 Tensor Products and Convolutions

It is crucial to possess a good theory of convolutions and ‘point-wise multiplica-tions’. These operations are special cases of bilinear maps of vector-valued distri-butions. In order to handle such bilinear operations we prove in Subsection 1.5a basic extension theorem that allows to carry over the desired results from scalardistributions to the vector-valued setting. For its proof we need a considerableamount of preparation, namely rather deep results from the theory of locally con-vex spaces as well as approximation and subtle continuity theorems for concretespaces of distributions. These preparations occupy Subsections 1.1 to 1.4.

Having established the basic extension theorem in Subsection 1.5, it is nottoo difficult to define point-wise multiplications, tensor products, and convolutionsfor vector-valued distributions. This is done in Subsections 1.6–1.9.

Although the proofs and techniques of this appendix are rather heavy, thefinal results are very simple to state and very satisfactory. Indeed, given the correctinterpretation, all known rules for scalar distributions carry over to the vector-valued setting.

1.1 Locally Convex Topologies

For the reader’s convenience we collect in this subsection some more advancedtopics1 from the theory of LCSs that we shall need below. Hereafter, we often usethese facts without further mention.

The Uniform Boundedness Principle

An LCS is barreled if each absolutely convex, closed, and absorbing subset is aneighborhood of zero. Every Frechet space, hence every Banach space, is barreled([Hor66, III.6]). Every reflexive LCS is barreled. In fact, an LCS is reflexive iff itis barreled and bounded subsets are relatively weakly compact.

Let E and F be LCSs. A set A ⊂ L(E, F ) is equicontinuous if for each neigh-borhood V of zero in F there exists a neighborhood U of zero in E such thatA(U) ⊂ V for all A ∈ A. Equivalently, A is equicontinuous iff to each continuousseminorm q on F there exists a continuous seminorm p on E such that

q(Ae) ≤ p(e) , e ∈ E , A ∈ A .

The set A is uniformly bounded if for each bounded subset B of E there existsa bounded set C ⊂ F such that A(B) ⊂ C for all A ∈ A. It is easily seen thateach equicontinuous set is uniformly bounded. Finally, A is point-wise bounded ifA(e) := Ae ; A ∈ A is bounded in F for each e ∈ E, that is, if A is bounded

1Recall the ‘Notations and Conventions’ of Volume I as well as the facts collected in Subsec-tions III.4.1 and III.4.2.

Tensor Products and Convolutions 371

in Ls(E, F ). Since L(E, F ) → Ls(E,F ), every bounded subset of L(E,F ) is point-wise bounded. The uniform boundedness principle asserts that each point-wisebounded subset of L(E,F ) is equicontinuous, hence uniformly bounded, if E isbarreled (e.g., [Jar81, Proposition 11.1.1] or [Sch71, Theorem III.4.2]). In particu-lar, it follows that

supA∈A

‖Ae‖ < ∞ for each e ∈ E =⇒ supA∈A

‖A‖ < ∞ , (1.1.1)

provided E and F are Banach spaces.Suppose that E is barreled. Let (Aα) be a net in L(E, F ) that is bounded

in Ls(E,F ), and assume that Ae := lim Aαe exists in F for each e ∈ E. Then theBanach–Steinhaus theorem asserts that A ∈ L(E,F ) and that Aα → A, uniformlyon compact sets (e.g., [Jar81, Theorem 11.1.3]). If E and F are Banach spaces and(Aj) is a sequence in L(E, F ) such that (Aje) converges in F for each e ∈ E, thenit follows that

‖A‖ ≤ lim infj→∞

‖Aj‖ < ∞ (1.1.2)

(e.g., [Yos65, Corollary II.1.2]).

Hypocontinuity

Let G be a third LCS and let b : E × F → G be bilinear. Clearly, b is separately

continuous if b(·, f) ∈ L(E, G) for each f ∈ F and b(e, ·) ∈ L(F, G) for each e ∈ E.Of course, every continuous bilinear map is separately continuous. A separatelycontinuous bilinear map is continuous if E and F are Frechet spaces (e.g., [Rud73,Theorem 2.1.7] or [Hor66, Theorem 4.7.1]).

The map b is hypocontinuous if b(·, f) : E → G is continuous, uniformly withrespect to f in bounded subsets of F , and b(e, ·) : F → G is continuous, uniformlywith respect to e in bounded subsets of E. If b is hypocontinuous, then it isseparately continuous and b(A,B) is bounded in G if A is bounded in E and B isbounded in F . Moreover, b is continuous on A× F and on E × B, and uniformlycontinuous on A× B (e.g., [Hor66, Propositions 4.7.2 and 4.7.3] or [Sch71, III.5.3]).Consequently, every hypocontinuous bilinear map is sequentially continuous.

Every separately continuous bilinear map is hypocontinuous if E and F arebarreled (e.g., [Hor66, Theorem 4.7.2] or [Sch71, III.5.2]). Of course, a continuousbilinear map is hypocontinuous.

The following lemma will be useful for proving the hypocontinuity of certainbilinear maps.

1.1.1 Lemma Suppose that F is barreled. Then the bilinear map

L(E,F )× L(F,G) → L(E, G) , (S, T ) 7→ TS

is hypocontinuous.

372 Appendix

Proof Let q be a continuous seminorm on G and let B be a bounded subset of E.Let T ⊂ L(F,G) be bounded. It follows from the uniform boundedness principlethat there exists a continuous seminorm p on F such that

q(Tf) ≤ p(f) , f ∈ F , T ∈ T .

Consequently,

supe∈B

q(TSe) ≤ supe∈B

p(Se) , S ∈ L(E, F ) , T ∈ T ,

which shows that(S 7→ TS) ∈ L(L(E, F ),L(E,G)

),

uniformly with respect to T in bounded subsets of L(F, G).

This proves one half of the statement. The proof for the other half is similar. ¥

Montel Spaces

Recall that an LCS is a Montel space if it is barreled and bounded subsets arerelatively compact. Since the unit-ball of a Banach space is relatively compact iffit is finite-dimensional, there are no infinite-dimensional Banach-Montel spaces.Every Montel space is reflexive (e.g., [Hor66, 3, §9]). The dual of a Montel spaceis a Montel space as well (e.g, [Hor66, Proposition 9 in Section 3, §9]).

Strict Inductive Limits

Let E be a vector space and let Eα ; α ∈ A be a family of subspaces such that:

(i) Each Eα is a Frechet space.

(ii) If Eα ⊃ Eβ , then Eα induces the original topology on Eβ .

(iii) There exists a cofinal increasing sequence (En) in Eα ; α ∈ A , that is,En ⊂ En+1 and to each Eα there is an En with En ⊃ Eα.

(iv)⋃

α Eα = E.

Then there exists a finest locally convex Hausdorff topology τ on E such thatEα → E for each α ∈ A, the (strict) inductive limit topology or LF topology inducedby Eα ; α ∈ A . The LCS (E, τ) is denoted by

lim−→

Eα = lim−→α

Eα

and said to be an LF space. Every LF space is complete and barreled, and induceson each Eα its original topology. A subset B of E is bounded iff B ⊂ Eα for someα ∈ A and B is bounded in Eα. Let F be an LCS. Then a linear map T : E → F iscontinuous iff T |Eα is continuous from Eα into F for each α ∈ A. This is the case iffeach T |Eα is bounded. Thus an LF space E is bornological, that is, every bounded


linear map from E into an LCS is continuous. Clearly, every Frechet space, henceevery Banach space, is an LF space (for all this see [Jar81, IV], [Hor66, II.12],[Sch71, II.6–II.8]).

It follows from (iii) that our inductive limits are countable. Thus, in principle,we could have restricted ourselves to the consideration of sequences (Ek) only,instead of admitting possibly uncountable families Eα ; α ∈ A . However, thegiven formulation is well adapted to the concrete spaces we have in mind. In thosecases, uncountable families of Frechet spaces occur naturally. Then we do not haveto select a particular sequence and keep repeating that the topology is independentof that particular sequence.

Smooth Functions

Let X be a nonempty open subset of Rn and let E = (E, |·|) be a Banach spaceover K. Recall from Subsection III.4.1 that

E(X,E) :=(C∞(X, E), pm,K ; m ∈ N, K b X

)

is a Frechet space, where the seminorms pm,K are defined by

pm,K(u) := max|α|≤m

‖∂αu‖∞,K . (1.1.3)

Test Functions

Given K b X, let

DK(X, E) :=

u ∈ E(X, E) ; supp(u) ⊂ K

. (1.1.4)

Then DK(X,E) is a closed linear subspace of E(X, E), and⋃

KbX

DK(X,E) =

u ∈ E(X,E) ; supp(u) b X

.

Let

Xk :=

x ∈ X ; dist(x, Xc) > 1/k ∩ kBn , k ∈

qN , (1.1.5)

where dist(x, ∅) := ∞. Then

Xk b Xk+1 ,⋃

k

Xk = X . (1.1.6)

Hence(DXk

(X, E))

is a cofinal increasing sequence inDK(X,E) ; K b X

.

ThusD(X, E) := lim−−→

KbX

DK(X,E) ,

the space of E-valued test functions, is an LF space (cf. Subsection III.1.1).

374 Appendix

Rapidly Decreasing Smooth Functions

Also recall from Subsection III.4.1 that the Schwartz space of smooth rapidlydecreasing E-valued functions on Rn is defined by

S(Rn, E) :=(S(Rn, E), qk,m ; k, m ∈ N ) ,

where

qk,m(u) := supx∈Rn

|α|≤m

(1 + |x|2)k |∂αu(x)| . (1.1.7)

It is a Frechet space.

Slowly Increasing Smooth Functions

Finally, we recall that OM (Rn, E) is the space of slowly increasing smooth func-tions on Rn. This means that u ∈ OM (Rn, E) iff u ∈ E(Rn, E) and, given α ∈ Nn,there exist mα ∈ N and cα > 0 such that

|∂αu(x)| ≤ cα(1 + |x|2)mα , x ∈ Rn . (1.1.8)

Moreover, OM (Rn, E) is given the topology induced by the family of seminorms2

u 7→ ‖ϕ∂αu‖∞ , ϕ ∈ S(Rn) , α ∈ Nn , (1.1.9)

so that it is an LCS.

Suppose u ∈ S(Rn, E). Then

‖ϕ∂αu‖∞ ≤ ‖ϕ‖∞ q0,|α|(u) , ϕ ∈ S(Rn) , α ∈ Nn .

This shows that

S(Rn, E) → OM (Rn, E) . (1.1.10)

Spaces of Vector-Valued Distributions

To simplify the writing we agree to put

F(X, E) := F(Rn, E) if F ∈ S,OM . (1.1.11)

Then, as usual,

F(X) := F(X,K) , F ∈ D, E ,S,OM ,

2For a proof of the fact that these seminorms are well-defined we refer to Proposition 1.6.1.


if no confusion seems likely, and

F′(X, E) := L(F(X), E

)(1.1.12)

so thatF′(X) = F(X)′

for F ∈ D, E ,S,OM. (Recall that L is always given the bounded convergencetopology.)

If u ∈ D′(X,E), then, as a rule, we write u(ϕ) for the value of u at ϕ ∈ D(X).However, if u is a scalar distribution, that is, u ∈ D′(X), then we continue todenote u(ϕ) by 〈u, ϕ〉.

1.1.2 Theorem Let F ∈ D, E ,S,OM. Then F(X) and F′(X) are complete Montelspaces, hence reflexive.

Proof Every LF space is complete and bornological. The dual of a bornologicalLCS is complete (e.g., [Sch71, IV.6.1]). Hence F(X) and F′(X) are complete forF ∈ D, E ,S. The fact that F(X) is a Montel space for F ∈ D, E ,S is well-known(e.g., [Hor66, Examples 3, 4, and 6 in Section 3, §9]). Thus F′(X) is a Montel spacein these cases as well. The assertion for F = OM follows from [Gro55, II.4.4]. ¥

1.2 Convolutions

In Subsection III.4.2 we have already stated the definition of the convolution of avector-valued distribution and a scalar test function, as well as some of its basicproperties. In this subsection we present proofs for some of the less obvious proper-ties. In addition, we extend the definition to include convolutions of a vector-valuedand a scalar distribution.

Convolutions of Distributions and Test Functions

Let E := (E, |·|) be a Banach space. Recall from Subsection III.4.2 that, given

(u, ϕ) ∈ F′(Rn, E) × F(Rn) , F ∈ D, E ,

the convolution, u ∗ ϕ, of u and ϕ, is defined by

u ∗ ϕ(x) := u(τxϕ

) , x ∈ Rn . (1.2.1)

As already noted, the usual scalar proof (e.g., [Hor83, Theorem 4.1.1]) carries overto the present situation to show that u ∗ ϕ ∈ E(Rn, E) and

∂α(u ∗ ϕ) = ∂αu ∗ ϕ = u ∗ ∂αϕ , α ∈ Nn . (1.2.2)

376 Appendix

Moreover,

supp(u ∗ ϕ) ⊂ supp(u) + supp(ϕ) (1.2.3)

so that

u ∗ ϕ ∈ D(Rn, E) if (u, ϕ) ∈ E ′(Rn, E) ×D(Rn) . (1.2.4)

It is obvious that convolution is bilinear. For the reader’s convenience we provenow that it is hypocontinuous.

1.2.1 Proposition Convolution is a bilinear and hypocontinuous mapping :(i) D′(Rn, E)×D(Rn) → E(Rn, E);(ii) E ′(Rn, E)× E(Rn) → E(Rn, E);(iii) E ′(Rn, E)×D(Rn) → D(Rn, E).

Proof To simplify the notation we put D := D(Rn) and E := E(Rn).(1) Let Kr := rBn for r > 0. Then u ∈ D′(Rn, E), if and only if u belongs to

L(DKr , E) for each r > 0.Let A be a bounded subset of D′(Rn, E). Given r > 0, the uniform bound-

edness principle implies the existence of k ∈ N such that

|u(ϕ)| ≤ cpk,Kr(ϕ) , ϕ ∈ DKr

, u ∈ A .

Observe that

x ∈ Kρ , ϕ ∈ DKr =⇒ τxϕ

∈ DKr+ρ (1.2.5)

and

pk,Kr+ρ(τxϕ

) = pk,x−Kr (τxϕ

) = pk,Kr (ϕ) (1.2.6)

for ρ > 0. Consequently,

pKρ(u ∗ ϕ) = supx∈Kρ

|u(τxϕ

)| ≤ cpk,Kr+ρ(τxϕ

) = cpk,Kr (ϕ) (1.2.7)

for ϕ ∈ DKr and u ∈ A. Now we obtain from (1.2.2) that

pj,Kρ(u ∗ ϕ) ≤ cpk+j,Kr (ϕ) , ϕ ∈ DKr , u ∈ A , j ∈ N , ρ > 0 .

This shows that

(ϕ 7→ u ∗ ϕ) ∈ L(DKr , E(Rn, E))

, u ∈ A , r > 0 . (1.2.8)

Hence (ϕ 7→ u ∗ ϕ) ∈ L(D, E(Rn, E)), uniformly with respect to u in bounded sub-

sets of D′(Rn, E).


Now let B be a bounded subset of D. Then there exists r > 0 such that B iscontained and bounded in DKr

. It follows from (1.2.7) and (1.2.2) that

pj,Kρ(u ∗ ϕ) = max

|α|≤jsup

x∈Kρ

|u ∗ ∂αϕ(x)| ≤ c supψ∈C

|u(ψ)| , (1.2.9)

whereC :=

τx(∂αϕ)

; x ∈ Kρ, |α| ≤ j, ϕ ∈ B

is a bounded subset of D, thanks to (1.2.5) and (1.2.6). Consequently,

(u 7→ u ∗ ϕ) ∈ L(D′(Rn, E), E(Rn, E))

,

uniformly with respect to ϕ in bounded subsets of D. This proves the hypoconti-nuity of (i).

(2) Let A be a bounded subset of E ′(Rn, E) = L(E , E). Then the uniformboundedness principle implies the existence of r > 0 and k ∈ N such that

|u(ϕ)| ≤ cpk,Kr (ϕ) , ϕ ∈ E , u ∈ A . (1.2.10)

Thus, given ρ > 0 and x ∈ Kρ,

|u(τxϕ

)| ≤ cpk,Kr (τxϕ

) ≤ cpk,Kr+ρ(ϕ) , ϕ ∈ E , u ∈ A .

Hence, by (1.2.2), given ρ > 0,

pj,Kρ(u ∗ ϕ) ≤ pk+j,Kr+ρ(ϕ) , j ∈ N , ϕ ∈ E , u ∈ A ,

which shows that (ϕ 7→ u ∗ ϕ) ∈ L(E , E(Rn, E)), uniformly with respect to u in

bounded subsets of E ′(Rn, E).

If B is a bounded subset of E , then estimate (1.2.9) is valid, where C is now abounded subset of E . Hence (u 7→ u ∗ ϕ) ∈ L(E ′(Rn, E), E(Rn, E)

), uniformly with

respect to ϕ in bounded subsets of E . Thus (ii) is hypocontinuous as well.

(3) Suppose that A is a bounded subset of E ′(Rn, E). Then we infer from(1.2.10) that u(ϕ) = 0 for ϕ ∈ D(Rn\Kr) and u ∈ A. Thus supp(u) ⊂ Kr foru ∈ A. Hence we deduce from (1.2.3) and (1.2.8) that

(ϕ 7→ u ∗ ϕ) ∈ L(DKρ ,DKr+ρ(Rn, E))

, u ∈ A , ρ > 0 .

Since DKr+ρ(Rn, E) → D(Rn, E), we see that

(ϕ 7→ u ∗ ϕ) ∈ L(DKρ ,D(Rn, E))

, u ∈ A , ρ > 0 .

Consequently, (ϕ 7→ u ∗ ϕ) ∈ L(D,D(Rn, E)), uniformly for u in bounded subsets

of E ′(Rn, E).

378 Appendix

If B is a bounded subset of D, then we deduce from (1.2.9) and (1.2.3) that

(u 7→ u ∗ ϕ) ∈ L(E ′(Rn, E),D(Rn, E))

,

uniformly with respect to ϕ ∈ B. Hence (iii) is also hypocontinuous. ¥

Translation-Invariant Operators

It is easy to verify that

u(ϕ) = u ∗ ϕ

(0) = u

∗ ϕ(0) (1.2.11)

and

(u ∗ ϕ)

= u

∗ ϕ

(1.2.12)

for u ∈ F′(Rn, E) and ϕ ∈ F(Rn), where F ∈ D, E.

The following theorem gives an important characterization of convolutions.

1.2.2 Theorem Suppose that F ∈ D, E and T ∈ L(F(Rn), C(Rn, E)

)and that

T commutes with translations :

T (τxϕ) = τxT (ϕ) , x ∈ Rn , ϕ ∈ F(Rn) .

Then there exists a unique u ∈ F′(Rn, E) such that

T (ϕ) = u ∗ ϕ , ϕ ∈ F(Rn) .

Proof It is trivial that reflection is a toplinear automorphism of F(Rn). Thus thecontinuity hypothesis implies that

u :=[ϕ 7→ (Tϕ

)(0)] ∈ F′(Rn, E) .

Consequently, we infer from the commutativity hypothesis that

(Tϕ)(x) = τ−x(Tϕ)(0) = T (τ−xϕ)(0) = u((τ−xϕ)

)= u(τxϕ

) = (u ∗ ϕ)(x)

for x ∈ Rn; thus Tϕ = u ∗ ϕ for ϕ ∈ F(Rn). ¥

Convolutions of Two Distributions

Now suppose that u ∈ D′(Rn, E) and v ∈ D′(Rn) and u or v has compact sup-port. Then

(ϕ 7→ u ∗ ϕ) ∈ L(D(Rn), E(Rn, E))


and(ϕ 7→ v ∗ ϕ) ∈ L(D(Rn),D(Rn)

)if supp(v) b Rn

by Proposition 1.2.1. The same arguments show that(ϕ 7→ u ∗ (v ∗ ϕ)

)∈ L(D(Rn), E(Rn, E))

.

Thus (III.4.2.15) and Theorem 1.2.2 guarantee the existence of a unique distribu-tion u ∗ v ∈ D′(Rn, E), the convolution of u and v, such that

(u ∗ v) ∗ ϕ = u ∗ (v ∗ ϕ) , ϕ ∈ D(Rn) . (1.2.13)

It is clear that convolution is bilinear. The next proposition shows that it ishypocontinuous as well.

1.2.3 Proposition Convolution is a bilinear and hypocontinuous mapping :

(i) E ′(Rn, E)×D′(Rn) → D′(Rn, E).

(ii) D′(Rn, E)× E ′(Rn) → D′(Rn, E).

(iii) E ′(Rn, E)× E ′(Rn) → E ′(Rn, E).

Proof Let A and B be bounded subsets of D′ := D′(Rn) and D := D(Rn), respec-tively. Since reflection is obviously a toplinear automorphism of D and of D′, itfollows from Proposition 1.2.1 and the boundedness properties of hypocontinuousmaps that

C :=

(u ∗ ϕ

)

; u ∈ A, ϕ ∈ B

is a bounded subset of E . Hence, given v ∈ E ′(Rn, E) and u ∈ A,

supϕ∈B

|v ∗ u(ϕ)| = supϕ∈B

|(v ∗ u) ∗ ϕ

(0)| = supϕ∈B

|v ∗ (u ∗ ϕ

)(0)|

= supϕ∈B

∣∣v((u ∗ ϕ

)

)∣∣ ≤ supψ∈C

|v(ψ)| .

This shows that (v 7→ v ∗ u) ∈ L(E ′(Rn, E),D′(Rn, E)), uniformly with respect

to u in bounded subsets of D′.

Let (uα) be a net in D′ converging to zero. Then uα ∗ ϕ

→ 0 in E , uni-formly with respect to ϕ in bounded subsets of D, thanks to Proposition 1.2.1(i).Consequently, Proposition 1.2.1(ii) guarantees that v ∗ (uα ∗ ϕ

) → 0 in E(Rn, E),uniformly with respect to ϕ in bounded subsets of D and to v in bounded sub-sets of E ′(Rn, E). Hence (v ∗ uα)(ϕ) = (v ∗ uα) ∗ ϕ

(0) → 0 in E, uniformly withrespect to ϕ in bounded subsets of D and to v in bounded subsets of E ′(Rn, E).This shows that (u 7→ v ∗ u) ∈ L(D′,D′(Rn, E)

), uniformly with respect to v in

bounded subsets of E ′(Rn, E). Hence (i) is hypocontinuous.

The hypocontinuity of (ii) and (iii) follows by modifying the above argumentsin an obvious way. ¥

380 Appendix

Elementary Properties of Convolutions

1.2.4 Remarks (a) Let δ ∈ E ′(Rn) be the Dirac distribution, that is,

〈δ, ϕ〉 := ϕ(0) , ϕ ∈ E(Rn) .

Then u ∗ δ = u for u ∈ D′(Rn, E).

Proof From (1.2.1) it is obvious that δ ∗ ϕ = ϕ for ϕ ∈ E(Rn). Hence

(u ∗ δ) ∗ ϕ = u ∗ (δ ∗ ϕ) = u ∗ ϕ , ϕ ∈ D ,

by (1.2.13). ¥

(b) Suppose that u ∈ D′(Rn, E) and v ∈ D′(Rn) and that u or v has compactsupport. Then

∂α+β(u ∗ v) = ∂αu ∗ ∂βv , α, β ∈ Nn .

Proof By repeatedly applying (1.2.2) we see that

∂α+β(u ∗ v) ∗ ϕ = (u ∗ v) ∗ ∂α+βϕ = u ∗ (v ∗ ∂α+βϕ)

= u ∗ ∂α(∂βv ∗ ϕ) = ∂αu ∗ (∂βv ∗ ϕ) = (∂αu ∗ ∂βv) ∗ ϕ

for ϕ ∈ D(Rn). ¥

(c) Let u ∈ D′(Rn, E) and v ∈ D′(Rn) such that u or v has compact support. Then

τa(u ∗ v) = τau ∗ v = u ∗ τav , a ∈ Rn ,

and(u ∗ v)

= u

∗ v

.

Proof This follows easily from (1.2.13) and (III.4.2.15), or (1.2.12), respectively.

(d) If ϕε ; ε > 0 is a mollifier, then ϕε → δ in E ′(Rn) as ε → 0. Consequently,ϕε ∗ u → u in D′(Rn, E) as ε → 0 for u ∈ D′(Rn, E).

Proof Given ψ ∈ E(Rn),

〈ϕε − δ, ψ〉 =∫

Rn

ϕε(x)ψ(x) dx − ψ(0) =∫

Rn

ϕ(y)(ψ(εy) − ψ(0)

)dy .

Hence, by the mean-value theorem,

|〈ϕε − δ, ψ〉| ≤ sup|y|≤1

|ψ(εy)− ψ(0)| ≤ ε sup|y|≤1

|∂ψ(y)| ≤ εp1,Bn(ψ)

for 0 < ε ≤ 1. This shows that 〈ϕε − δ, ψ〉 → 0 as ε → 0, uniformly with respectto ψ in bounded subsets of E(Rn), that is, ϕε → δ in E ′(Rn) as ε → 0. The secondpart of the assertion now follows from Proposition 1.2.3 and (a). ¥


Convolutions of Temperate Distributions

Next we turn to the case of temperate (also called: tempered) distributions. Forthis we recall the following facts.

1.2.5 Lemma The translation group acts strongly continuously on S(Rn, E) andon S ′(Rn, E).

Proof Theorem VII.3.3.1 in the main part of this volume. ¥

It is clear that (ϕ 7→ ϕ

) ∈ L(S(Rn, E)). Thus, given u ∈ S ′(Rn, E) and ϕ be-

longing to S(Rn), we can again define the convolution, u ∗ ϕ, by (1.2.1), since theright side defines a continuous E-valued function of x ∈ Rn by Lemma 1.2.5.

In order to prove continuity properties of this convolution map we need thefollowing technical result.

1.2.6 Lemma OM (Rn, E) is a complete LCS.

Proof Let (aβ) be a Cauchy net in OM (Rn, E). From OM (Rn, E) → E(Rn, E) itfollows that (aβ) is a Cauchy net in E(Rn, E). Since the latter space is complete,there exists a ∈ E(Rn, E) such that aβ → a in E(Rn, E). Given α ∈ Nn, ϕ ∈ S(Rn),and ε > 0, there exists β0 such that

‖ϕ∂αaβ − ϕ∂αaγ‖∞ < ε , β, γ ≥ β0 .

Thus, since aγ → a in E(Rn, E),

‖ϕ∂αaβ − ϕ∂αa‖∞ ≤ ε , β ≥ β0 .

Similarly,‖ϕ∂αaγ‖∞ ≤ ‖ϕ∂αaβ0‖∞ + ε , γ ≥ β0 ,

implies ‖ϕ∂αa‖∞ < ∞. Hence a ∈ OM (Rn, E) and aβ → a in OM (Rn, E). ¥

1.2.7 Proposition Convolution is a bilinear and

(i) continuous map: S(Rn, E)× S(Rn) → S(Rn, E);

(ii) hypocontinuous map: S ′(Rn, E) × S(Rn) → OM (Rn, E).

Proof (1) Given u ∈ S(Rn, E), it follows from ∂αu ∈ BUC(Rn, E) for α ∈ Nn

and from (III.4.2.10) and (III.4.2.19) that u ∗ v ∈ E(Rn, E) for v ∈ S(Rn) ⊂ L1.Note that

|x|2` ≤ (|x− y| + |y|)2` =2∑

j=0

(2`j

)|x − y|j |y|2`−j

382 Appendix

implies

|x|2` |u ∗ v(x)|

≤2∑

j=0

(2`j

) ∫|x − y|j |u(x − y)| |y|2`−j (1 + |y|2)n |v(y)| dy

(1 + |y|2)n

≤ c(`)q`,0(u)q`+n,0(v)

for u, v ∈ S(Rn). This proves (i).(2) Given ϕ ∈ S(Rn, E) and k, m ∈ N, we find

qk,m(τxϕ) = supy∈Rn

|α|≤m

(1 + |x + y|2)k|∂αϕ(y)| ≤ 2k(1 + |x|2)kqk,m(ϕ) (1.2.14)

for x ∈ Rn, thanks to the trivial inequality 1 + |x + y|2 ≤ 2(1 + |x|2)(1 + |y|2). Ifu ∈ S ′(Rn, E) and ϕ ∈ D(Rn), then u ∗ ϕ ∈ E(Rn, E) by Proposition 1.2.1. Let Ube a bounded subset of S ′(Rn, E). Then there exist k,m ∈ N with

|u(ϕ)| ≤ cqk,m(ϕ) , u ∈ U , ϕ ∈ S(Rn) .

Thus, for α ∈ Nn, we deduce from (1.2.14) that

|∂α(u ∗ ϕ)(x)| = |(u ∗ ∂αϕ)(x)| =∣∣u(

τx(∂αϕ))∣∣≤ cqk,m

(τx(∂αϕ)

) ≤ c (1 + |x|2)kqk,m+|α|(ϕ)

for ϕ ∈ D(Rn) and u ∈ U . Consequently, given a bounded subset B of S(Rn),

supψ∈B

‖ψ∂α(u ∗ ϕ)‖∞ ≤ ck,m supψ∈B

qk,0(ψ)qk,m+|α|(ϕ)

for u ∈ U and ϕ ∈ B ∩ D(Rn). This shows that, for u ∈ S ′(Rn, E), the linear map

D(Rn) → OM (Rn, E) , ϕ 7→ u ∗ ϕ ,

is continuous with respect to the topology induced on D(Rn) by S(Rn), uniformlywith respect to u in bounded subsets of S ′(Rn, E). Since D(Rn) is dense in S(Rn)and OM (Rn, E) is complete, it follows that

(ϕ 7→ u ∗ ϕ) ∈ L(S(Rn),OM (Rn, E))

,

uniformly with respect to u in bounded subsets of S ′(Rn, E).Now let B and C be bounded subsets of S(Rn) and observe that, given

α ∈ Nn, the image of the map

Rn × B × C → S(Rn) , (x, ψ, ϕ) 7→ ψ(x)τx(∂αϕ)


is contained in a bounded subset D of S(Rn), since (1.2.14) implies

qk,m

(ψ(x)τx(∂αϕ)

) ≤ ck,mqk,m(ψ)qk,m+|α|(ϕ) , x ∈ Rn , k ∈ N .

Thus

supψ∈B

‖ψ∂α(u ∗ ϕ)‖∞ = supψ∈B

supx∈Rn

∣∣u(ψ(x)τx(∂αϕ)

)∣∣ ≤ supχ∈D

|u(χ)| ,

which shows that

(u 7→ u ∗ ϕ) ∈ L(S ′(Rn, E),OM (Rn, E))

,

uniformly with respect to ϕ in bounded subsets of S(Rn). This proves the assertedhypocontinuity. ¥

1.3 Approximations

It is the main purpose of this subsection to show that tensor products of the formD(X)⊗ E are dense in F(X, E), where F stands for one of the letters D, E , S, OM ,D′, E ′, and S ′. This is a very useful approximation result that allows to reducemany results for vector-valued distributions to the corresponding scalar versions. Itwill be of particular importance in later subsections for extending the operations ofpoint-wise multiplication or of convolution to the case of two vector-valued factors.

Multiplications

Let E and Ej , j = 0, 1, 2, be Banach spaces. If no confusion seems likely, then wedenote the norms in these spaces simply by |·|. We also suppose that

E1 × E2 → E0 , (x1, x2) 7→ x1q x2 (1.3.1)

is a multiplication. Recall from (II.1.1.4) that this means that (1.3.1) is a contin-uous bilinear map of norm at most 1.

1.3.1 Examples The following maps are multiplications:

(a) Ordinary multiplication in a Banach algebra.

(b) Multiplication with scalars: K× E → E, (α, x) 7→ αx.

(c) The duality pairing E′ × E → K, (x′, x) 7→ 〈x′, x〉.(d) The evaluation map L(E1, E0) × E1 → E0, (A, x) 7→ Ax.

(e) Compositions L(E1, E2)× L(E0, E1) → L(E0, E2), (S, T ) 7→ ST .

(f ) Convolution in each one of the cases (III.4.2.18)–(III.4.2.22).

384 Appendix

(g) If b ∈ L(E1, E2; E0) and b 6= 0, then

E1 × E2 → E0 , (x1, x2) 7→ 1‖b‖b(x1, x2)

is a multiplication. This shows that it is no restriction assuming that the norm ofa multiplication is bounded by 1.

It is the advantage of this normalization that we do not have to drag alongnorm constants. ¥

Leibniz’ Rule

We also recall that, given any nonempty set S, point-wise multiplication

ES1 × ES

2 → ES0 , (a1, a2) 7→ a1

q a2

induced by (1.3.1) is defined by

a1q a2(s) := a1(s) q a2(s) , s ∈ S .

1.3.2 Lemma Let p ∈ K[X1, . . . , Xn] be a polynomial of degree at most k in n in-determinates and let U ⊂ Rn be a nonempty open subset of Rn. Then, putting

p(β) := ∂βp , β ∈ Nn ,

the generalized Leibniz rule,

p(∂)(a1q a2) =

∑

β

1β!

(∂βa1) q p(β)(∂)a2 ,

holds for aj ∈ Ck(U,Ej), j = 1, 2.

Proof From the obvious ‘product rule’

∂j(a1q a2) = ∂ja1

q a2 + a1q ∂ja2 , 1 ≤ j ≤ n , (1.3.2)

we deduce by induction that

p(∂)(a1q a2) =

∑

β

∂βa1q qβ(∂)a2 , aj ∈ Ck(U,Ej) , j = 1, 2 , (1.3.3)

where qβ ∈ K[X1, . . . , Xn] and qβ = 0 for |β| > k. Given yj ∈ Ej and ξ, η ∈ Rn, weput a1 := e〈ξ,·〉y1 and a2 := e〈η,·〉y2. Since

e−〈ζ,·〉q(∂)e〈ζ,·〉 = q(ζ) , ζ ∈ Rn , q ∈ K[X1, . . . , Xn] , (1.3.4)


it follows from (1.3.3) that

p(ξ + η)y1q y2 =

∑

β

ξβqβ(η)y1q y2 .

Since this is true for every choice of y1 and y2,

p(ξ + η) =∑

β

ξβqβ(η) , ξ, η ∈ Rn .

By differentiating this identity with respect to ξ and putting ξ = 0 it follows that∂βp(η) = β! qβ(η). This proves the assertion. ¥

Letting p(ξ) := ξα in Lemma 1.3.2, we obtain the standard Leibniz rule:

∂α(a1q a2) =

∑

β≤α

(αβ

)∂βa1

q ∂α−βa2 , aj ∈ C |α|(U,Ej) , j = 1, 2 . (1.3.5)

Approximation by Test Functions

After these preparations we can prove a useful approximation theorem for vector-valued distributions. For this we recall convention (1.1.11) and that X is a non-empty open subset of Rn.

1.3.3 Proposition Suppose that F ∈ D, E ,S and u ∈ F′(X, E). Then there existsa sequence (uj) in D(X,E) such that uj → u in F′(X, E).

Proof (1) First suppose that F ∈ D, E. Denote by (Xk) a sequence of nonemptyrelatively compact open subsets of Rn such that Xk ⊂ Xk+1 and

⋃Xk = X. For

each k ∈ N fix χk ∈ D(X) with χk |Xk = 1. Then χku ∈ E ′(X, E) ⊂ E ′(Rn, E). De-note by ψε ; ε > 0 a mollifier such that ψ1 is even and put

uk := (χku) ∗ ψ1/k , k ∈qN .

Then uk ∈ D(X, E) by Proposition 1.2.1. Given ϕ ∈ D(X),

uk(ϕ) =((χku) ∗ ψ1/k

) ∗ ϕ

(0) = (χku) ∗ (ψ1/k ∗ ϕ

)(0)

= χku ∗ (ψ

1/k ∗ ϕ)

(0) = u(χk(ψ1/k ∗ ϕ)

),

where we used (1.2.11), (1.2.12), and the evenness of ψ1/k. Since supp(ϕ) ⊂ Xj forsome j ∈ N, it follows from (III.4.2.10) and (III.4.2.25) that ψ1/k ∗ ϕ → ϕ in D(X).Also χk → 1 in E(X). Consequently,

χk(ψ1/k ∗ ϕ) → ϕ in D(X) if ϕ ∈ D(X) , (1.3.6)

386 Appendix

so that uk(ϕ) → u(ϕ) in E. This means that

uk → u in Ls

(D(X), E)

. (1.3.7)

(2) Next we choose ϕ ∈ E(X). Then the estimate

|∂α(ψε ∗ ϕ)(x) − ∂αϕ(x)| = |ψε ∗ ∂αϕ(x) − ∂αϕ(x)|=

∣∣∣∫

ψε(y)[∂αϕ(x − y)− ∂αϕ(x)

]dy

∣∣∣

=∣∣∣∫

ψ(y)[∂αϕ(x− εy)− ∂αϕ(x)

]dy

∣∣∣≤ sup

|y|<1

|∂αϕ(x− εy)− ∂αϕ(x)|

(1.3.8)

for x ∈ X and 0 < ε < dist(x, ∂X) shows that ψ1/k ∗ ϕ → ϕ in E(X). Thus

χk(ψ1/k ∗ ϕ) → ϕ in E(X) if ϕ ∈ E(X) , (1.3.9)

so that uk(ϕ) → u(ϕ) in E for u ∈ E ′(X,E). Consequently,

uk → u in Ls

(E(X), E)

if u ∈ E ′(X, E) . (1.3.10)

(3) Now suppose that ϕ ∈ S(Rn). Choose any sequence (χk) in D(Rn) satis-fying χk |(kBn) = 1 and supp χk ⊂ (k + 1)Bn such that

supk

‖∂αχk‖∞ < ∞ , α ∈ Nn . (1.3.11)

For example, putχk := ψε ∗ χ(k+1/2)Bn , k ∈ N ,

for a fixed ε ∈ (0, 1/4). Then, using Leibniz’ rule and (1.3.8), it is easily seen that

χk(ψ1/k ∗ ϕ) → ϕ in S(Rn) . (1.3.12)

Consequently, uk(ϕ) → u(ϕ) in E if u ∈ S ′(Rn, E), which means that

uk → u in Ls

(S(Rn), E)

if u ∈ S ′(Rn, E) . (1.3.13)

(4) Altogether, (1.3.7), (1.3.10), and (1.3.13) show that

uk → u in Ls

(F(X), E

)if u ∈ F′(X,E)

for F ∈ D, E ,S. Now, by the Banach–Steinhaus theorem, uk(ϕ) → u(ϕ) in E,uniformly for ϕ in compact subsets of F(X). Since F(X) is a Montel space itfollows that uk → u in L(

F(X), E)

= F′(X, E). This proves the assertion. ¥


1.3.4 Remark The arguments leading to (1.3.9) and (1.3.12) remain valid if F(X) isreplaced by F(X, E). This proves that

D(X, E)d

→ E(X, E) and D(Rn, E)d

→ S(Rn, E) ,

with sequential density, since the continuity of these injections is clear. This hasalready been stated in (III.4.1.2) and (III.4.1.5), respectively. ¥

Our next proposition shows that D(Rn, E) is dense in OM (Rn, E) as well.

1.3.5 Proposition D(Rn, E)d

→ OM (Rn, E).

Proof Suppose that K b Rn. Then, given ϕ ∈ S(Rn) and α ∈ Nn,

‖ϕ∂αu‖∞ ≤ ‖ϕ‖∞ pK,0(∂αu) ≤ ‖ϕ‖∞ p|α|,K(u) , u ∈ DK(Rn, E) .

This shows that DK(Rn, E) → OM (Rn, E). Hence D(Rn, E) → OM (Rn, E).Now fix u ∈ OM (Rn, E), as well as ϕ ∈ S(Rn) and α ∈ Nn. There exist m ∈ N

and cα > 0 such that

|∂βu(x)| ≤ cα(1 + |x|2)m , x ∈ Rn , β ≤ α . (1.3.14)

Let (χk) be a sequence as in (1.3.11). Then, by Leibniz’ rule,

∂α(χku) − ∂αu =∑

β<α

(αβ

)∂α−βχk∂βu + (χk − 1)∂αu .

Hence we infer from (1.3.11) and (1.3.14) that∥∥ϕ

(∂α(χku) − ∂αu

)∥∥∞ ≤ cqm+1,0(ϕ)(1 + k2)−1 , k ∈ N .

Since χku ∈ D(Rn, E), it follows that D(Rn, E) is dense in OM (Rn, E). ¥

The preceding approximation results can be used to prove a number of usefuldensity results. For this we make use of the following elementary facts that willoften be used without further mention:

Density by Iteration

Let A, B, and C be topological spaces. It is obvious that

Ad

→ Bd

→ C =⇒ Ad

→ C (1.3.15)

and

Ad⊂ C , A ⊂ B ⊂ C =⇒ B

d⊂ C . (1.3.16)

Implication (1.3.16) is also true if d stands for ‘sequentially dense’.

388 Appendix

Approximation by Tensor Products

We recall from Subsection V.2.4 that, given F ∈ D, E ,S,OM, the tensor productF(X)⊗ E is defined by putting first

ϕ⊗ e(x) := ϕ(x)e , ϕ ∈ F(X) , e ∈ E , x ∈ X , (1.3.17)

and then

F(X) ⊗ E := span

ϕ⊗ e ; ϕ ∈ F(X), e ∈ E

, (1.3.18)

where the linear hull is taken in EX . Analogously, we define the tensor productF′(X) ⊗ E by

(u ⊗ e)(ϕ) := u(ϕ)e , u ∈ F′(X) , e ∈ E , ϕ ∈ F(X) , (1.3.19)

and

F′(X) ⊗ E := span

u⊗ e ; u ∈ F′(X), e ∈ E

, (1.3.20)

now the span being taken in EF(X). It is obvious that F(X) ⊗ E ⊂ F(X, E) andF′(X) ⊗ E ⊂ F′(X, E). In the following approximation theorem these tensor prod-ucts are given the corresponding subspace topologies.

1.3.6 Theorem

(i) D(X) ⊗ Ed

→ D(X, E)d

→ E ′(X, E)d

→ D′(X,E);

(ii) D(X) ⊗ Ed

→ E(X) ⊗ Ed

→ E(X,E)d

→ D′(X, E);

(iii) E ′(X) ⊗ Ed

→ E ′(X,E);

(iv) D′(X) ⊗ Ed

→ D′(X, E);

(v) D(Rn) ⊗ Ed

→ S(Rn) ⊗ Ed

→ S(Rn, E)d

→ S ′(Rn, E)d

→ D′(Rn, E);

(vi) S(Rn) ⊗ Ed

→ S ′(Rn)⊗ Ed

→ S ′(Rn, E);

(vii) S(Rn) ⊗ Ed

→ OM (Rn)⊗ Ed

→ OM (Rn, E)d

→ S ′(Rn, E);

(viii) S(Rn) ⊗ Ed

→ W kp (Rn)⊗ E

d→ W k

p (Rn, E), k ∈ N, 1 ≤ p < ∞;

(ix) D(X) ⊗ Ed

→ Ck(X) ⊗ Ed

→ Ck(X,E), k ∈ N;

(x) D(X) ⊗ Ed

→ C0(X) ⊗ Ed

→ C0(X,E).

Proof From Remark 1.3.4 and Proposition V.2.4.1 we know that

D(X) ⊗ Ed

→ D(X, E)d

→ E(X,E) . (1.3.21)

Proposition 1.3.3 implies D(X, E)d

→ E ′(X, E) and D(X, E)d

→ D′(X, E), sincethe continuity of these injections is obvious. This and (1.3.16) give (i). From


D(X)d

→ E(X) it follows that D(X) ⊗ Ed

→ E(X)⊗ E. Now (ii) is obtained from

(i) and (1.3.16). Since D(X)d

→ E ′(X) by (i), D(X) ⊗ Ed

→ E ′(X) ⊗ E. Thus (iii)follows from (1.3.15) and (i). Analogous arguments prove (iv)–(viii), where onehas to use Proposition 1.3.5 for (vii), and Theorem V.2.4.3 and

S(Rn, E)d

→ W kp (Rn, E) , 1 ≤ p < ∞ ,

for (viii). Details are left to the reader.

(ix) Given K = K b L = L b X, choose ϕ ∈ D(X) with ϕ |L = 1. Then ϕubelongs to Ck(X, E) for u ∈ Ck(X, E), and supp(ϕu) b X with ϕu |L = u. Bymollifying ϕu we find v ∈ D(X,E) satisfying v |K = u. This gives

D(X,E)d

→ Ck(X,E) .

Now the assertion is obvious, as is (x). ¥

Approximation by Polynomials

Finally, we use this approximation theorem to prove the separability of the spacesunder consideration, provided E is separable. For this we need a preparatorylemma that will be useful in Subsection 1.8 as well.

1.3.7 Lemma Every element of D(X,E) is the limit in E(X,E) of a sequence of(restrictions of ) polynomials (in n indeterminates and with coefficients in E).

Proof Define w ∈ S(Rn) by

w(x) := (4π)−n/2e−|x|2/4 , x ∈ Rn . (1.3.22)

Then w is analytic and everywhere positive. Since w(x) =∏n

j=1 w(xj), it followsfrom Fubini’s theorem that ‖w‖1 = ‖w‖n

L1(R). But, again by Fubini’s theorem,

(∫

Re−x2/4 dx

)2

=∫

R2e−(x2+y2)/4 d(x, y)

= 2π

∫ ∞

0

re−r2/4 dr = −4πe−r2/4∣∣∞0

= 4π .

Hence ‖w‖1 = 1, and, putting

wt(x) := t−n/2w(x/√

t)

= (4πt)−n/2e−|x|2/(4t) , x ∈ Rn , t > 0 , (1.3.23)

it follows that wt ; t > 0 is an approximate identity. Thus, given ϕ ∈ D(X,E),we infer from (III.4.2.10) and (III.4.2.25) that w1/k ∗ ϕ → ϕ in E(X, E).

390 Appendix

Let k ∈qN be fixed and put

f(z) :=∫

Rn

w1/k(z − y)ϕ(y) dy , z ∈ Cn .

It is easily verified that f is well-defined and analytic. Hence f can be repre-sented by its Taylor series, that is easily seen to converge uniformly on com-pact subsets of Cn towards f . This implies that the sequence of Taylor polynomi-als of w1/k ∗ ϕ = f |Rn converges in E(Rn, E) towards w1/k ∗ ϕ, which gives theassertion. ¥

Separability

1.3.8 Proposition If E is separable, then D(X,E) is also separable.

Proof It is an obvious consequence of Theorem 1.3.6(i) that it suffices to provethat D(X) is separable. Thus let P be the set of all polynomials in n indeterminateswith rational coefficients (in K). Define (Xk) by (1.1.5). If Xk 6= ∅, fix ϕk ∈D(Xk+1)with ϕk |Xk = 1, and put ϕk := 0 if Xk = ∅. Then S := ϕkp ; p ∈ P , k ∈ N isa countable subset of D(X). It is an easy consequence of Lemma 1.3.7 and theproperties of the LF topology of D(X) that S is dense in D(X). ¥

1.3.9 Corollary Suppose that

F ∈ D′, E , E ′,S,S ′,OM , C0, Ck,W k

p ; 1 ≤ p < ∞, k ∈ N ,

and that X = Rn if F = W kp . Then F(X, E) is separable, if E is separable.

Proof Theorem 1.3.6 implies D(X, E)d

→ F(X,E). Hence the assertion followsfrom Proposition 1.3.8. ¥

1.4 Topological Tensor Products and the Kernel Theorem

In the first part of this subsection we collect some facts about tensor productsof LCSs. This theory is easily accessible in standard books on linear functional anal-ysis and topological vector spaces, in particular, in [Jar81], [Sch71], and [Tre67].Thus we are rather brief and do not give proofs. In the second part we prove aversion of the kernel theorem. This general abstract theorem will be of fundamentalimportance for defining bilinear operations on vector-valued distributions.

Algebraic Tensor Products

Let V and W be vector spaces. A tensor product of V and W is a pair (T, β)consisting of a vector space T and a bilinear map β : V × W → T such that


(i) T = span(im(β)

),

(ii) β(V ×W) is linearly independent in T , if V and W are linearly independentin V and W , respectively.

It can be shown that there exists a tensor product of V and W and that it isunique, except for vector space isomorphisms. If (T, β) is a tensor product of Vand W , then we put

V ⊗ W := T , v ⊗ w := β(v, w) , (v, w) ∈ V × W ,

which is justified by ‘uniqueness’.

1.4.1 Remarks (a) The tensor product has the following important universality

property: if U is a vector space and b : V × W → U is bilinear, there exists aunique linear map B : V ⊗ W → U such that the diagram

V × W V ⊗ W

U

b B

β-

@@@R

¡¡¡ª

is commutative.

Proof Let V and W be bases of V and W , respectively. Then β(V,W) is a basisof V ⊗ W by (i) and (ii) above. Define B on β(V,W) by

B(v ⊗ w) := b(v, w) , (v, w) ∈ V ×W ,

and extend it linearly. Then B has the desired property. ¥

(b) Let Vj and Wj be vector spaces and Aj ∈ Hom(Vj ,Wj) for j = 1, 2. Then thereexists a unique

A1 ⊗ A2 ∈ Hom(V1 ⊗ V2,W1 ⊗ W2) ,

the tensor product of A1 and A2, such that

(A1 ⊗ A2)(v1 ⊗ v2) = A1v1 ⊗ A2v2 , (v1, v2) ∈ V1 × V2 .

Proof Since (v1, v2) 7→ A1v1 ⊗ A2v2 is a bilinear map from V1×V2 into W1⊗W2,the assertion follows from (a). ¥

(c) Let V1, V2, and V3 be vector spaces. There exists a linear isomorphism

V1 ⊗ (V2 ⊗ V3) → (V1 ⊗ V2) ⊗ V3 (1.4.1)

such that

v1 ⊗ (v2 ⊗ v3) 7→ (v1 ⊗ v2) ⊗ v3 . (1.4.2)

This means that tensor products are (canonically) associative (so that parenthesescan be omitted).

392 Appendix

Proof Let Vj be a basis of Vj . Define (1.4.1) by (1.4.2) on the basis V1 ⊗ (V2 ⊗ V3)of V1 ⊗ (V2 ⊗ V3) and extend it linearly. ¥

Basic Examples

1.4.2 Examples (a) Let Km×n be the vector space of all (m× n)-matrices withentries in K and let

Km ×Kn → Km×n , (x, y) 7→ [xjyk]1≤j≤m1≤k≤n

.

Then Km ⊗Kn = Km×n. Note that

x⊗ y = xy> , x, y ∈ Km ×Kn ,

if Km and Kn are identified with Km×1 and Kn×1, respectively, and a> ∈ Ks×r

denotes the transposed of a ∈ Kr×s for r, s ∈qN. Furthermore,

x⊗ y = 〈y, ·〉x , (x, y) ∈ Km ×Kn ,

if Km×n and (Kn)′ are identified with L(Kn,Km) and Kn, respectively.

(b) Let S be a nonempty set, V a vector space, and Φ(S) a vector subspace of KS .Define a bilinear map

Φ(S) × V → V S , (ϕ, v) 7→ ϕ⊗ v

by ϕ⊗ v(s) := ϕ(s)v. Then

Φ(S)⊗ V := span

ϕ⊗ v ; ϕ ∈ Φ(S), v ∈ V

is the tensor product of Φ(S) and V in V S , that is, the span lies in V S (cf. Sub-sections V.2.4 and 1.3).

Proof Let ϕ1, . . . , ϕn be linearly independent in Φ(S) and v1, . . . , vn linearlyindependent in V . Suppose that there exist ξjk ∈ K such that

∑jk ξjkϕj ⊗ vk = 0.

Put ψk :=∑

j ξjkϕj so that∑

k ψk ⊗ vk = 0. Then∑

ψk(s)vk = 0 for each s ∈ S.This implies ψk = 0 for k = 1, . . . , n by the linear independence of v1, . . . , vn. Sincethe ϕj are linearly independent in Φ(S), we see that each ξjk is zero. ¥

(c) Suppose that F ∈ D, E ,S,OM. Then F(X)⊗ E, defined by (1.3.17) and(1.3.18), is the tensor product of F(X) and E in F(X, E). Similarly, if F ∈ D, E ,Sthen F′(X) ⊗ E, defined by (1.3.19) and (1.3.20), is the tensor product of F′(X)and E in F′(X, E).

Proof This follows from (b) by putting S := X and S := F(X), respectively. ¥


(d) Let E be a Banach space, S and T nonempty sets, and Φ(S) and Ψ(T,E)vector subspaces of KS and ET , respectively. Define a bilinear map

Φ(S) × Ψ(T,E) → ES×T , (ϕ,ψ) 7→ ϕ⊗ ψ

by ϕ⊗ ψ(s, t) := ϕ(s)ψ(t). Then

Φ(S) ⊗ Ψ(T, E) := span

ϕ⊗ ψ ; ϕ ∈ Φ(S), ψ ∈ Ψ(T, E)

is the tensor product of Φ(S) and Ψ(T, E) in ES×T .

Proof Identify ES×T and (ET )S by means of[(s, t) 7→ u(s, t)

] ←→ [s 7→ u(s, ·)] .

Then the assertion follows from (b). ¥

Projective Tensor Products

Let now F := (F,P) and G := (G,Q) be LCSs. For (p, q) ∈ P ×Q we put

p⊗π q(z) := inf ∑

p(fj)q(gj) ; z =∑

fj ⊗ gj

, z ∈ F ⊗ G .

This defines a seminorm on F ⊗ G, the tensor product of the seminorms p and q. Infact, it is the Minkowski functional of the absolutely convex hull of Bp ⊗ Bq whereBp := [p < 1]. It has the property

p ⊗π q(f ⊗ g) = p(f)q(g) , (f, g) ∈ F × G . (1.4.3)

Thus the family of the tensor product seminorms is separating. Consequently, itdefines a locally convex Hausdorff topology on F ⊗ G, the projective topology, and

F ⊗π G :=(F ⊗ G, p⊗π q ; p ∈ P , q ∈ Q)

is the projective tensor product of F and G. We denote by F∼⊗ G the completion of

the LCS F ⊗π G.

Nuclear Maps and Spaces

A linear map N from the LCS F into a Banach space E is nuclear if there exist anequicontinuous sequence (f ′j) in F ′, a bounded sequence (bj) in E, and a summablesequence (λj) in K such that

Nf =∑

j

λj〈f ′j , f〉bj , f ∈ F . (1.4.4)

The LCS F is nuclear if each continuous linear map from F into a Banach spaceis nuclear. It is conuclear if its dual is nuclear.

394 Appendix

For the following theorem we recall convention (1.1.11).

1.4.3 Theorem Let X be a nonempty open subset of Rn and suppose that F belongsto D, E ,S,OM. Then F(X) is nuclear and conuclear.

Proof If F ∈ D, E ,S, this can be found in [Tre67] (also cf. [Jar81] and [Sch71]).As for F = OM , we refer to [Gro55, II.4.4]. ¥

Projective Tensor Products and Maps of Finite Rank

Lastly, we define an injective linear map by

τ : F ⊗ G → L(F ′, G) , f ⊗ g 7→ 〈·, f〉g , (1.4.5)

the canonical injection. Then we prepare for the proof of the abstract kernel theoremby deriving a series of lemmas.

1.4.4 Lemma Let F be reflexive and nuclear. Then τ is a toplinear isomorphismfrom F ⊗π G onto the subspace of L(F ′, G) of maps of finite rank.

Proof (1) Given a continuous seminorm q on G and a bounded subset B′ of F ′,

supf ′∈B′

q(τz(f ′)

) ≤∑

j

supf ′∈B′

|〈f ′, fj〉| q(gj) , z =∑

fj ⊗ gj ∈ F ⊗ G . (1.4.6)

Sincef 7→ p(f) := sup

f ′∈B′|〈f ′, f〉|

defines a continuous seminorm on F ′′ = F , it follows from (1.4.6) that

supf ′∈B′

q(τz(f ′)

) ≤ p⊗π q(z) , z ∈ F ⊗ G .

Henceτ ∈ L(

F ⊗π G,L(F ′, G))

.

It is obvious that τz has finite rank for z ∈ F ⊗ G. Conversely, suppose thatT ∈ L(F ′, G) has finite rank, and let g1, . . . , gm be a basis of im(T ). Then

Tf ′ =m∑

j=1

ξj(f ′)gj , f ′ ∈ F ′ ,

whereξj(f ′) = 〈g′j , T f ′〉G = 〈T ′g′j , f

′〉F ′ , f ′ ∈ F ′ ,


the g′j ∈ G′ satisfying 〈g′j , gk〉 = δjk. Since T ′ ∈ L(G′, F ) by the reflexivity of F , itfollows that fj := T ′g′j ∈ F and T =

∑mj=1〈·, fj〉F gj . Hence

T = τz with z :=m∑

j=1

fj ⊗ gj ∈ F ⊗ G .

This shows that τ is a bijection from F ⊗ G onto the subspace of L(F ′, G) of finiterank operators.

(2) Now let p and q be continuous seminorms on F and G, respectively. GivenT ∈ L(F ′, G) of finite rank, let z := τ−1T , and let

∑fj ⊗ gj be a representation

of z in F ⊗ G. Denote by

Bp :=

f ′ ∈ F ′ ; |〈f ′, f〉| ≤ 1 for f ∈ Bp

the polar of Bp. Then

p⊗ε q(z) := sup(f ′,g′)∈Bp×Bq

∣∣∣∑

j

〈f ′, fj〉〈g′, gj〉∣∣∣

= supf ′∈Bp

supg′∈Bq

∣∣⟨g′, τz(f ′)⟩∣∣ = sup

f ′∈Bpq(τz(f ′)

) (1.4.7)

by the bipolar theorem. Since F is reflexive, Bp is weakly compact by the Alaoglu-Bourbaki theorem. Hence it is weakly bounded and thus, thanks to Mackey’stheorem, bounded in F ′ (cf. [Hor66], [Jar81], [Tre67], or [Sch71] for these standardtheorems from the theory of LCSs).

Observe that p ⊗ε q, as defined in (1.4.7), is a seminorm on F ⊗ G. In thetheory of LCSs it is shown that the family p ⊗ε q ; p ∈ P , q ∈ Q is a separatingfamily of seminorms on F ⊗ G defining the ‘injective tensor product topology’ onF ⊗ G. Since F is nuclear, this injective tensor product topology coincides with theprojective tensor product topology of F ⊗ G (e.g., [Tre67, Theorem 50.1(f)]). Thuswe infer from (1.4.7) that τ−1 ∈ L(R, F ⊗π G), where R is the linear subspaceof L(F ′, G) of finite rank operators. ¥

Approximation by Maps of Finite Rank

Let p be a continuous seminorm on F . Then ker(p) is a closed linear subspace of F .Hence Fp := F/ ker(p) is a normed vector space with respect to the quotient norm

x := xp := x + ker(p) 7→ p(x) := inf

p(y) ; y ∈ x

. (1.4.8)

Observe that p(x) = p(y) for y ∈ x. Let q be a second continuous seminorm on Fsuch that q ≥ p. Then ker(q) ⊂ ker(p) implies that xq 7→ xp is a well-defined con-tinuous linear map from Fq into Fp of norm at most one. Let Fp be the completionof Fp. Then we consider xq 7→ xp to be a linear map from Fq into Fp, the canonical

map Fq → Fp.

396 Appendix

1.4.5 Lemma Suppose that F is nuclear and p is a continuous seminorm on F .Then there exists a continuous seminorm q ≥ p on F such that the canonical mapFq → Fp is nuclear.

Proof Since F is nuclear, the quotient map πp : F → Fp is nuclear. Hence πp hasa representation of the form (1.4.4) where (bj) is a bounded sequence in Fp.Consequently,

p(f) = p(πpf) ≤ c supj|〈f ′j , f〉| =: q(f) , f ∈ F , (1.4.9)

with c :=∑

j |λj | supj ‖bj‖. The equicontinuity of the sequence (f ′j) implies thatq is a continuous seminorm on F . By (1.4.8) and (1.4.9), each f ′j defines naturallya continuous linear form h′j on Fq of norm at most 1/c. Thus the canonical map

Fq → Fp , fq 7→∑

j

λj

⟨h′j , fq

⟩Fq

bj

is nuclear. ¥

By means of the preceding lemma we can give sufficient conditions for linearoperators to be approximable by operators of finite rank.

1.4.6 Lemma If F is nuclear, then the maps of finite rank are dense in L(F, G).

Proof Let p be a continuous seminorm on F . Lemma 1.4.5 guarantees the exis-tence of a continuous seminorm q ≥ p on F such that the canonical map S : Fq → Fp

is nuclear. Hence there exist a summable sequence (λj) inK and bounded sequences(f ′j

)and

(fj

)in (Fq)′ and Fp, respectively, such that

Sf =∑

j

λj

⟨f ′j , f

⟩fj , f ∈ Fq .

Thus, letting

Sn :=n∑

j=0

λj

⟨f ′j , ·

⟩fj , n ∈ N ,

it is obvious that

Sn → S in L(Fq, Fp

). (1.4.10)

Observe that the diagram

F F

Fq Fp

S

id -

-

πq πp

? ?(1.4.11)


is commutative, where πq and πp are the quotient maps. Choose fj ∈ fj and put

Tnf :=n∑

j=0

λj

⟨f ′j , πqf

⟩fj , f ∈ F .

Then, thanks to (1.4.10) and (1.4.11),

p(Tnf − f) → 0 , n →∞ , (1.4.12)

uniformly with respect to f in bounded subsets of F .Now let R ∈ L(F, G) and a continuous seminorm r on G be given. Then

p := r R is a continuous seminorm on F . Put Rn := R Tn. It follows from(1.4.12) that r

((Rn − R)f

) → 0 as n →∞, uniformly with respect to f in boundedsubsets of F . Since Rn ∈ L(F, G) has finite rank, the assertion follows. ¥

Completeness of Spaces of Linear Operators

In our last preparatory lemma we give conditions for L(F ′, G) to be complete.

1.4.7 Lemma Let F and G be complete and let F be reflexive. Then L(F ′, G) iscomplete.

Proof Let (Tα) be a Cauchy net in L(F ′, G). Since G is complete and (Tαf ′) isa Cauchy net in G, there exists T ∈ Hom(F ′, G) such that Tαf ′ → Tf ′ in G foreach f ′ ∈ F ′. Given g′ ∈ G′,

〈f ′, T ′αg′〉F = 〈g′, Tαf ′〉G → 〈g′, T f ′〉G = g′ T (f ′) (1.4.13)

for each f ′ ∈ F ′, thanks to F ′′ = F . Observe that (T ′αg′) is a Cauchy net in F .

Hence we deduce from (1.4.13) and the completeness of F that g′ T ∈ F foreach g′ ∈ G′.

Suppose that g′αw∗−→ g′. Then

g′α T (f ′) = 〈g′α, T f ′〉G → 〈g′, T f ′〉G , f ′ ∈ F ′ .

This shows that tT ∈ L(G′w∗ , Fw), where tT is the algebraic dual of T . Let q be

a continuous seminorm on G. Then Bq is w∗-compact by the Alaoglu-Bourbakitheorem. Hence K := tT (Bq) is weakly compact as well, thus weakly bounded.Consequently, K is bounded in F . Hence V ′ := K is a neighborhood of zeroin F ′. Observe that f ′ ∈ V ′ iff |〈f ′, tTg′〉F | = |〈g′, T f ′〉G| ≤ 1 for g′ ∈ Bq , that is,iff Tf ′ ∈ (Bq), where, given C ′ ⊂ G′,

C ′ :=

g ∈ G ; |〈g′, g〉| ≤ 1 for g′ ∈ C ′

is the polar of C ′ in G. By the bipolar theorem (Bq) = Bq. Hence

sup

q(Tf ′) ; f ′ ∈ V ′ ≤ 1 . (1.4.14)

398 Appendix

Since V ′ is the polar of K, it is absolutely convex and w∗-closed. Thus it is weaklyclosed by the reflexivity of F . Hence it is closed. Denoting by p the Minkowskifunctional of V ′, it follows that f ′ ∈ V ′ iff p(f ′) ≤ 1. From this and (1.4.14) weinfer that

q(Tf ′) ≤ p(f ′) , f ′ ∈ F ′ ,

which shows that T ∈ L(F ′, G), thanks to the fact that p is a continuous semi-norm on F ′. ¥

The Abstract Kernel Theorem

After these preparations we can prove the abstract kernel theorem in a form thatis most suitable for our purposes. Recall that τ is defined in (1.4.5).

1.4.8 Theorem Let F and G be complete LCSs such that F is reflexive, nuclear,and conuclear. Then τ is (that is, extends to) a toplinear isomorphism

τ : F∼⊗ G → L(F ′, G) , (1.4.15)

the canonical isomorphism.

Proof It follows from Lemma 1.4.7 that L(F ′, G) is complete. Since F is conu-clear, we infer from Lemma 1.4.6 that the linear subspace R of maps of finiterank is dense in L(F ′, G). Since F is also nuclear, Lemma 1.4.4 guarantees thatτ is a toplinear isomorphism from F ⊗π G onto R. Now the assertion is an easyconsequence of the density of F ⊗π G in F

∼⊗ G and a well-known result aboutcontinuous extensions of continuous linear maps (e.g., [Jar81, Theorem 3.4.2]). ¥

Tensor Product Characterizations of Some Distribution Spaces

As a first application of this general theorem we can prove the following character-izations for spaces of vector-valued distributions. They are the basis for definingbilinear maps of vector-valued distributions in the next subsection.

1.4.9 Theorem Let E be a Banach space and F ∈ D, E ,S,OM. Then

F(X)∼⊗ E = F(X, E) ∼= L(

F′(X), E)

andF′(X)

∼⊗ E ∼= F′(X,E) = L(F(X), E

),

where ∼= denotes the canonical toplinear isomorphism.

Proof From Theorem 1.1.2 we know that F(X) and F′(X) are complete andreflexive. Furthermore, F(X) and F′(X) are both nuclear and conuclear, thanks to


Theorem 1.4.3. Hence it follows from Theorem 1.4.8 that, given F ∈ D, E ,S,OM,

τ : F(X)∼⊗ E → L(

F′(X), E)

andτ : F′(X)

∼⊗ E → L(F(X), E

)= F′(X,E)

are toplinear isomorphisms. This proves the second assertion and part of thefirst one. It remains to show that F(X)

∼⊗ E = F(X, E). Since F(X) ⊗ E is densein F(X, E) by Theorem 1.3.6, we have to verify that F(X, E) induces on F(X) ⊗ Ethe projective topology.

Suppose that F ∈ E ,S,OM and let p be one of the seminorms (1.1.3),(1.1.7), or (1.1.9), respectively. Then

p(ϕ⊗ e) = p(ϕ) |e| , ϕ ∈ F(X) , e ∈ E ,

where on the left side p is the seminorm on F(X,E), and on the right side p denotesthe corresponding seminorm on F(X). From this we easily deduce that F(X, E)induces on F(X) ⊗ E a topology that is weaker than the projective tensor prod-uct topology.

Conversely, let∑

ϕj ⊗ ej be a representation of z ∈ F(X) ⊗ E. Then, givenϕ′ ∈ F′(X) and e′ ∈ E′,

∑〈ϕ′, ϕj〉〈e′, ej〉 =

⟨e′,

∑〈ϕ′, ϕj〉ej

⟩=

⟨e′,

⟨ϕ′,

∑ϕj ⊗ ej

⟩⟩.

Consequently (cf. (1.4.7)),

p ⊗ε |·|E(z) = supϕ′∈B0

p

|〈ϕ′, z〉|E ≤ p(z)

by the bipolar theorem. This shows that the topology induced by F(X, E) onF(X)⊗ E is stronger than the injective tensor product topology. Since F(X) isnuclear, the latter coincides with the projective topology.

Finally, given K b X, the last string of arguments shows that DK(X, E) in-duces on DK(X) ⊗ E the projective topology. Since D(X, E) induces on DK(X, E)its original topology, it follows that D(X, E) induces on DK(X) ⊗ E the projectivetopology. From

D(X) ⊗ E = lim−−→KbX

DK(X)⊗ E

(e.g., [Jar81, Corollary 4 to Theorem 15.5.3]) it follows that D(X, E) induces onD(X)⊗ E the projective tensor product topology. Thus, since D(X) ⊗ E is densein D(X, E) by Theorem 1.3.6, it follows that D(X)

∼⊗ E = D(X,E). ¥

400 Appendix

Suppose that F and G are Frechet spaces such that F is nuclear. Then

(F∼⊗ G)′ = F ′ ∼⊗ G′ (1.4.16)

by means of the duality pairing induced by

〈f ′ ⊗ g′, f ⊗ g〉 := 〈f ′, f〉〈g′, g〉 (1.4.17)

for (f, g) ∈ F × G and (f ′, g′) ∈ F ′ × G′ (e.g., [Sch71, Theorem IV.9.9]). By meansof this fact we can identify the dual of F(X, E) for F ∈ E ,S.

1.4.10 Corollary Let E be a Banach space and suppose that F ∈ E , E ′,S,S ′. Then

F(X,E)′ =(F(X)

∼⊗ E)′ = F′(X)

∼⊗ E′ = F′(X, E′)

by means of the duality pairing induced by (1.4.17). If E is reflexive, then F(X, E)is reflexive as well.

In Theorem 1.7.5 below it is shown that this duality coincides with the oneintroduced in Subsection VI.1.3.

There are various versions of the abstract kernel theorem in the textbookliterature (e.g., [Jar81, Theorem 21.5.9], [Tre67, Section 50]). However, it is alwaysassumed that F and G are Frechet spaces. Since we need to apply Theorem 1.4.8in the special case that F = OM (Rn) and since OM (Rn) is not metrizable, thisassumption does not fit our purposes. For this reason we have included a completeproof of Theorem 1.4.8.

1.5 Extending Bilinear Maps

In this subsection we prove a general extension theorem for bilinear maps ontensor products. For this we need a technical lemma guaranteeing a suitable kindof uniformity for equicontinuous sets of nuclear maps.

1.5.1 Lemma Let F be a nuclear LCS and E a Banach space. Suppose that T isan equicontinuous subset of L(F, E). Then there exist a summable sequence (λj)in K, an equicontinuous sequence (f ′j) in F ′, and a bounded map

T → B(N, E) , T 7→ (ej(T )

)j∈N

such thatTf =

∑

j

λj〈f ′j , f〉ej(T ) , f ∈ F , T ∈ T .


Proof By the equicontinuity of T , the set⋂

T−1(BE) ; T ∈ T is a closedabsolutely convex neighborhood of zero. Let p be its Minkowski functional. Thenp is a continuous seminorm on F and

|Tf |E ≤ p(f) , f ∈ F , T ∈ T . (1.5.1)

The nuclearity of F implies, thanks to Lemma 1.4.5, the existence of a continuousseminorm q ≥ p on F such that the canonical map S : Fq → Fp is nuclear. Hencethere exist a summable sequence (λj) in K and bounded sequences

(f ′j

)and (gj)

in (Fq)′ and Fp, respectively, such that

Sf =∑

j

λj

⟨f ′j , f

⟩gj , f ∈ Fq . (1.5.2)

It is easy to verify that for each T ∈ T there exists T ∈ L(Fp, E

)such that the

diagram

F Fq

E Fp

T

πq-

¾

T S? ?

is commutative, where πq is the quotient map. Hence, letting f ′j := (πq)′f ′j andej(T ) := T gj , the assertion follows since (1.5.1) and (1.4.8) imply

∥∥T∥∥ ≤ 1. ¥

Now we are in a position to prove the following general extension theorem forbilinear maps. Throughout the remainder of this subsection we assume that Ej ,j = 0, 1, 2, are Banach spaces and

E1 × E2 → E0 , (e1, e2) 7→ e1q e2 (1.5.3)

is a multiplication. Furthermore, given an LCS F , we put F (E) := L(F ′, E) and weremind the reader of the definition in Subsection II.1.1 of point-wise multiplicationinduced by (1.5.3).

1.5.2 Proposition Let Fj, j = 0, 1, 2, be LCSs such that(i) F1 is reflexive, complete, nuclear, and conuclear.(ii) F0(E0) is complete.

Suppose that there is given a hypocontinuous bilinear map

F1 × F2(E2) → F0(E2) , (f1, v) 7→ f1 ¡ v . (1.5.4)

Define a bilinear map

¡ q : (F1 ⊗ E1)× F2(E2) → F0(E0)

402 Appendix

by

(f1 ⊗ e1, v) 7→ e1q (f1 ¡ v) , (e1, f1) ∈ E1 × F1 , (1.5.5)

and by bilinear extension. Then ¡ q possesses a unique hypocontinuous bilinearextension

¡ q : (F1∼⊗ E1) × F2(E2) → F0(E0) , (u, v) 7→ u ¡ q v .

Proof Uniqueness follows immediately from the density of F1 ⊗ E1 in F1∼⊗ E1.

Given u :=∑

fj ⊗ ej ∈ F1 ⊗ E1 and v ∈ F2(E2),

u ¡ q v =∑

ejq (fj ¡ v) ∈ F0(E0) . (1.5.6)

Let B′0 be a bounded subset of F ′

0. Then

p(w) := supf ′∈B′

0

|wf ′|E0 , w ∈ F0(E0) ,

andp2(v) := sup

f ′∈B′0

|vf ′|E2 , v ∈ F0(E2) ,

define continuous seminorms on F0(E0) and F0(E2), respectively. Hence

p(u ¡ q v) ≤∑

j

p2(fj ¡ v) |ej |E1. (1.5.7)

Let B2 be a bounded subset of F2(E2). Then (1.5.7) and the hypocontinuity of(1.5.4) imply the existence of a continuous seminorm p1 on F1 such that

p(u ¡ q v) ≤∑

j

p1(fj) |ej |E1, v ∈ B2 .

Since p1 is independent of the particular representation of u, it follows that

p(u ¡ q v) ≤ (p1 ⊗π |·|)(u) , u ∈ F1 ⊗ E1 , v ∈ B2 .

This shows that

(u 7→ u ¡ q v) ∈ L(F1 ⊗π E1, F0(E0)

), (1.5.8)

uniformly with respect to v in bounded subsets of F2(E2).Given v ∈ F2(E2), there exists a unique continuous extension

Uv ∈ L(F1

∼⊗ E1, F0(E0))

(1.5.9)


of (1.5.8), thanks to assumption (ii). Then[(u, v) 7→ u ¡ q v := Uv(u)

]: (F1

∼⊗ E1)× F2(E2) → F0(E0) (1.5.10)

is an extension of (1.5.5) that is trivially linear in the first variable. For u ∈ F1∼⊗ E1

choose a net (uα) in F1 ⊗ E1 such that uα → u in F1∼⊗ E1. Then, given λj ∈ K

and vj ∈ F2(E2) for j = 1, 2,

u ¡ q (λ1v1 + λ2v2) = limα

[uα ¡ q (λ1v1 + λ2v2)

]

= λ1 limα

(uα ¡ q v1) + λ2 limα

(uα ¡ q v2)

= λ1(u ¡ q v1) + λ2(u ¡ q v2) .

Thus (1.5.10) is bilinear and it remains to show that it is hypocontinuous.The uniformity assertion contained in (1.5.8) implies that the family of lin-

ear operators (1.5.8) is equicontinuous if v stays in bounded subsets of F2(E2).Thus, given a closed neighborhood V0 of zero in F0(E0) and a bounded subset B2

of F2(E2), there exists a neighborhood V1 of zero in F1 ⊗π E1 such that u ¡ q v ∈ V0

for (u, v) ∈ V1 × B2. Consequently,

Uv(V1) ⊂ V0 , v ∈ B2 ,

where V1 is the closure of V1 in F1∼⊗ E1. This shows that Uv ; v ∈ B2 is equicon-

tinuous in L(F1

∼⊗ E1, F0(E0)). Hence u ¡ q v → 0 in F0(E0) as u → 0 in F1

∼⊗ E1,uniformly with respect to v in bounded subsets of F2(E2).

Now let B be a bounded subset of F1∼⊗ E1. Thanks to assumption (i) and

Theorem 1.4.8,

F1∼⊗ E1

∼= L(F ′1, E1) = F1(E1) . (1.5.11)

Put B1 := τ(B), where τ is the isomorphism of (1.5.11). Then B1 is boundedin F1(E1). Since F ′

1 is barreled, being the dual of a reflexive space, hence re-flexive, the uniform boundedness principle shows that B1 is equicontinuous. ThusLemma 1.5.1 guarantees the existence of a summable sequence (λj) inK, a boundedsequence (fj) in F1, and a bounded set C in E1 such that each u ∈ B1 has a rep-resentation u =

∑j λjfj ⊗ ej , where ej ∈ C and the series converges in F1

∼⊗ E1.Hence, given v ∈ F2(E2), it follows from (1.5.9), (1.5.10), and (1.5.6) that

u ¡ q v =∑

j

λjejq (fj ¡ v) .

Consequently,

p(u ¡ q v) ≤∑

j

|λj | p2(fj ¡ v) |ej |E1, u ∈ B1 , v ∈ F2(E2) . (1.5.12)

404 Appendix

From the hypocontinuity of (1.5.4) and the boundedness of the sequence (fj) in F1

we deduce the existence of a continuous seminorm q on F2(E2) such that

p2(fj ¡ v) ≤ q(v) , v ∈ F2(E2) , j ∈ N . (1.5.13)

Since ej ∈ C and C is bounded in E1, we infer from (1.5.12) and (1.5.13) that

p(u ¡ q v) ≤ cq(v) , v ∈ F2(E2) , u ∈ B1 .

This proves that(v 7→ u ¡ q v) ∈ L(

F2(E2), F0(E0))

,

uniformly with respect to u in bounded subsets of F1∼⊗ E1. Thus the map (1.5.10)

is hypocontinuous. ¥

By specializing we deduce from the preceding proposition the following fun-damental extension result:

1.5.3 Theorem Let Fj, j = 0, 1, 2, be reflexive, complete, nuclear, and conuclearLCSs. Suppose that there are a bilinear map

F1 × F2 → F0 , (f1, f2) 7→ f1 ¯ f2 (1.5.14)

and a hypocontinuous bilinear map

F1 × (F2∼⊗ E2) → F0

∼⊗ E2 , (f1, v) 7→ f1 v (1.5.15)

such that

f1 (f2 ⊗ e2) = (f1 ¯ f2) ⊗ e2 , fj ∈ Fj , j = 1, 2 , e2 ∈ E2 . (1.5.16)

Then there exists a unique hypocontinuous bilinear map

¯ q : (F1∼⊗ E1)× (F2

∼⊗ E2) → F0∼⊗ E0 , (u, v) 7→ u ¯ q v (1.5.17)

satisfying

(f1 ⊗ e1) ¯ q (f2 ⊗ e2) = (f1 ¯ f2) ⊗ (e1q e2) (1.5.18)

for fj ∈ Fj and ej ∈ Ej, j = 1, 2.

Proof Since Ej ⊗ Fj is dense in Ej∼⊗ Fj for j = 1, 2, there exists at most one

separately continuous bilinear map ¯ q from (F1∼⊗ E1) × (F2

∼⊗ E2) to F0∼⊗ E0

satisfying (1.5.18).


Thanks to Theorem 1.4.8 we can identify Fj∼⊗ Ej with Fj(Ej) by means of

the respective canonical isomorphisms. Then it follows from (1.5.16) that, givenf ′0 ∈ F ′

0 and v := f2 ⊗ e2 ∈ F2 ⊗ E2,[e1

q (f1 v)](f ′0) = e1

q [(f1 ¯ f2) ⊗ e2

](f ′0) = e1

q 〈f ′0, f1 ¯ f2〉e2

= 〈f ′0, f1 ¯ f2〉(e1q e2) =

[(f1 ¯ f2) ⊗ (e1

q e2)](f ′0)

for f1 ∈ F1 and e1 ∈ E1. Hence

e1q (f1 v) = (f1 ¯ f2) ⊗ (e1

q e2) (1.5.19)

for f1 ⊗ e1 ∈ F1 ⊗ E1 and v := f2 ⊗ e2 ∈ F2 ⊗ E2. By Proposition 1.5.2 there ex-ists a unique hypocontinuous bilinear extension (1.5.17) of (1.5.15) satisfying

(f1 ⊗ e1) ¯ q v = e1q (f1 v) , f1 ⊗ e1 ∈ F1 ⊗ E1 , v ∈ F2

∼⊗ E2 .

Thus (1.5.19) implies (1.5.18). ¥

The results of this subsection are due to L. Schwartz. In fact, they arespecial cases of much more general theorems given in [Schw57b, chap. II] (alsosee [Schw57a]).

General Hypothesis

Throughout the remainder of this section we suppose that

E, F , and Ej, j = 0, 1, 2, are Banach spaces andE1 × E2 → E0 , (e1, e2) 7→ e1

q e2

is a multiplication. Moreover, X is a nonempty open subset of Rn

and convention (1.1.11) is effective.

(1.5.20)

1.6 Point-Wise Multiplication

As a first application of the general extension result of Theorem 1.5.3 we definepoint-wise multiplication of a vector-valued smooth function with a vector-valueddistribution and study some of its properties.

It is an immediate consequence of Leibniz’ rule that for each m ∈ N point-wisemultiplication induced by (1.5.20),

Cm(X,E1) × Cm(X, E2) → Cm(X, E0) , (a1, a2) 7→ a1q a2 , (1.6.1)

is well-defined, bilinear and continuous.

406 Appendix

Given a ∈ E(X) and v ∈ D′(X, E), we recall that av ∈ D′(X, E) is defined by

av(ϕ) := v(aϕ) , ϕ ∈ D(X) . (1.6.2)

It is easily verified that

(ϕ 7→ aϕ) ∈ L(D(X))

, a ∈ E(X) .

A Characterization of OM

First we prove a characterization of OM (Rn, E) that extends (III.4.1.9). In addi-tion, it explains the name ‘space of multipliers’ for OM (Rn, E).

1.6.1 Proposition Suppose that a ∈ E(Rn, E). Then

a ∈ OM (Rn, E) iff (ϕ 7→ ϕa) ∈ L(S(Rn),S(Rn, E))

.

Proof Given a ∈ OM (Rn, E) and k, m ∈ N, Leibniz’ rule and (1.1.8) imply

qk,m(ϕa) ≤ c max|α|≤m

∑

β≤α

supx∈Rn

(1 + |x|2)k |∂βϕ(x)| |∂α−βa(x)| ≤ cq`,m(ϕ) (1.6.3)

for ϕ ∈ S(Rn), where ` := k + maxmα−β ; β ≤ α, |α| ≤ m and where mα−β ∈ Nare such that there exists cα−β satisfying |∂α−βa(x)| ≤ cα−β(1 + |x|2)mα−β forx ∈ Rn. This shows that

(ϕ 7→ ϕa) ∈ L(S(Rn),S(Rn, E))

. (1.6.4)

Conversely, let (1.6.4) be true. Then, given α ∈ N, there exist k,m ∈ N anda positive constant c such that

‖∂α(ϕa)‖∞ ≤ q0,|α|(ϕa) ≤ cqk,m(ϕ) , ϕ ∈ S(Rn) .

Fix ψ ∈ D(Rn) such that ψ equals 1 near zero. Since τxψ ∈ D(Rn) ⊂ S(Rn) forx ∈ Rn, it follows that

|∂αa(x)| =∣∣∂α

((τxψ)a

)(x)

∣∣ ≤ cqk,m(τxψ)

≤ c sup|α|≤my∈Rn

(1 + |x− y|2)k |∂αψ(y)| ≤ c (1 + |x|2)k (1.6.5)

for x ∈ Rn, thanks to the trivial inequality 1 + |x − y|2 ≤ 2(1 + |x|2)(1 + |y|2).Hence a ∈ OM (Rn, E). ¥

1.6.2 Corollary Suppose that a ∈ E(Rn). Then

a ∈ OM (Rn) iff (u 7→ au) ∈ L(S ′(Rn))

.

Proof For simplicity, we put OM = OM (Rn) etc. If a ∈ OM , it follows from Prop-osition 1.6.1 that (ϕ 7→ ϕa) ∈ L(S). Hence the dual of this linear map, which is


given by u 7→ au for u ∈ S ′, satisfies (u 7→ au) ∈L(S ′). Conversely, (u 7→ au) ∈L(S ′)implies (u 7→ au)′ ∈ L(S) by the reflexivity of S. Since the latter dual is the mapϕ 7→ ϕa for ϕ ∈ S, the assertion follows. ¥

The General Theorem

Now it is easy to prove the following technical lemma that will be used in the proofof the next theorem, one of the main results of this subsection.

1.6.3 Lemma Suppose that (F1, F2; F0) is one of the triplets (E ,D′;D′), (E , E ′; E ′),(S,OM ;S), (OM ,OM ;OM ), (OM ,S ′;S ′), (E , E ; E), or (E ,D;D). Then ‘point-wisemultiplication’

F1(X) × F2(X, E) → F0(X,E) , (a, u) 7→ au (1.6.6)

is a well-defined hypocontinuous bilinear map.

Proof (1) We see from (1.1.8) and (1.1.9) that B is bounded in OM (Rn, E) iff,given α ∈ Nn, there exist mα ∈ N and cα > 0 such that

|∂αu(x)| ≤ cα(1 + |x|2)mα , x ∈ Rn , u ∈ B . (1.6.7)

Hence (1.6.3) shows that

(a 7→ au) ∈ L(S(Rn),S(Rn, E))

,

uniformly with respect to u in bounded subsets of OM (Rn, E). Since the functiondefined by x 7→ (1 + |x|2)k∂βϕ(x) belongs to S(Rn) for ϕ ∈ S(Rn), k ∈ N, andβ ∈ Nn, it follows from (1.6.3) that, given k,m ∈ N and a ∈ S(Rn), there existsak,α ∈ S(Rn) such that

qk,m(au) ≤ c max|α|≤m

‖ak,α∂αu‖∞ .

It follows

(u 7→ au) ∈ L(OM (Rn, E),S(Rn, E))

, a ∈ S(Rn) . (1.6.8)

Thus, letting Mu := (a 7→ au) ∈ L(S(Rn),S(Rn, E)), we see from (1.6.8) that

(u 7→ Mu) ∈ L(OM (Rn, E),Ls

(S(Rn),S(Rn, E)))

.

Consequently, S(Rn) being a Montel space, the Banach–Steinhaus theorem guar-antees that

(u 7→ Mu) ∈ L(OM (Rn, E),L(S(Rn),S(Rn, E)

)),

that is, the map u 7→ au is continuous from OM (Rn, E) into S(Rn, E), uniformlywith respect to a in bounded subsets of S(Rn). This proves the assertion for thecase (F1, F2; F0) = (S,OM ;S).

408 Appendix

(2) Given a ∈ OM (Rn) and u ∈ OM (Rn, E), it follows from Leibniz’ rule that

‖ϕ∂α(au)‖∞ ≤ c∥∥∥ϕ

∑

β≤α

∂βa∂α−βu∥∥∥∞

, ϕ ∈ S(Rn) .

Thus (1.6.7) implies the existence of m ∈ N such that

‖ϕ∂α(au)‖∞ ≤ c∑

β≤α

‖(1 + |x|2)mϕ∂βa‖∞ , u ∈ B .

This shows that a 7→ au maps OM (Rn) continuously into OM (Rn, E), uniformlywith respect to u in bounded subsets of OM (Rn, E). Similarly, we see that u 7→ aumaps OM (Rn, E) continuously into itself, uniformly with respect to a in boundedsubsets of OM (Rn). This proves the assertion if (F1, F2; F0) = (OM ,OM ;OM ).

(3) Since supp(au) ⊂ supp(a) ∩ supp(u) for a ∈ E(X) and u ∈ D(X, E), Leib-niz’ rule and the properties of LF spaces easily imply that point-wise multiplicationis separately continuous from E(X) ×D(X,E) into D(X, E). Now hypocontinuityin the case (E ,D;D) follows from the fact that E(X) and D(X,E) are barreled.

(4) If (F1, F2; F0) = (E , E ; E), then the assertion follows from (1.6.1).(5) Suppose

(F1, F2; F0) ∈(E ,D′;D′), (E , E ′; E ′), (Om,S ′;S ′)

and set F := F2 = F0. By Theorem 1.4.9,

F(X, E) = L(F′(X), E

),

and reflexivity guarantees F′(X) ∈ D(X), E(X),S(X). Hence it follows from

step (3), resp. step (4), that

Ma := (ϕ 7→ aϕ) ∈ L(F′(X)

), a ∈ F1(X) , (1.6.9)

and

(a 7→ Ma) ∈ L(F1(X),L(

F′(X))

(1.6.10)

if (F1, F) = (E ,D′), resp. (F1, F) = (E , E ′).If (F1, F) = (Om,S ′), then (1.6.9) and (1.6.10) are implied by an obvious

modification of step (1).Consider the bilinear map

L(F′(X), F′(X)

)× L(F′(X), E

) → L(F′(X), E

), (M,u) 7→ uM .

Lemma 1.1.1 guarantees that it is hypocontinuous. From this and since, by (1.6.10),the map a 7→ Ma map is bounded on bounded sets, we deduce that

F1(X) × L(F′(X), E

) → L(F′(E), E

), a 7→ uMa = Mau

is hypocontinuous. This proves the assertion if (1.6.9) applies. ¥


Now we can extend point-wise multiplication to the case that both factorsare vector-valued.

1.6.4 Theorem There exists a unique hypocontinuous bilinear map

E(X, E1) ×D′(X, E2) → D′(X, E0) , (a, u) 7→ a q u , (1.6.11)

called point-wise multiplication induced by (1.5.20), such that

(ϕ⊗ e1) q (ψ ⊗ e2) = ϕψ ⊗ (e1q e2) (1.6.12)

for a := ϕ⊗ e1 ∈ D(X) ⊗ E1 and u := ψ ⊗ e2 ∈ D(X)⊗ E2. It restricts to a hypo-continuous bilinear map

F1(X, E1) × F2(X, E2) → F0(X, E0) ,

where (F1, F2; F0) is any one of the triplets (E , E ′; E ′), (S,OM ;S), (OM ,OM ;OM ),(OM ,S ′;S ′), (E , E ; E), or (E ,D;D).

Proof Let (F1, F2; F0) be as stated or (F1, F2; F0) = (E ,D′;D′), and write Fj

for Fj(X), j = 0, 1, 2. Then the spaces Fj are reflexive, complete, nuclear, andconuclear by Theorems 1.1.2 and 1.4.3. Moreover3,

Fj∼⊗ Ejt = Fj(X, Ej) , j = 0, 1, 2 ,

thanks to Theorem 1.4.9. Define the bilinear maps (1.5.14) and (1.5.15) by point-wise multiplication. It is easily verified that (1.5.16) is true and it follows fromLemma 1.6.3 that the map (1.5.15) is hypocontinuous. Hence Theorem 1.5.3 guar-antees the existence of a unique hypocontinuous bilinear map

F1(X, E1) × F2(X, E2) → F0(X, E0) , (a, u) 7→ a q u (1.6.13)

satisfying (1.6.12) for a := ϕ⊗ e1 ∈ F1(X) ⊗ E1 and u := v ⊗ e2 ∈ F2(X) ⊗ E2.The assertion is now an obvious consequence of the almost trivial fact that the sep-arately continuous bilinear map (1.6.13) is uniquely determined by its restrictionto the subspace

(D(X) ⊗ E1

)× (D(X)⊗ E2

)which, by Theorem 1.3.6, is dense

in F1(X, E1)× F2(X,E2). ¥

Basic Properties of Multiplications

1.6.5 Remarks (a) Since E2 × E1 → E0, (e2, e1) 7→ e1q e2 is a multiplication as

well, the assertions of Theorem 1.6.4 are ‘symmetric’ with respect to E1 and E2,that is, the roles of E1 and E2 can be interchanged. This fact will often be employedin the following, usually without further mention.

3Here and below we identify F(X)∼⊗ E with F(X, E) by means of the canonical isomorphism

of Theorem 1.4.9.

410 Appendix

(b) Point-wise multiplication induced by (1.5.20), as defined in Theorem 1.6.4,coincides on regular distributions with point-wise multiplication in the usual sense.In other words, if a ∈ E(X,E1) and u ∈ L1,loc(X,E2) then a q u ∈ L1,loc(X, E0) and

a q u(·) = a(·) q u(·) . (1.6.14)

Observe that this justifies the use of the name ‘point-wise multiplication’ for thebilinear map of Theorem 1.6.4.

Proof Since L1,loc(X, E2) →D′(X, E2), it follows from Theorem 1.6.4 and an ob-vious estimate that both sides of (1.6.14) are separately continuous bilinear maps

E(X, E1) × L1,loc(X, E2) → D′(X, E0) .

Theorem 1.3.6(viii) immediately implies

D(X) ⊗ E2d

→ L1,loc(X, E2) . (1.6.15)

Thus we deduce from (1.6.12) that both sides of (1.6.14) coincide on the densesubspace

(D(X) ⊗ E1

)× (D(X)⊗ E2

)of E(X, E1)× L1,loc(X, E2). Hence the as-

sertion follows. ¥

(c) Leibniz’ rule is valid in the general case as well. More precisely, if p is a poly-nomial in n indeterminates then

p(∂)(a q u) =∑

β

1β!

(∂βa) q p(β)(∂)u (1.6.16)

for a ∈ E(X, E1) and u ∈ D′(X,E2), where p(β) := ∂βp. In particular:

∂α(a q u) =∑

β≤α

(αβ

)(∂βa) q ∂α−βu (1.6.17)

for a ∈ E(X, E1) and u ∈ D′(X,E2).

Proof It follows from ∂α ∈ L(E(X, E1))

and p(β)(∂) ∈ L(D′(X, E2))

that bothsides of (1.6.16) define separately continuous bilinear maps:

E(X, E1) ×D′(X, E2) → D′(X, E0) .

Thanks to Lemma 1.3.2 and to property (1.6.12) they coincide on the linear sub-space

(D(X)⊗ E1

)× (D(X) ⊗ E2

). Since, by Theorem 1.3.6(ii) and (v), this sub-

space is dense in E(X,E1)×D′(X,E2), they coincide everywhere. ¥

(d) If (a, u) ∈ E(X, E1) ×D′(X, E2), then

supp(a q u) ⊂ supp(a) ∩ supp(u) .

Proof Let K := supp(a) and denote by Kε :=

x ∈ X ; dist(x,K) < ε

the openε-neighborhood of K in X for ε > 0. Choose ψε ∈ E(X) with supp(ψε) ⊂ Kε and


ψ |K = 1. Since E(X) ⊗ E1 is dense in E(X,E1), there exists a sequence (aj)in E(X) ⊗ E1 converging in E(X,E1) towards a. Then bj := ψεaj ∈ E(X)⊗ E1

and supp(bj) ⊂ Kε. It is easily verified that supp(bjq u) ⊂ Kε ∩ supp(u) =: Mε

and thatD′

Mε(X,E0) :=

v ∈ D′(X,E0) ; supp(v) ⊂ Mε

is a closed linear subspace of D′(X, E0). Now we deduce from Theorem 1.6.4 thatbj

q u → a q u in D′(X, E0), hence in D′Mε

(X, E0). Consequently, supp(a q u) ⊂ Mε

for each ε > 0, which implies the assertion. ¥

(e) Let E3, E4, and E5 be further Banach spaces and suppose that there aremultiplications

E1 × E2 E2 × E3

E0 × E3 E1 × E4

E5

? ?

HHHj©©©¼

all denoted by q , that are associative, that is,

(e1q e2) q e3 = e1

q (e2q e3) , ej ∈ Ej , j = 1, 2, 3 . (1.6.18)

Then point-wise multiplication is associative as well, when defined, that is,

(u1q u2) q u3 = u1

q (u2q u3) , uj ∈ Fj(X, Ej) , j = 1, 2, 3 , (1.6.19)

where (F1, F2, F3) =(E , E ,D′), (OM ,OM ,S ′).

Proof It follows from Theorem 1.6.4 that both sides of (1.6.19) define separatelycontinuous trilinear maps

3∏

j=1

Fj(X,Ej) → F3(X,E5)

that coincide on the dense linear subspace∏3

j=1 D(X) ⊗ Ej , hence everywhere. ¥

(f ) Suppose that F ∈ D, E ,S,OM ,D′, E ′,S ′. Then the map

E1 × F(X, E2) → F(X,E0) , (e1, u) 7→ e1q u := (1⊗ e1) q u (1.6.20)

is bilinear and continuous, and

∂α(e1q u) = e1

q ∂αu , α ∈ Nn , e1 ∈ E1 , u ∈ F(X,E2) . (1.6.21)

Proof Since 1⊗ e1 ∈ OM (Rn, E1) → E(Rn, E1), it follows from Theorem 1.6.4and Remark (a) that (1.6.20) is a well-defined separately continuous bilinear map.

412 Appendix

Moreover, given u = v ⊗ e2 ∈ F(X)⊗ E2 and ϕ ∈ F′(X),

(e1q u)(ϕ) =

[(1⊗ e1) q (v ⊗ e2)

](ϕ) = 〈v, ϕ〉F′e1

q e2 = e1q (〈v, ϕ〉F′e2

)

= e1q u(ϕ) .

Thus the density of F(X)⊗ E2 in F(X,E2) implies

(e1q u)(ϕ) = e1

q u(ϕ) , e1 ∈ E1 , u ∈ F(X, E2) , ϕ ∈ F′(X) . (1.6.22)

Consequently, given a bounded subset B of F′(X),

supϕ∈B

|(e1q u)(ϕ)|E0 ≤ |e1|E1 sup

ϕ∈B|u(ϕ)|E2 , e1 ∈ E1 , u ∈ F(X, E2) .

This proves the continuity of (1.6.20), thanks to Theorem 1.4.9. Lastly, (1.6.21) isa consequence of (c). ¥

(g) Let F ∈ D, E ,S,OM ,D′, E ′,S ′ and consider the multiplication

L(E, F ) × E → F , (T, e) 7→ T q e := Te . (1.6.23)

Then it follows from (f) that (1.6.23) induces a continuous bilinear map

L(E,F )× F(X,E) → F(X,F ) , (T, u) 7→ T q u := Tu .

In particular, each T ∈ L(E, F ) induces a continuous linear map via (1.6.23),

(u 7→ T q u) ∈ L(F(X,E), F(X, F )

). (1.6.24)

We denote it by T as well and call it the linear map induced by T ∈ L(E, F ) via point-

wise multiplication. Moreover, (1.6.21) implies ∂α T = T ∂α for α ∈ Nn, that is,the diagram

F(X, E) F(X, F )

F(X, E) F(X, F )T

T -

-

∂α ∂α

? ?

is commutative for α ∈ Nn. If T ∈ L(E, F ) is injective, then (1.6.24) is injective aswell. Consequently,

i : E → F implies i : F(X, E) → F(X, F ) .

Proof We have to prove only the injectivity assertion. Thus suppose that Tu = 0for some u ∈ F(X,E) and that T ∈ L(E, F ) is injective. Then we obtain from(1.6.22) that T

(u(ϕ)

)= 0 for each ϕ ∈ F′(X). Hence u(ϕ) = 0 for each ϕ ∈ F′(X),

that is, u = 0. ¥


(h) Suppose that E1 = E2 =: E and multiplication (1.5.20) is symmetric, that is,e1

q e2 = e2q e1 for ej ∈ E. Then a q u = u q a for a ∈ E(X,E) and u ∈ D′(X, E).

Proof This is an immediate consequence of (b), the density of D(X, E) ×D(X, E)in E(X,E)×D′(X, E), and the separate continuity of point-wise multiplication. ¥

1.6.6 Corollary Suppose that (E, q) is a [commutative] Banach algebra and thatF ∈ D, E ,S,OM. Then F(X, E) is a locally convex [commutative] algebra withrespect to point-wise multiplication, a multiplication algebra. Multiplication is con-tinuous if F ∈ D, E ,S, and hypocontinuous if F = OM . If E possesses a unit, e0,then E(X, E) and OM (Rn, E) possess a unit as well, namely 1⊗ e0.

Proof This follows from Theorem 1.6.4, Remarks 1.6.5(e) and (h), appropriatecontinuous injections given in Theorem 1.3.6, and the fact that hypocontinuousmaps are continuous on barreled spaces. ¥

1.7 Scalar Products and Duality Pairings

In this subsection we show, in particular, that, given F ∈ D, E ,S, there exists aunique hypocontinuous bilinear map

F′(X, E′) × F(X, E) → K (1.7.1)

that extends in a natural way the duality pairing F′(X) × F(X) → K. In fact, weconsider more general situations that will be needed in the remaining subsections.

1.7.1 Lemma Let F ∈ D, E ,S,D′, E ′,S ′. Then the bilinear map

F(X, E) × F′(X) → E , (u, ϕ) 7→ u(ϕ) (1.7.2)

is hypocontinuous.

Proof Since F′′ = F, it follows from Theorem 1.4.9 that F(X, E) = L(F′(X), E

).

Hence the map (1.7.2) is well-defined and continuous in u, uniformly on boundedsubsets of F′(X). Thanks to the fact that F′(X) is barreled, bounded subsetsof L(

F′(X), E)

are equicontinuous by the uniform boundedness principle. Thus(1.7.2) is continuous in ϕ, uniformly with respect to u varying in bounded subsetsof F(X, E). ¥

It is now easy to prove the following general existence theorem that willimply, in particular, the desired extension (1.7.1).

1.7.2 Theorem Suppose that F ∈ D, E ,S,D′, E ′,S ′. Then there exists a uniquehypocontinuous bilinear map

F′(X,E1) × F(X, E2) → E0 , (u′, u) 7→ 〈u′ q u〉F , (1.7.3)

414 Appendix

the scalar product induced by the multiplication (1.5.20), such that⟨(ϕ⊗ e1) q (ψ ⊗ e2)

⟩F

= 〈ϕ, ψ〉D(e1q e2) (1.7.4)

for ϕ,ψ ∈ D(X) and ej ∈ Ej, j = 1, 2.

Proof Put F0 := K, F1 := F′(X), and F2 := F(X). Then the Fj are reflexive, com-plete, nuclear, and conuclear LCSs. Consequently, Theorem 1.4.9 guarantees thatF2

∼⊗ E2 = F(X, E2), and it is trivially true that F0 ⊗ E0 = F0∼⊗ E0 = E0. Define

the bilinear maps (1.5.14) and (1.5.15) by f1 ¯ f2 := 〈f1, f2〉F2 and f1 v := v(f1),respectively. Then Lemma 1.7.1 shows that (1.5.15) is hypocontinuous. Since it isobvious that condition (1.5.16) is satisfied, Theorem 1.5.3 gives the assertion. ¥

1.7.3 Remark Suppose that u ∈ L1,loc(X,E1) and v ∈ D(X,E2). Then

〈u q v〉D =∫

X

u(x) q v(x) dx . (1.7.5)

Proof Since L1,loc(X, E1)d

→ D′(X, E1), Theorem 1.7.2 implies that the map

L1,loc(X, E1) ×D(X,E2) → E0 , (u, v) 7→ 〈u q v〉D (1.7.6)

is bilinear and separately continuous. From(1.6.15) and Theorem1.3.6 we infer that(D(X) ⊗ E1

)× (D(X)⊗ E2

) d⊂ L1,loc(X, E1) ×D(X, E2) . (1.7.7)

It is obvious that (1.7.5) is true for u ∈ D(X)⊗ E1 and v ∈ D(X)⊗ E2. Hence theassertion follows from (1.7.7) and the separate continuity of the map (1.7.6). ¥

Parseval’s Formula

As a first application of Theorem 1.7.2 we obtain a natural extension of Parseval’sformula for the Fourier transform.

1.7.4 Proposition Parseval’s formula is valid for vector-valued distributions, thatis,

〈u q ϕ〉S = (2π)−n⟨u q

ϕ⟩S , u ∈ S ′(Rn, E1) , ϕ ∈ S(Rn, E2) . (1.7.8)

Proof Thanks to Theorem 1.7.2 and the fact that the Fourier transform and re-flection are toplinear automorphisms, it follows that both sides of (1.7.8) are sep-arately continuous bilinear maps from S ′(Rn, E1) × S(Rn, E2) into E0. By (1.7.4)and the well-known Parseval formula for scalar distributions the two expressionson either side of (1.7.8) coincide on the dense linear subspace

(S ′(Rn) ⊗ E1

)× (S(Rn) ⊗ E2

).

Hence they are equal. ¥


Duality Pairings

By specializing the above results to the case where (1.5.20) is a duality pairing weobtain the desired map (1.7.1).

1.7.5 Theorem Let E be a Banach space and let F ∈ D, E ,S. Then there existsa unique hypocontinuous bilinear map

F′(X, E′)× F(X,E) → K , (u′, u) 7→ 〈u′, u〉F(X,E) ,

the duality pairing between F′(X, E′) and F(X,E), such that

〈u′, u〉F(X,E) =∫

X

⟨u′(x), u(x)

⟩E

dx (1.7.9)

for u′ ∈ D(X,E′) and u ∈ D(X, E).

Proof Let E1 := E′, E2 := E, and E0 := K, and put e1q e2 := 〈e1, e2〉E . Then

the assertion follows from Theorem 1.7.2 and Remark 1.7.3. ¥

1.7.6 Remarks (a) It is also true that

〈u′, u〉D(X,E) =∫

X

⟨u′(x), u(x)

⟩E

dx , (u′, u) ∈ L1,loc(X,E′) ×D(X, E) ,

and

〈u′, u〉S(Rn,E) =∫

Rn

⟨u′(x), u(x)

⟩E

dx , (u′, u) ∈ Lp(Rn, E′) × S(Rn, E) ,

where 1 ≤ p < ∞.

Proof The first part of the assertion is a consequence of Remark 1.7.3. The secondone follows from

D(Rn, E′)d

→ Lp(Rn, E′)d

→ S ′(Rn, E′) , 1 ≤ p < ∞ ,

from D(Rn, E)d

→ S(Rn, E), and from the separate continuity of the bilinear map

(u′, u) 7→∫

Rn

⟨u′(x), u(x)

⟩E

dx

on Lp(Rn, E′) × S(Rn, E). ¥

(b) Theorem 1.7.5 contains a more precise description of the duality pairing ofCorollary 1.4.10. ¥

416 Appendix

1.8 Tensor Products of Distributions and Kernel Theorems

It is the purpose of this subsection to define tensor products of arbitrary vector-valued distributions. These results are of interest for their own sake. In addition,they will be the basis for an important extension of convolutions of vector-valueddistributions given in the next subsection. Recall convention (1.1.11).

Approximation by Tensor Products

1.8.1 Theorem Suppose that F ∈ D, E ,S,D′, E ′,S ′ and Xj ⊂ Rnj , j = 1, 2, areopen. Then D(X1)⊗D(X2, E) is sequentially dense in F(X1 × X2, E).

Proof (1) Let ϕ ∈ D(X1 × X2, E) be given and put Kj := prj

(supp(ϕ)

), where

prj : Rn1 × Rn2 → Rnj , j = 1, 2 ,

are the canonical projections. Then Kj b Xj and there exist ψj ∈ D(Xj) withψj |Kj = 1. It follows that ψ1 ⊗ ψ2 ∈ D(X1 × X2) and ψ1 ⊗ ψ2 | supp(ϕ) = 1.Thanks to Lemma 1.3.7 there exists a sequence (pk) of polynomials on Rn1 × Rn2

with coefficients in E, converging in E(X1 × X2, E) towards ϕ. Then the sequence((ψ1 ⊗ ψ2)pk

)k∈N lies in D(X1) ⊗D(X2, E) and converges in E(X1 ×X2, E) to-

wards ϕ. Since the supports of (ψ1 ⊗ ψ2)pk are contained in a fixed compact set,the convergence takes place in D(X1 × X2, E). This proves the claim if F = D.

(2) Let F ∈ E ,S. Then we know from Remark 1.3.4 that D(X1 × X2, E) issequentially dense in F(X1 × X2, E). Now the assertion follows from step (1).

(3) Suppose F ∈ D′, E ′,S ′. Proposition 1.3.3 guarantees that D(X1 ×X2, E)is sequentially dense in F(X1 × X2, E). Thus applying step (1) gives the state-ment. ¥

1.8.2 Corollary If F ∈ D, E ,S,D′, E ′,S ′, then F(X1)⊗ F(X2, E) is sequentiallydense in F(X1 × X2, E).

Proof This is now obvious from

D(X1)⊗D(X2, E) ⊂ F(X1) ⊗ F(X2, E) ⊂ F(X1 × X2, E)

and from observation (1.3.16). ¥

Tensor Products of Distributions

As a preparation for defining tensor products of distributions we establish thefollowing technical result.

1.8.3 Lemma Suppose that F ∈ D, E. If ϕ ∈ F(X1 × X2) and u1 ∈ F′(X1, E),then [

x2 7→ u1

(ϕ(·, x2)

)] ∈ F(X2, E)


and∂αu1

(ϕ(·, x2)

)= u1

(∂α2 ϕ(·, x2)

), α ∈ Nn2 .

Proof If ϕ ∈ D(X1 × X2), let Kj := prj

(supp(ϕ)

). Then supp(ϕ) ⊂ K1 × K2 and

Kj b Xj . From this it follows easily that, in each case,[x2 7→ ϕ(·, x2)

] ∈ F(X2, F(X1)

).

Now, using u1 ∈ F′(X1, E) = L(F(X1), E

), it is not difficult to derive the asserted

properties. ¥

Suppose that F ∈ D, E and let ϕ ∈ F(X1 × X2) and uj ∈ F′(Xj , Ej) begiven. Then we put

u1

(ϕ(x1, x2)

):=

[x2 7→ u1

(ϕ(·, x2)

)](1.8.1)

and

u2

(ϕ(x1, x2)

):=

[x1 7→ u2

(ϕ(x1, ·)

)]. (1.8.2)

In other words, the symbolic notation uj

(ϕ(x1, x2)

)means that the distribution uj

acts on the entry following it as a function of the variable xj , the other variablebeing a free parameter.

It follows from Lemma 1.8.3 and Theorem 1.7.2 that, given uj ∈ F′(Xj , Ej),j = 1, 2, and ϕ ∈ F(X1 × X2),

⟨u1

q u2

(ϕ(x1, x2)

)⟩F

,⟨u2

q u1

(ϕ(x1, x2)

)⟩F

(1.8.3)

are well-defined elements of E0.

Now we are prepared for the proof of the main result of this subsection.

1.8.4 Theorem Suppose that F ∈ D, E and Xj are open in Rnj for j = 1, 2.Given uj ∈ F′(Xj , Ej), j = 1, 2, there exists a unique distribution u1 ⊗ q u2 be-longing to F′(X1 × X2, E0), the tensor product of u1 and u2 with respect to the mul-

tiplication (1.5.20), satisfying

u1 ⊗ q u2(ϕ1 ⊗ ϕ2) = u1(ϕ1) q u2(ϕ2) , ϕj ∈ F(Xj) , (1.8.4)

such that the map

F′(X1, E1) × F′(X2, E2) → F′(X1 × X2, E0) , (u1, u2) 7→ u1 ⊗ q u2 (1.8.5)

is bilinear and hypocontinuous. It can be evaluated by Fubini’s rule:

u1 ⊗ q u2(ϕ) =⟨u1

q u2

(ϕ(x1, x2)

)⟩F

=⟨u1

(ϕ(x1, x2)

) q u2

⟩F′ (1.8.6)

for ϕ ∈ F(X1 × X2).

418 Appendix

Proof Put X0 := X1 ×X2 and Fj := F′(Xj), j = 0, 1, 2. Then the Fj satisfythe hypotheses of Theorem 1.5.3 and Fj

∼⊗ Ej = F′(Xj , Ej), j = 0, 1, 2, by Theo-rem 1.4.9. Define the maps (1.5.14) and (1.5.15) by

ϕ 7→ f1 ¯ f2(ϕ) :=⟨f1, f2

(ϕ(x1, x2)

)⟩F

andϕ 7→ f1 v(ϕ) := v

(f1

(ϕ(x1, x2)

))

for ϕ ∈ F(X0), respectively. It follows from Lemma 1.8.3 that they are well-defined.From that lemma we also infer that

Tf1 :=[ϕ 7→ f1

(ϕ(x1, ·)

)] ∈ L(F(X0), F(X2)

). (1.8.7)

Thus, since f1 v = v Tf1 , we obtain from Lemma 1.1.1 that the map (1.5.15) ishypocontinuous. Given fj ∈ Fj , ϕj ∈ F(Xj), j = 1, 2, and e2 ∈ E2, it is clear that

[f1 (f2 ⊗ e2)

](ϕ1 ⊗ ϕ2) = 〈f1, ϕ1〉F〈f2, ϕ2〉Fe2 =

[(f1 ¯ f2)⊗ e2

](ϕ1 ⊗ ϕ2) .

Since F(X1) ⊗ F(X2) is dense in F(X0) by Corollary 1.8.2, we infer that condition(1.5.16) is satisfied. Now the first part of the assertion follows from Theorem 1.5.3.

From Lemma 1.8.3 we deduce that[ϕ 7→ uj

(ϕ(x1, x2)

)] ∈ L(F(X0), F(X3−j , E3−j)

), uj ∈ F′(Xj , Ej) , j = 1, 2 .

Hence, given ϕ ∈ F(X0), Theorem 1.7.2 guarantees the existence of hypocontinuousscalar products

⟨u1

q u2

(ϕ(x1, x2)

)⟩F

and⟨u1

(ϕ(x1, x2)

) q u2

⟩F′ satisfying

⟨(f1 ⊗ e1) q (f2 ⊗ e2)

(ϕ(x1, x2)

)⟩F

=⟨f1, f2

(ϕ(x1, x2)

)⟩F(e1

q e2)

and⟨(f1 ⊗ e1)

(ϕ(x1, x2)

) q (f2 ⊗ e2)⟩

F′ =⟨f1

(ϕ(x1, x2)

), f2

⟩F′(e1

q e2)

for uj := fj ⊗ ej ∈ F′(Xj)⊗ Ej , j = 1, 2, respectively. Choosing ϕ := ϕ1 ⊗ ϕ2 inF(X1) ⊗ F(X2), we see that

⟨u1

q u2

(ϕ(x1, x2)

)⟩F

=⟨u1

(ϕ(x1, x2)

) q u2

⟩F′

= 〈f1, ϕ1〉F〈f2, ϕ2〉F(e1q e2)

=(〈f1, ϕ1〉Fe1

) q (〈f2, ϕ2〉Fe2

)

= u1(ϕ1) q u2(ϕ2) = u1 ⊗ q u2(ϕ1 ⊗ ϕ2) .

(1.8.8)

From u1 ⊗ q u2 ∈ F(X0, E0), from (1.8.7), and from Theorem 1.7.2 we infer thateach one of the trilinear maps

F(X0)× F′(X1, E1)× F′(X2, E2) → E0


sending (ϕ, u1, u2)

to u1 ⊗ q u2(ϕ) , to⟨u1

q u2

(ϕ(x1, x2)

)⟩F

, or to⟨u1

(ϕ(x1, x2)

) q u2

⟩F′ ,

respectively, is separately continuous. By (1.8.8) they are all equal on the lin-ear subspace

(F(X1) ⊗ F(X2)

)× (F′(X1)⊗ E2

)× (F′(X2) ⊗ E2

),

which is dense by Theorem 1.3.6 and Corollary 1.8.2. Hence they are all equal,that is, Fubini’s rule is valid. ¥

Basic Properties

1.8.5 Remarks (a) If uj ∈ D′(Xj , Ej), j = 1, 2, then

supp(u1 ⊗ q u2) ⊂ supp(u1)× supp(u2)

with equality, provided e1q e2 6= 0 whenever ej ∈

qEj .

Proof Let Aj := supp(uj). If supp(ϕ) ⊂ Rn × Ac2, then

u2

(ϕ(x1, x2)

)= 0 , x1 ∈ X1 ,

hence u1 ⊗ q u2(ϕ) = 0 by Fubini’s rule. Similarly, we see that u1 ⊗ q u2(ϕ) = 0if supp(ϕ) ⊂ Ac

1 × Rn2 . Consequently, supp(u1 ⊗ q u2) ⊂ A1 × A2. On the otherhand, given (x1, x2) ∈ A1 × A2 and any product neighborhood U1 × U2 of (x1, x2)in X1 × X2, there exist ϕj ∈ D(Uj) such that uj(ϕj) 6= 0. Hence

u1 ⊗ q u2(ϕ1 ⊗ ϕ2) = u1(ϕ1) q u2(ϕ2) 6= 0

if e1q e2 6= 0 for ej ∈

qEj . ¥

(b) Given uj ∈ D′(Xj , Ej),

∂α11 ∂α2

2 (u1 ⊗ q u2) = (∂α11 u1)⊗ q (∂α2

2 u2) , αj ∈ Nnj . (1.8.9)

Proof It follows from Theorem 1.8.4 and the continuity properties of the deriv-ative that both sides of (1.8.9) define separately continuous bilinear maps fromD′(X1, E1)×D′(X2, E2) into D′(X1 × X2, E0). From (1.8.4) it is obvious thatthey coincide on the dense linear subspace

(D′(X1) ⊗ E1

)× (D′(X2)⊗ E2

), hence

everywhere. ¥

(c) Let the associativity hypotheses of Remark 1.6.5(e) be satisfied and let X3 bea nonempty open subset of Rn3 . Then

(u1 ⊗ q u2) ⊗ q u3 = u1 ⊗ q (u2 ⊗ q u3) (1.8.10)

for uj ∈ D′(Xj , Ej) and j = 1, 2, 3.

420 Appendix

Proof By Theorem 1.8.4 both sides of (1.8.10) define separately continuous tri-linear maps

3∏

j=1

D′(Xj , Ej) → D′(X1 × X2 × X3, E5) .

By (1.8.4), Remark 1.4.1(c), and Example 1.4.2(c) they coincide on the denselinear subspace

∏3j=1 D′(Xj) ⊗ Ej , hence everywhere. ¥

(d) If uj ∈ L1,loc(Xj , Ej), then u1 ⊗ q u2 ∈ L1,loc(X1 × X2, E0) and

u1 ⊗ q u2(x1, x2) = u1(x1) ⊗ q u2(x2) for a.a. (x1, x2) ∈ X1 × X2 .

Proof This follows easily from (1.8.6) and the classical Fubini theorem. ¥

(e)(Distributivity) Suppose that

E1 × E2 E3 × E4 E1 × E3 E2 × E4

E0 × E5 E6 × E7

E8

BBBN

£££°

BBBN

£££°

HHHj©©©¼

are multiplications between Banach spaces, all denoted by q . Also suppose that

(e1q e2) q (e3

q e4) = (e1q e3) q (e2

q e4) , ej ∈ Ej , j = 1, . . . , 4 ,

and aj ∈ E(Xj , Ej) and uj ∈ D′(Xj , Ej+2), j = 1, 2. Then

(a1 ⊗ q a2) q (u1 ⊗ q u2) = (a1q u1) ⊗ q (a2

q u2) . (1.8.11)

Proof First we note that, thanks to (d), a1 ⊗ q a2 ∈ E(X1 × X2, E0). Hence The-orems 1.8.4 and 1.6.4 guarantee that both sides of (1.8.11) are well-defined quadri-linear separately continuous maps from

E(X1, E1)× E(X2, E2) ×D′(X1, E3) ×D′(X2, E4)

into D′(X1 × X2, E8). It follows from (1.8.4) and (d) that they coincide on thedense linear subspace

(E(X1) ⊗ E1

)× (E(X2) ⊗ E2

)× (D′(X1)⊗ E3

)× (D′(X2) ⊗ E4

),

hence everywhere. ¥


(f ) Let Fj , j = 0, 1, 2, be Banach spaces and suppose that

F1 × F2 → F0 , (f1, f2) 7→ f1q f2

is a multiplication. Also suppose that Tj ∈ L(Ej , Fj), j = 0, 1, 2, satisfy

(T1e1) q (T2e2) = T0(e1q e2) , ej ∈ Ej , j = 1, 2 ,

that is, the diagram

E1 × E2 E0

F1 × F2 F0

q

q

T1 T2 T0

-

-? ? ?

is commutative. Then, given F ∈ D, E, the diagram

F′(X1, E1) × F′(X2, E2) F′(X1 × X2, E0)

F′(X1, F1) × F′(X2, F2) F′(X1 × X2, F0)⊗ q

⊗ q

T1 T2 T0

-

-? ? ?

is commutative as well.

Proof We have to show that

(T1u1) ⊗ q (T2u2) = T0(u1 ⊗ q u2) , uj ∈ F′(Xj , Ej) , j = 1, 2 .

From Remark 1.6.5(g) and Theorem 1.8.4 we infer that both sides of this equal-ity define separately continuous bilinear maps that coincide on the dense linearsubspace

(F′(X1) ⊗ E1

)× (F′(X2)⊗ E2

). Thus they are equal. ¥

Examples

Finally, we include a few simple examples demonstrating the usefulness of ten-sor products.

1.8.6 Examples (a) The Dirac distribution δ ∈ E ′(Rn) on Rn is the n-fold ten-sor product of the ‘one-dimensional’ Dirac distribution δ ∈ E ′(R). Thus, in sym-

bolic notation,

δ(x) = δ(x1) ⊗ δ(x2)⊗ · · · ⊗ δ(xn) , x = (x1, . . . , xn) ∈ Rn ,

where δ(y) := δ ∈ E ′(Rm) for y ∈ Rm and ⊗ is the tensor product induced by thestandard multiplication in K.

422 Appendix

(b) Suppose that n = n1 + n2 with 0 ≤ n1 ≤ n− 1. Let X1 be open in Rn1 andlet X2 be a nonempty open cube in Rn2 . Also suppose that u ∈ D′(X1 × X2, E)satisfies ∂ju = 0, n1 + 1 ≤ j ≤ n. Then u = u1 ⊗ 1X2 for some u1 ∈ D′(X1, E),where D′(0, E)

:= E. Lastly, u ∈ D′(X, E) is constant if ∂ju = 0 for 1 ≤ j ≤ nand X is connected.

Proof (1) First we consider the case n1 = 0 and n = 1. Thus X is a nonemptyopen interval in R and u ∈ D′(X, E) satisfies ∂u = 0. Fix any ϕ0 ∈ D(X) with〈1, ϕ0〉 = 1. Then, given ϕ ∈ D(X), put ψ := ϕ− 〈1, ϕ〉ϕ0 ∈ D(X). Let

χ(x) :=∫ x

−∞ψ(t) dt , x ∈ R ,

and observe that χ ∈ D(X). Consequently,

0 = (∂u)(χ) = −u(∂χ) = −u(ψ) = 〈1, ϕ〉u(ϕ0) − u(ϕ) ,

which shows that u = u(ϕ0)1X , that is, u is constant.(2) If Yj are open subsets of Rmj for j = 0, 1, 2 and Y0 = Y1 × Y2, then

1Y0 =1Y1 ⊗1Y2 . Using this fact and induction, it suffices to consider the case n2 = 1.Given ϕ1 ∈ D(X1), it is easily verified that

uϕ1 :=(ϕ2 7→ u(ϕ1 ⊗ ϕ2)

) ∈ D′(X2, E) . (1.8.12)

By assumption, ∂uϕ1 = 0. Hence (i) implies the existence of e(ϕ1) ∈ E such thatuϕ1 = e(ϕ1)1X2 . Consequently,

u(ϕ1 ⊗ ϕ2) = e(ϕ1)〈1X2 , ϕ2〉 , ϕ2 ∈ D(X2) . (1.8.13)

Fix ϕ2 ∈ D(X2) with 〈1X2 , ϕ2〉 = 1 and note that

u1 :=(ϕ1 7→ u(ϕ1 ⊗ ϕ2)

) ∈ D′(X1, E) .

By (1.8.13), we see that u1(ϕ1) = e(ϕ1) so that

u(ϕ1 ⊗ ϕ2) = u1(ϕ1)〈1X2 , ϕ2〉 , ϕj ∈ D(Xj) .

Now the first assertion follows from Theorem 1.8.4.(3) If u ∈ D′(X,E) satisfies ∂ju = 0 for 1 ≤ j ≤ n, then we infer from (ii)

that each point x ∈ X has a neighborhood Ux in X such that u |Ux = ex1Ux forsome ex ∈ E. If Ux ∩ Uy 6= ∅, then it follows from

ex〈1, ϕ〉 = u(ϕ) = ey〈1, ϕ〉 , ϕ ∈ D(Ux ∩ Uy) ,

that ex = ey. Let x and y be any two points in X. Since X is connected there isa continuous path γ : [0, 1] → X connecting x and y. Since γ

([0, 1]) ⊂ ⋃

x∈X Ux

and γ([0, 1]

)is compact and connected, we can find a finite chain U0, U1, . . . , Um


of these neighborhoods covering γ([0, 1]

)and satisfying Uj−1 ∩ Uj 6= ∅. Then the

above considerations imply that ex = ey for x, y ∈ X, so that u = e ⊗ 1X for somee ∈ E. ¥

(c) Let Θ := χR+ ∈ L1,loc(R) be the Heaviside function. Then

χP := χR+ ⊗ · · · ⊗ χR+︸︷︷︸n

is the characteristic function of the natural positive cone P := (R+)n of Rn. Since∂Θ = δ, we deduce from Remark 1.8.5(b) and from (a) that

∂1 · · · ∂nχP = ∂χR+ ⊗ · · · ⊗ ∂χR+ = δ(x1) ⊗ · · · ⊗ δ(xn) = δ(x) .

(d) Suppose that n = n1 + n2 with 0 ≤ n1 ≤ n− 1, and that u ∈ E ′(Rn, E) has itssupport in Rn1 × 0. Then there exist m ∈ N and uα ∈ E ′(Rn1 , E) for α ∈ Nn2

with |α| ≤ m such thatu =

∑

|α|≤m

uα ⊗ ∂αδ(x2) .

This representation is unique.

Proof (1) For clarity, we write ∂F for the Frechet derivative. Suppose ϕ ∈ E(Rd)for some d ∈

qN and let m ∈ N. By Taylor’s theorem,

ϕ(x) =m∑

k=0

1k!

∂kF ϕ(0) [x]k + ψ(x) [x]m

=∑

|α|≤m

∂αϕ(0)α!

xα + ψ(x) [x]m ,

(1.8.14)

where ψ(x) is a symmetric m-linear form, depending smoothly on x ∈ Rd, suchthat

limx→0

ψ(x) = 0 , (1.8.15)

and [x]k = [x, . . . , x] with k entries (e.g., [AmE06, Section VII.5]). Consequently,by Leibniz’ rule, using the obvious point-wise multiplication,

∣∣∂jF

(ψ(·) [·]m)

(x)∣∣ ≤ c

j∑

i=0

|∂j−iF ψ(x)| |x|m−i

≤ c

j∑

i=0

εm−i max|x|≤ε

|∂j−iF ψ(x)|

(1.8.16)

for |x| ≤ ε and 0 ≤ j ≤ m.

424 Appendix

(2) Assume u ∈ E ′(Rn, E) with

supp(u) ⊂ Rn1 × 0 . (1.8.17)

There exist m ∈ N and K b Rn such that

|u(ϕ)| ≤ cpm,K(ϕ) , ϕ ∈ E(Rn) . (1.8.18)

Let ϕε ; ε > 0 be a mollifier on Rn2 . Then, putting ` := n2 and

ψε := ϕε/4 ∗ χ(ε/2)B` , ε > 0 ,

it is easily verified that ψε ∈ D(Rn2) satisfies ψε(x) = 1 for x ∈ (ε/4)B`, has itssupport in εB`, and

‖∂αψε‖∞ ≤ c(α, n2)ε−|α| , α ∈ Nn2 , ε > 0 . (1.8.19)

Thus, given ϕj ∈ E(Rnj ), j = 1, 2, we see that ϕ1 ⊗ (1 − ψε)ϕ2 ∈ E(Rn1 × Rn2)has its support in Rn1 × (Rn2)

q. Consequently,

u(ϕ1 ⊗ ϕ2) = u(ϕ1 ⊗ ψεϕ2) + u(ϕ1 ⊗ (1 − ψε)ϕ2

)= u(ϕ1 ⊗ ψεϕ2) , ε > 0 ,

thanks to (1.8.17). Hence we infer from (1.8.18)

|u(ϕ1 ⊗ ϕ2)| ≤ cpm,K(ϕ1 ⊗ ψεϕ2)

≤ c max|α1|+|α2|≤m

pK1(∂α1ϕ1)pεB`

(∂α2(ψεϕ2)

), (1.8.20)

where K1 is the canonical projection of K into Rn1 .Now suppose that ∂αϕ2(0) = 0 for |α| ≤ m. Using (1.8.19) and Leibniz’ rule,

we deduce from (1.8.16) that

max|x2|≤ε

∣∣∂α2(ψεϕ2)(x2)∣∣ ≤ c

∑

β≤α2

ε|β|−|α2|∑

γ≤β

εm−|γ| max|x2|≤ε

∣∣∂|β|−|γ|F ψ(x2)∣∣

≤ c∑

β≤α2

∑

γ≤β

ε|β−γ|+|α2| max|x2|≤ε

∣∣∂|β|−|γ|F ψ(x2)∣∣

for |α2| ≤ m− |α1|. Now we infer from (1.8.15) that

limε→0

pεB`

(∂α2(ψεϕ2)

)= 0 , |α2| ≤ m− |α1| .

Consequently, by (1.8.20),

u(ϕ1 ⊗ ϕ2) = 0 for ϕj ∈ E(Rnj ) with ∂αϕ2(0) = 0 , |α| ≤ m . (1.8.21)

Given any ϕ2 ∈ E(Rn2), we obtain from (1.8.14) and (1.8.15) that

ϕ2(x2) =∑

|α|≤m

1α!

∂αϕ2(0)xα2 + χ(x2) , x2 ∈ Rn2 ,


where χ ∈ E(Rn2) satisfies ∂αχ(0) = 0 for |α| ≤ m. Thus (1.8.21) implies

u(ϕ1 ⊗ ϕ2) = u(ϕ1 ⊗

∑

|α|≤m

1α!

∂αϕ2(0)xα2

)=

∑

|α|≤m

(−1)|α|uα(ϕ1)∂αϕ2(0)

for ϕj ∈ E(Rnj ), where

uα(ϕ1) :=(−1)|α|

α!u(ϕ1 ⊗ xα

2 ) , |α| ≤ m .

Now Theorem 1.8.4 gives

u =∑

|α|≤m

uα ⊗ ∂αδ(x2)

with uα ∈ E ′(Rn1 , E).(3) Finally, suppose that there are m ∈ N and uα ∈ E ′(Rn1 , E) such that

u =∑

|α|≤m

uα ⊗ ∂αδ(x2) .

By replacing m and m by maxm, m, we can assume that m = m. Then

0 =∑

|α|≤m

(uα − uα) ⊗ ∂αδ(x2)(ϕ1 ⊗ xβ2 ) = (−1)|β|β! (uβ − uβ)(ϕ1)

for |β| ≤ m and ϕ1 ∈ E(Rn1). This proves the asserted uniqueness. ¥

It should be noted that the proofs of the preceding examples are adaptionsof corresponding results for scalar distributions (e.g., [Hor83]).

Topological Tensor Products of Distributions

Now we consider the special case of tensor products of a scalar and a vector-valueddistribution.

1.8.7 Theorem Suppose that F ∈ E ,S. Then

F(X1)∼⊗ F(X2, E) = F(X1 × X2, E) .

Proof Thanks to Corollary 1.8.2 it suffices to show that F(X1 × X2, E) induceson F(X1)⊗ F(X2, E) the projective topology.

Given a continuous seminorm p on F(X1 × X2, E), belonging to the families(1.1.3) and (1.1.7) respectively, it is easily verified that there exist continuous

426 Appendix

seminorms r on F(X1) and s on F(X2, E), respectively, such that

p(ϕ⊗ ψ) ≤ r(ϕ)s(ψ) , ϕ ∈ F(X1) , ψ ∈ F(X2, E) .

For a representation∑

ϕj ⊗ ψj of z ∈ F(X1) ⊗ F(X2, E) it follows that

p(z) ≤∑

p(ϕj ⊗ ψj) ≤∑

r(ϕj)s(ψj) .

This implies that p(z) ≤ r ⊗π s(z) for z ∈ F(X1) ⊗ F(X2, E). Thus F(X1 × X2, E)induces on F(X1) ⊗ F(X2, E) a topology weaker than the projective topology.

Conversely, let r and s be continuous seminorms on F(X1) and F(X2, E), re-spectively, belonging to the respective families (1.1.3) and (1.1.7), and let

∑ϕj ⊗ψj

be a representation of z ∈ F(X1) ⊗ F(X2, E). Observe that∑

〈ϕ′, ϕj〉〈ψ′, ψj〉 =⟨ϕ′,

⟨ψ′,

∑ϕj ⊗ ψj

⟩⟩, ϕ′ ∈ F′(X1) , ψ′ ∈ F′(X2, E) .

Hence, cf. (1.4.7),

r ⊗ε s(z) = supϕ′∈B0

r

ψ′∈B0s

∣∣∣∑

〈ϕ′, ϕj〉〈ψ′, ψj〉∣∣∣ = r

(s(∑

ϕj ⊗ ψj

))

by the bipolar theorem, where Br is the open r-unit-ball in F(X1) and Bs is theopen s-unit-ball in F(X2, E). From this we infer the existence of a continuous semi-norm p on F(X1 × X2, E) such that r ⊗ε s(z) ≤ p(z) for z ∈ F(X1)⊗ F(X2, E).Thus F(X1 × X2, E) induces on F(X1) ⊗ F(X2, E) a topology that is stronger thanthe injective tensor product topology. Since F(X1) is nuclear, the latter coincideswith the projective topology (e.g. [Tre67, Theorem 50.1(f)]). ¥

1.8.8 Corollary If F ∈ E ,S, then

F′(X1)∼⊗ F′(X2, E

′) = F(X1 × X2, E)′ .

Proof From Corollary 1.4.10 and Theorem 1.8.7 we know that

F′(X1 × X2, E′) = F(X1 × X2, E)′ =

(F1(X1)

∼⊗ F(X2, E))′

.

Hence the assertion follows from (1.4.16) and by applying Corollary 1.4.10 oncemore. ¥

It should be remarked that

D(X1)∼⊗ D(X2) 6= D(X1 × X2)

as topological vector spaces, although these spaces are equal as vector spaces.Moreover,

D′(X1)∼⊗ D′(X2) 6=

(D(X1)∼⊗ D(X2)

)′

(cf. [Schw57b, chap. I, p. 95]).


Kernel Theorems

After these preparations we can prove some ‘vector-valued kernel theorems’:

1.8.9 Theorem Let E be a Banach space and let Xj be open in Rnj , j = 1, 2. Then

E(X1 × X2, E) ∼= L(E ′(X1), E(X2, E))

,

E ′(X1 × X2, E′) ∼= L(E(X1), E ′(X2, E

′))

,

S(Rn1 × Rn2 , E) ∼= L(S ′(Rn1),S(Rn2 , E))

,

S ′(Rn1 × Rn2 , E′) ∼= L(S(Rn1),S ′(Rn2 , E′))

,

where ∼= denotes the canonical toplinear isomorphism.

Proof These assertions are immediate consequences of Theorems 1.4.8 and 1.8.7,and of Corollary 1.8.8. ¥

For completeness we recall the ‘classical’ Schwartz kernel theorem for scalardistributions.

1.8.10 Theorem D′(X1 × X2) ∼= L(D(X1),D′(X2))

by means of the canonicaltoplinear isomorphism.

Proof For example, [Hor83] or [Tre67]. ¥

1.8.11 Remark It may be useful to reinterprete the Kernel Theorems 1.8.9 and1.8.10. Suppose, for instance, that k ∈ E ′(X1 × X2, E) and define K by the relation

(Kϕ1)(ϕ2) = k(ϕ1 ⊗ ϕ2) , ϕj ∈ E(Xj) , j = 1, 2 . (1.8.22)

Then Theorem 1.8.9 guarantees that

K ∈ L(E(X1), E ′(X2, E))

. (1.8.23)

Conversely, for any K satisfying (1.8.23) there exists a unique k ∈ E ′(X1 × X2, E)such that (1.8.22) is true. Moreover, the map

E ′(X1 × X2, E) → L(E(X1), E ′(X2, E))

, k 7→ K (1.8.24)

is a toplinear isomorphism. The distribution k is said to be the kernel of the map K,and K is associated to the kernel k. In symbolic notation the isomorphism (1.8.24)is written as

Kϕ1 =∫

X1

k(x1, ·)ϕ1(x1) dx1 , ϕ1 ∈ E(X1) .

In this case (1.8.22) takes the suggestive form∫

X2

(Kϕ1)(x2)ϕ2(x2) dx2 =∫

X1×X2

k(x1, x2)ϕ1(x1)ϕ2(x2) d(x1, x2) .

428 Appendix

The kernel k ∈ E ′(X1 × X2, E) is said to be regularizing if its associated map Kcan be extended from D(X1) to a continuous linear map of E ′(X1) into E(X2, E).Thanks to Theorem 1.8.9 this is the case iff k ∈ E(X1 × X2, E). Similar definitionsand conventions apply to tempered distributions and to the classical case of theSchwartz kernel theorem 1.8.10. ¥

1.9 Convolutions of Vector-Valued Distributions

In Subsection 1.2 we have already defined convolutions of a vector-valued and ascalar distribution. In this subsection we extend these definitions to the case oftwo vector-valued distributions. In addition, we weaken the support restrictions.

The Basic Theorem

1.9.1 Theorem Suppose that either

u1 ∈ D′(Rn, E1) and u2 ∈ E ′(Rn, E2) ,

oru1 ∈ S(Rn, E1) and u2 ∈ S ′(Rn, E2) .

Then there exists a unique distribution in D′(Rn, E0) or OM (Rn, E0), respectively,the convolution of u1 and u2 with respect to the multiplication (1.5.20), written u1 ∗ q u2,such that

(v1 ⊗ e1) ∗ q (v2 ⊗ e2) = (v1 ∗ v2) ⊗ (e1q e2) (1.9.1)

for v1, v2 ∈ D(Rn), ej ∈ Ej, and j = 1, 2, and such that the ‘convolution maps’(u1, u2) 7→ u1 ∗ q u2 are bilinear and hypocontinuous:

D′(Rn, E1) × E ′(Rn, E2) → D′(Rn, E0) (1.9.2)

and

S(Rn, E1) × S ′(Rn, E2) → OM (Rn, E0) , (1.9.3)

respectively. In addition, the convolution maps restrict to hypocontinuous bilin-ear maps:

E ′(Rn, E1)× E ′(Rn, E2) → E ′(Rn, E0) ,

E(Rn, E1)× E ′(Rn, E2) → E(Rn, E0) ,

D(Rn, E1)×D′(Rn, E2) → E(Rn, E0) ,

D(Rn, E1)× E ′(Rn, E2) → D(Rn, E0) ,

S(Rn, E1)× S(Rn, E2) → S(Rn, E0) .

Proof Let (F1, F2; F0) be either of the triplets (D′, E ′;D′), (S,S ′;OM ), (E ′, E ′; E ′),(E , E ′; E), (D,D′; E), (D, E ′;D), or (S,S;S), and put Fj := Fj(Rn), j = 0, 1, 2.


Then the Fj are reflexive, complete, nuclear, and conuclear by Theorems 1.1.2and 1.4.3. Moreover, Fj

∼⊗ Ej = Fj(Rn, Ej) by Theorem 1.4.9. Define the maps(1.5.14) and (1.5.15) by f1 ¯ f2 := f1 ∗ f2 and f1 v := f1 ∗ v, respectively. FromPropositions 1.2.1, 1.2.3, and 1.2.7 we know that they are well-defined and that(1.5.15) is hypocontinuous. Definition (1.2.1) of convolution immediately impliesthat condition (1.5.16) is satisfied. Hence Theorem 1.5.3 guarantees the existenceof a unique hypocontinuous bilinear map

F1(X, E1) × F2(X, E2) → F0(X,E0) , (u1, u2) 7→ u1 ∗ q u2 (1.9.4)

satisfying (1.9.1) for uj ∈ Fj(X) and ej ∈ Ej , j = 1, 2. By Theorem 1.3.6 thelinear subspace

(D(X) ⊗ E1

)× (D(X)⊗ E2

)is dense in F1(X, E1) × F2(X, E2).

Thus the map (1.9.4) is determined by its restriction to this subspace. This provesthe theorem. ¥

1.9.2 Remarks (a) It should be observed that Theorem 1.9.1 is ‘symmetric in E1

and E2’, that is, the roles of E1 and E2 can be interchanged. This fact will be usedthroughout, usually without further mention.

(b) Suppose that either uj ∈ D′(Rn, Ej), j = 1, 2, and that u1 or u2 has compactsupport, or u1 ∈ S ′(Rn, E1) and u2 ∈ S(Rn, E2). Then

u1 ∗ q u2(ϕ) = u1 ∗ q (u2 ∗ ϕ

)(0) =⟨u1

q (u

2 ∗ ϕ)⟩D (1.9.5)

for ϕ ∈ D(Rn).

Proof We can assume that u2 ∈ E ′(Rn, E2) in the first case. Then by Theorems1.7.2 and 1.9.1 each one of the three expressions in (1.9.5) defines a separatelycontinuous trilinear map

D′(Rn, E1) × E ′(Rn, E2) ×D(Rn) → E0

in the first case, and

S ′(Rn, E1) × S(Rn, E2) ×D(Rn) → E0

in the second one. Suppose that uj = vj ⊗ ej ∈ D(Rn) × Ej , j = 1, 2. Then wededuce from (1.9.1), (1.2.11), and (1.2.13) that

u1 ∗ q u2(ϕ) = (v1 ⊗ e1) ∗ q (v2 ⊗ e2)(ϕ) = v1 ∗ v2(ϕ)(e1q e2)

= v1 ∗ (v2 ∗ ϕ

)(0)(e1q e2) =

[v1 ∗ (v2 ∗ ϕ

)]⊗ (e1

q e2)(0)= u1 ∗ q (u2 ∗ ϕ

)(0) .

Moreover, by these calculations, (1.2.11), Remark 1.2.4(c), and (1.7.4),

u1 ∗ q u2(ϕ) = v1 ∗ (v2 ∗ ϕ

)(0)(e1q e2) =

⟨v1, (v2 ∗ ϕ

)

⟩D(e1

q e2)

= 〈v1, v

2 ∗ ϕ〉D(e1q e2) =

⟨(v1 ⊗ e1) q [(v

2 ∗ ϕ) ⊗ e2

]⟩D

=⟨u1

q (u

2 ∗ ϕ)⟩D .

430 Appendix

Thus all three terms in (1.9.5) agree on the dense linear subspace(D(Rn) ⊗ E1

)× (D(Rn)⊗ E2

)×D(Rn) ,

and so everywhere. ¥

Lp Functions with Compact Supports

Given K b X, we put

Lp,K(X, E) :=

u ∈ Lp(X, E) ; supp(u) ⊂ K

, 1 ≤ p ≤ ∞ .

Then Lp,K(X, E) is a closed linear subspace of the Banach space Lp(X, E), hencea Banach space as well. Moreover,

⋃

KbX

Lp,K(X, E) =

u ∈ Lp(X,E) ; supp(u) b X

.

Thus, recalling (1.1.5) and (1.1.6),

Lp,c(X, E) := lim−−→KbX

Lp,K(X, E) , 1 ≤ p ≤ ∞ , (1.9.6)

is a well-defined LF space, the space of E-valued Lp functions with compact support.It is obvious that

Lp,c(X,E) → Lp(X,E) → L1,loc(X, E) → D′(X, E) (1.9.7)

for 1 ≤ p ≤ ∞, and it is not difficult to see that

D(X, E)d

→ Lp,c(X,E) , 1 ≤ p < ∞ , (1.9.8)

by using Theorem 1.3.6(viii) with k = 0.

Convolutions of Regular Distributions

Our next proposition shows that the convolution product of regular distributionsis again a regular distribution, if defined.

1.9.3 Proposition Suppose that u1 ∈ L1,loc(Rn, E1) and u2 ∈ L1,c(Rn, E2). Thenu1 ∗ q u2 ∈ L1,loc(Rn, E0) and

u1 ∗ q u2(x) =∫

Rn

u1(x − y) q u2(y) dy =∫

Rn

u1(y) q u2(x − y) dy (1.9.9)

for a.a. x ∈ Rn. Moreover, the convolution map (u1, u2) 7→ u1 ∗ q u2 is bilinear andcontinuous :

L1,loc(Rn, E1) × L1,c(Rn, E2) → L1,loc(Rn, E0) . (1.9.10)

Proof We deduce from Theorem 1.9.1 and (1.9.7) that the convolution map isseparately continuous from L1,loc(Rn, E1) × L1,c(Rn, E2) into D′(Rn, E0).


Suppose that K, K2 b Rn. Then, letting K1 := K − K2 b Rn, Tonelli’s the-orem implies

∫

K

∣∣∣∫

Rn

u1(x− y) q u2(y) dy∣∣∣ dx ≤

∫

K

∫

K2

|u1(x − y)| |u2(y)| dy dx

≤ ‖u1‖1,K1 ‖u2‖1,K2

for uj ∈ L1,loc(Rn, Ej) with supp(u2) ⊂ K2. From this we deduce that

(u1, u2) 7→∫

Rn

u1(· − y) q u2(y) dy

is a bilinear and continuous map

L1,loc(Rn, E1)× L1,c(Rn, E2) → L1,loc(Rn, E0) → D′(Rn, E0) .

By (III.4.2.17) and (1.9.1) we infer that both sides of (1.9.9) coincide on the lin-ear subspace L1,loc(Rn, E1) ×

(D(Rn)⊗ E2

), which is dense by (1.9.8). Thus the

assertion follows. ¥

Tensor Products and Convolutions

Given the hypotheses of Proposition 1.9.3, we deduce from (1.9.9) by Fubini’stheorem, a change of variables, and Remark 1.8.5(d) that

u1 ∗ q u2(ϕ) =∫

Rn

∫

Rn

ϕ(x)u1(x − y) q u2(y) dy dx

=∫

Rn×Rn

ϕ(x + y)u1(x) q u2(y) d(x, y) = u1 ⊗ q u2

(ϕ(x1 + x2)

)

for ϕ ∈ D(Rn), where in the last term we use our symbolic notation, meaning thatuj acts on the function following it as a variable of xj ∈ Rn. The next theoremshows that this formula holds in general.

1.9.4 Theorem Suppose that one of the distributions uj ∈ D′(Rn, Ej), j = 1, 2,has compact support. Then

u1 ∗ q u2(ϕ) = u1 ⊗ q u2

(ϕ(x1 + x2)

), ϕ ∈ D(Rn) .

Proof By Remark 1.9.2(b) and Fubini’s rule for tensor products

u1 ∗ q u2(ϕ) =⟨u1

q (u

2 ∗ ϕ)⟩D =

⟨u1

q u

2(τx1ϕ

)⟩D =

⟨u1

q u2

(τx1ϕ

(−x2))⟩

D=

⟨u1

q u2

(ϕ(x1 + x2)

)⟩= u1 ⊗ q u2

(ϕ(x1 + x2)

)

for ϕ ∈ D(Rn). ¥

432 Appendix

Theorem 1.9.4 enables us to define convolutions in certain cases where neitheru1 nor u2 has compact support. For this we recall that a mapping between metricspaces is proper, if preimages of compact sets are compact. Given nonempty closedsubsets A1, . . . , Am of Rn, we say (A1, . . . , Am) satisfies condition (Σ) if the map

A1 × · · · × Am → Rn , (x1, . . . , xm) 7→ x1 + · · · + xm

is proper.

1.9.5 Remarks (a) A proper continuous mapping is closed, that is, it maps closedsets onto closed sets. Hence if (A1, . . . , Am) satisfies condition (Σ) then the setA1 + · · · + Am is closed in Rn.

(b) If A1, . . . , Am are closed in Rn and all but one are compact, then (A1, . . . , Am)satisfies condition (Σ).

(c) Let Γ be a closed convex proper cone in Rn. (A cone Γ is proper if it does notcontain a full line, that is, if Γ ∩ (−Γ) = 0.) Then the m-tuple (Γ, . . . , Γ) satisfiescondition (Σ). If H :=

y ∈ Rn ; 〈η, y〉 ≥ 0

is a closed half-space with interior

normal η lying in the interior of the dual cone Γ′ :=

ξ ∈ Rn ; 〈ξ, x〉 ≥ 0, x ∈ Γ,

then the m-tuple (Γ, . . . , Γ, H) satisfies condition (Σ). Note that int(Γ′) 6= ∅.Proof First we assume that m = 2. Thus let

((xj , yj)

)be a sequence in Γ × B,

where B ∈ Γ, H, such that xj + yj → z in Rm. Then, by the local compactnessand the closedness of Γ and B, it suffices to prove that (xj) and (yj) are boundedsequences. If this is not the case, then clearly both sequences have to be unbounded.Thus, by passing to a suitable subsequence we can assume that |xj | → ∞ andxj/|xj | → x ∈

qΓ. Hence yj/|xj | → −x ∈

qB. If B = Γ, then this is impossible since

Γ is proper. If B = H, then it follows that 0 < 〈η, x〉 since η ∈ int(Γ′) and x ∈qΓ,

and 〈η, x〉 ≤ 0 since −x ∈ H, which is again a contradiction. The general case isderived by an obvious induction argument. ¥

In the following, the distributions u1, . . . , um are said to have convolutive

supports or to satisfy condition (Σ) if their supports satisfy condition (Σ).

Let uj ∈ D′(Rn, Ej), j = 1, 2, satisfy condition (Σ), and let X be a boundedopen subset of Rn. Then there exists ρ := ρX > 0 such that

(xj ∈ supp(uj) , x1 + x2 ∈ X

)=⇒ (

xj ∈ ρBn , j = 1, 2)

. (1.9.11)

Thus, given

ψj ∈ D(Rn) with ψj |ρBn = 1 , j = 1, 2 , (1.9.12)

the convolution (ψ1u1) ∗ q (ψ2u2) is well-defined, since ψjuj ∈ E ′(Rn, Ej). More-over, for χj ∈ D(Rn) with χj |ρBn = 1 it follows from Remarks 1.8.5(a) and (e),


the fact that ψj − χj vanish on ρBn, and from (1.9.11) that, given ϕ ∈ D,[(ψ1u1) ∗ q (ψ2u2)− (χ1u1) ∗ q (χ2u2)

](ϕ)

=[(ψ1u1)⊗ q (ψ2u2) − (χ1u1) ⊗ q (χ2u2)

](ϕ(x1 + x2)

)

=[(ψ1 − χ1)u1 ⊗ q (ψ2u2) + (χ1u1)⊗ q (ψ2 − χ2)u2

](ϕ(x1 + x2)

)

=[

(ψ1 − χ1) ⊗ ψ2

](u1 ⊗ q u2) +

[χ1 ⊗ (ψ2 − χ2)

](u1 ⊗ q u2)

(ϕ(x1 + x2)

)

= (u1 ⊗ q u2)[

(ψ1 − χ1)(x1)ψ2(x2) + χ1(x1)(ψ2 − χ2)(x2)]ϕ(x1 + x2)

= 0 .

This shows that (ψ1u1) ∗ q (ψ2u2) defines a distribution in D′(X, E0), independentlyof the choice of the ψj satisfying (1.9.12). From this and the fact that a distributionis determined by its values on the open subsets of Rn (e.g., [Hor83, Theorem 2.2.4]and note that the proof carries over to the E-valued case) we see that

given uj ∈ D′(Rn, Ej), j = 1, 2, having convolutive supports,there exists a unique distribution u1 ∗ q u2 ∈ D′(Rn, E0), theconvolution of u1 and u2 with respect to multiplication (1.5.20),satisfying

u1 ∗ q u2 |X = (ψ1u1) ∗ q (ψ2u2) |X , X = X b Rn ,where ψj satisfies (1.9.12) for j = 1, 2.

(1.9.13)

Basic Properties

In the following remarks we collect some of the most important properties ofconvolutions.

1.9.6 Remarks Unless explicit restrictions are given, we suppose that eitheruj ∈ D′(Rn, Ej) for j = 1, 2, with convolutive supports, or that u1 ∈ S ′(Rn, E1)and u2 ∈ S(Rn, E2).

(a) It is an obvious consequence of (1.9.13) and Remark 1.9.2(b) that

u1 ∗ q u2(ϕ) = u1 ∗ (u2 ∗ ϕ

)(0) =⟨u1

q (u

2 ∗ ϕ)⟩D

for ϕ ∈ D(Rn).

(b) If uj ∈ L1,loc(Rn, Ej) satisfy condition (Σ), then u1 ∗ q u2 ∈ L1,loc(Rn, E0) and

u1 ∗ q u2(x) =∫

Rn

u1(x − y) q u2(y) dy =∫

Rn

u1(y) q u2(x − y) dy , a.a. x ∈ Rn .

Proof Thanks to (1.9.13) we can assume that uj has compact support for j = 1, 2.Then the assertion follows from Proposition 1.9.3. ¥

434 Appendix

(c)(Associativity) Let the associativity hypotheses of Remark 1.6.5(e) be satisfiedand suppose that either uj ∈ D′(Rn, Ej), j = 1, 2, 3, have convolutive supports,or u1 ∈ S ′(Rn, E1) and uk ∈ S(Rn, Ek), k = 2, 3. Then

u1 ∗ q (u2 ∗ q u3) = (u1 ∗ q u2) ∗ q u3 . (1.9.14)

Proof If condition (Σ) is satisfied,4 then it suffices, by (1.9.13), to prove theassertion for uj ∈ E ′(X, Ej), j = 1, 2, 3. Then Theorem 1.9.1 implies that bothsides of (1.9.14) define separately continuous trilinear maps

E ′(Rn, E1)× E ′(Rn, E2) × E ′(Rn, E3) → E ′(Rn, E0)

if condition (Σ) is satisfied, and

S ′(Rn, E1) × S(Rn, E2) × S(Rn, E3) → OM (Rn, E0)

otherwise. Thanks to (1.9.1) and (1.2.13) they coincide on the dense linear subspace∏3j=1 D(X)⊗ Ej . Thus they are equal. ¥

(d) (Commutativity) Suppose that E1 = E2 =: E and multiplication (1.5.20) issymmetric. Then u1 ∗ q u2 = u2 ∗ q u1.

Proof By (1.9.13) we can assume that uj ∈ E ′(Rn, E) if condition (Σ) is satisfied.Thus, by Theorem 1.9.1 and the density of D(Rn, E) in E ′(Rn, E) and in S ′(Rn, E)and S(Rn, E), respectively, we can assume that uj ∈ D(Rn, E). Now the assertionfollows from (b). ¥

(e)(Distributivity) Let the distributivity hypotheses of Remark 1.8.5(e) be satis-fied and suppose that uj ∈ D′(Rn, Ej) as well as vj ∈ D′(Rn, Ej+2), j = 1, 2, haveconvolutive supports. Then

(u1 ∗ q u2) ⊗ q (v1 ∗ q v2) = (u1 ⊗ q v1) ∗ q (u2 ⊗ q v2) . (1.9.15)

Proof First we show that uj ⊗ q vj ∈ D′(Rn × Rn, Ej+5) satisfy condition (Σ). ByRemark 1.8.5(a)

supp(uj ⊗ q vj) ⊂ supp(uj) × supp(vj) , j = 1, 2 . (1.9.16)

Let X ⊂ Rn be bounded and open. Then there exists ρ > 0 such that xj ∈ supp(uj)and yj ∈ supp(vj) with x1 + x2 and y1 + y2 belonging to X imply xj , yj ∈ ρBn.Thus, if (x1, y1) + (x2, y2) ∈ X × X, it follows from (1.9.16) that

(xj , yj) ∈ ρBn × ρBn ⊂ ρ1B2n , j = 1, 2 ,

for some ρ1 > 0. Since every bounded open subset of R2n is contained in a productset X × X, we see that uj ⊗ vj satisfy condition (Σ). Hence both sides of (1.9.15)are well-defined.

4We leave it to the reader to verify that the convolutions on either side of (1.9.14) are well-defined if (Σ) is satisfied (also cf. (f)).


Given ψj , ψj+2 ∈ D(Rn), it follows from Remark 1.8.5(e) that

(ψjuj) ⊗ q (ψj+2vj) = (ψj ⊗ ψj+2)(uj ⊗ q vj) , j = 1, 2 .

Using this and (1.9.13) it is easily seen that it suffices to prove the assertion foruj ∈ E ′(Rn, Ej) and vj ∈ E ′(Rn, Ej+2), j = 1, 2. Now Theorems 1.8.4 and 1.9.1 to-gether with a density argument show that we can assume that uj ∈ D(Rn, Ej) andvj ∈ D(Rn, Ej+2). Thus, by (b) and Remark 1.8.5(d), the left side of (1.9.15) equals∫

Rn

u1(x1 − y1) q u2(y1) dy1q∫

Rn

v1(x2 − y2) q v2(y2) dy2 , (x1, x2) ∈ Rn × Rn .

By Fubini’s theorem and (1.8.11) this product takes the form∫

R2n

(u1(x1 − y1) q v1(x2 − y2)

) q (u2(y1) q v2(y2))d(y1, y2) ,

which is the right side of (1.9.15). ¥

(f )(Support Theorem) Suppose that uj ∈ D′(Rn, Ej), j = 1, 2, have convolutivesupports. Then supp(u1 ∗ q u2) ⊂ supp(u1) + supp(u2).

Proof From Remark 1.9.5(a) we know that C := supp(u1) + supp(u2) is closedin Rn. Let ϕ ∈ D(Cc). Then the support of the function (x1, x2) 7→ ϕ(x1 + x2)does not meet supp(u1)× supp(u2), hence not supp(u1 ⊗ q u2) by Remark 1.8.5(a).Thus (ψ1u1) ∗ q (ψ2u2)(ϕ) = (ψ1u1) ⊗ q (ψ2u2)

(ϕ(x1 + x2)

)= 0 for all ψj ∈ D(X),

j = 1, 2. Consequently, (1.9.13) implies u1 ∗ q u2(ϕ) = 0, that is, the assertion. ¥

(g) ∂α+β(u1 ∗ q u2) = ∂αu1 ∗ q ∂βu2, α, β ∈ Nn.

Proof First suppose that u1 and u2 satisfy condition (Σ). Since

supp(∂γu) ⊂ supp(u) , u ∈ D′(Rn, E) , γ ∈ Nn ,

it follows that ∂αu1 and ∂βu2 satisfy condition (Σ). Hence ∂αu1 ∗ q ∂βu2 is well-defined. Given ϕ ∈ D(Rn),

∂α+β(u1 ∗ q u2)(ϕ) = (−1)|α|+|β|(u1 ∗ q u2)(∂α+βϕ) .

Let X be a bounded open subset of Rn containing supp(ϕ) and choose ψj ∈ D(Rn)with ψj |ρXBn = 1. Then

(−1)|α|+|β|(u1 ∗ q u2)(∂α+βϕ) = (−1)|α|+|β|(ψ1u1) ∗ q (ψ2u2)(∂α+βϕ)

= (−1)|α|+|β|(ψ1u1) ⊗ q (ψ2u2)(∂α+βϕ(x1 + x2)

)

= ∂α(ψ1u1) ⊗ q ∂β(ψ2u2)(ϕ(x1 + x2)

)

= (ψ1∂αu1) ⊗ q (ψ2∂

βu2)(ϕ(x1 + x2)

)

= (ψ1∂αu1) ∗ q (ψ2∂

βu2)(ϕ)

= ∂αu1 ∗ q ∂βu2(ϕ) ,

thanks to Remark 1.8.5(b), Leibniz’ rule, and the properties of ψj .

436 Appendix

If u1 ∈ S ′(Rn, E1) and u2 ∈ S(Rn, E2), then we can restrict ourselves as usualto the case uj ∈ D(Rn) ⊗ Ej . Then the assertion follows from Remark 1.2.4(b). ¥

(h) τa(u1 ∗ q u2) = (τau1) ∗ q u2 = u1 ∗ q τau2, a ∈ Rn, and

(u1 ∗ q u2)

= u

1 ∗ q u

2 .

Proof It is easily verified that these convolutions are all well-defined. If u1 and u2

satisfy condition (Σ), we deduce from (1.9.13) that we can assume that u1 and u2

have compact supports. Thus in either case it suffices by Theorem 1.9.1 to considerthe case where uj ∈ D(Rn) ⊗ Ej . Now the assertion follows from Remark 1.2.4(c). ¥

(i) Suppose that a ∈ E1 and u ∈ D′(Rn, E2). Then

∂α[(δ ⊗ a) ∗ q u]

= (∂αδ ⊗ a) ∗ q u = a q ∂αu = ∂α(a q u)

for α ∈ Nn.

Proof Since ∂βδ ⊗ a ∈ E ′(Rn, E1), we see that the above convolutions are well-defined. Thanks to (g) and Remark 1.6.5(f) we can assume that α = 0. By The-orems 1.6.4 and 1.9.1 it suffices to consider u ∈ D(Rn) × E2. Then the assertionfollows from (1.9.1) and Remark 1.2.4(a). ¥

Convolution Algebras

Given a nonempty subset K of Rn, we put

D′K(Rn, E) :=

u ∈ D′(Rn, E) ; supp(u) ⊂ K

(1.9.17)

and, as usual, D′K(Rn) := D′

K(Rn,K), if no confusion seems likely. We also set

D′+(E) := D′

R+(R, E) . (1.9.18)

Note that D′K(Rn, E) is a closed linear subspace of D′(Rn, E).

In the following theorem we collect some of the properties of convolutions ina particularly important setting.

1.9.7 Theorem Let Γ be a proper closed convex cone in Rn and let

F ∈ D,S, E ′,D′Γ .

Then convolution is a well-defined hypocontinuous bilinear map

F(Rn, E1)× F(Rn, E2) → F(Rn, E0) , (u1, u2) 7→ u1 ∗ q u2

which possesses the associativity and commutativity properties of Remarks 1.9.6(c)and (d), respectively. In particular, if (E, q) is a [commutative] Banach algebra,


then(F(Rn, E), ∗ q) is a [commutative] algebra, a convolution algebra. If (E, q) has

a unit e0 and F ∈ E ′,D′Γ, then

(F(Rn, E), ∗ q) has a unit as well, namely δ ⊗ e0.

Proof Everything, except the hypocontinuity of the convolution map if F = D′Γ,

follows easily from Theorem 1.9.1, the injection D(Rn, E2) → E ′(Rn, E2), and Re-marks 1.9.5(c), 1.9.6(f), and 1.2.4(a). Recall that D′

Γ(Rn, E) is a closed linearsubspace of D′(Rn, E). Thus, if (uα) is a net in D′

Γ(Rn, E1) converging to zero,it converges to zero in D′(Rn, E1). Let V be a bounded subset of D′

Γ(Rn, E2),hence of D(Rn, E2). Then, given any bounded subset B of D(Rn), there existsK := KB b Rn such that supp(ϕ) ⊂ K for ϕ ∈ B. Hence condition (Σ) guaran-tees the existence of ρ > 0 such that x, y ∈ Γ with x + y ⊂ K implies x, y ∈ ρBn.Thus, letting ψj ∈ D(Rn) satisfy ψ |ρBn = 1, it follows from (1.9.13) that

uα ∗ q v(ϕ) = (ψ1uα) ∗ q (ψ2v)(ϕ) , ϕ ∈ B , v ∈ V . (1.9.19)

Note that ψ2v(χ) = v(ψ2χ) and ψ2χ ∈ Dsupp(ψ2)(Rn) for χ ∈ E(Rn) imply thatψ2v ; v∈ V is bounded in E ′(Rn, E2). Now the hypocontinuity of the map (1.9.2)and (1.9.19) entail uα ∗ q v(ϕ) → 0 in E0, uniformly with respect to v ∈ V andϕ ∈ B. Consequently, uα ∗ q v → 0 in D′(Rn, E0), hence in D′

Γ(Rn, E0), uniformlywith respect to v in bounded subsets of D′

Γ(Rn, E0). This proves, by symmetry,the asserted hypocontinuity. ¥

1.9.8 Remarks (a) Let Γ be a proper, closed, and convex cone in Rn. ThenD(Rn), S(Rn), E ′(Rn), and D′

Γ(Rn) are commutative convolution algebras. More-over, E ′(Rn) and D′

Γ(Rn) possess the unit δ.

(b) Since D(Rn, E) and S(Rn, E) are barreled, the convolution product is, in fact,continuous on these spaces.

(c) Suppose that uj ∈ D′+(Ej) are regular distributions. Then u1 ∗ q u2 ∈ D′

+(E0)is a regular distribution as well, and

u1 ∗ q u2(t) =∫ t

0

u1(t− s) q u2(s) ds , a.a. t ∈ R+ .

Proof This is an easy consequence of Remarks 1.9.6(b) and (f). ¥

Convolutions of Bounded and Integrable Functions

For the reader’s convenience we now collect the properties of convolution on certainBanach spaces of vector-valued convolutions, thus extending, in particular, asser-tions (III.4.2.19)–(III.4.2.22) (note that (III.4.2.18) is a special case of (III.4.2.21)).

1.9.9 Theorem Suppose that (F1, F2; F0) is any one of the triplets

(BUC, L1; BUC), (C0, L1; C0), (Lp, L1; Lp), (L∞, L1; BUC), (Lq, Lq′ ; C0) ,

438 Appendix

where 1 ≤ p < ∞ and 1 < q < ∞. Then the convolution with respect to the multi-plication (1.5.20) extends from F1(Rn, E1)×D(Rn, E2) to a multiplication

F1(Rn, E1)× F2(Rn, E2) → F0(Rn, E0) .

It is given by

u1 ∗ q u2(x) =∫

Rn

u1(x − y) q u2(y) dy =∫

Rn

u1(y) q u2(x − y) dy (1.9.20)

for a.a. x ∈ Rn.

Proof Since F1(Rn, E1) → L1,loc(Rn, E1) ∩ S ′(Rn, E1) (see (VII.1.2.1)), we inferfrom (1.9.3) that

u1 ∗ q u2 ∈ OM (Rn, E0) , u1 ∈ F1(Rn, E1) , u2 ∈ D(Rn, E2) ,

and Proposition 1.9.3 guarantees that u1 ∗ q u2 is given by the above integralin this case. Since D(Rn, E2) is dense in F2(Rn, E2), it suffices to show thatu1 ∗ q u1 ∈ F0(Rn, E0) and that the estimate

‖u1 ∗ q u2‖F0(Rn,E0) ≤ ‖u1‖F1(Rn,E1) ‖u2‖F2(Rn,E2)(1.9.21)

is valid for u1 ∈ F1(Rn, E1) and u2 ∈ D(Rn, E2). Then the assertion follows bycontinuous extension.

If u1 ∈ Lr(Rn, E1), 1 ≤ r ≤ ∞, and u2 ∈ D(Rn, E2), then

|u1 ∗ u2(x)| ≤∫

Rn

|u1(x − y)| |u2(y)| dy , x ∈ Rn ,

and the classical scalar Young inequality imply

‖u1 ∗ q u2‖Lr(Rn,E0) ≤ ‖u1‖Lr(Rn,E1) ‖u2‖L1(Rn,E2)(1.9.22)

and

‖u1 ∗ q u2‖L∞(Rn,E0) ≤ ‖u1‖Lr(Rn,E1) ‖u2‖Lr′ (Rn,E2)(1.9.23)

(cf. (III.4.2.20)–(III.4.2.22)). If uj ∈ D(Rn, Ej) for j = 1, 2, we know from the nextto last line of Theorem 1.9.1 that u1 ∗ q u2 ∈ D(Rn, E0). Since D(Rn, E1) is densein F1(Rn, E1) if F1 belongs to C0, Lp, Lq and since C0(Rn, E0) is a closed linearsubspace of L∞(Rn, E0) containing D(Rn, E0), we infer from (1.9.22) and (1.9.23)that the assertion is true for F1 ∈ C0, Lp, Lq.

If u1 ∈ BUC(Rn, E1) and u2 ∈ D(Rn, E2), then, given ε > 0, there existsδ > 0 such that |u1(x) − u1(y)| ≤ ε for x, y ∈ Rn with |x − y| ≤ δ. Hence

|u1 ∗ q u2(x) − u1 ∗ q u2(y)| ≤∫

Rn

|u1(x− z) − u1(y − z)| |u2(z)| dz ≤ ε ‖u2‖1

for x, y ∈ Rn with |x− y| ≤ δ. This shows that u1 ∗ q u2 ∈ BUC(Rn, E0) in thiscase. Hence the assertion follows for F1 = BUC from (1.9.22) and the fact thatBUC(Rn, E0) is a closed linear subspace of L∞(Rn, E0).


Finally, suppose that u1 ∈ L∞(Rn, E1) and u2 ∈ D(Rn, E2). Then we inferfrom Remark 1.9.6(h) and (1.9.22) that

‖τa(u1 ∗ q u2) − u1 ∗ q u2‖∞ = ‖u1 ∗ q (τau2 − u2)‖∞ ≤ ‖u1‖∞ ‖τau2 − u2‖1

for a ∈ Rn. Since the translation group is strongly continuous on L1(Rn, E2), itfollows that u1 ∗ q u2 ∈ BUC(Rn, E0) in this case. Thus the assertion holds forF1 = L∞ as well. ¥

The Convolution Theorem

Lastly, we extend the convolution theorem to vector-valued distributions.

1.9.10 Theorem (u1 ∗ q u2) = u1q u2 for u1 ∈ S ′(Rn, E1) and u2 ∈ S(Rn, E2).

Proof Since F is a toplinear automorphism of S(Rn, E) and of S ′(Rn, E), itfollows from Theorems 1.6.4 and 1.9.1, together with

S(Rn, E) → OM (Rn, E) → S ′(Rn, E) ,

that the bilinear maps (u1, u2) 7→ (u1 ∗ qu2) and (u1, u2) 7→ u1q u2 are well-defined

and hypocontinuous from S ′(Rn, E1)× S(Rn, E2) into S ′(Rn, E0). Thanks to

(u⊗ e) = u⊗ e , u ⊗ e ∈ S ′(Rn) ⊗ E ,

we see from (1.6.12), (1.9.1), and the scalar convolution theorem that they coincideon the dense linear subspace

(S ′(Rn)⊗ E1

)× (S(Rn) ⊗ E2

), hence everywhere. ¥

1.9.11 Remarks (a) Let

O′C(Rn, E) := F−1OM (Rn, E) :=

u ∈ S ′(Rn, E) ; Fu ∈ OM (Rn, E)

,

endowed with the unique locally convex topology such that

F ∈ Lis(O′

C(Rn, E),OM (Rn, E))

.

Then it follows from Theorem 1.9.10, the continuity properties of F , and Theo-rem 1.6.4 that we can define a hypocontinuous bilinear map

S ′(Rn, E1)×O′C(Rn, E2) → S ′(Rn, E0) , (u1, u2) 7→ u1 ∗ q u2 , (1.9.24)

the convolution with respect to multiplication (1.5.20), by putting

u1 ∗ q u2 := F−1(u1q u2) , u1 ∈ S ′(Rn, E1) , u2 ∈ O′

C(Rn, E2) . (1.9.25)

We leave it to the interested reader to carry over the properties of Remarks 1.9.6to this more general case.

440 Appendix

(b) Given a ∈ OM (Rn, E1), we put

a(D)u := F−1a qFu := F−1(a q u) , u ∈ S ′(Rn, E2) .

Then the isomorphism properties of the Fourier transform and Theorem 1.6.4imply that

a(D) ∈ L(S ′(Rn, E2),S ′(Rn, E0))

.

Moreover,a(D)u = F−1(a) ∗ q u , u ∈ S(Rn, E2) ,

thanks to the Convolution Theorem 1.9.10. Motivated by these facts we put

a(D)u := F−1(a qF)u := F−1(a) ∗ q u (1.9.26)

whenever a ∈ S ′(Rn, E1) and u ∈ S ′(Rn, E2) are such that the convolution prod-uct on the right side is well-defined. In these general situations a(D) is said tobe a translation-invariant (pseudodifferential) operator with symbol a (related to themultiplication (1.5.20)). Of course, this is motivated by Theorem 1.2.2.

It should be observed that this definition is consistent with the definitionof Fourier multipliers in Subsection VI.3.4 of the main text (see, in particular,Theorem VI.3.4.1. ¥

Vector Measures and the Riesz Representation Theorem 441

2 Vector Measures and the Riesz Representation The-

orem

In connection with duality theory and Besov spaces we need some explicit infor-mation on C0(Rn, E)′. For this reason we present in this section some elements ofthe theory of vector measures and prove a generalization of the well-known Rieszrepresentation theorem.

Measures of Bounded Variation

Throughout this section E := (E, |·|) is a Banach space, X is a σ-compact metriz-able space, and BX denotes the Borel σ-algebra of X. By an E-valued vector

measure µ on X we mean a σ-additive map µ : BX → E satisfying µ(∅) = 0. Forsuch a vector measure µ we define the variation |µ| : BX → R+ ∪ ∞ by

|µ| (B) := supπ(B)

∑

A∈π(B)

|µ(A)| , B ∈ BX ,

where the supremum is taken over all partitions π(B) of B into a finite number ofpair-wise disjoint Borel subsets. Then µ is said to be of bounded variation if

‖µ‖BV := |µ| (X) < ∞ .

The set of all E-valued vector measures on X of bounded variation is denoted by

MBV (X,E) :=(MBV (X, E), ‖·‖BV

).

It is clear that MBV (X, E) is a normed vector space with the obvious linearstructure (as a vector subspace of EBX ).

Given µ ∈MBV (X, E), it follows from |µ(B)| ≤ |µ| (B) for B ∈ BX that|µ| is a positive Borel measure on X. We denote its completion, defined on theσ-algebra A|µ|, consisting of all A ⊂ X of the form A = B ∪ N where B ∈ BX andN is a subset of a |µ|-null set, again by |µ|. Then |µ| is a positive finite Radonmeasure on X.

2.0.1 Remark Denote by dx a positive Radon measure on X. Then, given anyfunction u ∈ L1(X, dx,E),

(u dx)(B) :=∫

B

u dx , B ∈ BX ,

defines an E-valued measure of bounded variation and ‖u dx‖BV = ‖u‖1 (cf. The-orem 4(iv) on p. 46 in [DU77] or [Lan69, Theorem XI.9]). Hence

(u 7→ u dx) : L1(X, dx, E) →MBV (X, E)

is a linear isometry. Occasionally, it will be convenient to identify u with u dx, thatis, to consider L1(X, dx, E) as a closed linear subspace of MBV (X, E). ¥

442 Appendix

Integrals with Respect to Vector Measures

Let E0, E1, and E2 be Banach spaces and suppose that

E1 × E2 → E0 , (e1, e2) 7→ e1q e2 (2.0.1)

is a multiplication. Let B(X,E1) be the closure in B(X, E1) of the linear subspaceS(X, E1) of all simple functions

u =∑

B∈π

χB ⊗ eB , eB ∈ E1 , (2.0.2)

where π := π(X) runs through all partitions of X. If u ∈ S(X,E1) is given by(2.0.2) and µ ∈MBV (X, E2), then we put

∫u dµ :=

∫

X

u dµ :=∑

B∈π

eBq µ(B) ∈ E0 .

It follows that

S(X, E1) ×MBV (X, E2) → E0 , (u, µ) 7→∫

u dµ

is a well-defined bilinear map satisfying∣∣∣∫

u dµ∣∣∣E0

≤∑

B∈π

|eB | |µ(B)| ≤∑

B∈π

|eB | |µ| (B) ≤ ‖u‖∞ ‖µ‖BV . (2.0.3)

Hence it possesses a unique continuous bilinear extension of norm at most oneover B(X,E1) ×MBV (X,E2). Note that C0(X, E1) is a closed linear subspaceof B(X,E1). Thus, by restriction, we obtain a well-defined multiplication

C0(X,E1)×MBV (X,E2) → E0 , (u, µ) 7→∫

u dµ :=∫

X

u dµ (2.0.4)

and∫

Xu dµ is said to be the integral of u over X with respect to µ (and the multi-

plication (2.0.1)). In fact, from (2.0.3) we deduce that∣∣∣∫

u dµ∣∣∣E0

≤∫

|u| d |µ| ≤ ‖u‖∞ ‖µ‖BV (2.0.5)

for u ∈ C0(X,E1) and µ ∈MBV (X, E2).

Now suppose that there are further Banach spaces E3, E4, and E5, andmultiplications satisfying the associativity assumption of Remark 1.6.5(e). Thenit follows from the definition of

∫u dµ that, for each u of the form (2.0.2),

∫u dµ q e =

∑

B∈π

eBq (µ(B) q e) ∈ E5 , e ∈ E3 . (2.0.6)


Note that, given e ∈ E3,

µ q e := µ(·) q e : BX → E4 , B 7→ µ(B) q e

is σ-additive. Moreover,

|µ q e(B)| ≤ |µ(B)| |e| ≤ |µ| (B) |e| , B ∈ BX ,

implies that µ q e is an E4-valued vector measure on X of bounded variation,and that

MBV (X, E2) × E3 →MBV (X,E4) , (µ, e) 7→ µ q e (2.0.7)

is a multiplication. From this and (2.0.6) we infer that

(∫u dµ

)q e =

∫u d(µ q e) (2.0.8)

and that

C0(X, E1) ×MBV (X, E2) × E3 → E5 , (u, µ, e) 7→∫

u d(µ q e) (2.0.9)

is a trilinear multiplication, that is, a continuous trilinear map of norm at most 1.Similarly as above, we see that (2.0.1) induces a multiplication

E1 ×MBV (X, E2) →MBV (X, E0) , (e, µ) 7→ e q µ . (2.0.10)

Thus, givenϕ =

∑

B∈π

αBχB ∈ S(X,K) ,

it follows from

ϕ⊗ e =∑

B∈π

αBχB ⊗ e =∑

B∈π

χB ⊗ (αBe) ∈ S(X, E1)

for e ∈ E1 that∫

ϕ⊗ e dµ =∑

B∈π

αBe q µ(B) =∫

ϕd(e q µ) .

From this we deduce that∫

ϕ⊗ e dµ =∫

ϕd(e q µ) (2.0.11)

for ϕ⊗ e ∈ C0(X) ⊗ E1 and µ ∈MBV (X, E2).

444 Appendix

In the following, we leave it to the reader to identify the choices for the spacesE0, . . . , E5 and the multiplications that we are using in concrete situations.

Vector Measures as Distributions

Now suppose that X is open in Rn. Since D(X) is dense in C0(X), it follows thatthe multiplication

C0(X) ×MBV (X, E) → E , (ϕ, µ) 7→∫

ϕdµ

is completely determined by its restriction to D(X) ×MBV (X, E). Put

Tµϕ :=∫

ϕdµ , (ϕ, µ) ∈ D(X) ×MBV (X, E) .

Then D(X) → C0(X) implies

(µ 7→ Tµ) ∈ L(MBV (X, E),D′(X, E))

. (2.0.12)

Suppose that Tµ = 0 in D′(X, E) for some µ ∈MBV (X, E). Then we infer from(2.0.8) that

〈e′, Tµϕ〉E =∫

ϕd〈e′, µ〉E = 0 , ϕ ∈ D(X) , e′ ∈ E′ .

Since 〈e′, µ〉E ∈MBV (X) := MBV (X,K), this implies 〈e′, µ〉E = 0 for each e′ be-longing to E′. Thus

⟨e′, µ(B)

⟩E

= 0 for e′ ∈ E′ and B ∈ BX , which gives µ = 0.This shows that (2.0.12) is an injection. Thus the following convention is justified.

Let X be a nonempty open subset of Rn. Then we identify µ ∈MBV (X, E)with Tµ ∈ D′(X, E) so that

MBV (X,E) → D′(X, E) (2.0.13)

and

µ(ϕ) =∫

X

ϕ dµ , µ ∈MBV (X,E) , ϕ ∈ D(X) . (2.0.14)

Convolutions Involving Vector Measures

Observe that S(Rn)d

→ C0(Rn) implies

MBV (Rn, E) → S ′(Rn, E) . (2.0.15)

Hence we know from Proposition 1.2.7 that the convolution product ϕ ∗ µ is a well-defined element of OM (Rn, E) for ϕ ∈ S(Rn) and µ ∈MBV (Rn, E). Moreover,

(ϕ ∗ µ)(x) = µ(τxϕ

) =∫

Rn

ϕ(x − y) dµ(y) , x ∈ Rn , (2.0.16)


for ϕ ∈ S(Rn) and µ ∈MBV (Rn, E). Hence, by (2.0.5),

|ϕ ∗ µ(x)|E ≤∫

Rn

|ϕ(x− y)| d |µ| (y) = |ϕ| ∗ |µ| (x) , x ∈ Rn ,

so that, by Fubini’s theorem,

‖ϕ ∗ µ‖1 ≤ ‖ϕ‖1 ‖µ‖BV , ϕ ∈ S(Rn) , µ ∈MBV (Rn, E) . (2.0.17)

2.0.2 Proposition Convolution is a well-defined multiplication

L1(Rn)×MBV (Rn, E) → L1(Rn, E) .

Proof This follows from (2.0.17) by continuous extension. ¥

If E = K then, thanks to Remark 2.0.1, Proposition 2.0.2 reduces to the well-known fact that L1(Rn) is an ideal in the convolution algebra of all K-valued Borelmeasures (e.g., [Pet83, Theorem I.4.5]).

2.0.3 Lemma Suppose ϕ ∈ L1(Rn), u ∈ C0(Rn, E), and µ ∈MBV (Rn, E′). Then5

∫

Rn

ϕ ∗ u dµ =∫

Rn

u d(ϕ

∗ µ) =∫

Rn

〈ϕ

∗ µ, u〉E dx = 〈ϕ

∗ µ, u〉 . (2.0.18)

Proof From Theorem 1.9.9, Remark 2.0.1, Proposition 2.0.2, and (2.0.4) we inferthat each one of the maps that send

(ϕ, u, µ) ∈ L1(Rn) × C0(Rn, E) ×MBV (Rn, E′)

into one of the integrals in (2.0.18) is a trilinear K-valued multiplication. Thus,since C0(Rn) ⊗ E is dense in C0(Rn, E) by Theorem 1.3.6(x), it suffices to con-sider the case where u = ψ ⊗ e ∈ C0(Rn) ⊗ E. Since ϕ ∗ (ψ ⊗ e) = (ϕ ∗ ψ) ⊗ e, wededuce from (2.0.11), (2.0.16), and Fubini’s theorem that

∫ϕ ∗ u dµ =

∫ϕ ∗ ψ d〈µ, e〉E =

∫∫ϕ(x − y) d〈µ, e〉E(x)ψ(y) dy

=∫ (

ϕ

∗ 〈µ, e〉E)ψ dx .

(2.0.19)

Moreover, thanks to (2.0.8),

ϕ

∗ 〈µ, e〉E(x) =∫

ϕ

(x − y) d〈µ, e〉E(y) =⟨∫

ϕ

(x− y) dµ(y), e⟩

E

=⟨ϕ

∗ µ(x), e⟩

E,

5Recall the definition of 〈·, ·〉 in Subsection VII.1.2.

446 Appendix

so that the last integral in (2.0.19) equals∫ ⟨

ϕ

∗ µ(x), ψ(x) ⊗ e⟩

Edx =

∫〈ϕ

∗ µ, u〉E dx = 〈ϕ

∗ µ, u〉 .


The Riesz Representation Theorem

Finally, we prove a generalization of the classical (scalar) Riesz representationtheorem.

2.0.4 Theorem Let X be a σ-compact metrizable space and let E be a Banachspace. Then

C0(X,E)′ = MBV (X,E′)


〈µ, u〉C0 :=∫

X

u dµ , µ ∈MBV (X,E′) , u ∈ C0(X,E) .

Proof Suppose that µ ∈MBV (X, E′). Then it follows from (2.0.4) that

(u 7→

∫u dµ

) ∈ C0(X, E)′

and∣∣∣∫

u dµ∣∣∣ ≤ ‖µ‖BV ‖u‖∞ , u ∈ C0(X,E) . (2.0.20)

Conversely, let w ∈ C0(X,E)′ be fixed. Then, given any e ∈ E, the scalar Rieszrepresentation theorem (e.g., [Rud70, Theorems 2.14, 6.2, and 6.19]) implies theexistence of a unique regular K-valued Radon measure µe on X satisfying

w(ϕ⊗ e) =∫

ϕdµe , ϕ ∈ C0(X) , (2.0.21)

and

‖µe‖BV = sup‖ϕ‖∞≤1

∣∣∣∫

ϕdµe

∣∣∣ = sup‖ϕ‖∞≤1

|w(ϕ⊗ e)| ≤ ‖w‖C′0|e|E .

Since, by uniqueness, the map E → (BX → K), e 7→ µe is linear, it follows that⟨µ(B), e

⟩:= µe(B) , B ∈ BX , (2.0.22)

defines a map µ : BX → E′ that is easily seen to be finitely additive.


In order to show that µ has bounded variation we first consider a familyO1, . . . ,Om of pair-wise disjoint open subsets of X. For each j ∈ 1, . . . , mwe choose ϕj ∈ C0(X) with supp(ϕj) ⊂ Oj and ‖ϕj‖∞ ≤ 1, as well as ej ∈ BE .Then, putting

εj := sign w(ϕj ⊗ ej) ∈ Kand using (2.0.21),

m∑

j=1

∣∣∣∫

Oj

ϕj dµej

∣∣∣ =m∑

j=1

|w(ϕj ⊗ ej)| =m∑

j=1

εjw(ϕj ⊗ ej)

= w( m∑

j=1

ϕj ⊗ εjej

)≤ ‖w‖C′

0.

Consequently,

m∑

j=1

∣∣µej

∣∣ (Oj) ≤ ‖w‖C′0

, ej ∈ BE , 1 ≤ j ≤ m . (2.0.23)

Now let π := B1, . . . , Bm be an arbitrary partition of X into Borel sets.Then, given ε > 0, we find for each j ∈ 1, . . . ,m an element ej ∈ BE such that

|µ(Bj)|E′ ≤∣∣⟨µ(Bj), ej

⟩E

∣∣ + ε/m . (2.0.24)

The regularity of µej implies the existence of a compact subset Kj of Bj with∣∣µej

∣∣ (Bj) ≤∣∣µej

∣∣ (Kj) + ε/m . (2.0.25)

Since Kj ∩Kk = ∅ for j 6= k we find pair-wise disjoint open sets O1, . . . ,Om in Xsuch that Kj ⊂ Oj for 1 ≤ j ≤ m. Now we infer from (2.0.23)–(2.0.25) that

m∑

j=1

|µ(Bj)|E′ ≤m∑

j=1

∣∣⟨µ(Bj), ej

⟩E

∣∣ + ε ≤m∑

j=1

∣∣µej

∣∣ (Bj) + ε

≤m∑

j=1

∣∣µej

∣∣ (Kj) + 2ε ≤m∑

j=1

∣∣µej

∣∣ (Oj) + 2ε ≤ ‖w‖C′0+ 2ε .

Hence µ is of bounded variation.

Let now (Bj) be a sequence in BX such that Bj ∩Bk = ∅ for j 6= k, and putB :=

⋃Bj . Then it is clear that

∑

j

|µ(Bj)| ≤ ‖µ‖BV < ∞ .

448 Appendix

Thus⟨∑

j µ(Bj), e⟩

Eis well-defined for e ∈ E and

⟨∑

j

µ(Bj), e⟩

E=

∑

j

⟨µ(Bj), e

⟩E

=∑

j

µe(Bj) = µe(B) =⟨µ(B), e

⟩E

(2.0.26)

by the σ-additivity of µe. Since (2.0.26) is true for every e ∈ E, and E separatesthe points of E′, it follows that µ(B) =

∑j µ(Bj). Now (2.0.11), (2.0.21), (2.0.22),

and the density of C0(X) ⊗ E in C0(X,E) imply

〈w, u〉C0 =∫

u dµ , u ∈ C0(X,E) .

From this and (2.0.20) the assertion follows. ¥

The above proof of the generalized Riesz representation theorem is due toW. Arendt (personal communication).

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List of Symbols

Special Symbols448

448∗ q . . . . . . . . . . . . . . . . . . . . . . . . . . 428, 439sθ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281/rθ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28[`, d, ν] . . . . . . . . . . . . . . . . . . . . . . . . . . . 32[η] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43[m, ν] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

xi; xi . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7〈x〉 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.=η

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

[[·]] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33ω

qi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332

σt . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34, 37σν

t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43f ∼ g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39k(ν) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54t(j, q) . . . . . . . . . . . . . . . . . . . . . . . . . . . 300

Sets

J∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4JK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Sϑ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299Hd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299K∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Kd

k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Sd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5∂jK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5∂r(j)K . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42dK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334dIK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

Maps

t q ξ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33rQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44λy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146MQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188〈·, ·〉 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85〈〈·, ·〉〉 . . . . . . . . . . . . . . . . . . . . . . . . . . 20, 21〈u′, u〉S . . . . . . . . . . . . . . . . . . . . . . . . . . . 16〈u′ qu〉F . . . . . . . . . . . . . . . . . . . . . . . . . 413⊗ . . . . . . . . . . . . . . . . . . . . . . . . . . 388, 391⊗ q. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417⊗π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393⊗ε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395∼⊗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

Topological Concepts

BQ(x, t) . . . . . . . . . . . . . . . . . . . . . . . . . 188SQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44BX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299nS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

Operator Families

Hkz(Z, E) . . . . . . . . . . . . . . . . . . . . . . . . . 45

Mη(Rd, E) . . . . . . . . . . . . . . . . . . . . . . . 57P L(E1, E0); κ, ϑ . . . . . . . . . . . . . . . . 55Ps Z,L(E); κ . . . . . . . . . . . . . . . . . . . . 66Ps Z,L(E1, E0); κ, ϑ . . . . . . . . . . . . . 58

Linear Operators

1X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34T s

q,[η] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271δ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380(RK, EK) . . . . . . . . . . . . . . . . . . . . . . . 8, 14γk

j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333γ∂H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300γj

∂H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300

457© Springer Nature Switzerland AG 2019H. Amann, Linear and Quasilinear Parabolic Problems, Monographsin Mathematics 106, https://doi.org/10.1007/978-3-030-11763-4

458 List of Symbols

γdK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334∂n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354∂j

n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300∂k

nj. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

~γk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301~γ ~mI

I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

Spaces of Linear Operators

L(E0, E1; E2) . . . . . . . . . . . . . . . . . . . 224

Point-Wise Defined Functions

B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78BC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79BC∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80BC

m/νη . . . . . . . . . . . . . . . . . . . . . . . . . . 272

Cs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257C

m/ν0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Cs/ν0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

BUC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79BUCm/ν . . . . . . . . . . . . . . . . . . . . . . . . . 80BUCs/ν . . . . . . . . . . . . . . . . . . . . . . . . . 157BUC

m/νη . . . . . . . . . . . . . . . . . . . . . . . . 273

D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373OM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374`r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20`sr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

bρ/ν

pppp∞,surj . . . . . . . . . . . . . . . . . . . . . . . . . . 345

bucs/ν . . . . . . . . . . . . . . . . . . . . . . . . . . . 157c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19cs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258c0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20cs0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

cc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Integrable Functions

Bt/νr . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Bs/νq,r;η . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

Hs/νq;η . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

L∗r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

Lp,K . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430Lp,c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430Lq,ρ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

Wm/ν∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

Wm/ν

q;η . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

Wm/ν

q . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Ws/ν

q . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

Ws/νp . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

`s/νr X . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

cs/ν0 X . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Distributions

Bsq,r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

Bs/νq,r . . . . . . . . . . . . . . . . . . . . . . . 104, 129

bs/νq,r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

C−m/ν0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Fs/νq,r . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

Hs/νq . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

Ws(~k,p)/ν

ppppp . . . . . . . . . . . . . . . . . . . . . . 326

D′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256D′

+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436D′

K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436MBV . . . . . . . . . . . . . . . . . . . . . . . . 82, 441O′

C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439S ′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11B

s/νq,r . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Fs/νq . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

Ft/νq,B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360

Fs/ν

q,~γk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

Fs/νq,r (H+,H−; E) . . . . . . . . . . . . . . . . 354

Fs/νq (H+,H−; E) . . . . . . . . . . . . . . . . 357

∂Fs(~k,q)/ν

ppppq . . . . . . . . . . . . . . . . . . . . . . 301

∂Ft/ν

ppppq . . . . . . . . . . . . . . . . . . . . . . . . . . . 300

Differential Operators

∇ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77∂α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11∂ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77∂a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146B(Cm). . . . . . . . . . . . . . . . . . . . . . . . . . . 355a(D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440

Fourier Transforms, Multipliers, andConvolutions

Jz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Jz

η . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Hz(Z, E) . . . . . . . . . . . . . . . . . . . . . . . . . 56M(Rd, E) . . . . . . . . . . . . . . . . . . . . . . . . 54Ma(Rd, E) . . . . . . . . . . . . . . . . . . . . . . 171

List of Symbols 459

Mi(Rd, E) . . . . . . . . . . . . . . . . . . . . . . 171m(D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52u ∗ ϕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375

Interpolation Theory

J(θ, E0, E1) . . . . . . . . . . . . . . . . . . . . . 212

Norms and Seminorms

Λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43‖·‖s,q,J . . . . . . . . . . . . . . . . . . . . . . . . . . 257‖·‖s,q,r,J . . . . . . . . . . . . . . . . . . . . . . . . 258‖·‖∗s/ν ,p,r . . . . . . . . . . . . . . . . . . . . . . . . 149|||·|||m/ν ,q;η . . . . . . . . . . . . . . . . . . . . . . 272|||·|||m/ν ,r . . . . . . . . . . . . . . . . . . . . . . . . . 79

|||·|||∗s/ν ,p,r . . . . . . . . . . . . . . . . . . . . . . . 154|||·|||s/ν,p,r . . . . . . . . . . . . . . . . . . . . . . . 150[[·]]θ,p,r;i . . . . . . . . . . . . . . . . . . . . . . . . . 154[·]θ,q,J . . . . . . . . . . . . . . . . . . . . . . . . . . . 257[·]θ,q,r,J . . . . . . . . . . . . . . . . . . . . . . . . . 257

[·]δϑ,∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . 159[·]s,p,r . . . . . . . . . . . . . . . . . . . . . . 149, 213N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39|µ| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441pm,K . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373qk,m . . . . . . . . . . . . . . . . . . . . . . . . . . 5, 374

Index

adapted, Ω-, 67adjoint corner, 4admissible

ν-, 67ν– Banach space, 173

annihilator, 94augmentation, parametric, 43

Banachν-admissible – space, 173– space of bounded functions, 78

barreled, 370Besov space, 104

–Holder space, 155–Slobodeckii space, 155Fourier multiplier theorem for –s,

113little, 107very little, 107

Besselν-anisotropic – kernel, 53– potential space, 168

bornological, 372

clustering, d-, 33cofinal, 372condition (Σ), 432

distributions satisfying –, 432cone, 42

dual, 432proper, 432

conuclear, 393convolution, 428, 439

– of distributions, 379core, 137corner

adjoint, 4face of a, 5general, 6

standard, 4type of a, 5

covering, ν-dyadic, 67

derivativedirectional, 146distributional, 11jump of a normal, 357

dilation associated with a weight sys-tem, 33

Dirac distribution, 380directional derivative, 146distribution

– satisfying condition (Σ), 432Dirac, 380tempered, 10

dual exponent, 21duality

– pairing, 11, 20, 21, 415distributional – pairing, 85

embeddingRellich–Kondrachov - theorem, 255

equivalentη-uniformly, 58, 273uniformly, 273

Euclidean ν-quasinorm, 40extension

point-wise, 224trivial, 8

face of a corner, 5Fatou property, 200Fourier multiplier, 52

– space, 54– theorem for Besov spaces, 113

Fubini’s rule, 417function

Banach space of bounded –s, 78

© Springer Nature Switzerland AG 2019H. Amann, Linear and Quasilinear Parabolic Problems, Monographsin Mathematics 106, https://doi.org/10.1007/978-3-030-11763-4

461

462 Index

Heaviside, 423space of bounded and uniformly con-

tinuously differentiable –s, 80

Hardy inequalities, 312Heaviside function, 423Holder, generalized – inequality, 217Holder space, 157

– scale, 157Besov–, 155little, 157

homogeneousν– of degree z, 39positively z–, 39

inductive limit topology, 372inequality

generalized Holder, 217Hardy –ies, 312Sobolev, 217

integral with respect to vector measures,442

interface, 353

jump, 355, 357– of a normal derivative, 357

kernelν-anisotropic Bessel, 53– theorem, 398regularizing, 428

Kondrachov, Rellich– embedding theo-rem, 255

Kronecker symbol, 19

Leibniz rule, 385, 410linear representation, 34locally convex direct sum, 19

map– associated to a kernel, 427canonical, 395proper, 432

matrix, transposed, 392measure

vector, 441weighted counting, 27

metric, ν-parabolic, 42

multiplication, 47, 383– algebra, 413point-wise, 384, 409

multiplierFourier, 52Fourier – space, 54Fourier – theorem for Besov spaces,

113universal point-wise – space, 226

normquotient, 26uniformly equivalent, 273

normal– boundary operator, 344– inner (unit), 299

ν-admissible, 67– Banach space, 173

ν-dyadic– covering, 67– partition of unity, 68

ν-starshaped, 67ν-homogeneous weight system, 33ν-parabolic

– metric, 42– quasinorm, 42

nuclear, 393

operatornormal boundary, 344normal transmission, 356trace, 283translation-invariant, 440transmission, 355

order sequence, 348

pairr-c, 4r-e, 9, 14restriction-extension, 9, 14retraction-coretraction, 4universal retraction-coretraction, 104

pairingdistributional duality, 85duality, 11, 20, 21, 415

parametric augmentation, 43partition of unity, ν-dyadic, 68

Index 463

point-wise– extension, 224– multiplication, 384, 409– multiplier space, 226– restriction, 7

polar set, 94product

injective tensor, 395projective tensor, 393scalar – induced by multiplication,

414tensor, 388, 390, 417

proper, 432property

– (α), 171Fatou, 200

pull-back, 11

quasinormν-, 39Euclidean ν-, 40natural ν-, 39, 43ν-parabolic, 42

quasisphere, Q-, 44quotient norm, 26

Rellich–Kondrachov embedding theorem,255

restriction-extension pair, 9, 14retraction, Q-, 44retraction-coretraction

pair, 4universal – pair, 104

ruleFubini’s, 417Leibniz, 385, 410

scaleHolder space, 157little Holder space, 157Sobolev–Slobodeckii space, 157very little Holder space, 157

Schwartz kernel theorem, 427

SlobodeckiiBesov – space, 155Sobolev– space scale, 157space, 157

Sobolev– space of negative order, 86–Slobodeckii space scale, 157anisotropic Lq – space, 84inequality, 217

space– of Lp functions with compact sup-

port, 430– of bounded and uniformly contin-

uously differentiable functions,80

– of tempered distributions, 10– of test functions, 373anisotropic Lq Sobolev, 84Banach – of bounded functions, 78Besov, 104Besov – on corners, 129Besov–Holder, 155Besov–Slobodeckii, 155Bessel potential, 168Fourier multiplier, 54Holder, 157Holder – scale, 157LF –, 372little Besov, 107little Holder, 157Montel, 372ν-admissible Banach, 173Schwartz, 374Slobodeckii, 157Sobolev – of negative order, 86Sobolev–Slobodeckii – scale, 157universal point-wise multiplier, 226very little Besov, 107very little Holder, 157weighted sequence, 26

standard corner, 4starshaped, ν-, 67support, convolutive, 432

tempered distribution, 10

tensor product, 388, 390, 417

theoremBanach–Steinhaus, 371convolution, 439Fourier multiplier, 113kernel, 398

464 Index

Rellich–Kondrachov embedding, 255Schwartz kernel, 427

topologyinductive limit, 372injective tensor product, 395LF –, 372projective tensor product, 393

trace operator, 283translation semigroup, 146translation-invariant operator, 440transmission operator, 355

normal, 356trivial

– extension, 8– weight system, 42

type– of a boundary operator, 343– of a corner, 5

uniformly– equivalent, 273– surjective, 345η-, 58, 273

universal– point-wise multiplier space, 226– r-e pair, 15– retraction-coretraction pair, 104-ity property, 391

variation, 441bounded, 441

vector measure, 441

weight system, 32dilation associated with a, 33non-reduced, 33ν-homogeneous, 33parabolic, 33reduced, 33trivial, 42

weighted– counting measure, 27– sequence space, 26

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