Sobolev Spaces on Domains

TEUBN ER -TEXTEzur Mathematik Band 137

Victor I. Burenkov

Sobolev Spaceson Domains

B. G. Teubner Stuttgart Leipzig

TEUBNER-TEXTE zur Mathematlk Band 137

V. I. Burenkov

Sobolev Spaces on Domains

TEUBNER-TEXTE zur Mathematik

Herausgegeben vonProf. Dr. Jochen Broning, BerlinProf. Dr. Herbert Gajewski, BerlinProf. Dr. Herbert Kurke, BerlinProf. Dr. Hans Triebel, Jena

Die Reihe soil sin Forum fOr BeitrAge zu aktuellen Problemstellungen derMathematik sein. Besonderes Anliegen let die Verbffentlichung von Darstellun-gen unterschiedlicher methodischer AnsAtze, die das Wechselsplel zwischenTheorie and Anwendungen sowle zwischen Lehre and Forschung reflektleren.Thematlache Schwerpunkte sind Analysis, Geometrie and Algebra.

In den Texten sollen sich sowohl Lebendigkeit and Originalitat von SpezieNor-lesungen and Seminaren ale such Diskussionsergebnlsse aus Arbeitsgruppenwiderspiegein.

TEUBNER-TEXTE erscheinen In deutscher odor englischer Sprache.

Sobolev Spaceson Domains

By Prof. Dr. Victor I. Burenkov

University of Wales, Cardiff

M B. G.Teubner Stuttgart Leipzig 1998

Preface

The book is based on the lecture course "Fnction spaces", which the authorgave for more than 10 years in the People's Friendship University of Russia(Moscow). The idea to write this book was proposed by Professors H. Triebeland H.A. SchmeiBer in May-June 1993, when the author gave a short lecturecourse for post-graduate students in the Friedrich-Schiller University Jena.

The initial plan to write a short book for post-graduate students was trans-formed to wider aims after the work on the book had started. Finally, the bookis intended both for graduate and post-graduate students and for researchers,who are interested in applying the theory of Sobolev spaces. Moreover, themethods used in the book allow us to include, in a natural way, some recentresults, which have been published only in journals.

Nowadays there exist numerous variants and generalizations of Sobolevspaces and it is clear that this variety is inevitable since different problemsin real analysis and partial differential equations give rise to different spaces ofSobolev type. However, it is more or less clear that an attempt to develop atheory, which includes all these spaces, would not be effective. On the otherhand, the basic ideas of the investigation of such spaces have very much incommon.

For all these reasons we restrict ourselves to the study of Sobolev spacesthemselves. However, we aim to discuss the main ideas in detail, and in such away that, we hope, it will be clear how to apply them to other types of Sobolevspaces.

We shall discuss the following main topics: approximation by smoothfunctions, integral representations, embedding and compactness theorems, theproblem of traces and extension theorems. The basic tools of investigation willbe mollifiers with a variable step and integral representations.

Mollifiers with variable step are used both for approximation by smoothfunctions and for extension of functions (from open sets in R" in Chapter 6and from manifolds of lower dimensions in Chapter 5). All approximationand extension operators constructed in these chapters are the best possible in

6 PREFACE

the sense that the derivatives of higher orders of approximating and extendingfunctions have the minimal possible growth on approaching the boundary.

Sobolev's integral representation is discussed in detail in Chaper 3. It isused in the proofs of the embedding theorems (Chapter 4) and some essen-tial estimates in Chapter 6. An alternative proof of the embedding theorems,without application of Sobolev's integral representation, is also given.

The direct trace theorems (Chapter 5) are proved on the basis of someelementary identities for the differences of higher orders and the definition ofNikol'skii-Besov spaces in terms of differences only.

The author pays particular attention to all possible "limiting" cases, includ-ing the cases p = oc in approximation theorems, p = 1 in embedding theoremsand p = 1, oc in extension theorems.

There are no references to the literature in the main text (Chapters 1-6):all relevant references are to be found in Chapter 7, which consists of briefnotes and comments on the results presented in the earlier chapters.

The proofs of all statements in the book consist of two parts: the ideaof the proof and the proof itself. In some simple or less important cases theproofs are omitted. On the other hand, the proofs of the main results aregiven in full detail and sometimes alternative proofs are also given or at leastdiscussed. The one-dimensional case is often discussed separately to providea better understanding of the origin of multi-dimensional statements. Alsosharper results for this case are presented.

It is expected that the reader has a sound basic knowledge of functionalanalysis, the theory of Lebesgue integration and the main properties of thespaces L9(SI). It is desirable, in particular, that he/she is accustomed to ap-plying Holder's and Minkowski's inequalities for sums and integrals. The bookis otherwise self-contained: all necessary references are given in the text orfootnotes. Each chapter has its own numeration of theorems, corollaries, lem-mas, etc. If you are reading, say, Chapter 4 and Theorem 2 is mentioned, thenTheorem 2 of Chapter 4 is meant. If we refer to a theorem in another chapter,we give the number of that chapter, say, Theorem 2 of Chapter 3.

For more than 30 years the author participated in the famous seminar "Thetheory of differentiable functions of several variables and applications" in theSteklov Institute of Mathematics (Moscow) headed at different times by Pro-fessors S.L. Sobolev, V.I. Kondrashov, S.M. Nikol'skiT, L.D. Kudryavtsev andO.V. Besov. He was much influenced by ideas discussed during its work and,in particular, by his personal talks with Professors S.M. Nikol'skiI and S.L.Sobolev.

It is a pleasure for the author to express his deepest gratitude to the partic-

PREFACE 7

ipants of that seminar, to his friends and co-authors, with whom he discussedthe general plan and different parts of the book.

I am grateful to my colleagues in the University of Wales Cardiff: ProfessorW.D. Evans, with whom I have had many discussions, and Mr. D.J. Harris,who has thoroughly read the manuscript of the book.

I would also like to mention Dr. A.V. Kulakov who has actively helped intyping the book in 'IBC.

Finally, I express my deepest love, respect and gratitude to my wife Dr.T.V. Tararykova who not only typed in TIC a considerable part of the bookbut also encouraged me in all possible ways.

Moscow/Cardiff, November 1997 V.I. Burenkov

Contents

Notation and basic inequalities 11

1 Preliminaries 151.1 Mollifiers ... . ... .. .. .. ..... .... ... ...... 15

1.2 Weak derivatives .. .. .. ....... .... . .. .. .... 18

1.3 Sobolev spaces (basic properties) ..... ... . .. . ..... 28

2 Approximation by infinitely differentiable functions 392.1 Approximation by C000-functions on R" . .... .. . .. .... 392.2 Nonlinear mollifiers with variable step .. . ..... . ...... 422.3 Approximation by C°°-functions on open sets .. . ....... 472.4 Approximation with preservation of boundary values .. .... 562.5 Linear mollifiers with variable step ...... . . . .. .. ... 602.6 The best possible approximation with preservation of boundary

values . . ... . .. .. . . . . . . ... ... .. . .. .. ... 73

3 Sobolev's integral representation 813.1 The one-dimensional case ... .. ..... ..... .. ..... 813.2 Star-shaped sets and sets satisfying the cone condition ..... 923.3 Multidimensional Taylor's formula ... .. . . . ......... 1003.4 Sobolev's integral representation .... ...... ... ..... 1043.5 Corollaries .. .. ......... .. . ...... . .. .. ... 111

4 Embedding theorems 1194.1 Embeddings and inequalities .................... 1194.2 The one-dimensional case .... . ..... ...... ...... 1274.3 Open sets with quasi-resolvable, quasi-continuous, smooth and

Lipschitz boundaries .......... ..... . .... .. .. 1484.4 Estimates for intermediate derivatives . ..... .. ....... 1604.5 Hardy-Littlewood-Sobolev inequality for integral of potential type177

10 CONTENTS

4.6 Embeddings into the space of continuous functions ....... 1814.7 Embeddings into the space Lq . .............. . ... 186

5 Trace theorems 1975.1 Notion of the trace of a function ... .. ... .. ..... ... 1975.2 Existence of the traces on subspaces ... ........ .... . 201

5.3 Nikol'skil-Besov spaces .......... .. .... .. .... . 202

5.4 Description of the traces on subspaces . ...... .. ... . . 214

5.5 Traces on smooth surfaces ... ....... .... .. .... . 238

6 Extension theorems 2476.1 The one-dimensional case ........... ...... . .. .. 2476.2 Pasting local extensions .......... . .. . ... . .. .. 2646.3 Extensions for sufficiently smooth boundaries .. .. . . . ... 2696.4 Extensions for Lipschitz boundaries .. .. . . . . .. .. . . .. 271

7 Comments 289

Bibliography 296

Index 311

Notation and basic inequalities

We shall use the following standard notation for sets:N the set of all natural numbers,1o-- the set of all nonnegative integers,Z the set of all integers,R - the set of all real numbers.C - the set of all complex numbers.Rlo = Nlo x x No - the set of multi-indices (rs is the natural number

nwhich will be used exclusively to denote the dimension),

R" = llt lR,n

B(x, r) - the open ball of radius r > 0 centered at the point x E R",'11 (S2 C R") - the complement of SZ in R",S2 (52 C Rn) - the closure of 12,S2 (S2 C R") - the interior of 0,126 (9 c R", 6 > 0) - the 6-neighborhood of 0 (S26 = U:en B(x, 6)),126={xE9 :list(x,812)>6} (S2CR",d>0) (for each 0CR"

16={xES2:dist(x,8S2)>6}).For a E 1%, a 96 0, we shall write:

D° f the (ordinary) derivative of the function f of order aand

DW f \B +,, q)W

the weak derivative of the function f of order a

(see section 1.2).For an arbitrary nonempty set fl C R" we shall denote by:

C(S2) - the space of functions continuous on 0,C6(SZ) - the Banach space of functions f continuous and bounded on 11

with the norm

III 11C(n) = sup If (x) I,

12 NOTATION

C(S2) - the Banach space of functions uniformly continuous and boundedon 12 with the same norm.

For a measurable nonempty set 11 C R" we shall denote by:LD(S2) (1 < p < oo) - the Banach space 1 of functions f measurable 2 on

) such that the norm

Of IIL,(n) _ (J I J I9dx) 1 < 00,n

L,°(S2) - the Banach space of functions f measurable on S2 such that thenorm

If IIL-(n) = ess sup If (x)I = inf - sup If W1 < 00sEn w:measw=0 yEn\w

(in the case in which meas 1 > 0 3 ; if meas S2 = 0, then we setIIf IIL (n) = 0).

For an open nonempty set f2 C R" we shall denote by:L, (QZ) (1 < p < oo) - the set of functions defined on SZ such that for

each compact K C SZ f E Ln(K),5C1 (fl) (l E N) - the space of functions f defined on 0 such that Va E 1V$

where Ial = al + . + a" _ l and b'x E 11 the derivatives(D° f)(x) exist and D* f E C(O),

Cb(9) (l E N) - the Banach space of functions f E Cb(fl) such thatVa E N where j al = I and Vx E SZ the derivatives (D° f)(x) existand DI f E Cb(1l), with the norm

IIf IIo(n) = If 11c(n) + E II D°f IIc(n),IaI=t

i As usual when saying a "Banach space" we ignore here the fact that the conditionIIfIIL,(n) = 0 is equivalent to the condition f - 0 on Cl (i.e., f is equivalent to 0 on fl e=:mess {x E Cl : f (x) # 0) = 0) and not to the condition f = 0 on Cl. To be strict we ought tocall it a "semi-Banach space" (and it will be necessary to keep this fact in mind in Section4.1) or consider classes of equivalent functions instead of functions. The same applies to thespaces and W,(fl) below.

2 "Measurable" means "measurable with respect to Lebesgue measure." All the integralsthoughout the book are Lebesgue integrals.

s We need to do so because otherwise if meal f i = 0, then by the convention sup 0 = -oowe have ess sup If (x) I = -oo.

:enIf Cl C R" is an open set, then for f e C(fl) IIfIIc(n) = IIfIIL.,(n)-

a fk - f in La (fl) as k - oo means that for each compact K C fl fl, -i f in L,(K).

NOTATION 13

C'(Sl) (1 E N) - the Banach space of functions f E C(Q) such thatVa E 1`l0` where Jai = I and dx E Sl the derivatives (D° f)(x)exist and D* f E C(S1), with the same norm.

00

C°°(Sl) = n o(Q) -- the space of infinitely continuously differentiable1=0

functions on Sl,Co (Sl) - the space of functions in C' (0) compactly supported in Q,tit'n(Q) (1 E N, I < p < oo) - Sobolev space, which is the Banach space

of functions f E L,(Sl) such that Va E t where jai = I the weakderivatives DW f exist on Q and Dc ,,,f E L,(1), with the norm

Ill lw;(n) = JIDWfllr.,(n)101=1

(see Section 1.3),

wl,(Q) (I E N.1 < p < cc) the semi-normed Sobolev space, which is thesemi-Banach space of functions f E L;a(St) such that Va E Nwhere Jai = 1 the weak derivatives Dw f exist on Q andDW f E L,(Sl), with the semi-norm

IllIlw,(n) = E IIDwfIILP(n)Ia1=1

W, (f2) (1 E N, I < p < oo) - the Banach space of functions f E Lp(11)such that Va E No' where I al < l the weak derivatives Dw f existon Sl and D. ('f E L,(Sl), with the norm

IllIIfpP(n) = E IIDwf lILo(n)I°I<I

(see Sections 2.3 and 4.4).(11')o(0) (1 E N, 1 < p < oe) the space of functions in W,(Q) compact-

ly supported in Qand, finally,

WW(I) (I E N, 1 < p < oo) - the closure of Co (S2) in WW (Sl). 6

Further notation will be introduced in the text.

(see Section 1.3),

6 In general, if Z(fl) is a space of functions defined on an open set fl C 1°, then Z0(f))will denote the space of all functions in Z(f)) compactly supported in fl and Z(f)) -- theclosure of C01(0) in the topology of Z(fl) (if Co (f)) C Z(fl)).

14 BASIC INEQUALITIES

Let 1l C W' be a measurable set and 1 < p < oc.

Holder's inequality. Suppose that 1 + 1 = 1, i.e., p' = i1 for 1 < p < oo,p'=ocforp=1andp'=1forp=oo. If f ELp(12) and9ELp,(Q),thenfg E L1(1l) and

llf 9UL,(n) <_ IlfIIL,(,z)11911L,,(+2)-

Minkowski's inequality. If f, g E Lp(12), then f + g E Lp(U) and

If +911L,(n) 5 IIfIIL,(n) + II9111.,,(s:)-

Minkowski's inequality for integrals. In addition. lot .4 C RW" be asurable set. Suppose that f is measurable on A x 11 and f (.. y) E L,(1?) foralmost all y E A. Then

IIf f y) dyllL,(n) 5 J y) dyh,(n) dyA A

if the right-hand side is finite.

Similar inequalities hold for finite and infinite sums. Let Uk, bk E C. Then

Inkbk1 5 Iaklp)° (FIbkI'/)k=1 k=1 k=1

and` !

Iak + bklp) < ( Iaklp)1 + ( lbklp) P.k=1 / k=1=1 k=1

Here s E N or s = oo. (If p = oo, one should replace (E laklp), by sup lakl.)k k

Throughout the book we shall often use these basic inequalities (withoutadditional comments).

Chapter 1

Preliminaries

1.1 MollifiersLet w he a kernel of mollification, i.e..

fwdx=1.w E C, (R"), supp w C B(0,1), (1.1)

z^

For 6 > 0 and Vx E R" we set w5(x) = a!,w(s).

Definition 1 Let S2 C R" be a measurable set and 6 > 0. For a function fdefined on 11 and such that f E L1(S2 fl B) for each ball B, the operator A6A6.1z (a mollifier with step (or radius) 6) is defined by the equality !: `dx E R"

(A6f)(.r) = (wd * fo)(a-) = II ( y).f (y)dy = f fo(x - 6>) w(z)dz.

B(0,1)

(1.2)

We recall that for each function f under consideration A6f E CO°(R"),VnEl

on R" and

D°A6f = 6-IQI (D°y) i *.fo (1.3)

supp A6f C (supp f )J. (1.4)

1 Here and in the sequel fe denotes the extension of f by zero outside Cl: fo(x) = f(x)

for r E fl and fo(x) = 0 for x E `fl.

16 CHAPTER 1. PRELIMINARIES

We note also that on S26

r(A6f)(x) = (w6* f)(x) =

Jf(x - bz)w(z)dz,

B(o,1)

and We E No"

D°A6f = a-1°1(Dow)6 * f.

If 5l C lR is an open set and f E Ll°`(S1), then A6f E C°°(S2a) and

A6f -a f a.e.2 on S2

as d -a 0+ (if f E C(Q). then the convergence holds everywhere on 5l). ForI <p<ooandVf ELp(c1)

IIA6fIILp(a,) 5 1- 1If1IL,(n). (1.6)

Moreover, for each measurable set G C R'

IIA6fII1,,(o) <_ C (1.7)

Here c = 11wlIL,(a^) (c = 1 for a nonnegative kernel w; if, in addition, thefunction f is nonnegative, then IIA6fIIL,(R^) = IIfIIL,(n))

Furthermore, 3II Asf - f IIL,(sa) <- c w(d, f)L,(n), (1.8)

wherew(6, f)L,(n) = sup IIfo(x + h) - f (x) II L,(n)

NOis the modulus of continuity of the function f in Lp(Q).

From (1.8) it follows that for 1 < p < oc and V f E Lp(S2)

A6f -a f in Lp(S2) (1.9)

as 6 -+ 0+. For p = oo for any kernel of mollification this relation in generaldoes not hold.

From (1.9) it follows that for 1 < p < 00

IIA6f IIL,(n) -} IIf IIL,(a) (1.10)

2 a.e.= almost everywhere.3 See also Lemma 12 of Chapter 5.

1.1. MOLLIFIERS 17

as b - 0+. We note that for nonnegative kernels w this relation holds also forp = oo. (If the kernel w changes its sign, this relation in general does not hold.For example, if n = 1, w(x) < 0 on (-1, 0) and w(x) > 0 on (0.1), then dd > 0we have IIA6(sgnx)IIL-(R) = IIWIIL,(R) > 1.)

If SZ C ]R", then the function 4 77 = A;; constructed with the help of anonnegative kernel is a function of "cap-shaped" type, i.e.,

rIEC°°(IR"), 0<,i<1, i=1on52, supp77C526, (1.11)

and

I (D°rl)(x)J < c°b-1°',

where c° depends only on n and (k.If the function f satisfies the Lipschitz condition on IR", i.e., if for some

it/>0anddx,yElR"

If (x) - f(y)I <- MIX - yI,

then `db>0and`dx,yER"

I (A6f)(x) - (A6f)(y)I <- cMIx - yl.

(1.12)

(1.13)

Thus for nonnegative kernels, in which case c = 1, the mollifier A6 completelypreserves the Lipschitz condition. If (1.12) holds for all x, y E 52, where 52 C RIis an open set, then (1.13) holds on 526.

The mollifier Aa defined by (1.2) with the kernel of mollification w'(x) _w(-x) replacing w(x) is the conjugate of the mollifier A6 in L2(S1). In particular,if the kernel w is real-valued and even, then the mollifier A6 is a self-adjointoperator on L2(52).

Finally, we note that for a measurable set 1 C R" and for any function fsuch that f E L, (52 (1 B) for each ball B

A6A.rf = A7A6f on

In particular,

06+,.

A6A, = A.rA6 on Li `(1R" ).

4 Here and in the sequel XG denotes the characteristic function of a set G.


1.2 Weak derivativesWe shall start with the following observation for the one-dimensional case andfor an open interval (a, b), -oo < a C(a, b) 5

is a closed operator in C(a, b), i.e., if fk E C' (a, b), k E N, f, g E C(a, b) and

fk - f, -4 g in C(a, b)

ask oo, 5 then f E C' (a, b) and 9- = g on (a, b).Suppose now 1 LP "(a, b) (1.14)

is not closed in Lp (a, b).

Example 1 Let (a, b) _ (-1,1) and dx E (-1,1) set f (x) _ Ix 1, fk(x) _(x2 + k)'/2, k E N. Then fk Jxi, f' -> sgnx even in Lp(-1,1), but Jxi VC'(-1, 1) (and lxi' does not exist on the whole interval (-1,1)).

Idea of the proof. This follows easily by direct calculation. O

For this reason it is natural to study the closure of the operator (1.14) inLp (a, b). This is one approach leading to a generalization of the notion ofdifferentiation.

On the other hand if f E C' (a, b) and V E C' (a, b), then

b bf f c'dx = - f f'wdx.a a

This equality can also be naturally used to generalize the notion of differentia-tion, since for some functions (e.g., f (x) = jxj) the ordinary derivative does not

5 Here and in the sequel we shall write for brevity C(a, b), C(a, b), Lp(a, b), L' (a, b) etcinstead of C((a,b)),C((a,b)),Lp((a,b)),L',L',*'((etc.

6 By fk -+ f in C(a, b) we mean that Ilfk - f Ilcia.pl - 0 as k - oo for each dosed interval[a, 0] C (a, b). This definition is similar to that of convergence in LP (a, b) (see footnote 5on page 12).

1.2. WEAK DERIVATIVES 19

exist on (a, b), but a function g E Li°C(a, b) exists (in Example I g(s:) = sgnx)such that dcp E Co (a, b)

b b

f f p dx f gydx.a a

These approaches lead to strong, weak respectively, extensions of the clif-ferentiation operator.

We give now the corresponding definitions for the multidimensional caseand for differentiation of arbitrary order.

Definition 2 Let Q C R" be an open set. n E No. (t 36 0 and f. g E L;"` (S?).The function g is a weak derivative of the function f of order (t on S2 (bri(ffetg=Dwf) if

V, E Co (S2) ffD.dz _J

grd.r. (1.13)

ft !i

Lemma 1 Let 11 C R" be an open set, a E K, a 36 0. Moreover, let f be afunction defined on n, which dx E fl has an (ordinary) derivative (D°f)(r)and D* f E C(f1). Then D* f = Dw f .

Idea of the proof. By integrating by parts al times with respect. to the variahlesx,, j = 1, ..., n, show that

/`d(p E Co (S2) f f D°cpdx = (-1)i°i

JD°f cpdx. (1.16)

fl ft

(One may assume without loss of generality that S2 is bounded and considerinstead of f the extended function fo on a cube (-a, a)" D 0.) 0

Remark 1 The assumption about the continuity of DI f in Lemma 1 is es-sential. For example, the ordinary derivative of the function f (x) = x2 sin(x 36 0; f (0) = 0), which exists everywhere on R, is not a weak derivative of fon R because it is not locally integrable on a (See also Example 4.)

From Definition 2 it follows that if g = D.af and the function h is equivalenttog on fl, then h = D* .f also. Thus the weak derivative is not uniquely defined.The following lemma shows that it is the only way in which uniqueness fails.


Lemma 2 Let 11 C R" be an open set, a E No, a 9& 0, f, g, h E L'OC(S2) andg=Dwf , h=Dwf on Q. Theng - h on Q.

Idea of the proof. Use the main lemma of the calculus of variations. 0

Remark 2 Because of this nonuniqueness, the notation g = Dw f in Definition2 (which is not to be interpreped as equality of the functions g and Dw f) needssome explanation. To be strict, the binary relation = Dw on Ll01 is introduced:"g = Dw f " means "g is a weak derivative of the function f of order a on P".We also use Dw f for any weak derivative of the function f of order a on 9.Thus, for example, the assertion "the function f has a weak derivative Dw f"means "the function, denoted by Dw f, is a weak derivative of the function f oforder a on W. From this point of view the relation Dw fl +Dwf2 = DO(f1+f2)means the following: if each of Dw fk, k = 1, 2, is a weak derivative (i.e., anyof the weak derivatives) of the function fk, then the function Dw fl + D. *f2 isa weak derivative of the function fl + f2. Finally, we assume that D: f = gmeans g = Dw f . This will allow us to rewrite the above relation in the moreusual form Dw(f1 + f2) = Dwf1 +Dwf2

Remark 3 Note that if a function f E L;OC(c) has a weak derivative Dw f onS2, then automatically D, *f E L0 c(c).

Example 2 (n = 1, S2 = R) JxI , = sgnx.

Idea of the proof. This was discussed above. 0

Example 3 Let n = 1 and f E L10C(R), then, as is known from the theory ofx

Lebesgue integral, the function f f (y)dy is locally absolutely continuous 7 on0

X

R and (f f (y)dy)' = f (x) for almost all x E R. There can, of course, exist ana

x E R, for which either the derivative does not exist or exists but is different

from f (x). On the other hand, V f E L1OC(R) we have (f f (y)dy)1, = f (x) ona

R.

7 We recall that the function g is absolutely continuous on the closed interval [a.#] ifYe > 0 there exists d > 0 such that for each finite collection of disjoint intervals (af,#j) C

(a, $), j = 1, ..., a, satisfying E (01 - a j) < d one has t If (i3i) - f (aj)I < e.The function gJ-1 J=1

is locally absolutely continuous on the open set f1 C R if it is absolutely continuous on eachclosed interval [a,#) C A.


Idea of the proof. Integrate by parts. This is possible since f f (y)dy is locallya

absolutely continuous on R.

Example 4 Suppose that n > 2, 1 E N, the function f E C'(R" \ {0}), a Eand jai = 1. Then the weak derivative Dw f exists on R" if, and only if, the(ordinary) derivative D* f lies in Ll (B(0,1) \ {0}). If n = 1, then this statementholds for f E C'(R \ {0}) fl C'-'(R).

In particular, for n > 1, p E R and Va E Non, a 34 0, the weak derivativeD"(Jxj1) exists on R" if, and only if, either y > l - n, or p is a nonegative eveninteger < l - n.

Idea of the proof. For n > 2 integrate by parts, excluding the origin. For n = 1use Definition 4 below and the properties of absolutely continuous functions.

Example 5 (n = 1, 0 = R) The weak derivative (sgnx)V, does not exist on R.

Idea of the proof. Suppose that g E L' `(R) is a weak derivative. By integratingby parts show that VV .E CO '(R) f gcpdx = 2V(0). Taking V(x) = xo(x) with

R

arbitrary 0 E CO -(R), prove that f xg(x)O(x)dx = 0. Thus g - 0, which leadsR

to a contradiction.

Remark 4 For each f E L"°(S2) the derivative D* f exists in the sense of thetheory of distributions, i.e., as a functional in D'(St):

V E Co (ft) f fD°Wdx.n

In Example 5 (sgnx)' = 26(x), where 6 is the Dirac 6-function. From the pointof view of the theory of distributions the weak derivative D,f of a functionf E LlOc(f) exists if, and only if, the distributional derivative D* f is a regulardistribution, i.e., a functional represented by a function g E L;°c(Q):

VV E CO (0) (D°f, (p) = fgiPdx.n

This function g (defined up to equivalence on 11) is a weak derivative of thefunction f of order a on 0.


Definition 3 Let Il C R" be an open set, a E K, a 0 and f, g E LI°'(1).The function g is a weak derivative of the function f of order a on (2 (brieflyg = D,° f) if there exist t/ik E Coo (Q), k E N, such that

?Gk -} f, D°4/Jk -+ g in L"(0) (1.17)

as k --roo.

Theorem 1 Definitions 2 and 3 are equivalent.

Idea of the proof. 2 3. In (1.15) write for f and pass to the limit ask -a oc. 3 2. For k E N let Xk be the characteristic function of theset {x E 92 : jxj < k, dist (x.852) > k}. Functions Wk E C°°((2) (and evenWk E Co ((2)) are constructed in the following way: I'k = At(fXk), where .44is a mnollifier as in Section 1.1. 0

Definition 4 Let 51 C R be an open set, I E N and f, g E L]°c((2). The functiong is a weak derivative of the function f of order l on Q (briefly g = DI. f - f( V)

if there is a function h equivalent to f on 51, which has a locally absolutely con-tinuous (1-1)-th ordinary derivative 0-1> and such that its ordinary derivative0) is equivalent to g. (Recall that heel exists almost everywhere on 11.)

Theorem 2 In the one-dimensional case Definitions 2, 3 and 4 are equivalent.

Idea of the proof. It is enough to consider the case in which 51 = (a, b).4 2. Since V-1) is locally absolutely continuous on (a, b), it is possible

V, E C.o ((2) to integrate by parts l times:

b b b b

f f(pi'idx = f hp(')dx = (-1)e f hlolcpdx = (-1)i fa a a a

3 4. Let I = 1. Since Vik -+ f in Lr(a, b) as k -+ oo there existsa subsequence k, and a set G C (a, b) such that meas [(a, b) \ G] = 0 and?ik, (x) -a f (x) as s - oo for each x E G. Choose z E G and pass to the limit in

the equality iPk, (x) = t(ik,(z)+ f t/JjF, (y)dy. Then f (x) = f (z) + f g(y)dy =_ h(x)

for each x E G. By the properties of absolutely continuous functions thefunction h (which is defined on (a, b) and equivalent to h) is locally absolutelycontinuous on (a, b) and g - h'.


If 1 > 1, then apply the averaged Taylor's formula (3.15) with a < a < x <,6 < b to the functions Ok,. Write it in the form

Ok. (x) = f P(x, y)Ok.(y)dy + (1 11)! J (x - y)'-' ,(u)du)'Ok?(y)dya ° a

1 (x - y)'-' (J w(u)du),Ok. (y)dy(l - 1). fx Y

and argue as above. (Here p E C([a, b] x [a, b]), Vy E [a, b] y) is a polyno-mial of order less than or equal to l - 1 and w E Co (a, i3).) 0

The notion of a weak derivative, as the notion of an ordinary derivative,is a local notion in the following sense. If the function g E L1OO(Q) is a weakderivative of the function f E L1OC(1l) of order a E M, a 0, on l locally, i.e.,Vx E SZ there exists a neighbourhood Ux of x such that g is a weak derivativeof f of order a on Ux, then 8 g is a weak derivative of f of order a on St.

For an open set ) C R and a E M, a 96 0, let us denote by Ga(f) thedomain of the operator Dti, i.e., the subset of L1OC(0) consisiting of all functionsf E LiOO(Q), for which the weak derivatives Dw f exist on 12. We note that theweak differentiation operator

Dw : G,,(fl) -+ L'-(Q)

is closed, i.e., if the functions fk E Ga(ff) and the functions f,g E L' (Sl) aresuch that

fk -+ f in L1Oc(S2), Dwfk - g in L1OC(f ),

8Indeed, consider for an arbitrary w E Co (ft) a finite open covering {U.y }k=1 of suppVand the corresponding partition of unity (Ok}k-1, i.e., a family of fuctions >/,k E Co (U,),

which are such that E +yk = I on supp ip. (See Lemma 3 of Section 2.2.) Then ,p = E ,*kk=1 k=1

on Ii and

f fD°Vdx=E f fD°(k)dx=(-1)10IE f gw1,kdx=(-1)1°I f g,pdz.n k=1U. k=1U. n


then f E G°(S2) and Dc,,, f = g. The operator DO, considered as operator

Dw : G°(S2) fl Lp(5I) -a Lp(S2),

where 1 < p < oo, is also closed. In order to prove these statements it is enoughto write fk for f in (1.15) and let k -- oo.

Lemma 3 (Weak differentiation under the integral sign) Let SZ C W& be anopen set, A C R' a measurable set and let a E N, a 0 0. Suppose that thefunction f is defined on 52 x A, for almost every y E A f y) E L'°"(12)and there exists a weak derivative Dw f y) on Q. Moreover, suppose thatf, DW f E L1(K x A) for each compact K C 0. Then on S2

D (f f (x, y)dy) = f(Df)(x,v)dv. (1.18)

A A

Remark 5 According to Remark 2 formula (1.18) means the following: if fora function denoted by D. "f and defined on 0 x A for almost every y E A thefunction (DO y) is a weak derivative of order a of f y) on 0, then thefunction f (D. *f)(., y)dy is a weak derivative of order a off f y)dy on Sl.

A A

Idea of the proof. Use Definition 2 and Fubini's theorem.Proof. For all W E Co (S2) the functions f (x, y)(D°cp)(x) and (DO f)(x, y)w(x)belong to L1(Sl x A), because, for example,

f M f IfIdxdy < oo,

nxA supp,pxA

where M = manx I Therefore, starting from Definition 2, we canuse Fubini's theorem twice to change the order of integration and deduce thatVlp E Co (0)

f (f(1x,y)dy)1P(x)dx = f (f(x,y)'P(x)dx)dyn A A n

_ (-001 f (f f(x,y)(D°co)(x)dx)dy = (-1)1-l f (f f(x,y)dy)(D°'p)(x)dxA n n A

and (1.18) follows.


Lemma 4 (Commutativity of weak differentiation and the mollifiers) Let S2 CR" be an open set, a E No, a # 0, f E L"'(Q) and suppose that there exists aweak derivative D. f on Q. Then Vd > 0

D°(A6 f) = A5(Dw f) on Sta. (1.19)

Idea of the proof. Use Lemma 3.Proof. Recall that A6(Dwf) E C°°(116) (see Section 1.1). Moreover, Vx E Q

(A6f)(x) = 1 f(x - bz)w(z)dz.B(O,1)

Furthermore, D" (f ( - bz)) = (D" f) ( - bz), on i6i which follows from Defi-nition 2.

For (x, z) E S26 x B(0, 1), let F(x, z) = f (x - dz)w(z) and G(x, z) _(Dw f)(x - bz)w(z). Then for each compact K C Q6 the functions F, G be-long to L1(K x B(0,1)), because they are measurable on S26 x B(0,1) 9 and,for example,

f r rJ (J If(x-&z)w(z)Idz)dx<MJ ( J If(x-bz)Idz)dxK B(0,1) K B(0,1)

= M f ( f If(y)Idy)dx < MJ (J if(y)Idy)dxK B(x,6) K K&

= M measK f If(y)Idy < oo.

Kd

Here M = max Iw(z)I and K6 C i (because K C Z6). Now (1.19) follows from,,ER"

Lemmas 1 and 3: Vx E SZ

D°((Aef)(x)) = DW( f f (x - dz)w(z)dz) = f D.-(f (x - 6z))w(z)dz

B(0,1) B(0,1)

f (Dwf)(x - dz)w(z)dz = (A6(Dnif))(x)B(0,1)

9 We use the following fact from the theory of measurable functions: if a function g ismeasurable on a measurable set E C R", then the function G, defined by G(x, y) = g(x - y)is measurable on the measurable set {(x,y) E R2" : x - y E E} C R2".


Corollary 1 For ) = R"D.A6 = A6Dw. (1.20)

Corollary 2 If ry E N and ry> a,10 then

D''(A6f) = Dwf on 526. (1.21)

Idea of the proof. Use Lemma 4.Proof. Using the properties of mollifiers (Section 1.1), we can write

D"(A6f) = D"-°(D°(A6f)) = D"-°(A6(DWf))

= 810I-I1l(D'r-Qw)6 * Dwf

on S26 (we note that Dw f E L'" (n))-

Example 6 If 1 C R" is an open set, 52 96 R", then (1.20) does not hold on2, because, for f = 1 on 52, A6(D° f) = 0 on 52 and Dw(A6 f) 0 on 52 \ Q6.

In Definition 2 the weak derivative is defined directly (not by induction asie ordinary derivative). Therefore the question arises as to whether a weakerivative DWf, where Q < a, i3 # a, exists, when a weak derivative D' Ofxists. In general the answer is negative as the following example shows.

Example 7 Set d(xl, x2) E R2 A XI, x2) = sgn x1 + sgn x2. Then derivatives(e )w and (e )°, do not exist (see Example 2, while (e- )w = 0 on R2.

Idea of the proof. Direct calculation starting with Definition 2.

Nevertheless, in some important cases we can infer the existence of deriva-tives of lower order.

Lemma 5 Let 52 C R" be an open set, I E N, I > 2, f E Ll°`(52) and supposethat for some j = l,n a weak derivative (4)w exists on 52. Then Vm E N

satisfying m < 1 a weak derivative (gym )w also exists on 52.

10 Here and in the sequel y > a means that ' 3 > a3 for j =fin. We note also thatj = 1,n means j E {1,...,n}.


Idea of the proof. Apply the inequality

IafIIL,(Q) c' (IIfllL,(Q) + la f L,(Q)

where f E C'(Q), Q is any open cube with faces parallel to the coordinateplanes, which is such that Q C Sl and cl > 0 is independent of f. (See footnote3 in Section 3.1.)Proof. For sufficiently large k E N the functions fk = A, f E C' (Q). By (1.5)

and Lemma 4 fk - f in L1(Q) and = A ( ) a in LI (Q). Moreover,

11

&-fk - ma II < C, (04 - fall L,(Q) + IIfk _ O'f'

II /axe axe L,(Q) BxJ 8xt L,(Q)

Consequently,

lim j, -'11

= 0.k,a-,oo Ox ' 8xJ L, (Q)

Because of the completeness of L1(Q) there exists a function gQ E L1(Q) suchthat -* gQ in L1(Q) as k -+ oo. Since fk -4 f in L1(Q) as well, byDefinition 3 it follows that gQ is a weak derivative of order I with respect to x;on Q.

We note that if Q1 and Q2 are any intersecting admissible cubes then gQ, _gQ, almost everywhere on Q, n Q2 , since both gQ, and gQ, are weak derivativesof f on Q1nQz. Consequently, there exists a function g E Lla(Il) such thatg = gq almost everywhere on each admissible cube Q and g is a weak derivativeof f on Q. Hence, by Section 1.2 g is a weak derivative of f of order I withrespect to x3 on SI.

Lemma 6 Let n > 2, SI C R" be an open set, I E N, I > 2, f E L;°°(O) andsuppose that Va E N satisfying lal = I a weak derivative DO Of exists on S1.Then V O E N satisfying 0 < 1,31 0 isindependent of f, and the proof of Lemma 5.


Proof. The above inequality, by induction, follows from the inequality consid-ered in the proof of Lemma 5. For, if Q = (a, b)", then

IID1fIIL,(Q) =II...IIOx" (a

={...,gxp

c, \II .+0f IL(Q) + II8x- -...-Of292 ... II L,(Q))'

< ... < c2 (11fIIL,(Q) + IID°fI1L1(Q))101=1

The rest is the same as in the proof of Lemma 5.By writing fk for f in this inequality and taking limits we see that it is

possible to replace here the ordinary derivatives D° f, D' f by the weak onesDW f , DI If respectively. 11

Lemma 7 Let n > 2, 1Z C ]l2" be an open set, I E N, l > 2, f E L,0C(Q)and suppose that Vi E {1,...,n} a weak derivative (u),,, exists on fl Then

V,3 E N satisfying 0 < 1,31 < l a weak derivative DO .f also exists on Q. For101 = I in general a weak derivative Dwf does not exist, but if, in addition,for some p > 1 ( -00 $ ' ) w w ELr(cl), then a weak derivative Dw f does exist for

1131=1.

Idea of the proof. This statement is a corollary of Theorem 9 of Chapter 4.

1.3 Sobolev spaces (basic properties)Definition 5 Let 11 C B" be an open set, I E N, 1 < p < oo. The functionf belongs to the Sobolev space Wp(i2) if f E Lp(f), if it has weak derivatives

1 Moreover, starting by the appropriate inequality in footnote 3 of Chapter 3, by the sameargument it follows that

M(IIfIIL,(q) + E IIDWfIIL,(Q)),101_1

where 1 < p < co and M is independent of f. This inequality holds also for fl = a". Thisfollows by replacing Q by Qo + k, where Qo = {z E R" : 0 < x j < 1, j = 1, ...,n) and k re Z",raising these inequalities to the power p, applying to the right-hand side Holder's inequalityfor sums, adding all of them and raising to the power p. For more general open sets suchinequalities will be proved in Section 4.4.

1.3. SOBOLEV SPACES (BASIC PROPERTIES) 29

Dw f on S2 for all a E 1 satisfying I al = I and

IIfIIW,(n) = IIfllL,(n) + > IIDwfIIL,(tl) < 00. (1.22)

Remark 6 In the one-dimensional case this definition is by Definition 4 equiv-alent to the following. The function f is equivalent to a function h on S2, forwhich the (ordinary) derivative h('-') is locally absolutely continuous on S2 and

IIf16L(n) = IIfIIr.9(11) + IIfW"IILP(n) = IIhIIL,(n) + 00.

Moreover, if S2 = (a, b) is a finite interval, the limits lim h(x) and lim h(x)

exist and one may define h on [a, b] by setting h(a) and h(b) to be equal tothose limits. Then h('), s = 1, ..., l - 1, exist and h('-') is absolutely continuouson [a, b]. This follows from the Taylor expansion

'-e-t h(3+k) xh(s)(x) = E

k! o) (x - xo)k +(l - s - 1)! J (x - u)'--'h(')(u) du,

k-oxo

where x, xo E (a, b) and s = 1, ...,1 - 1. Since h(1) E L,(a, b), henceh(') E Li(a,b), the limits lim h(x) and lim h(x) exist. Consequently, the

x+a+ x-4b-right derivatives h(')(a) and the left derivatives h(')(b) exist and h(')(a) _lim h(x), h(')(b) = lim h(x). Finally, since h('-')(x) = h('-')(xo) +

x-+a+ z-ib-

f h(') (u) du for all x, xo E [a, b] and h(') E LI (a, b), it follows that h('-'> isxoabsolutely continuous on [a, b].

Remark 7 By Lemma 6 D° f exists also for lal < I. Moreover, D. *f ELpi (S2), but in general D. Of ¢ Lp(c) (see Section 4.4).

Theorem 3 Let SZ C R" be an open set, l E N, 1 < p.-5 oo. Then Wpl(S2) is aBanach space. 12

Idea of the proof. Obviously WW(c) is a normed space. To prove complete-ness, starting with the Cauchy sequence { fk}kEN in WP(S1) , deduce using thecompleteness of Lp(c) that there exist f E Lp(c) and f, E Lp(11), wherea E N, lal = 1, such that fk -r f and Dw fk -+ f, in Lp(c). From the closed-ness of the weak differentiation it follows that f, = DW Of. Hence fk -+ f inWD(12). D

12 See footnote 1 on page 12. The same refers to the spaces L,(() in Remark 9 below.


Remark 8 Norm (1.22) is equivalent to

Ilfllwo(n) = (1 (Itlp+ IDwfIP) )n 101=1

for 1 0 are independent of f . This follows, with c3, c4 depending onlyon n, p and 1, from Holder's and Jenssen's inequalities for finite sums. If p = 2,then W21(S2) is a Hilbert space with the inner product

(f,9)ww(n) = f (f9+ E DwfDwg) dxn IQI=1

and Ilf 11(')W,l n) is a Hilbert norm, i.e., Ilf IIwt(n) _ (f,Let us consider the weak gradient of order I

nfo ) )((w - 8xi, axi, w i,....,y=1

Thenn

I

f12oa = ` I ( f )I2 = I

f12l!IDL. a ... a '!i =1 xis xis w a'

101=1

and norm (1.22) is equivalent to

Ilfllw;(n) _ (f (IfIP + IawfI) dx) `.n

We also note that for even I Vf E C000(f))

f IV1fI2dx=fI1if12dx,n n

where A is the Laplacian. Hence, for such f,

IIfII(2) = (f (If I2+Io{f12)dx)12.

n


We shall also need the following variant of Sobolev spaces.

Definition 6 Let 12 C R" be an open set, l E N, 1 < p < oo. The function fbelongs to the semi-normed Sobolev space w,(f2) if f E L, 0c(12), if it has weakderivatives D: f on fl for all a E l satisfying I a I= I and

IIfIIwi,(o) IID*fIILP(n) < oo. (1.23)1a,=(

The space wp(fZ) is also a complete space (the proof is similar to the proofof Theorem 3). Thus w,(1l) is a semi-Banach space, because the conditionIf llwP(n) = 0 is equivalent to the following one: on each connected component

of an open set ) f is equivalent to a polynomial of degree less than or equalto l - 1 (in general different polynomials for different components).

Remark 9 Let Sl C R" be a bounded domain and B be a ball such thatR C 12, 1 E N, 1 < p < oo. We denote by LP' L (0) the Banach space, which isthe set w,(l), equipped with the norm

IIf1IL;(n) = IllIIL,(B) + IllIIw,(n)

(It is a norm, because if If IIL;(n) = 0, then from If Ilwt(n) = 0 it follows thatf is equivalent to a polynomial of degree less than or equal to I - 1, and fromIll IIL,(B) = 0 it follows that f - 0 on Q.) For different balls with closurein I? these norms are equivalent. (This will follow from Section 4.4). Onecan replace IllIIL,(B) by Of IIL,(B) and the corresponding norms will again beequivalent. Note that by definition L,(SZ) = WP(1l) 13.

Remark 10 Clearly Wp(Q) C w,(fl). In general WW (fl) 96 w,(12), but locallythey coincide, i.e., for each open set G with compact closure in SZ Wp(Sl)Io =wp(SZ)IG. This will follow from the estimates in Section 4.4. In that section theconditions on fl also will also be given ensuring that WA()) = wp(fl).

Remark 11 The semi-norm II' IIW;(an)(in contrast to the norm II Ilwa(Rn)) pose-sses the following homogenity property: Vf E w,(R") and Ve > 0

IIf(ex)IIw;(R) = e'-""PIIf(x)IIw;(s.)

Is Here and in the sequel for function spaces ZI (fl), Z2(fl) the notation ZI (fl)Z'(fl), ZI(fl) C Z2(fl) means equality, inclusion respectively, of these spaces consideredonly as sets of functions (see also Section 4.1).


Moreover, V f E WD (R")

If (Ex)IIWw(R") ' e'"/PIIf (x)IILp(R")

ase->0+andjIf(ex)jjW;(R") ^

e)-n/PIll(x)jjw,(R")

ase-++oo.The number 1- n/p, which is called the differential dimension of the spaces

Wp(S2) and wp(Q), plays an important role in the formulation of the propertiesof thebe spaces (see Chapters 4,5) 19. It will also appear in the next statement.

Example 8 Let n, I E N, µ, v E R, 1 < p < oc. Denote by 1%,e the set ofall nonnegative even integers. Then jxjµilogIxIj" E Wo(B(0,1/2)) if, and onlyif, 1xIµIlogjxj I" E w,(B(0,1/2)) and if, and only if, the following conditions onthe parameters are satisfied. If 1 l - n/p, v ERor µ=l-n/p, v<-1/p and in thecase µENo,,:v=0orµ>1-n/p, vERorµ=l-n/p, v<l-1/p.lfp=oo,then inthecaseµ§ENo,,:µ>1, vEllfor IA =l, v < 0 and in the case p E No,, : v = 0 orµ>1, vERorp=1, v<1.Inparticular, for1<p<oo

1) jxI" E Wp(B(0,1/2)) if, and only if, either p 0 No,e and µ > I - n/p, orµ E 1No,e;

2) IloglxIj' E Wy(B(0,1/2)) where l = n/p if, and only if, v < 1 - 1/p.

14 Let Z(R") be a semi-normed space of functions defined on R". One may define thedifferential dimension of the space Z(R") as a real number p posessing the following property:V f E Zo(R") there exist ee, c5, co > 0 such that VE > co

cae°IIf(x)IIZ(R") <- IIf(E2)IIZ(R") <- cse'IIf(Z)IIZ(R")

If the semi-norm II is homogenuous, i.e., for some v E R Vf E Z(R") and VE > 0then the differential dimension of Z(RI) is equal to Y. The

differential dimension of Ln(R") is equal to -n/p, the differential dimensions of both W, (R")and wy(R") are equal to 1- n/p (which follows from the above relations).

This notion may be usefull when obtaining the conditions on the parameters necessary forvalidity of the inequality

II/IIZ,(R") <- cr IIfIIZ,(R-),

where cy > 0 does not depend on f. From this inequality it follows that the differentialdimension of Z1(R") is less than or equal to the differential dimension of Z2(R" ). If, inaddition, both of the semi-norms II IIZ,(R") and II IIz,(R") are homogenuous, then theirdifferential dimensions must coincide.


Idea of the proof. Apply Example 4. Let M = No for n = 1 and M = 1%,e forn > 1. Prove by induction that Va E N, a 0, and Vx E R", x 54 0,

lal

( lDa(IxlµlloglxlI") = Ixlu-I0I E Pk,a\

xI/ jlogjxjjv-k'

k=o

where Pk,a are polynomials of degree less than or equal to Jai, P0,a A 0 ando=0forpVMorpEM, Jai A;a=1forpEM, Jai >p, v540 (thecase in which p E M, jai > p, v = 0 is trivial: Da(jxjµjlogjxjj") = 0). Deducethat VxER,x00,

jDa(IxjµjlogjxllV)I c8 Ixlµ-Iall loglxlj O,

where c8 > 0 does not depend on x. Moreover, if n > 2, then for some t E R",where I{I=1,e>0andb'xEK-{xER":x00, I11-eI<E}

jDa(jxjµjlogjxjjv)j ? c9 jxja-1Q1jlogjxjI"-O,

where cg > 0 does not depend on x. Finally, use that for some clo, c,, > 0I/2 1/2

f g(axl)dx = cio f 9(p)p"-idp, f g(IxD)dx = cii f g(p)p"-idp.B(0,1/2) 0 B(0,1/2)nK 0

Example 9 Let 1 < p < oo. Under the suppositions of Example 8jxjµ(logjxj)" E Wo(eB(0, 2)) if, and only if, p < -n/p, v E R or p = -n/p, v <-1/p. On the other hand, (xjµ(logjxj)" E wp(CB(0, 2)) and if, and only if, inthecase p0No,e:p<1-n/p, vERorp=l-n/p, v<-1/p and in thecase.ENo,e:v=0orp<l-n/p, vERorp=l-n/p, v<1-1/p. Forp = oo the changes are similar to Example 8.

Let F f denote the Fourier transform of the function f : for f E L1(R" )and V R"

(21r)-"' f e-;='{ f (x)dx; (1.24)

for f E L2(R")R"

Ff = lim F(fxk), (1.25)

where Xk is the characteristic function of a ball B(0, k) and the limit is takenin L2(R"). It exists for each f E L2(R") and

JIFf jIL2(R^) = IIf IIL2(R") (1.26)

(Parseval's equality).


Lemma 8 For all I E N and f E WW (R")

IlowflDL,(R.) = II

and

IlfllW;(R') =11(1 + IIL,(R').

(1.27)

(1.28)

Idea of the proof. For f E L1(R") n WZ(R") starting with Definition 4 provethat F(Dwf)({) = (il;)°(Ff)(1;) on R". To obtain (1.27) and (1.28) apply(1.26) and the identity

1'2-1 ('1) ... (fin) = ICI21.

Ial=( Inl=t

Lemma 9 Let c2 C R" be an open set, M > 0 and suppose that Vx, y E S2

If(x)-f(y)1<-MIx-y I. (1.29)

Then f E w;o(S2), the gradient (vf)(x) exists for almost every x E S2 and

I v f(x) 1< M a.e. on 12. (1.30)

If, in addition, S2 is a convex set, then the condition (1.29) is equivalent to thefollowing: f E C(c) n wl (S2) and (1.30) holds.00

Idea of the proof. Let j E { I_-, n}, x = (x0), x;), x(>> _ (xl, ..., x;_1 i x;+l,...,xn), c0) = Prxj=oc C R"-1 and dx0) E 0) 5201(x(')) = Pro,Sinlz(,) CR, where lz(,) is a straight line parallel to the axis Ox; and passing throughthe point (x0),0). Deduce from (1.29) that for almost every x; E 120)(x0))there exists L, (x) = s

,j(x0>, x;) and ( f.L (x) ( < M. Integrating by parts

(which is possible because dx0> E c0) the function f (x0), ) is locally absolutelycontinuous on 520)(x0))) show that the ordinary derivative a (existing thusalmost everywhere on 0) is a weak derivative (E )w on a

If 12 is convex, then to obtain the converse result use Lemma 4 and (1.7) toprove that Vx, y E S2 and 0 < 6 < dist ([x, y], 8f2) the following inequalities forthe mollifier A6 with a nonnegative kernel are satisfied is

I(Asf)(x) - (Aof)(y)I -< II V A6fIIC(Iz,vuIx - yI

15 When writing II v 911c(c) we mean that

- L99 11

I I V 911c(c) = I I I v 91 IIC(c) =1 2 ) 1 1 2

(c)fJ-1

(II v 911L.,(o) is understood in a similar way).


= II A6 vw f Ilc([x, ]) Ix - y1:5 II V. f II L-([x,y)6) I x - yl

< II V. f IIL (n)IX - y1 = II v f Y15 MIX - yl(note also that for f E C(Sl) fl wl(Q) the gradient of exists a.e. on Q andv f = vw f on Sl ). Now it is enough to pass, applying (1.5), to the limit as6->0+.

Corollary 3 If SZ C R" is a convex open set, then g E w.(Sl) if, and only if,it is equivalent to a function f satisfying (1.29) with some M > 0. (Given afunction g, the function f is defined uniquely.)

Moreover, denote by M' the minimal possible value of M in (1.29). ThenV 9 M' and, hence,

M' < II9IIw- (n) < n M*.

Idea of the proof. The first statement is just a reformulation of Lemma 9 forthe case of convex open sets. The second one follows from the definitions ofII9IIwi (n) and vwg.

Lemma 10 (Minkowski's inequality for Sobolev spaces) Let S1 C W" be anopen set and A C Wn a measurable set, l E N, 1 < p < oc. Moreover, supposethat f is a function measurable on Sl x A and that f y) E WP(I) for almostevery y E A. Then

pn)dy (1.31)11f f(x,y)dyllwo (n)

S fIlf(x,y)llw,(

A A

(the norm Ilf (x, y) II wo(n) is calculated with respect to x).

Idea of the proof. Use Lemma 3 and Minkowski's inequality for Lp(Sl).Proof. Let the right-hand side of (1.31) be finite, then by Holder's inequalityfor each compact K C 11

f ( f if (x, y)ldx)dy < oo and 1(1 IDwf(x, y)Idx)dy < ooA K A K

Va E N where Ial = I. Hence by Fubini's theorem the function f, beingmeasurable on K x A, belongs to Ll(K x A). Now the inequality (1.31) followsfrom Lemma 3 and Minkowski's inequality for Lp(SQ):

11f f(x, y)dy11Wp(n) = 11f f (x, y)dy11Lp(n) + E 11Dw f f (x, y)dyllLv(n)


f IIf(x,y)IIL,(n)dy+E f II(Dwf)(x,y)IIL,(n)dy= f IIf(x,y)Ilwp(n)dy.A I°I=1 A A

Lemma 11 (Multiplication by Co -functions) Let S2 C R" be an open set,I E N, I 0 such that V f E W,(0)

114 16"(f)) <_ (1.32)

Idea of the proof. Use Lemma 6, Leibnitz' formula and the Ly-estimates of thederivatives of lower order.Proof. Let a E R satisfy Ial = 1. By Lemma 6 d$ E N where 1,3 1<- 1 thereexist Du, f , therefore on S1 Leibnitz' formula 16 holds:

()Do-sDf. (1.33)o<p<° Q

Let Q, C Q. j = 1...., s. be open cubes with faces parallel to the coordinateb

planes such that supp W C U Q,. Then, applying twice the inequality in foot-j=1

note 11. we get

IID°W)IIL,(n) <- 2'maxIID'c Iic(supp,,) I1D1 fI1 L,(suppsv)I0I<I

x x

< 2' E IID°0IL,(Q,))I7I5' i=1 I0I51 i=1

<- M IIIPIICI(n) 11f 16"n,

where Al depends only on 1,11 and supp W. (See also Lemma 15 of Chapter4.)0

Lemma 12 Let 11 C R" be an open set, l E 1V, 1 < p < oo. Then the E Cu (s1)andVf E w,(S1) Wf E w,(S2).

Idea of the proof. Since locally w,(S1) and Wp(S1) coincide (see Remark 9) andW is compactly supported in fl, it is enough to apply Lemma 11. The estimate(1.32) does not hold if Wy(11) is replaced by wp(fl). (Take any nontrivialpolynomial of degree less than or equal to 1 - I as f to verify this.)

n °i !

16 Here (p) _ , o a! = a1!...a"!; note that E (p) = j'[ E 21° .

o«<° s=1 31=o


Lemma 13 Let I E N, 1 < p < 00, rt E Co (R") be a function of "cap-shaped"type such that g = 1 on B(0,1) and Vs E N, Vx E R" 77,(x) = i (8) . ThenVf E Wp(R")

rl,f -4 f in WW(W) (1.34)

ass-4oo.

Idea of the proof. Use the definition of the norm in li n(R") and Leibnitz'formula.Proof. First of all Vg E Lp(R") where 1 < p < oc

1101, -1)gIIL,(Rn) <- IIgIIL,(<a(o,,)) 0

as s -i oc. From (1.33) it follows that Va E No where I R = I

IIDw(nef - f)IIL,(R")

511('i -1)DwfIIL,(Rn)+ (a,)

IIrlmDwf - D,°u,f II a! E IIDmf II/.,(a.. ).OG,9<n,tl#0

where M does not depend on f and s. By footnote 11 Dwf E Lp(R"), conse-quently we have (1.34).

Remark 12 For p = oo Lemma 13 does not hold, because, for instance, forf = 1 on R' Vs E N II r), f - f II 1. However, rl, f -* f a.e. inR" and II rl,f IIw;,(Rn) - If IIw' (Rn) as s - oo, which sometimes is enough forapplications.

It is well-known that if S1 C R" is a measurable set and 1 < p < oo, theneach function f E Lp(11) is continuous with respect to translation (__ continuousin the mean), i.e.,

lo II fo(x + h) - f(x) IIL,n= 0. (1.35)

The analogous result is valid for Sobolev spaces. We recall that for an openset S2 C R" the space (W1',)o(S2) is the set of all functions f E Wp(Q) compactlysupported in n.

38 CHAPTER I. PRELIMINARIES

Lemma 14 (Continuity with respect to translation for Sobolev spaces) Let9 c R" be an open set, l E Ill, 1 < p < oo. Then V f E W, (fl)

lim II f (x + h) - f (x) 0, (1.36)

where h E R", f2ih) _ {x E Q: x + h E S1}, and Vf E (W,)o(S2)

lim II fo(x + h) - f (x) Ilwa(n)= 0. (1.37)

Idea of the proof. Use the definition of the norm in Wp(R") and (1.35). 0Proof. (1.36) follows from (1.35) because

II f (x + h) - f (x) IIwn(n{,,})

=11 f (x + h) - f (x) IILp(na.}} + Ell (Dwf)(x + h) - (Dwf)(x) IILp(n{n})IQI=1

<_ II fo(x + h) - f (x) IILp(n) + Ell (Dwf)o(x + h) - (Dwf)(x) IILP(n)101=l

If f E (Wp)o(S2), then Va E N satisfying Ial = I we have (Da f )o = DO-(fo) onR", which easily follows from Definition 2, and thus fo E Wp(R"). Therefore

II fo(x + h) - f (x) ilw;(n)<-II fo(x + h) - fo(x) IIw;(Rn)

and (1.38) follows from (1.37). 0

Remark 13 In contrast to the situation in L,(1)-spaces the relation (1.37) isnot valid for all functions in W, )). For example, if n = 1, fa = (0, 1), f = 1,then on (0, 1) we have fo(x + h) - f (x) = -X(}-h,h)(x) 0 Wp(0,1) for everyh E (0, 1). Moreover, Lemma 14 does not hold for p = oo. For example, ifn = 1, 1 = 1, ) = (-1,1), f (x) = lxl, then

II f (x + h) - f (x) f,,(x + h) - f,,(x)

for every h E (0,1).

Chapter 2

Approximation by infinitelydifferentiable functions

2.1 Approximation by C01-functions on RnLet Aa be a mollifier with the kernel w defined in Section I.I. We start bystudying the properties of Aa in the case of Sobolev spaces.

Lemmal LetlEN. Thendf EWP(R") for t<p<0o

II Aaf IIW'(tn)<_ C II f IIWW(R-),

where c =II w IIL,(i4W)-Moreover, for 1 0+. For p = oo (2.1) is valid `df E C'(R"). If f E W;,(R"), thenin general Aaf -.+ f in W;,(R"), but in the case of nonnegative kernels ofmollification

Aaf -+ f in W.-'(W), IIAafIIw.(vn) - IIfIIW (R-)

as ' E - 0+.

(2.2)

' By footnote 11 of Chapter 1 it follows that A4 f - f in Wp (R" ), where m = 0,..., l if1 <p<oo andm=0,...,l-1 ifp=oo.

40 CHAPTER 2. APPROXIMATION BY C°°-FUNCTIONS

Idea of the proof. Apply (1.6), (1.8), (1.9), (1.10) and (1.20). 0Proof. Using the above properties we find that for 1 < p:5 00

II A6f IIW,(R")=11 Ajf IIL,(R") + E II A6Dwf IIL,(R")lot=1

S c(II f IIL,(R") + E Ii Dwf IIL,(R")) = c II f IIWP(R")Ia1=1

If1<p<oc,then

II A6f - f IIWI(R")= IIA6f - f IIL,(R") + E II A6(D°f) - D.Qf IIL,(R")-+ 0IaI=1

as 6-*0+.If p = oc, then the same argument works Vf E (R"). It follows from

(1.8), because w(6, 0 as 6 -* 0+ for these f. If f E W;°(R"), thenby (1.8)

IIA6f - f II W (R") = IIA6f - f IIL (R") +E II AoDwf - DwfIQI=1-1

<c(w(6,f)L.(R")+ F w(6,D*f)L.(R"))

By Corollary 7 of Section 3.3

W(&, 11f (x + h) - f 6 IIf IIw;,(R").

Similarly for IaI = I - 1

W(6, Dwf )L0 (Rn) <- 61IfIIwL(R")-

Consequently, w(6, DW 0 for IaI = I - 1 as 6 -+ 0 +. It also followsthat w(6, A- W) -- 0, since by footnote 11 of Chapter 1

IIlIIw;,(R") << MIIfIIwi (R"),

where M is indepent of f.The second statement of (2.2) follows from (1.10) with p = oo and (1.20).Finally by Remark 2 below it follows that for f E W.(R") in general

A6 -« f in W;°(R"). 0

2.1. APPROXIMATION BY Co -FUNCTIONS ON RN 41

Remark 1 If 11 is a proper open subset of R", 1 0

A6f -> f in Wp(0f) (2.3)

as 6 -* 0+. We next aim to construct more sophisticated mollifiers, which willallow us to prove the analogous assertion for l itself.

Lemma 2 Let I E N, 1 < p < oc. Then Co (R") is dense z in Wp(R")

Idea of the proof. Let f E W,(R") and rI s E N, have the same meaning as inLemma 9 of Chapter 1. Set spa = A 1(rja f ). Then V, E Co (R") and Va f inWP(R") as s -1 oo. 0Proof. By (2.1), (2.2) and (1.34)

II A;(rlaf) - f IIW;(R')<-II Al f - f IIWo(R") + II A1(nsf) - At f IIWW(R-)

<II A1f - f IIW;(R^) +C II ?7 f - f IIW;(R-)-1 0

as 6-+0+. 0

Remark 2 For p = oo Lemma 2 is not valid. The counter-example is simple:f = 1 on R". Moreover, C°°(R") also is not dense in W,,,,(R"). In order toprove this fact, for example, for n = 1 and I = 1, it is enough to considerthe function f (x) = IxIrI(x), where t1 is the same function as in Lemma 13 ofChapter 1. Then dip E C°°(R)

II f -'P II WJ(R) >- II f a - ' IIL00(-1,1)= II sgnx IIL.(-1,1) > 2

However, by Lemmas 1- 2 it follows that Co (R") is dense in W0O (R") in aweaker sense, namely,'df E W;° (R") functions W. E C' (W), s E N, exist suchthat

ova - f in woo i(R"), II'PsIIW;,(R") -' IIfIIw (R")

as8-100.

2 Thus *y(R") = W, (R"), where T pl(R") is the closure of Co (R") in WW (R").

42 CHAPTER 2. APPROXIMATION BY Coo-FUNCTIONS

2.2 Nonlinear mollifiers with variable stepWe start by presenting four variants of smooth partitions of unity, which willbe constructed by mollifying discontinuous ones.

Lemma 3 Let K C R" be a compact set, s E N, Slk (: R", k = 1, ..., s, beopen sets and

9

K C U Stk. (2.4)

k=1

Then functions 1Jik E Co (Slk), k = 1, ..., s, exist such that 0 < ok < 1 and

k=1

a

t tpk = 1 on K. (2.5)

Idea of the proof. Without loss of generality we may assume that the 1k are

bounded. There exists b > 0 such that K C G =_ 6 (S1k)a. Set Gk = (1k)6 \k=1

k-1 a

U (Slm)6 and consider the discontinuous partition of unity: E co,, = Xc onm=1 k=1

R". Mollifying it establishes the equality t A¢Xck = AIXc on R", whichk=1 ] 2

implies (2.5), where 'pk = Al Xck. (Here A6 is a mollifier with a nonnegativekernel.) 0

Lemma 4 Let SZ C R" be an open set and S1k C R", k E N, be bounded opensets such that

00

C S1k+1, k E N, U Slk = 0. (2.6)k=1

Then functions 1Pk E Co (Sl), k E,/'N, exist such that

Gk C 3upp Wk C Gk-1 U Gk U Gk+l , (2.7)

where Gk=Slk\Ilk- I (fork=0 we set ilk= 0),0<Iik<1 and

00

E tpk = 1 on S1. (2.8)k=1

2.2. NONLINEAR MOLLIFIERS WITH VARIABLE STEP 43

Idea of the proof. Starting again with the discontinuous partition of unity0Ek=1

Xck = 1 on ft, choose

ek = 4 dist (Gk, i9(Gk_l u Gk U Gk+1))

= i min{dist (f2k-1, Oak ), dist (f)k, O1k+l)) (2.9)

(ifQ R", then ek-a0as (k-+ oo) and set

ok - ( Ae._, Xc, on (nk)vt_, , (2.10)l Aek Xck on ft \ (000k,

where A6 is a mollifier with a nonnegative kernel w.So the characteristic function Xck is mollified with the step pk-1 "in the

direction of the set Gk_1 " and with the step pk "in the direction of the setGk+1". Let Gk = Gk u Gk U Gk , where

Gk = (1k-1)ek-1\ ck-1, Gk = G+k = k \

Then tik = 1 on Gk, supp t+I'k C CT- 1 U Gk u G1k+1, therefore, 7p,,, = 0 on Gkwhere m 96 k - 1, k, k + 1. Moreover, on Gk U Gk+1

00

E PPm = lOk + 1Gk+1 = Aek (Xck + XCk+,) = 1.M=1

Lemma 5 Let SZ C R' be an open set, n 0 Rn,

Gl = {x E 12 : dirt (x, Of) > 2-2}

and f o r k E N, k > 1, let

Gk = {x E SZ : 2-k-1 < dist (x, O1) < 2-k }

(fork < 0 Gk = 0) Then functions lPk E C°°(Sl), k E Z, exist (fork < 0 weset ik = 0) such that "0 < t/ik < 1,

Gk c supp y k c {x E fl : 82-k-1 < dist (x, OI) < 12-k }

C Gk-1 U Gk U Gk+1,00 00

1: Pk = 10, = 1 on 0 (2.11)ko-00 k=1

and Va E N there exists cQ > 0 such that Vx E R" and Vk E Z

kID°+'k(x)I .5 (2.12)


Idea of the proof. The same as in Lemma 4. Now the Ilk are defined via thek

Gk: Qk = U G," and ok = 2-k-3. Estimate (2.12) follows from the equalitym=-oo

D°V,k = 0k10i (D°W ek) * XGk on Q \ (Ilk_1)ek and the analogous equality on

(Dk)ek-i . 0

Remark 3 Sometimes it is more convenient to suppose that the functions &kin Lemmas 4 and 5 are defined on R" and supp'k C Il. (We shall use the samenotation ikk E Co (Il) in this case also). Then equality (2.8) can be written in

00the following form: E Ok = Xn (the same refers to equality (2.11)).k=1

Remark 4 There may exist an integer ko = k0(Il) > 1 such that Gk = 0 fork < ko (in this case we assume that '1k - 0) and (2.11) takes the form

00 00

E V)k=EOk=1 on Q. (2.13)k=-oo k=ko

For Il = R" we shall apply the following analogue of Lemma 5.

Lemma 6 For nonpositive k E Z let

Go={xEW':Ixj <1}, Gk={xER":2-k-'<IxI<2-k}, k<0.

Then functions k E Co (R" ), k E Z, exist (O k =_ 0 for k > 0) such that theproperties (2.7) and Vc E No (2.12) are satisfied, 0 < tPk < 1 and

00 0

E4 =E Vk = 1 on R". (2.14)k=-oo k=-oo

Idea of the proof. The same as in Lemma 5. 0

Remark 5 Note that in Lemmas 4 - 6

(supp Qek C (Gk-1 U Gk U Gk+1)ek (2.15)

Moreover, in the case of Lemma 4 for any arbitrarily small ryk > 0, k E N,one can construct functions 'kk, k E N, satisfying the requirements of Lemma 4such that

suppikk C (Gk)7k, Ok = 1 on (Gk),. (2.16)

2.2. NONLINEAR MOLLIFIERS WITH VARIABLE STEP 45

To do this it is enough to replace pk defined by (2.9) by

pk = min{a

dist (Gk, 8(Gk_l U Gk U Gk+1)),'Yk}.

In the case of Lemmas 5 and 6 for any fixed y > 0 one can constructfunctions yk, satisfying the requirements of those lemmas, such that

suPPiPk C (Gk)ry2-k, lPk = 1 on (Gk),2-k. (2.17)

Remark 6 From (2.15) it follows, in particular, that the multiplicity of thecovering {supp iik} in Lemmas 4-6 is equal to 2, i.e., Vx E Q there are atmost 2 sets supp tbk containing x and there exists x E Il such that thereare exactly 2 sets supp t/ik containing x. (From (2.7) it follows only that themultiplicity of this covering does not exceed 3.) Of course 2 is the minimalpossible value (if supp ,bk J Gk and the multiplicity of covering is equal to1, then iik = XGk). Moreover, from (2.15) it follows that for 6 E (0,'-g] themultiplicity of the covering {(supp

Vlk)62_k

} is also equal to 2.

In Chapter 6 we shall need a variant of Lemma 5 for 0 = {x E R" : x,, >W(x1,...,x"-1)}, where W is a function of class Lip 1 on R"-1, - that variantwill be formulated there.

Let Q C R" be an open set and Qk C S2, k E N, be bounded open sets,possessing the properties (2.6), Gk = ilk \ S1k_1. Suppose that pk is defined by(2.9) and {k}kEN is the partition of unity in Lemma 4 defined by (2.10).

Definition 1 Let b = {bk}kEN, where

0<dk<Ok (2.18)

and f E L'-(fl). Then VX E 11

00

(B7f)(x)_E(A6r(lkf))(x)_: J Ok(x-bkz)f(x-6kz)w(z)dz, (2.19)k=1 k=1B(0,1)

where w is a kernel of mollification defined by (1.1).

Remark 7 The functions ' k f E L1(S2), therefore, A6R (z/ik f) E C°° (R" ). Wenote that we assume that 1/ik(y) f (y) = 0 for all y 0 supp Ok even if y 0 St andf (y) is not defined (for this reason in contrast to (1.2) in (2.19)1Iik(x-dkz) f (x-


bkz) is written instead of (1bk)o(x-bkz)fo(x-8kz)). By (1.4), (2.18) and (2.15)it follows that

supp A6,: (i f) C (supp +Gk)6k C (Gk-1 U Gk U Gk+1)6k, (2.20)

therefore,

A6k(kf) E Co (1l) (2.21)

and the sum in (2.19) is finite. For, let `dx E f2 a number s = s(x) be chosensuch that x E G,. Then fork # s -1, s, s + 1 we have x E supp A6k (ipk f) and 3

B(X)+l

(B1f)(x) = EJ

iGk(x - akz)f(x - 6kz)w(z)dz. (2.22)

k=a(x)-1B(o,1)

For the same reason dm E N

m+1

BBf = E A6k(Okf) on G.. (2.23)

k=m-1

Moreover, by Remark 5, for any given ryk E (0, pk], k E N, a partition of unity{1 }kEN can be chosen in such a way that for all sufficiently small 6k, k E N,we have V f E Lt°C(I) and dm E N

B6f = A6,,,f on (Gm)7,,,. (2.24)

Lemma 7 Let 1 C R" be an open set and f E L10c(fl). Then for each d ={dk}kEN satisfying (2.18) B7f E C(11) and Va E NB

00

D"(BB f) = E D°(A6k (iyk f )) on 0- (2.25)k=1

Idea of the proof. Apply (2.21) and (2.23).0Proof. From (2.21) and (2.23) it follows that `dm E N and Va E Plo

m+1 00

D"(B8f) = E D"(A6k(Vkf)) _ ED"(A6k(,Pkf))k=m-1 k=1

on Gm. Hence Bgf E C°°(S2) and (2.25) holds on 12. 0

3 Moreover, from (2.20) it follows that Vx E (I in the sum (2.16) no more than 2 summandsare not equal to 0.

2.3. APPROXIMATION BY Coo-FUNCTIONS ON OPEN SETS 47

When applying the mollifiers Ba f we shall choose bk satisfying (2.18)depending on f. For this reason we call them nonlinear mollifiers with variablestep (though, of course, BI is a linear operator for fixed b). The variable-ness of the step follows from (2.19): t/x E Si the mollification is carried outwith the steps b,_1, b 6,+1 depending on x (and these steps tend to 0 asx approaches the boundary 8c). We can also say that By is, in some sense,a mollifier with piecewise constant step, because by (2.23) the same constantsteps bm_1i bm, bm+1 are used for the whole "strip" G,,,. Moreover, by (2.24)only one step bm is used for the whole "substrip" (Gm).r,,,.

Remark 8 In a number of cases it is more suitable to apply the mollifiers CC,which are similar to the mollifiers Bg and are defined by the equality: Vx E Il

00

00

(Cif)(x) = 1: 1N(x)(Aakf)(x) = ck(x) f f(x - Skz) w(z) dz. (2.26)k=1 k=1 B(0,1)

For instance, in contrast to the mollifiers B6, for f = 1 on Sl and arbitrarybk E (0, pk] we have Cg = 1 on Q. On the other hand, for CCf the equalitiesanalogous to (2.22), (2.23) and (2.24) are valid and Cg f E C°°(c).

Furthermore, if the kernel of mollification w is real-valued and even, thenthe operator CT is the adjoint of Bg in L2(ci), because V f, g E L2 (n)

c00 00 " l(BBf, 9) = L,(Aak ('+l)kf ), 9) _ I:(Okf, Ask9) = f, F, 1GkAak9 f = (f, Ca9)

k=1 k=1 k=1 /

(note that for these kernels w the operator Aa is self-adjoint in L2(fl) - seeSection 1.2).

2.3 Approximation by C°-functions on opensets

In this section our main aim is to prove the following statement.

Theorem 1 Let Cl C R" be an open set, I E N. Then CO' (0) f1 W1(Sl) is dense

in W,(Sl) where 1 < p < oo and C°°(Sl) f1 '(Cl)Cis dense in C°(11).


Moreover, Vf E W, ,(Q) if 1:5 p < oo and Vf E C((SZ) if p = oo there existfunctions pk E CO°(Sl) fl WP(fl) such that 4

Wk -4 f in WP '(0), m = 0,1, ...,1.

The set CI(I) fl WW(Q) is not dense in W;°(S)). However. df E W.(Sl)there exist functions Wk E COO (0) fl W;O(Sl), k E N, such that

Wk -> f in WW(Sl), m = 0,1, ...,1- 1, IIcvkHlw, (a) -+ IIf IIwd,(nl

ask-4 oo.

Later, in Section 2.6, we shall see that bf E W,(Sl) (or &(Q)) functionsgyp, E COO (11) fl WD(S1), gyp, E C°°(11) fl ?71 (11) respectively, exist, which dependlinearly on f, do not depend on p and are such that for 1 < p < oo

G, -Y f in Wy(1l), (2.27)

in C'(c) respectively. Moreover, the functions W. may possess additional usefulproperties.

We shall deduce the statement of this theorem, in the case in which1 < p < oc, from a much more general result, which holds for a wide classof semi-normed linear spaces Z(Sl) of functions defined on SZ with semi-normsII' 11z(n) such that Co (Sl) C Z(fl) C Li°°(Sl). Let Zo(fl) denote the subspace ofZ(Sl) that consists of all functions f E Z(fl), which are compactly supported inSZ. Moreover, let Z(SZ) denote the space of all functions f E Lr(12), whichare such that VW E C01(fl) we have cof E Z(Sl). From these definitions itfollows, in particular, that

Co (Sl) C Zo(fl) C (L()o(SI)

and

C°°(Sl) C Z(°`(Sl) C L10°(Sl).

"Under additional assumptions on fl (see Theorem 6 of Chapter 4), IIfIIw, (n) 5M II/IIw;(n), m = 1,..,1- 1, where M is independent of f, and this statement follows fromthe density of COO (0) nW.(fl) in W,(fl). However, for arbitrary opens sets it is not so (seeExamples 8 - 9 of Chapter 4), and this statement needs a separate proof. We also note thatit is possible that f ¢ W.'(11). In that case also ,k V Wa (fl) but f - wk E W, '(fl) andf -Ws -0in WD (Sl) as k-4oo.


Remark 9 For any I E No we have (C1)'a(c2) = (C,)'°°(c2) = C1(cl). More-over, for 1 < p:5 oo the following equivalent definition of the space (Wp)'-(Q)can be given: (WP)" (c2) = (f E Li°`(c1): for each open set G compactlyembedded into 1l f E Wp(G)}.

Theorem 2 Let Q C R" be an open set and suppose that the semi-Banachspace Z(n) satisfies the following conditions:

1) Co (cl) C Z(c2) C Li°L(1),

2) (Minkowski's inequality) if A C 1R is a measurable set and f is afunction measurable on c2 x A, then

II f f (x, y)dyl z(n)5 f IIf (x, y)Ilz(n)dy,

A A

3) if p E C0 00(Q) and f E Z(cl), then W f E Z(S2),

4) all functions f E Zo(S2) are continuous with respect to translation, i.e.,

u oIIfo(x +h) - f(x)IIz(n) = 0. (2.28)

Then C°°(Q) is dense in Z"(cu) (and, hence, C°°(Q) f1 Z(Q) is dense inZ(11)), i.e., V f E Z11(c2) functions cp, E C°°(c2) fl Z'-(c2), 8 E N, exist suchthat

cp, - f in Z(c2) (2.29)

ass -4 oo.

Idea of the proof. Apply Minkowski's inequality to the right-hand side of theequality

00(Baf)(x) - f(x) = E f (fk(x - bkz) - fk(x)) w(z) dz, (2.30)

k-1B(o,1)

where fk = I k f and the mollifier BI is constructed with the help of a nonneg-ative kernel of mollification, and prove the inequality

It Baf - f Itz(n)5 F, w(bk, fk)Z(n) (2.31)k=1


Here

w(b, f)z(n) = sup Ilfo(x + h) - f (x)llz(n)Ihj<6

is the modulus of continuity of the function f in Z(Q). (Compare with (1.8).)Using condition 4) choose dk in such a way that w(bk, fk) < e 2-k. ThenII Baf - f 11z(n) < e. 0Proof. 1. From 4) it follows that V f E Zo(SZ) there exists y = y(f) > 0such that Vh E R" satisfying IhI < 'y the function fo( + h) - f E Z(SZ).Let us suppose, in addition, that y < dist (suppf, 6Q), then suppfo( + h) C(suppf )Ihi C Sl and fo( + h) E Zo(SZ). For, first of all fo( + h) E Z'-(Q).Consider, furthermore, a function of "cap-shaped" type 'n E C0 00(Q) such thatn = 1 on (see Section 1.1), then by definition of Z'°°(1)rlfo(' + h) E Zo(SZ).

Let A(h) = Il fo(x+h) -f (x)IIz(n) for h E B(0, ry). Condition 4) means thatthe function A is continuous at the point 0. Moreover, A E C(B(0, ry)). Indeed,let u E B(0, ry). Then, by the continuity of the semi-norm, in order to provethat A(h) -+ A(u) as h i u it is enough to prove that fo(x + h) - f (x) -fo(x + u) - f (x) or fo(x + h) - fo(x + u) = go(x + h - u) - g(x) -r 0 ash - uwhere g(x) = fo(x + u). And this is valid because g E Zo(SZ).

2. Let us consider the mollifiers B1, which are constructed with the help"of any nonnegative kernel. Since E )k = 1 on SZ and f wdx = 1 we have00

k=1 B(0,1)

NnEi100

(Baf)(x) - f (x) = E,((A6k (Okf )) - Wk(x)f (x))k=1

00

_ F I (fk(x - 6kz) - fk(x))w(z)dz = Fk(x),

k=1 B(0,1) k=1

where fk = 'kf and Fk(x) = f (fk(x - akz) - fk(x)) w(z)dz.B(0,1)

By 3) and (2.15) we have that fk, Ft E Zo(S1) and supp fk, supp Fk CGk_1 U Gk U Gk+1. Applying Minkowski's inequality for infinite sums (whichholds besause of the completeness of the space Z(Il)) we have

00

II Baf - f Ilz(n) <_ IIFkllz(n)k=1

2.3. APPROXIMATION BY C°°-FUNCTIONS ON OPEN SETS 51

Now suppose that, in addition to (2.18),

bk < 2 y(fk), (2.32)

then the function Ilfk(x - bkz) - fk(x)II z(n) is continuous on B(0,1) and

f 11f,& - bkz) - fk(x)II z(n) w(z) dz < oc.

B(0,1)

Moreover, the function [fk(x - bkz) - fk(x)]w(z) is measurable on SZ x B(0,1)(see footnote 9 of Chapter 1). Therefore, we can apply Minkowski's inequalityin condition 2) and establish that

IIFkIIz(n) < f II fk(x - 5kZ) - fk(x)II Z(n) w(z) dz

B(0,1)

 0 by 4) choose 5k such that, in addition to (2.18) and (2.32),

kw(bk, fk)Z(n) < e 2 . (2.33)

(we note that (fk)o = fk). With this choice ofd = a(e, f) (depending on a andf) we have Bgf E C°°(SZ) and

II Baf - f 11z(n) < e. (2.34)

_Thus, Theorem 2 is proved (in (2.29) one can take cp, = Bra f , where S.a (;, f)). o

Remark 10 The functions gyp, in the given proof are constructed in such a waythat they depend on f, in general, nonlinearly. Moreover, they may depend,of course, on the space Z(f ). For example, in the case of Z(Q) = Wp(f2) theymay depend on n,1, p, SZ and f .

Idea of the proof of Theorem 1. The density of C°°(12) f1 WP(SZ) in W,(SZ)

where 1 < p < oo and of C- (Q) f1 C'(SZ) in C1(1l) follows directly by applyingTheorem 2 to Z(S2) = WW(S1), GV'(S1) respectively. In order to prove the second

52 CHAPTER 2. APPROXIMATION BY COO-FUNCTIONS

statement of the theorem take Z(1) = WP(sl), where WP(n) = n Wn (Sl),m=0

hence I I f Il wp(n) = E II f II wp (n) Note that by Corollary 14 of Chapter 4=0

(WP)°Oc(Q) D WP(sl). The case of the spaces C'(Sl) is similar.In the case of the spaces W;,(1) take open sets Ilk C Sl, k E N, which

are such that mess OSlk = 0. Consider any ryk, E (0, pk), k, s E N, suchthat lim ryk, = 0. Choose a partition of unity {0k}kEN in such a way that

8-400for any sufficiently small Sk,

Ba,f = A6,,,.f on (Gm)7.,,.,

where b, _ {6k,}kEN. This is possible by Remark 7. Moreover, assume that thekernel of mollification w is nonegative.Proof of Theorem 1. By Lemmas 10, 11 and 14 of Chapter 1 the conditionsof Theorem 2 are satisfied for Z(Sl) = WP(Q) where 1 < p < oo and for

Z(Sl) = C (a). Hence the first two statements of Theorem 1 follow fromTheorem 2.

However, if p = oo, then condition 4) of Theorem 2 is not satisfied, andTheorem 2 is not applicable. In this case we need a more sophisticated argu-ment. Let f E W.1(11) and m < I - 1. Then for any mollifier Ba, which isconstructed with the help of a nonnegative kernel, and m = 0, 1, ...,1 - 1 wehave

IIBaf - f G E '(6k, fk)w,TM,(n) :5 E J(bk,''kfo)W,m(R")k=1 k=1

By Lemma 11 of Chapter 1 Okfo E W;O(R") and as in the proof of Lemma 1,applying, in addition, footnote 11 of Chapter 1, we establish that

w(4,0kf0)W;(R^) :5

M1 is independent of f and k.Consequently, b's > 0 there exist akl) > 0, k E N, such that Wk E (0, akl))

we have w(bk, 1kfo)w;(R") < e 2-k, m = 0, ...,1 - 1, and hence

.(n) < s, m = 0, ...,1- 1.IIBaf - f 11w,,_

Furthermore, for Va E No satisfying j al = I by (2.25), Lemma 4 of Chapter1 and Leibnitz' formula we have

00

E (*)EAj,,(D--0,0kD.#f),0<p<a 0 k=1

2.3. APPROXIMATION BY C00-FUNCTIONS ON OPEN SETS 53

n 00where 0!(&"! , = n , °'' If 3 96 a, then E 0 on Q and

i=1 k=1by (1.8)

00 00

lIEA6.(D°-flakDwf) 1

flX `(A6,

(D°-9,bkDwf)-D°-,e1,kDB,f)II

Le(n)k=1 k=1

00 00

II A6. (D°-R?k D",f)-D°-6,iIjk Dwf IILm(f1) w(bk, D°-l3tPk Dwf)Le,(n)

k=1 k=1

00

< > w(bk, D°-dV k Dwfo)LW(R')k=1

Since by Lemma 11 of Chapter 1 D`011k Dw f0 E W.'-"l (R") as in the proof ofLemma 1 we establish that

w(bk, D°-p k L'w0 f0)Loo(R°) < M2bkIID°-1 Vk DwfolI W:181(R") ,

where M2 is independent of f and k.Consequently, there exist ak2) E (0, ak1)), k E N, such that dbk E (0, ak2)) we

havew(bk, D°-",Pk DwfO)Loo(R") < c 2-k-"(1 + E 1)-1

101=1

and, hence,

II E A6.(D°-O*k Dwf)IIL,o(n) < e 2-"(1 + E 1)-1.00

k=1 I°I=1

If Q = a, then since 10k, k E N, and the kernel of mollification is nonnegativewe have

00 00

II IIEIA6k(PkD,°of)IIIL-(n)k=1 k=1

00 00

<- Sk=1 k=1

00

< III + E(A6k'Ok - *k)IIL,.,(n)IIDwf IIL (n)k=1

54 CHAPTER 2. APPROXIMATION BY C00-FUNCTIONS

< (1+EIIA6k7Gk-')kIIL (n))IIDwfIIL..(n)k=100

< (1 + (J (bk+ Y k) II Dwf IIk=1

Since ?k E C000(0), as in the proof of Lemma 1 we have

w(6k,Ok)L (n)) W(6k,Ok)L (R")) 6k

and it follows as above that there exist ak3) E (0, ak2)), k E N, such that Wk E3

(0+ ak )00

IIA6k(VkDwf)IIL (n) < (1+e)IIDwfIIL.,(n)k=1

(This inequality also holds for cr = 0.)Thus, if bk E (0, ak3)), then

Il Baf 11w, (n) < e + (1 + E)IIfIIw;,(n)

(We have applied the equality E (;) = 2".)0<0<a

In particular, if 6k, E (0, ak3)), k, 8 E N, then

IIBi.fllw;,(n) <e+(1+e)IIfllw;,(n)

On the other hand by construction of the mollifier b, and by Lemma 4 ofChapter 1

IIBa,fllw, (n) >_ IIB8,f llw;,((Gk).,k,) = IIA6k.fllw;,((C4)ryk.)

IIA6k,fIIL..((Gk),k,) + II 4. DwfIIL.,((Gk),k, ).

By relation (1.9) for p = oo there exist a(4) E (0, ak3)), k E N, such that for6k, E (0, ak4))

IIBJ,fIIW'(n) IIfIIW;,((Gk),,) 2

and, hence,

IIBa.fIlwa,(n) ? sup IIfIIW.((Gk),k,) - z = IIf IIkEN W., (kU (G,,),,,.) - z


Since meas (U 09%) = 0 we have00k=1

Ilfllw;,(n) = IIfIIWW,(n\U 8O) IIfIIWW(Uk (Gk)., )

Consequently, there exists s E N such that

Ul(Gk),,) >- IllIIW;,(n) z'k

Thus, W > 0, there exist s E N and 5ks E (0,ak4)) such that

IlBd,f - fIIw,1(n) <e

and

IIfUIW,(n) -s < IIBa.fllw.,(n)and the statement of Theorem 1 in the case p = oo follows.

Corollary 1 Let 11 C R" be an open set, l E No Then C°°(S2) is dense in(WP)'0°(&1) where 1 < p < oo and in C'(S2).

Idea of the proof. Apply Theorem 2 to Z'«(S2) _ (WW)'°°(SZ) and Z(SZ) _

C'(Sl).

Remark 11 If p = oo, then C°°(SI) fl W/,(SZ) is not dense in W;°(1l) (seeRemark 2).

Remark 12 The crucial condition in Theorem 2 is condition 4). It can beproved that under some additional unrestrictive assumptions on Z(Sl) the den-sity of C°°(Il) in Z'°°(n) (or the density of C°°(a)flZ(SZ) in Z(Sl)) is equivalentto condition 4).

Remark 13 Theorem 2 is applicable to a very wide class of spaces Z(fl),which are studied in the theory of function spaces. We give only one example.Consider positive functions ao, as E C(fl) (a E 1%, lal = 1) and the weightedSobolev space Wp.{°o)(S2) characterized by the finiteness of the norm

IlaofIIL,(n)+ lla.DwfIIL,(n)

By Theorem 2 it follows that C°°(fl) fl WP(°Ql(S2) is dense in this space for1 < p < oo without any additional assumptions on weights ao and a.. Suchgenerality is possible due to the fact that the continuity with respect to trans-lation needs to be proved only for functions in this weighted Sobolev space,which are compactly supported in 0.


Now we give one more example of an application of Theorem 2, in whichthe spaces Z'°`(SI) (and not only Z(SZ)) are used.

Example 1 Let SI C R" be an open set, then dµ E C(SI) and de > 0 thereexist µE E C°°(Q) such that Vx E SI we have µ(x) < p,(x) < µ(x) + e.

To prove this it is enough to set µe = Bg(,u + Z) with 3 = b(z, µ + a) inthe proof of Theorem 2 for Z(S)) = C(SI) (hence, Zb°`(SI) = C(SI)) and applyinequality (2.34).

2.4 Approximation with preservation ofboundary values

In Theorem I it is proved that for each open set SI C R" and Vf E W'(SI)(1 < p < oo) functions W. E C°°(i) n WW(SI), s E N, exist such that (2.27)holds. In this section we show that it is possible to choose the approximatingfunctions cp, in such a way that, in addition, they and their derivatives of ordera E No satisfying jal < 1 have in some sense the same "boundary values" as theapproximated function f and its corresponding weak derivatives. The problemof existence and description of boundary values will be discussed in Chapter5. Here we note only that for a general open set SI C R" it may happen thatthe boundary values do not exist and even for "good" 0 boundary values ofweak derivatives of order a satisfying lad = 1, in general, do not exist. For thisreason in this section we speak about coincidence of boundary values withoutstudying the problem of their existence - we treat the coincidence as the samebehaviour, in some sense, of the functions f and cp, (and their derivatives)when approaching the boundary of SI.

Theorem 3 Let SZ C R" be an open set, I E N, 1 0 (2.35)

ass -* oo. For p = oo this assertion is valid V f E C'(SI).

Corollary 2 Let SI C R" be an open set, fI 34 R" and I E N. Then V f E U1 (fl)functions cp, E COD(A) n s E N, exist, which depend linearly on f and

2.4. PRESERVATION OF BOUNDARY VALUES 57

are such that, besides (2.27) where p = oo, s

D*v.Ion = D°f Ian, (al < 1. (2.36)

Idea of the proof. Choose any positive p E C(12) such that lim µ(y) = ooy-rx,yEf

forall xE812.Proof. One may set, for example, p(x) = dist (x, 812)-1. For a continuousfunction II - 11C(n) = II - IIL,(n), therefore, from (2.35) it follows that for someM>OVsENandVyE0

ID°w9(y) - D°f (y)I S M(.u(y))-1.

Passing to the limit as y -+ x E all, y E S2 we arrive at (2.36).

Corollary 3 Let 12 C R" be an open set, 1 E N, 812 E C1 and 1 < p < oc.Then V f E W(1l) functions cp, E C°°(Q) n WW(Q), s E N, exist such that,besides (2.27),

D°wal an = Dwf Ian, j al < 1 - 1. (2.37)

Idea of the proof. Take again µ(x) = dist (x, 812)-'. By Chapter 5 it is enoughto consider the case, in which 12 = R+ = {x E R" : x" > 01 and µ(x) = x;1.In this case the statement follows by Lemma 13 of Chapter 5.

Remark 14 The function p in Theorem 3 can have arbitrarily fast growthwhen approaching M. Let, for instance, u(x) = g(Q(x)), where e(x) =dist (x, 80) and g E C((0, oo)) is any positive, nonincreasing function. Thenforl<p<co

II Dwf - D°w,ll L,(n\n,) <_ (9(6))-' II (D0 f - D°w,)9(e)II L,(n) <- M(9(6))'-'

with some M > 0, which does not depend on s and 6. It implies that for afixed f E Wp(12) where 1 < p < oo one can find a sequence of approximating

5 We recall that Vf E G'(12), Vx E OQ and Va E IT satisfying jal < I thereexists lim D° f (y) and, thus, the functions D° f , which are defined on n can be ex-

I# Z'Ventended to ?I as continuous functions. It is assumed that D° f den are just restrictions to Oilof these extensions. The same refers to the functions gyp, E C°0(11), because by (2.27) where

p = oo we have gyp, E &(fl). From Theorem 8 below it follows, in particular, that w, can bechosen in such a way that they depend linearly on f.

Here by D,°, f Ian and D°w, Ion the traces of the functions Dl.f and D°,p, on 80 aredenoted (in the sense of Chapter 5, they exist if lal < 1- 1). See also Theorem 9 below.


functions V which is such that, besides (2.27), Va E l satisfying Ial < 1 thenorm IIDwf - D' a$II c,(n\n,) tends to 0 arbitrarily fast as 6 -+ +0.

Thus, condition (2.35) with arbitrary choice of p means not only coincidenceof boundary values, but, moreover, arbitrarily close prescribed behaviour ofthe functions f and cp, and their derivatives of order a satisfying Ial < 1 whenapproaching the boundary O 1.

For unbounded fZ we have the same situation with the behaviour at infinity.Choosing positive µ E C(Q) growing fast enough at infinity, we can constructthe functions cp, E C°°(1) f1 W,(SZ) such that IIDwf - D°WaII j,(n\B(o,*)) wherej al < 1 tends to 0 arbitrarily fast as r -+ +oo, i.e., D,f and D°cp, havearbitrarily close prescribed behaviour at infinity.

As in Section 2.4 we derive Theorem 3 from a similar result, which holdsfor general function spaces Z(SZ).

Theorem 4 In addition to the assumptions of Theorem 3, let the followingcondition be satisfied:

5) bf E C0 00(Q) there exists cy > 0 such that df E Z0(SZ)

Ikof Ilz(n) <- c,II f IIz(n)

Then Vp E C°°(SZ) and Ìf E Z(Q) functions gyp, E C00(11) f1 Zl°°(SZ), a E N,exist such that

W. - f in Z(Q) (2.38)

and

11(f - WW.)PIIz(n) -+0 (2.39)

as s -+ coo.

Idea of the proof. Starting with the equality that differs from (2.30) by thefactor p show, applying 5), that

00

II (BIf - f)µllz(n) <- E Ckw(6k, fk)z(n), (2.40)k.1

where the ck > 0 are independent of 6k. ElProof. In addition to the proof of Theorem 2, we must estimate the expression

00

(Baf - f )µ = E uFk.k=1

2.4. PRESERVATION OF BOUNDARY VIAL LIES 59

Recall that Fk E Zo(SZ) and supp Fk C Gk = Gk_, U Gk U Gk+1. Let us denoteby 77k E Co (SZ) a function of "cap-shaped" type, which is equal to 1 on Gk (seeSection 1.1), then 1Fk = µ71kFk. By condition 5) where cp = µ77k there existsck > 0 depending only on µ77x (and, thus, independent of bk), such that

IIIFkIIZ(n) < CkIIFkIIz(n) < CkW6,t(fk)Z(ft)

and (2.40) follows (without loss of generality we can assume that ck > 1).Choosing Ve > 0 positive numbers 5k in such a way that in this casew6,t(fk)z(n) < e2_kckl (instead of (2.33)), we establish, besides (2.34), theinequality II (B3f - f)µllz(n) < c.

Remark 15 From the above proof it follows that b'µ1i...,µm E C' (Q) (m EN) and Vf E Zb0C(Sl) functions cp, E C°°(Q) f1 Zf0C(c), s E N, exist such that

II(f - e)WIIz(f:) -> 0, i = 1,...,in.

as s -+ oo. (Theorem 4 corresponds to m = 2, it, = 1, µ2 = µ.)

Idea of the prof of Theorem 3. Apply Theorem 4 and Remark 15 to thespace Z(Q) = WW(c) and to a set of the weight functions (D'µ,)I,I<,, whereµ, E C°°(c) and IµI < µl on Q.Proof of the Theorem 3. The existence of the function µ, follows by Example1. By Remark 15 bf E (W,)(0C(S2) 7 functions gyp, E C°°(c), s E N, exist such

that ip, -+ f in WI (fl) and dry E N satisfying Iyl < 1.

11(f - w$)D'pIII `wP(n) + 0

as s -* oo. Hence, Va E N satisfying Ial < I

IIDW((f - co$)D'µl)II L,(n) -+ 0. (2.41)

Applying "inverted" Leibnitz' formula 8, we have

(Dwf - D°v,)µl = (-1)191 (c) DW((f -,pa)D---'{ll)0<p<o

We recall that this space was also considered in the proof of Theorem 1.s For n = 1 and ordinary derivatives it has the form

f(k)9 = E (-1)m (m) (f9(k m))(m)M=0 \ /

and is easily proved by induction or by Leibnitz' formula.


and from (2.34) it follows that Vf E (Wp)1oc(Q) and Va E 1 satisfying Ial < l

II L,(11) <_ II(Dwf - D°Ve)121IIL,(n) -+ 0

as s -+ oo. Finally, as in the proof of Theorem 1, it is enough to note thatW,(c) C (W,)1oc(cl).

2.5 Linear mollifiers with variable stepWe start by studying the mollifiers A6 having kernels with some vanishingmoments. The main property of the mollifier A6 is that Yf E LP(I) where1<p<00

IIA6f - IIIL,(n) = 0(1) (2.42)

as 6 -3 0+. As for the rate of convergence of A6f to f, in general, it can bearbitrarily slow. However, under additional assumptions on f one can havemore rapid convergence.

Lemma 8 Let S2 C Rn be an open set, 1 <p:5 00, 6 > 0 and G C S2 be apleasurable set such that G6 C SZ. Then Vf E wp(Q)

IIA6f - f II L,(c) <- c1bIIf (2.43)

where

cl = Maxi=l,...,n(2.44)

Idea of the proof. For f E C°O(Sl) apply Taylor's formula and Minkowski'sinequality. For f E wp(f) approximate f by Aryf and pass to the limit asry-r0+.Proof. For f E C°°(12)

IIA6f - f II L,(c) = II f (f (x - 6z) - f (x)) w(z) dz 114(G)

9(0,1)

= II

(

/f f of (x - t6z)(-6z;)w(z)) dt dz IIL,(c)B(0,1) o

1 n

<6 f J(II_(x_toz)ilafIz;w(z)Idt dz

ax; L (0) 19(0,1) 0

2.5. LINEAR MOLLIFIERS WITH VARIABLE STEP 61

<5 max f Iziw(z)Idzi=1,...,n

i=1B(0,1)

of8xi C151IfIIw;(G6).

L, (GS)

=

Now let f E wP(Sl) and first suppose that p = dist (G6, 8Sl) > 0. Then for0 < y < P we have that Af E C°°(f) on G6 C Sly and A,(A6 f) = A6(A.1 f )on G (see Section 1.1). Consequently,

II A,(A6f - f)IIL,(G) = II A6(A,f) - A,f II L,(G)

5c15IIAf11w;(G6)=C15EII

axjf 11 L,(C6)-C15jl1 Ary(8xj)wII Lv(G6)

by Lemma 4. Passing to the limit as y - 0+ (see (1.10)) we obtain (2.43).If dist (G6, 811) = 0, we choose measurable sets Gk, k E N, such that

Gk C Sl, Gk C Gk+1 and kJ Gk = G. Inequality (2.43) is already proved fork=1

Gk replacing G. Passing to the limit as k -+ oo we obtain (2.43) in this casealso.

Estimate (2.43) is sharp as the following examples show.

Example 2 Let for some j E { 1, .., n} f zjw(z) dz 3k 0 and 0 < meas Sl < oo.

ThenR

IIA6xj - xjIIL,(G) = c26, c2 = I fz,w(z)dz I(meas G)o > 0.

R

Remark 16 This example shows also that for some kernels of mollification clis the best possible constant in inequality (2.43). Let us choose j = 1, ..., n,such that Ilzjw(z)IIL,(R) = iMax

Ilziw(z)IIL,(R) Moreover, let G be a bounded

measurable set such that 0 < mess G = mess G < oo. Then

sup 5-' sup II A6f - f IIL,(G) II f II W'(G6)6>O:G6cn IUII.'Imoo

>- 6i 6 'IIA6xj -xjIIL,(G)IIxjII;P(G6)

=If zjw(z) dz 16lli+m I

messmess

G G6' PL

= I

zjw(x) dz

R R

Thus, if, in addition to (1.1), w(z) < 0 if zj < 0 and w(z) > 0 if zj > 0, thencl is the best possible constant in inequality (2.43).


Example 3 Let n = 1, G = St = R, p = 2 and f E W2 (R). Then by theproperties of the Fourier transform

IIA6f - fIIL,(R) = (2ir)26II6-'((Fw)(6f) - (Fw)(0))(Ff)(f)IIL,(R).

We have6-1((Fw)(bf) - (Fw)(0)) -* (Fw)'(0)t

ash -0+ and

sup6-'I (Fw) (6f) - (Fw)(0)I m ax I (Fw)'(z)I

6>0

Therefore, by the dominated convergence theorem

II6-'((Fw)(6f) - (Fw)(0))(Ff)(f)IIL2(R)

- I(Fw)'(0)I IIe(Ff)(f)IIL,(R) = I Jzw(z)dz I IIf'IIL,(R).R

Hence, if f zw(z) dz 0 0 and f E W2 (1R) is not equivalent to zero, then forR

some c3 > 0 (independent of 6) and II A6f -f IIL,(R) >- c36 for sufficiently small J.

Let us make now a stronger assumption: f E WP(f) where I > 1. In thiscase, however, in general we cannot get an estimate better than

IIA6f - f II L,(c) = O(5)

(which is the same as for I = 1), if for some j E {1, ..., n} f z,w(z) dz # 0,R

as Examples 2 - 3 show. Thus, in order to obtain improvement of the rateof convergence of A6f to f for the functions f E Wp(f2) where 1 > 1, somemoments of the kernel of mollification need to be equal to zero.

Lemma9 LetnCW bean open set, 1<p<oo,IEN, 6>0andGC1be a measurable set such that G6 C 11. Moreover, assume that the kernel of themollifier A6 satisfies, besides (1.1), the following condition:

f z°w(z) dz = 0, a E ?V, 0 < Ial < 1-1, (2.45)

B(o,1)

where z° = zi' . z.*-. Then V f E wip(11)

II A6f - f II L,(o) <- c46'IIf Ilw,(6), (2.46)


where rc4 = max J

z°w(z) dz < IIwIIL,(a^). (2.47)

IaI=1 a!R^

Condition (2.45) is necessary in order that inequality (2.46) be valid for allf E WW(G) with some c4 > 0 independent off and J.

Idea of the proof. By condition (2.45) Yf E COO(1)

(Asf)(x) - f (x) = f (f (x - bz) - f (x)) w(z) dzB(o,1)

P x) - E(Daa)(z)

(-bz)a w(z) dz.f IaI«B(o,1)

Now multidimensional Taylor's formula (see section 3.3), Minkowski's inequal-ity and direct estimates (close to those which were applied in the proof ofLemma 8) imply (2.46). If f E w,(S2), then pass to the limit in the samemanner as in the proof of Lemma 8.

As for necessity of condition (2.45) for bounded G it is enough to takein (2.46) successively f (x) = xj,j = 1,...,n, f (x) = xjxk,j, k = 1, ..., n, ...,f (x) = , ji-i = 1, ..., n. If G is unbounded, then one needs tomultiply the above functions by a "cap-shaped" function 77 E Co (R"), whichis equal to 1 on a ball B such that mess B f1 G > 0. O

In the sequel we shall apply the following generalization of inequality (2.46).

Lemma 10 Let S2 C R" be an open set, 1 < p:5 oo, l E N, b > 0 and G C flbe a measurable set such that G6 C S2. Assume that the kernel of the mollifierAs satisfies, besides (1.1), condition (2.45). Then V f E w,(S)) and Va E N

IID°(Asf) - DafUL..,(o) <_C56'-I°1IIfIIwp(G

), Ial < 1, (2.48)

and

II Da(Aaf)II L,(C) 5 C66'-'Q1 11f1014(01), lal ? 1, (2.49)

where c5, c6 > 0 do not depend on f, 5, G and p. (For instance, one can setca = II(IIL,(n") and ce = max

Is1=IaI-1


Idea of the proof. Inequality (2.48) follows by Lemma 4 of Chapter 1 andinequality (2.46) applied to Dwf E w1 I°I(Q). Estimate (2.49) does not usecondition (2.45). It is enough to apply Young's inequality to the equality (see(1.21))

D°(A6f) = 6IRI-I7I(D°-Iw)6 * Dwf ,

where ry E too is such that 0 < ry < a and (rye = 1. 0

Let S2 be an open set and let the "strips" Gk be defined as in Lemma 5 ifi1 0 R° and as in Lemma 6 if 11 = )R". Moreover, let {ii*}kEZ be partitions ofunity constructed in those lemmas.

Definition 2 Let 0< 6< l E N and f E Ll-(f2) . Then dx E 0

(E6f)(x) _ (E6,lf)(x) _k=-oo

0 r

= E 'pk(x) J f (x - a2-IkIz) w(z) dz, (2.50)k=-oo

B(0,1)

where w is a kernel satisfying, besides (1.1), condition (2.45) 9.

Remark 17 For boundedfZ the operator E6 is a particular case of the operatorC1 by Remark 7. As in Section 2.2 in (2.50) in the last term we write f andnot fo, assuming that iik(x)g(x) = 0 if 'k(x) = 0 even if g(x) is not defined.(This can happen if dist (x, Of)) < 62-I1I). Since 0 < 6 < a we have

supp kkA62-Iklf C (Gk-, U Gk U Gk+1)62-IkI (2.51)

and

1ikAd2-IkI f E C°°(fl). (2.52)

(If SZ is bounded, then 0jtA62-IkI f E Co (fl).) As in the case of the operatorsBI and Cg the sum in (2.50) is finite. If Vx E fl the number s(x) is chosen insuch a way that x E G then

8(x)+1

(E6f) (x) = t,b (x) f f (x - 62-1klz) w(z) dz. (2.53)k=8(2)-1 B(0,1)

9 If I = 1, then there is no additional condition on the kernel w.


Moreover, Vm E Z

m+1

E6f = E 1kA62-iki f on Gm. (2.54)k=m-1

We call the E6 a linear mollifier with variable step. The quantity E6(x) is anaverage of ordinary mollifications with the steps 52-I3(s)I-1, 52-I4052-I3(=)I+1which (in the case 1196 R") tend to 0 as x approaches the boundary M. Againwe can say that the E6 is a mollifier with a piecewise constant step since thesteps of mollification, which are used for the "strip" Gm, namely 52-ImI-1

52-I-I, 52-1"'1+', do not depend on x E Gm.Moreover, by Remark 5 for any fixed -y > 0 we can choose a partition of

unity {z'k}kEz in such a way that, in addition to (2.54), Vm E Z

E6f = A62-1-If on (Gm)ry2-m.

Remark 18 Changing in (2.50) the variables x - 52-1kIz = y we find

(E6f)(x) = JK(xyio)f(y)dy,

where

n

K(x,y,5) _ E Vk(x)(82-jkj) "wk= -00

(E)Comparing these formulae with formula (1.2) we see that, similarly to themollifiers A6i the mollifiers E6 are linear integral operators, however, with amore sophisticated kernel K(x, y, 6) replacing 5-"w (Z-811)

-

The mollifier E6 inherits the main properties of the mollifier A6i but thereare some distinctions.

Lemma 11 Let S2 C R" be an open set and f E L1 `(S1). Then V6 E (0, 8]E6f EC0O(51) andVaE1g

00

D°(E6f) = E D°(OkA62-ikif) on 11. (2.55)k=-ao

Remark 19 In contrast to the mollifier E6 we could state existence and infinitedifferentiability of A6 f for f E Li00(11), in general, only on 526.


Idea of the proof. The same as for Lemma 7.

Lemma 12 Let 11 C R" be an open set and f E L,°`(11). Then

E8 f -+ f a.e. on it

ash->0+.

(2.56)

Idea of the proof Apply (2.54) and the corresponding property of the mollifierA6.

Lemma 13 Let 11 C R' be an open set and 1 < p:5 oo. Then db E (0, 8] andf E Lp(il)

IIEaf IIL,(n) <_ 2c7 IIf IIL,(n) , (2.57)

where c7 = IIwIIL,(R-)

In order to prove this lemma we need the following two properties of Lp-spaces where 1 < p < oo.

1) If 11 C R' is a measurable set and Vx E 1 a finite or a denumerable sumE ak(x) of functions ak measurable on Il contains no more than x nonzero

k

summands, in other words, if the multiplicity of the covering {supp ak} doesnot exceed x, then

1

I I E ak II L,(n) <_ xl = E I Iak IIi,(n)k k

(2.58)

(This is a corollary of Holder's inequality.)2) If it = U Ilk is either a finite or a denumerable union of measurable sets

kilk and the multiplicity of the covering {ilk} does not exceed x, then for eachfunction f measurable on i1

IIfII,,(nk) <_ xaIIfIIL,(n) (2.59)k

In particular, if p = 1 and f = 1, then we have

E mess Slk < x mess il. (2.60)k


For p = oo these inequalities take the following form

II E akllL (n) <_ csup IIakIIL-(n), SUP Ill IIL (n)k k k

Idea of the proof of Lemma 13. Apply (2.58), (1.7) and (2.59).Proof of Lemma 13. By (2.7) and (2.15) V6 E (0, el

00 00

U suPP Ok = U (suPP Ok)62k=-oo k=-oo

and the multiplicities of the coverings {SUpp10k}kEZ and {(SUppok)62-IkI}kEZ

are equal to 2 (see Remark 6). Therefore, by (2.58) and (2.59)

00

IlE6fIIL,(n) = II VbkA62-I.IfIIL,(n)k=-oo

/ 00

<21 p( ao

IIA62I'll fll'L(SUpp 0,)=-00

00 yl< 2'-a IIfIIP 2c711fIIL,(n)=-oo

Now for an open set S2 C R" and Vx E fl we set p(x) = dist (x, 8Sl) ifSZ96 R" and 10 p= (1+Ixl)-1} if S2=IRS.

Lemma 14 Let 1l C Rn be an open set and 0 < 6 < Then1) for 1 < p < oo and `d f E Lp(S2)

E6f -+ f in LP(S2) (2.61)

2) for p = oo relation (2.61) holds V f E G (SZ),

io It is also possible to consider Q(x) = min{diet (x, tifl), (1 + IxI)-' }. However, in thatcase one must use a partition of unity constructed on the base of altered p and verify thatestimate (2.12) still holds.


3) for 1<p<oo,lENand Vf Ewp(S2)

II Eaf - f IIL,(n) 5 ca. 011f (2.62)

4) for 1 < p < oo, ! E N and df E w,'(52)

II (Ed - f)e 11IL,(n) <- c901fhIW',(n) , (2.63)

where c8 and c9 do not depend on f , 6, n and p.

Idea of the proof. To prove the statements 1) and 2) establish, by applying theproof of Lemma 13, that the series (2.50) converges in LP()) uniformly withrespect to 6 E (0, e] and use the corresponding properties of the mollifiers Aa.To prove (2.62) and (2.63) follow the proof of Lemma 13, applying inequality(2.48) with a = 0 instead of (1.7). In the case of inequality (2.62) apply, inaddition, the fact that there exist B1, B2 > 0 such that

B12-k

5 o(x) < B22-k on supp tpk (2.64)

dkEZif( R" andVk<0ifS2=R°. 0Proof. We start with the proof of inequality (2.62). If f E ww(11), then using,in addition, Minkowski's inequality for sums we find

00

IIEaf - fIIL,(n) = II E +Gk(Aa2-Iklf - f)IIL,(n)k=-ao

00

< 2' P IIA62-Iklf - fIILP(80PPtlk)k=-0o

i

< 2' Pcs61 (J-0Ez-lkl'PIIfIIWy((.PP

1bk)'2-Ikl )

a0

00

5 2' Pcs6l EIIDwhhL,((wPP16k)a=-Ikl)

k=-0o 1011=1

P\ s

1

< 21-ac,,a1E CC IIDwfIIPLP((.PP*k)62-Ikl)J

P

101=1 =-oo


< 2c5a' IID`f1IL,(n) = c86Ìlfllw;(a),

Ial=1

because the multiplicity of the covering {(supp1Pk)a2-"" }kEZ is equal to 2 (seeRemark 6).

In the case of inequality (2.63) we find with the help of (2.64) that

II (Eaf - f )P 'IILP(n) = II = 'iPk(Aa2-ikif - f)IILo(n)k=-oo

I0o_2' PBl' 21k1'pIIAa2-Iklf -

k=-oo

a

0

< 2'_0Bl'c56'11i11°

(k=-000

The rest is the same as above (c9 = 2Bi c5). 0

Lemma 15 Let Il C Rn be an open set, 1 E N and 0 < b < Then for eachpolynomial pl-1 of degree less than or equal to 1- 1

(Eapr-1)(x) = Pr-1(x), x E S2.

Idea of the proof. Apply multidimensional Taylor's formula (see Section 3.3) topt-1(x - 62-IkIZ) in (2.50) and use (2.45), (1.1) and (2.11) or (2.14). 0

Remark 20 In Lemmas 13 - 14 the property (2.45) of the kernel of mollifica-tion was not applied. It was applied in Lemma 15, but this lemma will not beused in the sequel. The main and the only reason for introducing this propertyis connected with the estimates of norms of commutators [Dw, Ea] f , which willbe given in Lemma 20 below. In its turn these estimates are based on Lemmas9-10, in which the mollifiers Aa with kernels satisfying the property (2.45)were studied.

Let us denote the commutator of the weak differentiation of first order andthe mollifier Ea in the following way:

(esf). (Ej) = [(a )m, EE] ( )w EE - Ea (k)w.

This operator is defined on (W1)1O1(0)


Furthermore, for I E N, l > 2, we define the operators b.0 (E6), wherea E I ` and jal =1, with the domain (Wi)'O°(12):

b.-(E6) = ((aI)w' ...(E6).

Lemma 16 Let 0 C R" be an open set, 1 E ICI, 0 < b < s and f E (Wi)i0c(Sl)Then Va E N' satisfying dal < I

00

(D°(E6)) f = E (D°,0k)A62-Ik1fk=-oo

Idea of the proof. Induction. For Ian = 1 by (2.55)

(((00l\ax,) (Ea))f = (A62-ikif+V)kaxj(A62-ikif))

w k=-oo

00 00

-kA62-iki (e ii) _ A62-ikifW 821

k=-oo k=-oo

on S2, because by (2.7) and (1.19) ikk (Ab2_j,,jf) = tPkAd2-iki (e )W on 11. E3371

Remark 21 For the mollifiers A6 we have (Dw(A6)) f =_ 0 but, only on 126 (seeSection 1.5), while for the mollifiers E6 in general (Dw(E6)) f $ 0 even on 526,but on the whole of S2 the quantity (Dw*(E6))f is in some sense small (because

00

E D°9fik = 0 on S2) and, as we shall see below, it tends to 0 in Lp(S2) fastk=-ooenough under appropriate assumptions on f .

Remark 22 On the base of Lemma 16 we define for Va E N satisfying a 96 0the operator E6°) with the domain L"(1) directly by the equality

Ea°if = E (D°*k)A62_Iki f.k=-00

(2.65)

Lemma 17 Let SZ C W' be an open set and 0 < b < Then Va E Plasatisfying a 96 0 and V f E Li«(12)

Es(*) f -s 0 a.e. on S2. (2.66)


m+1Idea of the proof. Since Vm E Z we have E(°) f = F, Da1PkA62-IkI f and

k=m-1m+1E Da0k = 0 on Gm, the relation (2.66) follows from (1.5). 0

k=m-1

Lemma 18 Let fZ C JW be an open set, 1 < p:5 oo and 0 < 6 < e. Then1) V f E w,(Q) and Va E N satisfying 0< IaI < 1

IIEea)f IIL°(n) <_ C1 blllf (2.67)

2) Vf E w,(11) and Va E N satisfying jai > 0

II(E6(a)f)P'a1-1IIL°(n) < clIa1llfllwp(!1), (2.68)

where c10, c11 > 0 do not depend on f, 6, 0 and p.

Idea of the proof. Starting for a 36 0 from the equality

EE°)f = = D0,0k(A62-141 f - f) (2.69)k=-oo

follow the proof of Lemma 13, apply estimate (2.12) of Lemmas 5 and 6 andinequality (2.48), in which a = 0, G = supp tpk and 6 is replaced by 62-A. Inthe case of inequality (2.68) apply, in addition, (2.63). OProof. For IaI < 1 from (2.69) it follows that

°IIE6a)fllL°(:z) < 21 ° IIDabk(A62-Iklf - f)IIL°(n))

k=-oo

I< 21 sca F, f IIL°(supp*h)

°

k=-oo

< 21 'c0c5d1 (E 21k1(1al-1)PIIfIIp )6r1k1)00

=-00 y k

001

r21 . Ca4g61 E IIf lip 69-1.1=- +°((sappd.) )


< 2c°c56'lifllwy(f) = c1061IIfIIwy(n)-

We have taken into account that the multiplicity of the covering{

(suPPOk)62ik'}kez is equal to 2 (see Remark 6).In the case of inequality (2.68)

1 a oft ° PII (E6 f)Q IIL,(n) < 2 n E IIQI D ''k(A62-ikif - f)IIL,(n)

(J-000

a

21-1 (max {Bj 1, B2}) II°I-A ca

(J-00021kh1P IIA62-ikif - f 11 i ,,,I v(

The rest is the same as above (c11 = 2c°cs (max {B1', B2})11°I-111 O

Lemma 19 Let I C R" be an open set, l E N and f E (Wi)1oc (S2). ThenV E N satisfying 0< f ai < I

D°(Esf) _ (;)E-(Df) (2.70)

and

[Dw, E6]f = (a)E6(--0)(D1Pf). (2.71)

0:50:50,0000

Idea of the proof. Apply (2.55), Leibnitz' formula, Lemma 4 of Chapter 1 andthe definition of the operator E6") (see Remark 22). 0

In the sequel we shall estimate D°(E6 f) and D°(E6 f) - D.*f with the helpof (2.71) and the following obvious identities:

D°(E6f) = [Dc, EB) f + E6(Dwf) (2.72)

and

D°(E6f) - D. ,f = [D,°n, E6] f + E6(D"f) - Dwf- (2.73)

Lemma 20 Let 11 C R?' be an open set, 1 0

II (D , )PIaI-ÌIL,(n)C1381-IaI+1

Ifllwy(R) +(2.75)

where c12, C13 > 0 do not depend on f , 8, SZ and p.

Idea of the proof. Starting from equality (2.71) apply inequalities (2.67), re-spectively (2.68), with 1- 1,61 replacing 1, a - p replacing a and DO ,,f replacingf. Take into consideration that plat-1 = 19la-P1-(1-IaI) and IQI < IaI - 1.

2.6 The best possible approximation withpreservation of boundary values

We start by studying some properties of the mollifiers E6 in Sobolev spaces.

Theorem 5 Let Q C R" be an open set, I E N, 0 < 8 < e and 1 0 does not depend on f , 8, 11 and p.

Idea of the proof. Apply (2.71) and Lemmas 13 and 20.Proof. By (2.72), (2.74) and (2.57)

IIE6f 11w;,(n) _ II DwE6f IIL,(n)IaI=1

E II[Dw, E6] f IIL,(n) + II EoDW f IIL,(n)IaI=1

<_ (c12IIf Ilwo(n) + 2C7II Dw f IIL,(n)) = c14IIf 11,4(n)IaI=1

Inequality (2.77) follows from (2.57) and (2.76).

Theorem 6 Let) C R" be an open set, 1 E N and 0 < 8 <1) If 1:5 p:5 oo, then df E Wp(11) and Va E N satisfying IaI 5 1

Da(E6f) -+ D.Of a.e. on 11 (2.78)


asb-r0+.2) If 1 f in Wo (S2), m = 0, ...,1, (2.79)

ash->0+.3) If p = oo, then Vf E W;°((2)

Ea f -> f in WW (S2), m = 0,..., l -1, (2.80)

as 6 -+ 0+ (if f E C1((t) then (2.79) holds).

Idea of the proof. Relation (2.78) follows from equalities (2.72), (2.71) andLemmas 14 and 20; relations (2.79) and (2.80) follow from (2.72) and Lemmas14 and 20.Proof. Let us prove (2.79). From (2.72), (2.74), (2.62) and, in the case m = 1,(2.61) it follows that

IIEaf - Ill w; (n) = II Eaf - f IIL,(n) + II D°Eaf - Dwf IIL,(n)IaI=m

<- IIEaf - f IIL,(n) + (II[D.*, Ea]f IIL,(n) + IIEa(Dwf) - Dwf IIL,(n)) -' 0IaI=m

asb-90+.The same argument works to prove (2.80). Since in this case m < 1, it is

enough to apply only inequalities (2.74) and (2.62).

Theorem 7 Let 11 C R" be an open set, I E N, 1 0 does not depend on f, 6, 11 and p.2) If Ial > 1, then V f E w,(1l)

II (Do(Eaf ))p101-'IIL,(n) <- c166' -IaI ilf Ilwp(n) , (2.82)

where c16 > 0 does not depend on f , 6, Sa and p.3) There exists an open set (1 such that for any e > 0 inequality (2.82) with

0101-1-E replacing OI01-1 does not hold.


Idea of the proof. Inequality (2.81) follows from equality (2.73) and the inequal-ities (2.74) and (2.62) with D. *f replacing f and I - IaI replacing 1. Inequality(2.82) follows from equality (2.72), inequality (2.75) and the inequality

II(Ea(Dwf))Plal-'IIL,(n)

5 0170'-1°l1IfIJt4(n) (2.83)

for IaI > 1, where c17 > 0 does not depend on f, b, SZ and p. In order toprove (2.83) apply the proof of inequality (2.63). The third statement will beconsidered in the proof of the Theorem 8 below. 0Proof. It is enough to prove (2.83). Applying Lemma 4 of Chapter 1 and theinequalities (2.63), (2.58), (2.49) and (2.59) we establish that

00II(E6(Dwf))P101-1IILP(n) = II

Pl°1-11kDaA62-Iklfll LP(n)k=-oo

W

< 21-=B2°"-1

2 B2°I-tc6at-IaIIif IIPw ((eup )6t-Ikl )

Pk=-oo P k

P

< 2B2°I-'Ca'-I°IIIfIIw4(n) _ C176`-°1I1J Iw,(n)

(For details see the proof of Lemma 14.) O

Theorem 8 I. Let SZ C R' be an open set, l E N and 1 < p < oo. Thendf E Wp(SZ) functions cp, E C0(S2) f1 Wp(I), s E N, exist, which dependlinearly on f and satisfy the following properties:

1) for1<p<oo

D°W, -4 DW f a.e. on 0, IaI < 1,

ass-+00,2) fort<p<00

(o, -+ f in Wp (f ), m = 0, ...,1, (2.84)

ass -aoo,3) for p = 00

w, -+ f in W.(0), m = 0,..., 1 - 1, (2.85)


as s -+ 00 (if f E &(S1), then relation (2.84) also holds),4) for1<p<oo

II(D°wf - D°W,)8'°'-'II L,(n) -10, IaI < 1, (2.86)

ass-100,5) for1<p<oo

°,sllj Wl(n), IaI > 1, (2.87)

where c°,, are independent of f, Sl and p.II. There exists an open set SI C R°, for which, given e > 0 and m > 1, a

function f E Wp(SI) exists such that , whatever are the functions gyp, E COO (11) flYtip(SI), s E N, satisfying property 4), for some V E IN satisfying IaI = m

II '°l -'-EII L,(n) = 00. (2.88)

Idea of the proof. The first part of Theorem 8 is an obvious corollary of Theo-rems 6 and 7: it is enough to take cp, = Ei f The second part will be provedin Remark 14 of Chapter 5.

Remark 23 The second part of Theorem 4 is about the sharpness of condition(2.87). We note that since in (2.87) p(y)IOH --10 as y approaches the boundaryBSI, the derivatives D°v,(y) where IaI > I can tend to infinity as y approachesBSI. By the second part of Theorem 8 for some SI C R" and f E WW (Q) for anyappropriate choice of gyp, some of the derivatives D°cp,(y) where IaI = m > l dotend to infinity as y approaches a certain point x E on. Indeed, for bounded11 from (2.88) it follows that for some Va E NB satisfying IaI = m, for somex E BSI and for some yk E Sl such that yk --- x as k -+ oo

lim (D°Ws)(yk)P(yk)1Q1-'_, = 00, (2.89)

i.e., (D°w,)(yk) tends to infinity faster, than d-101-(yk). (We note that thehigher order of a derivative is the faster is its growth to infinity.)

Remark 24 This reveals validity of the following general fact: if one wants tohave "good" approximation by COO-functions, in the sense that the boundaryvalues are preserved, then there must be some "penalty" for this higher quality.This "penalty" is the growth of the derivatives of higher order of the approx-imating functions when approaching the boundary. The "minimal penalty" isgiven by inequality (2.87).


Remark 25 By Theorems 6 and 7 the functions cp, = E1 f satisfy the state-ments of the first part of Theorem 8. Thus, by the statement of the second partof this theorem the mollifier E6 is the best possible approximation operator,preserving boundary values, in the sense that the derivatives of higher ordersof E6f have the minimal possible growth on approaching 852.

Now we formulate the following corollary of Theorem 8 for open sets withsufficiently smooth boundary, in which the preservation of boundary valuestakes a more explicit form.

Theorem 9 Let I E N,1 < p < oo and let 12 C R" be an open set with aC'-boundary (see definition in Section 4.3).

I. For each f E Wp(12) functions W. E C°°(12), s E N, exist, which dependlinearly on f and are such that

1) gyp, - f in Wp(12) as s -> oo,

" lal < 1- 1,2) D°welen = Dwf I.

3) IID°w,d°I-ÌIL,(n) < cc, j al > 1.

II. Given e > 0 and m > 1, a function f E WP(12) exists such that, whateverare the functions W. E C°°(12), s E N, satisfying 1) and 2), for some Va E I'satisfying Ial = m

IIL,(n) = cc. (2.90)

Idea of the proof. As in the proof of Corollary 3, by Lemma 13 of Chapter 5,propety 2) follows from relation (2.86). 0

The most direct application of Theorem 7, for the case in which p = oo, isa construction of the so-called regularized distance. We note that for an openset S2 C R", SZ # R1, the ordinary distance p(x) = dist (x, 812), x E 12, satisfiesLipschitz condition with constant equal to 1:

I p(x) - p(y)I 5 Ix - yI, x, y E S2. (2.91)

(This is a consequence of the triangle inequality.) Hence, by Lemma 8 ofChapter 1

p E wolo(12), Ivpl < 1 a.e. on 12. (2.92)

The simplest examples show (for instance, p(x) = 1- Ixi) for 12 = (-1, 1) C R)that in general the function p does not possess any higher degree of smoothnessthan follows from (2.91) and (2.92).


Theorem 10 Let Sl C R" be an open set, 11 36 R. Then Vl E (0,1) a functionea E C°°(11) (a regularized distance) exists, which is such that

(1 - 6)e(x) < ea(x) < e(x), x E 0, (2.93)

I Pa(x) - e6(y) 15 Ix - yI, x, y E Sl, (2.94)

I oea(x)I < 1 on fl (2.95)

and We E N satisfying IaI > 2 and Vx E Q

I (D°ea)(x)I 5 cool-lalg(x)1-IaI , (2.96)

where cc, depends only on a.

Idea of the proof. In order to construct the regularized distance it is naturalto regularize, i.e., to mollify, the ordinary distance. Of course, one needs toapply mollifiers with variable step. Set ea = aE"g and choose appropriatea, b > 0. Here EM is a mollifier defined by (2.50) where l = 1 and the kernel ofmollification w is nonnegative.Proof. First let Da = E6e. Since p E w;o(Q), from (2.81) and footnote 4 onthe page 12 it follows that

sup Ioa(x) - e(x)I e(x)-1 < c156'En

or bxESl(1 - c156)0(x) 5 Do(x) 5 (1 + ctsb)e(x),

where c15 > 0 depends only on n.Moreover, from (2.82) it follows that Va E N satisfying a 96 0

sup IDaoa(x)le(x)IaI-1 < c16b1-IaI

Zen

or Vx E Cl

IDao6(x)I 5 c1861-Iale(x)'-IaI

where c16 > 0 depends only on n and a.Furthermore, by definition of Ea and by (2.11) or (2.14)

Aa(x) -,Na(y) = E (0k(x)(Aaz-i,ie)(x) -Ok(y)(Aaa-i+ie)(y))k--o°

2.6. THE BEST POSSIBLE APPROXIMATION

00

k(x)((A62-Iklp)(X) - (A62-IklQ)(0)k=-oo00

+ E ('ibk(x) - Vk(y))((A62-IkIQ)(Y) - 9(y))k=-0o

Hence,

Io6(x) - o6(Y)I G V)k(x)I(A62-Iklp)(x) - (A62-I410)(y)Ik=-oo

79

+ E I)k(x) -'k(y)I f I A(y - 62-Iklz) - p(y)I w(z) dz.kES(x,y) B(0,1)

Here by (2.53) S(x, y) = {s(x) -1, s(x), s(x) + 1, s(y) -1, s(y), s(y) + 11. From(2.12) it follows that

n

I'+hk(x) -'+Gk(y)I = E(x9 - y3) f 8xk (x + t(y - x)) dt G c182kIx - yI ,=1 0

where c18 = ( E c2)1/2 with ca from (2.12) depends only on n. Now, applying

(1.13), (2.11) or (2.14), and (2.91) we have1- 1=1

IO6(x) - O6(y)1 :5 Ix - yI 1 + C38 E 2k(62-IkI) f Izi w(z) dzkES(x,y) B(0,1)

G (1 + 6c186)Ix - yI.

Finally, it is enough to set p6 = aEb6p, where, for instance, a = (1 + z)-' andb = min{cig , (6c18)0

Remark 26 The regularized distance can be applied to the construction oflinear mollifiers with variable step. It is quite natural to replace the constantstep 6 in the definition of the mollifiers A6 by the variable step bp(x), i.e., toconsider the mollifiers

(Hof)(x) = (A6Q(x)f)(x) = f f (x - 6p(x)z) w(z) dz

B(0,1)


for 0 < b < 1. (In this case B(x, bp(x)) C 11 for each x E ) and, therefore, thefunction f is defined at the point x - bp(x).) If p E CO°(11), it can be provedthat H6 f E C°°(S1) for f E L Oc(S1) and that Ha f -+ f a.e. on fl. This is so, forinstance, for 0 = R" \R'", 1 < m < n, in which case p(x) = (xm+I+...+xn)1/2However, as it was pointed out above "usually" gEC°°(SZ). This drawback canbe removed by replacing the ordinary distance p by the regularized distancep' = p4 with some fixed 0 < So < 1 (say, 6o = 1). We set

(H6f)(x) _ (A6 =)f)(x) = J f (x - be(x)z) w(z) dz.B(0,1)

Then Vf E L'°`(Sl) we have H6 f E C°°(Q) and lib f -p f a.e. on 11. Asfor results related to the properties of the derivatives D°H6 f , in this caseestimate (2.96) is essential. Some statements of Theorems 8-9 can be provedfor the operator H6 as well. The main difficulty, which arises on this way isthe necessity to work with the superposition f (x - 5p60(x)z). For this reasonthe mollifiers E6 with piecewise constant step are more convenient, because intheir construction superpositions are replaced by locally finite sums of products.Another advantage of the mollifiers with piecewise constant step is that it ispossible to choose steps depending on f . This is sometimes is convenient inspiteof the fact that the mollifiers become nonlinear. (See the proofs of Theorems1- 4 of this chapter and Theorems 5 - 7 of Chapter 5.)

Example 4 For each open set 0 C R" a function f E C°O(R") exists such thatit is positive on S1 and equal to 0 on R" \ 0. The function f can be constructedin the following way: f (x) = exp(-ee(=)) with some fixed b E (0,1). Theproperty (D° f)(x) = lim (D° f)(y) = 0 for x E 80 follows from (2.96).

y-4z,yEftThis function f possesses, in addition, the following property, which sometimesis of importance: Vy > 1 and da E N there exists c19 = cis(ry, a) > 0 suchthat dxER"

I (D°f)(x)11 < clef (x)

This also follows from (2.96).

Another application of a regularized distance for extensions will be given inRemark 17 of Chapter 6.

Chapter 3

Sobolev's integral representation

3.1 The one-dimensional caseLet -oo<a < b<oo,

w E L1(a,b), Jwdx=1 (3.1)

a

and suppose that the function f is absolutely continuous on [a, b]. Then thederivative f exits almost everywhere on [a, b], f E L1 (a, b) and Vx, y E [a, b] we

have f (x) = f (y) + f f'(u)du. Multiplying this equality by w(y) and integrating

with respect to y from a to b we get

b fbxf (x) f f (y)w(y) dy+ (J f'(u) du)w(y) dy

a a y

Interchanging the order of integration we obtain

b x x z b yv

f (f f(u) du)w(y) dy = f (f f'(u) du)w(y) dy - f ( f f'(u) du)w(y) dya V a V x z

z u b b b

= f (f w(y) dy) f'(u) du - f ( f w(y) dy)f'(u) du = f A(x, y)f'(y) dy,a a s u a

82 CHAPTER 3. SOBOLEV'S INTEGRAL REPRESENTATION

where

Hence Vx E (a, b)

V

fw(u)du, a<y<x<b,A(x, y) = a

- fw(u)du, a<x<y<b.V

b 6

f (x) = f f (y) w(y) dy + f A(x, y)f'(y) dy. (3.3)a a

This formula may be regarded as the simplest case of Sobolev's integral repre-sentation.

We note that A is bounded:

Vx, y E [a, b) jA(x, y)j < II)IIL.(a,b) (3.4)

and if, in addition to (3.1) w > 0, then 1

dx, y E [a, b] IA(x, y)j< A(b, b) = 1. (3.5)

Let us consider two limiting cases of (3.3). The first one corresponds tow = const, hence, Vx E (a, b) we have w(x) = (b - a)-1. Then Vx E [a, b]

6 x b

f (x) = b af f (y) dy + f b- a f'(y) dy - f b- a f'(y) dy. (3.6)

a a x

To obtain another limiting case we take w = +X(6_,6)), whereX(.,p) denotes the characteristic function of an interval (a, p), m E N andm > 2(b - a)-1. Letting mn -> oo we find: dx E [a, b]

b

f(x) = f(a)2

f(b) + 1f sgn(x - y)f'(y) dy. (3.7)

0

Of course both of formulas (3.6) and (3.7) can be deduced directly by integra-tion by parts or the Newton-Leibnitz formula.

1 Ifs is symmetric with respect to the point $4t, then Vy E [a, b] we have IA( °-, y)j < 1.

3.1. THE ONE-DIMENSIONAL CASE 83

Obviously, from (3.6) it follows that

b b

If(x)I <_bla f IfIdy+ f If'IdYa a

for all x E [a, b]. 2If f E (Wi)1-(a, b), then f is equivalent to a function, which is locally

absolutely continuous on (a, b) (its ordinary derivative, which exists almost ev-erywhere on (a, b), is a weak derivative f,', off - see Section 1.2). Consequently,(3.3), (3.6) and (3.8) hold for almost every x E (a, b) if f' is replaced by f,,.

Let now -oo < a < b < oo, xo E (a, b), I E N and suppose that thederivative f(1-1) exists and is locally absolutely continuous on (a, b). Thenthe derivative f (1) exists almost everywhere on (a, b), f (1) E L", (a, b) and byTaylor's formula with the remainder written in an integral form t/x, x0 E (a, b)

1-1

f (x) = E f ikk! (x - x0)k + (1 11)!

f (x - u)'-1 f i'l(u) duk_0

io

_ f lkk x0 (x - x°)k + ((1- 1)!! J(i - t)1-1 f<<l (x° + t(x - x°)) dt. (3.10)k=0 0

Theorem 1 Let 1 E N, -oo < a < a <,3< b < oo and

w E LI(R), suppw C [a,Q], fwdx = 1. (3.11)

R

Moreover, suppose that the derivative f (1-1) exists and is locally absolutely con-tinuous on (a, b).

Y By the limiting procedure inequality (3.8) can be extended to functions f, which are ofbounded variation on [a, b]: Vx E [a, b]

6

If(x)I <-b l a f Ifl dy+ a&f

a

One can easily prove it directly: it is enough to integrate the inequality I f (x)I < I f (y)I +If (z) - f (y)I <- If (y)I + ar f with respect to y from a to b.


Then Vx E (a, b)

b b

f(x) f fik'(y)(x - y)kw(y) dy + (j11)1 f(x - y)'-'A(x, y)fiì(y) dy

X=O a a(3.12)

0 bs

f f(k'(y)(x - y)kw(y) dy + (l11),

t (x - y)'-'A(x, y)f(')(y) dy,k=0

(3.13)where aZ = x, bs = a for xE (a, a]; as = a, b; for xE (a, 3); a= = asb= = x for x E [Q, b).

Idea of the proof. Multiply (3.10) with xo = y by w(y), integrate with respectto y from a to b and interchange the order of integration (as above).Proof. The integrated remainder in (3.10) takes the form in (3.12) after inter-changing the order of integration:

b

f (f (x - u)'-'f(')(u) du)w(y) dy = f w(y) (f (x - u)'-' f I>(u) du) dya y a y

b

if ( - u)' f (') (u) du) dy = - u)(f w(y) dy) f (u) dus a

b b b

- (x - u)'-' ( fw(y) dy) f (') (u) du = f (x - y)1-1A(x,

y)f (')(y) dys u a

Finally, since supp w C [a, p], it follows that A(x, y) = 0 if y E (a, as)U(b=, b)and, hence, (3.13) holds.

Remark 1 If in Theorem 1 a > -oo and f (1-1) exists on [a, b) and is absolutelycontinuous on [a, 61) for each b1 E (a, b), then equality (3.12) - (3.13) holds forx = a and a = a as well. To verify this one needs to pass to the limit as x -4 a+and a -+ a+, noticing that in this case f() E Li (a, bi) for each bi E (a, b). Theanalogous statement holds for the right endpoint of the interval (a, b).

If, in particular, -oo < a < b < oo, f (1-1) exists and is absolutely continu-ous on [a, b], then equality (3.12) - (3.13) holds Vx E [a, b] and for any interval(a,fl) C (a, b).


Remark 2 Suppose that -oo < a < b < oo, P-') exists on [a, b] and isabsolutely continuous on [a, b]. Then the right-hand side of (3.12) is actuallydefined for any x E R, if to assume that A(x, y) is defined by (3.2) for anyx E R and for any y E [a, b]. Since in this case for any x < a and for any

b

y E [a, b] we have A(x, y) f w(u) du, the right-hand side of (3.12) fora

x < a is the polynomial pa of order less than or equal to l - 1 such thatpakT(a) = f(') (a), k = 0, 1, ..., I - 1. Respectively, for x > b the right-hand sideis the polynomial pb of order less than or equal to I - 1, which is such thatpbk) (b) = f (L) (b), k = 0,1, ..., 1 - 1. Thus the function

1_1 b b

F(x) _ k1 f flki(y)(x - y)kw(y) dy + (l 11)1 f(x - y)'-'A(x, y)fi'i(y) dya

is an extension of the function f with preservation of differential properties,since F<'-1) is locally absolutely continuous on R. See also Section 6.1, wherethe one-dimensional extensions are studied in more detail.

Corollary 1 Suppose that l > 1, condition (3.11) is replaced by

w E C(1-2) (R), supp w c [a,#], Jwdx = 1R

and the derivative w(1-2) is absolutely continuous on [a, b].Then for the same f as in Theorem 1 Vx E (a, b)

f(x)= f(p 1-1 (-1); k,

k[(x-y)kw(y)](")f(y)dya

b.

+(l 11)1 J(x - y)1-1A(x,y)f(1)(y) dy

as

Idea of the proof. Integrate by parts. 0From (3.14) it follows, in particular, that

(3.14)

(3.15)

w(a) = ... = w(t-2)(a) = w(3) = ... = w(i-2)(Q) = 0. (3.16)


Corollary 2 Suppose that 1, m E N, m < 1. Then for the same f and w as inCorollary 1 Vx E (a, b)

9 1m-1 _f(-)(X)=f( E ( k!

+m

((x - y)kw(y)1(k+1fl))

f (y) dya k=0 /

bs

+(1 - m - 1)! ,f (x - y)'-m-lA(x, y)f(L)(y) dy. (3.17)

a,

Idea of the proof. Apply (3.15), with I - m replacing 1, to f (m) and integrateby parts in the first summand taking into account (3.16).

Remark 3 The first summand in (3.15) may be written in the following form:

a.(x-y)°wl°i(y))f(y)dy' a-= LS_! ) k(3.18)

a s-0 k=s

It is enough to apply Leibnitz' formula and change the order of summation inorder to see this.

By the similar argument the first summand of (3.17) may be written in thefollowing form:

9 1_1

f (E Qs,m(x - y) mw(s)(y))f (y) dy, (3.19)a s-m

where

= (-1)' 1 °-1 s + kvs,m

(s - m)! ( k ) . (3.20)

From (3.18) and (3.19) it is clearly seen that the first summand in (3.17) isa derivative of order m of the first summand of (3.15) and thus (3.17) can bedirectly obtained from (3.15) by differentiation. (In order to differentiate thesecond summand one needs to split the integral into two parts - see the proofof Theorem 1.)

Corollary 3 Let -oo < a < b < oo, l E N, m E N0, m < 1. Moreover, supposethat the derivative j(1_1) is absolutely continuous on (a, b]. Then 11x E (a, b]

b

flm)(x) dy + f ix-ylt-m-lIflll(y)Idy)


B 6

< cl (b - a)1-m-1((fi - a)-t Jii dy + f If ("I dy) (3.21)

at a

and, consequently,

b b

If(m)(x)I <_ ct((b-a)-m-' f IfI dy + f Ix-ylt-m-'If(o(y)I dy)a a

and 3

6 6

< c1 ((b - a)-m-' f If I dy + (b - a)t-m-' f If (1)1 dy) (3.22)

a a

0 b

If(-)(X)I <- C2(f IfI dy + fix - yI`-m-'If(')(y)I dy)° a

Q b

<c3( f IfI dy+ f If (°Idy), (3.23)a a

where cl > 0 depends only on 1, while c2, c3 > 0 depend on 1 and, in addition,depend on Q - a and b - a.

Idea of the proof. In (3.17) take w(x) = rµ( ), where x0 = , r = 2and u E C0 '10(R) is a fixed nonnegative function, for which supp µ C (-1, 11 andf µ dx = 1. In order to estimate the first summand in (3.17) apply (3.19) andRthe estimate iw(') (x)I < M r-'-' for m < s < I - 1, where M depends only on1. To estimate the second summand in (3.17) apply (3.5). 0

3 From (3.23) it follows, by Holder's inequality, that for I < p:5 00

Iif(m)IIL,(a,b) <- M1 (IIIIIL,(a.b) + Ilft`)IIL,(a,b)),

where Ml = c3(b - a), and, after additional integration, that

II Bx; IIL,(Q) Mz (IIfIIL.(Q) + II8 i IIL,(Q))'

where Q C lit" is any cube, whose faces are parallel to the coordinate planes, f E &(Q) andM3 > 0 is independent of f. These inequalities were used in the proof of Lemmas 5-6 ofChapter 1.


Remark 4 We note a simple particular case of the integral representation

(3.17): if w is absolutely continuous on [a, b], w(a) = w(b) = 0 and fwdx = 1,a

then for each f such that f' is absolutely continuous on [a, b], for all x E [a, b]

b b

f'(x) = - f w'(y)f (y) dy + f A(x, y)f"(y) dy.a a

It follows that

6 b

If'(x)I <- IIw IIL (a,b)f IfI dy + IIA(x,')IIL.(a,b) 111,9dy.

a a

Choosing w in such a way that is minimal we find

4 (b-a _ a+bw(x)

= (b- a)2 \ 2 - Ix 2 )

and, hence,

(3.24)

b 6

If'(x)I S (b 4a)2 Jutdy + 1-2 (minx- a,b- x)2) fIf"I dy. (3.25)

In particular

and

From (3.25) it follows that Vx E [a, b]

a a

b 6

If'(a)I,If'(b)I <- (b 4a)2a a

b b

If'(a2 b)I `- (b 4a)2 flfldy+if If"Idy.a a

bb

( la)2 f III dy+ f If"I dy).If'(x)I 5 4 (b (3.26)


This is a particular case of (3.22) with the minimal possible constant c, =4. The latter follows from setting f (y) = y - aZb. The same test-functionshows that the constant multiplying fa If I dy in (3.25), (3.26) also cannot bediminished even if the constant multiplying fQ If"I dy is enlarged.

We note that the constant muliplying fQ I f"I dy in (3.25) also cannot bediminished. 4 This can be proved in the following way. For a < x 0 consider the function 5 ga,=(y) = (x - y + b)+, y E [a, b], if a < x <2+6and ga,x (y) = (y - x + b)+, y E (a, b], if a2 < x < b. In (3.24) take f = A

2where Aa is a mollifier, and pass to the limit as b -+ 0 +.

Finally, as in the case of the integral representation (3.3), we consider alimiting case of (3.24). We write wm for w, where m E N, m > b?a, wm(x) _m(x - a)(b - a - L)-' for a < x < a + wm(x) _ (b -a - m)-` fora+m <x<b - mandwm(x)=m(b-x)(b-a-m) forb - m < x<b.Taking limits we get the equality

s b

=f(b)-f(a) y-a b-

-yfI(x) b - a +f b-af (y)dy- f b - af(y)dya x

(3.27)

Here x E [a, b] and f is such that f' exists and is absolutely continuous on [a, b].Again, as in the case of representations (3.6) and (3.7), (3.27) can be deduceddirectly.

Corollary 4 Let I E N, m E Nlo,1>2 and m<1-1.1. If -oo < a 0 depends only on t and

K(e) = e- . (3.29)

b

4 In contrast to the constant multiplying f If I dy it can be diminished if to enlarge appro-a

b

priately the constant multiplying fIfI dy - see Corollary 4.a

5Here and in the sequel a = a fora>0anda+=0fora<0.


2. If I = [a, oo) where -oo < a < oo, I = (-oo, b) where -oo 0 depends on I only.

Idea of the proof. In the first case for x E [a, b] apply (3.22) replacing [a, b]by any closed interval [a1, b1] C [a, b] containing x, whose length is equal to b,where 0 < b < b - a, and set clot-'"-' = e. The second case follows from thefirst one.

There is an alternative way of proving (3.28). Given a function f, it isenough to apply (3.22) to the functions fs,s, where 0 < 6 < b -a and x E [a, b],which are defined for y E [a,b] by fb,z(y) = f(x+b(')), change the variablesputting x + 6(y a) = z and set c16I-1-1 = e.

Corollary 5 If -oo < a < b < oo, I E N and fV) is absolutely continuous on[a, b], then there exists a polynomial p1_1(x, f) of degree less than or equal toI - I such that for each m E No, m < l and tlx E [a, b]

b

If(m)(x) -pi(_)(x,f)I S(b-a)1-'"-' JIfIdx. (3.31)(1-m-1).

Idea of the proof. It is enough to take Taylor's polynomial T...1 (x, f) asP1-1(x,f):

1-1 (k} xPI-1(x, f) = Ti-1(x, f) _ > f Is

o (x - xo)kk=0

with an arbitrary xo E [a, b], apply to f(m) Taylor's formula with I-m replacingI (with the same xo) and take into account that

(Ti-m-i (x, f (m)) = Tm) (x, f). (3.32)

It is also possible to take

1_1 1 b

Pl-1(x, f) = S,-1(x, f) _ kl f f(*) (y) (x - y)kw(y) dy,k-(1


where w satisfies (3.11) and is nonnegative. (Notice that this is the firstsummand in (3.12).) One must take into account that in this case alsoSi_m_1(x, f(m)) = S('i)(x, f). Both of the choices lead to (3.31) (in the sec-ond case according to (3.5)).

Theorem 2 Let I E N, -oo < a < a <,0 < b < oo, w satisfy condition (3.11)and f E (W11)b0c(a, b). Then for almost every x E (a, b)

1-1 R b:

f (x) _ I f fwki(y)(x - y)kw(y) dy +(I

11)!J (x - y)`-'A(x, y)fu,'(y) dy,Z- k!k=0 a.

(3.33)where a= and b= are defined in Theorem 1.

Idea of the proof. Set a(&) = max{a+b, -11, b(b) = min{b-b, 1) for sufficientlysmall S > 0, write (3.13) for A7 f E C°°(a(b), b(6)), where 0 < 7 < b, and passto the limit as y -y 0+.Proof. Since for sufficiently small 6 > 0 (a,,31 C (a(b), b(b)) and (a(b))= _a1, (b(S))x = by for each x E (a(S), b(b)), we have Vx E (a(b), b(b))

1-1J(1&f)(k)(y)(x(A,f)(x) _ - y)kw(y) dy

k=O a

eb.

f (x - y)' A(x, y)(A,f)(')(y) dy.(1- 1)!

By Lemma 5 of Chapter 1 f , ( k ) exists on (a, b) where k = 1, ..., l - 1, and byLemma 4 of Chapter 1 (A, f )iki = on (a(S), b(S)) where k = 1, ...,1.Consequently, Vx E (a(S), b(b))

a a

if (A,f(y)(x - y)kw(y) dy - f f(k) (y)(x - y)kw(y) dyIa a

d B

5 f I A,(fwkl)(y) - fwk)(y)I I (x - y)kw(y)I dy 5 M1 f fwkiI dy --> 0

a a

as 7 -+ 0+, where k = 1, ...,1- 1 and M1 is independent of y and x0.

92 CHAPTER 3. SOBOLEV'S INTEGRAL REPRESENTATIO.\'

Analogously, in view of (3.4), Vx E (a(b), b(b))

e. b.

lI (x - y)'-'A(x, y)(A,f)(`)(y) dy - J(x - y)'-'A(x, y)f.()(y) dyla.

b(6)

<MMf IA7(fw!))-f,iThdy->0a(6)

as ry -a 0+, where M2 is independent of ry and x.Finally, by (1.5) A, f -+ f almost everywhere on (a(b), b(b)). Thus

(3.33) is valid almost everywhere on (a(b), b(b)) and, hence, on (a, b) sinceU (a(6), b(6)) = (a,b). 06>0

Remark 5 By Theorem 2 it follows that if in Corollaries 1-2 f E (W1)'a(a, b)and in Corollaries 3-5 f E WW(a, b), then equalities (3.15), (3.17) and inequal-ities (3.21) - (3.23), (3.28), (3.30) and (3.31) hold almost everywhere on (a, b),if the ordinary derivatives f (1) and f (m) by the weak derivatives f,(`)

3.2 Star-shaped sets and sets satisfying thecone condition

A domain Sl C V is called star-shaped with respect to the point y E fl if`dx E fl the closed interval [x, y] c fl. A domain fl c R" is called star-shapedwith respect to a point if for some y E Sl it is star-shaped with respect to thepoint y. A domain fl C Rn is called star-shaped with respect to the ball 6

B C Sl if Vy E B and dx E Sl we have [x, y] C fl. A domain SZ C R" is calledstar-shaped with respect to a ball if for some ball B C S] it is star-shaped withrespect to the ball B. If 0 < d < diam B < diam Sl < D, we say that Cl isstar-shaped with respect to a ball with the parameters d, D.

We call the setV. = V:,B = U (x, y)

pEB

Recall that by "ball" we always mean "open bail".

3.2. STAR-SHAPED SETS AND THE CONE CONDITION 93

a conic body with the vertex x constructed on the ball B (if x E B, thenV, = B). A domain St star-shaped with respect to a ball B can be equivalentlydefined in the following way: Vx E Q the conic body V= C S2.

Let us consider now the conen-1 I

K=K(r,h)={xER":0<x,)1 <rh""<r}. (3.34)i=1

We say also that an open set S2 C R" satisfies the cone condition with theparameters r > 0 and h > 0 if Vx E 0 there exists 7 a cone Kx C 0 with thepoint x as vertex congruent to the cone K. Moreover, an open set Sl C R"satisfies the cone condition if for some r > 0 and h > 0 it satisfies the conecondition with the parameters r and h.

Example 1 The one-dimensional case is trivial. Each domain SZ = (a, b) C R

is star-shaped with respect to a ball (__ interval). An open set ft = 6 (ak, bk),k=1

where s E N or s = oo and (ak, bk) fl (am, bm) = 0 for k # m, satisfies the conecondition if, and only if, inf (bk - ak) > 0.

Example 2 A star (with arbitrary number of end-points) in R2 is star-shapedwith respect to its center and with respect to sufficiently small balls (= circles)centered at its center. It also satisfies the cone condition.

Example 3 A convex domain 11 C R" is star-shaped with respect to eachpoint y E fl and each ball B C C. A domain fZ is convex if, and only if, it isstar-shaped with respect to each point y E 0.

Example 4 The domain ci C R2 inside the curve described by the equationlxi If + lxs l' = 1 where 0 < y < 1 (the astroid for y = 2/3) is star-shaped withrespect to the origin, but it is not star-shaped with respect to any ball B c Cl.It does not also satisfy the cone condition.

Example 5 The union of domains, which are star-shaped with respect to agiven ball, is star-shaped with respect to that ball. The union (even of a finitenumber) of domains star-shaped with respect to different balls in general isnot star-shaped with respect to a ball. In contrast to it the union of a finitenumber of open sets satisfying the cone condition satisfies the cone condition.Moreover, the union of an arbitrary number of open sets satisfying the conecondition with the same parameters r and h satisfies the cone condition.

7"Vx E !2" can be replaced by "Vx E Il" or by "Vx E Oft" and this does not affect thedefinition.


Example 6 The domain 0 = {z E Rn : IxI7 < x < 1, Jxj < 1}, whereY = (xl, ..., for ry > 1 is star-shaped with respect to a ball and satisfiesthe cone condition. For 0 < y < 1 it is not star-shaped with respect to aball. Furthermore, it cannot be represented as a union of a finite number ofdomains, which are star-shaped with respect to a ball, and does not satisfy thecone condition.

Example 7 The domain 0 = {x E Rn : -1 < xn < 171-1, 171 < 1} satisfiesthe cone condition for each 'y > 0. It is not star-shaped with respect to a ball,but can be represented as a union of a finite number of domains, which arestar-shaped with respect to a ball.

Example 8 The domain SZ = ((XI, x2) E R2 : either - 2 < x1 < 1 and- 2 < x2 < 2, or 1 < x1 < 2 and - 2 < x2 < 1} is star-shaped with respectto the ball B(0,1). For 0 < b < %F2 - 1 the domain fl D B(0,1), but it is notstar-shaped with respect to the ball B(0,1). (It is star-shaped with respect tosome smaller ball.)

Lemma 1 An open set S2 C R" satisfies the cone condition if, and only if,there exist s E N, cones Kk, k = 1, ._a, with the origin as vertex, which aremutually congruent and open sets SZk, k = 1, ..., s, such that

,Uk=1

2)b'xEIk the cone' x+KkCQ.

Idea of the proof. Sufficiency is clear. To prove necessity choose a finite numberof congruent cones Kk, k = 1, ..., s, with the origin as a vertex, whose open-ings are sufficiently small and which cover a neighbourhood of the origin, andconsider the sets of all x E 1 for which x + Kt C Q. 0Proof. Necessity. Let fl satisfy the cone condition with the parameters r, h > 0.We consider the cone K(rl, h1) defined by (3.34), where h1 < h and r1 < r issuch that the opening of the cone K(rl, h1) is half that of the cone K(r, h).Furthermore, we choose the cones Kk, k = 1, ..., s, with the origin as a vertex,

which are congruent to K(rl, h1) and are such that B(0, h1) C 6 Kk. Hence,k=1

Vx E Sl the cone Ks of the cone condition contains x + Kk for some k. Denoteby Gk the set of all x E St, for which K. contains x + Kk. Finally, there existsb= > 0 such that Vy E B(x, 6) we have y + Kk C fl. Consequently, the open

8 Here the sign + denotes a vector sum. The cone z + Kk is a translation of the cone Kkand its vertex is x.


sets S2k = IJ B(x, a=), k = 1, ..., s, satisfy conditions 1) and 2).zEG4

Let a domain SZ C R" be star-shaped with respect to the point x0. For t; ES"'', where S"-' is the unit sphere in R", set w(t;) = sup{p > 0 : xo+pt; E S2}.Then

52={XER":x=xo+pt where DES"-'. 0<g<

Moreover, set R1 = inf lp(l; ), R2 = sup and for 1,71 E S"-' denote(ESn- (ES-1

by d(t;, rl) the distance between and i along the sphere S' 1, which is equalto the angle -y between the vectors 0 t and O q, where 0 is the origin.

Lemma 2 Let a bounded domain SZ C W be star-shaped with respect to thepoint xo E Q. Then it is star-shaped with respect to a ball centered at .ro if. andonly if, the function V satisfies the Lipschitz condition on S" -'. i.e.. for someM>0and9Vt,rlES"-1

I v(77)1:5 M d(C, n).

Idea of the proof. Sufficiency. Consider the conic surface C(C) with the pointf = xo + v(C)t; as vertex, which is tangent to the ball B(xo, r). Suppose that

0<p < oo} intersects C(t;) at two points a and e. Denote d = xo + Sincef, d E 852 it follows that f V bd and d V V'. Therefore, d E [a, e].Necessity. For fixed t; E S"'' consider two closed rotational surfaces L+ andL_ defined by the equations p = F* (q), where F* (q) = p(t;) ± M d(t;, rl). Thenthe boundary 852 lies between L+ and L_. Let the (n - 2)-dimensional sphereE be an intersection of L_ and the surface of the ball B(xo, R1). Considertwo conic surfaces, which both pass through E and whose verticies are x0, frespectively. Let 6 denote the angle at the vertex of the conic surface D.,,then 6 = 'P((M)-R. (We assume that M > 0 and p(t;) > R1, since the cases, inwhich M = 0 or R1, are trivial.) If 6 > do = arccos set r(l;) = R1.

9 Since

If - ql <- d(f, n) = 2 s in zkk - nl <-

2 IE - nIs

this condition is equivalent to: for some M1 > 0 and V(, q E S"

WE) - W(17)1:5 M1 It - I.


Otherwise, let r(t;) be such that the ball B(xo, r(f )) is tangent to Dj. Thenthe conic body with the point f as vertex constucted on the ball B(xo, r(t;))lies in Q. 0Proof. Sufficiency. Denote c = x0 + 4Q,7. Since d E [a, c] or d E [c, eJ we have

iv(f) - w(77)1:5 max {ICI, Ic-tI}.

Since I a-tI < II 'o we establish that

Iw(f) - V(77)1:5 Ic I = __(a > - w(f)

2w(()sin2 '0(07C-0-2)sin B-7 ein(9-ry - w(f) sin -ry d(f,+])Consequently,

Vt;, 7 E 51-1 such that -y < Q

I '(q) I = R2 ' (a-7)d(f, n).

Hence, given e > 0, there exists (e) > 0 such that V , r! E S"-1: -y < b(e) wehave

Iw(f) - W(17)1:5 (R2 cot,0+e)d(t;,7) = (RZ Rz - r2 +e)d(la,r7).

Now let t; and' be arbitrary points in S"-1, f 96 ri. We choose on the circle,centered at x0 and passing through f and 17, the points to = f 1 -< ... -<f,,,-j -< t;,,, = 17 such that all the angles between the vectors O6_1 and 01,i = 1, m, are less than 6(e). Then

m

Iw(f) - w('7)I <- F, W&-1) - wWIi=1

1e One can see that In I = I& I(oot (9 +'y) + tan J) while IccI = IbI(cot (,6 -1') - tan J),where 10. The inequality (aa$I < Icc-tj follows from the inequality

cot(p+-y)+tan2 <cot (fl--y)-tan g,

which is valid for all @ and ry satisfying 0 < y < ,9 < . This inequality is equvalent to

2 tan 2 <sin 27 2 sin try

2 sin (p - ') sin (p +)ry cos 2,y - c os 2,0'

to cos 2ry - cos 2,6 < 4 cost I cosy and to - cos 20 < 2 coe ti + 1, which is obvious since0< y< 12 .


<(Rr-2 RZ-r2+e)(RZ RZ-r2+E)d(,rl)

Passing to the limit as e -* 0+ we find that the Lipshitz condition is satisfiedwith

M=R2 Rz-r2.rNecessity. If 0 < 6 < 6g. then

(3.35)

p(t:)R1 sin6 = cp({)R1 sin6

(cp({) - Ri cos 6)2 + (RI sin 6)2 (tp(0 - RI)2 + sin26

= cp(C)R1 sin 6 2 w(t)R1 > 2 R2 =r0

62M2 + 4cp( )R1 sine M2 + yo(C)Ri a M + Rl

One can verify that for any point g E L_ \ B(xo, RI), g 5A f , the interval (g, f )lies "" in 52. Therefore, the conic body Vf with the point f as vertex constructedon the ball B(xo, ro) lies in St. Hence, SZ is star-shaped with respect to the ballB(xo,ro)

Remark 6 The constant M given by (3.35) is the minimal possible, because,for example, for any conic body V., defined by (3.34) we have

sup jw(f) tt- R2 - r2.d{,,7ES^-`,f#9 d(S, 17) r

If a domain f C R", which is star-shaped with respect to the ball B(xo, r),is unbounded, then set S' = {t E Sn-1 : ip(t) < oo}.

u Consider the curve L obtained by intersecting L_ \ B(xo, RI) by the two-dimensionalplane passing through g and the ray going from zo through f. Let this ray be the axis Oxof a Cartesian system of coordinates in this plane. Suppose that y = '(x) is a Cartesianequation of the curve L. We recall that its polar equation is g = W(f) - MIyI and note thatIyj < J. The part of the curve l_, for which 0 < y < 6, is convex and the part of L, forwhich -6 < y < 0, is concave since, for example, for 0 < y < 6

2M2 + (IG(f) - My)2((W(f) - My)sinjp+Mcosrp)3

< 0.

Hence, for any g E t_,g 0 f, the interval (g, f) lies in fl.


Corollary 6 Let an unbounded domain 11 C R" be star-shaped with respect tothe ball B(xo, r). Then S' is an open set (in Sn-1) and the function p satisfiesthe Lipschitz condition locally 12 on S'.

Idea of the proof. Note that if cp(l;) = oo fort E S"`1, then the whole semi-infinite cylinder, whose axis is the ray R(t) and whose bottom is the hyperball{x E B(xo, r) : X-ol 1 O}, is contained in Q. Deduce from this that S' isopen and apply Lemma 1.

Example 9 For the domain S2 C R2, that is obtained from the unit circleB(0,1) by throwing out the segment {x1 = 0, 1 < x2 < 1) and which is star-shaped with respect to origin, but is not star-shaped with respect to a ball, thefunction V is not even continuous.

Example 10 For the domain 11 = {xl,x2) E R2 : Ixlx2I < 1), which is star-shaped with respect to the origin, but is not star-shaped with respect to a ball,the function c' is locally Lipschitz on the set S' = S' \ ((0,±1), (±l,0)}.

Lemma 3 If a bounded domain S2 C R" is star-shaped with respect to a ball,then it satisfies the cone condition.

Idea of the proof. Let 11 be star-shaped with respect to the ball B(xo, r).Then S2 satisfies the cone condition with the parameters and r. (It followsbecause the cone KZ with the point x as vertex and with axis that of the conicbody L, which is congruent to the cone K (Rs , r), is contained in Q).

Now we give characterization of the open sets, which satisfy the conecondition with the help of bounded domains star-shaped with respect to a ball.

Lemma 4 1. A bounded open set 0 C Rn satisfies the cone condition if, andonly if, there exist s E N and bounded domains 11k, which are star-shaped with

a

respect to the balls Bk C Bk C Slk, k = I,-, s, such that S2 = U Stk.k=1

2. An unbounded open set S2 C Rn satisfies the cone condition if, and onlyif, there exist bounded domains S2k, k E N, which are star-shaped with respectto the balls Bk C Rk- C SZk, k E N, and are such that0

1) S2 = U f2k,k=I

11 I.e., Vt E S' there exist M(f) > 0 and v(f) > 0 such that Vq E S', for which If -ql < v(t)we have IW(f) -W(i)I 5 M(E)d(f,+)


2) 0 < inf diam Bk < sup diam Ilk < 00kEN kEN

and3) the multiplicity of the covering x({S2k}k 1) is finite.

Idea of the proof. Sufficiency. By Lemma 3 ) satisfies the cone condition withthe parameters c 26q' and c6, where 13

c6 = inf diam Bk, C7 = sup diam S2k,k=1,s k=TTs

s E N for bounded 12 and s = oo for unbounded Q.Necessijr. Consider for x E 52, in addition to the cone Kx, the conicbody K. with the point x as a vertex, which is constructed on the ballB(y(x), r1) inscribed into the cone Kx (here r1 = rh/(r + r2 + h2)) andthe conic body K= with the point z(x) = x + exas a vertex, whereex = 2 min{rl,dist (x, 8Q)}, which is constructed on the same ball B(y(x), r1).Then S2 = U K. Choose xk E R', k E N, such that R" = U B(xk, 11) and

xEfl kEN

the multiplicity of the covering 14 x({B(xk,Z)}kEN)

<- 2". Set

Wk = IlnB (1k,2 , Gk = U Ks .lEfl: y(x)Ewk

00

Then S2 = U Gk. Renumber those of Gk which are nonempty and denotek=1

them by 521, 522i .... 0

Proof. Necessity. Suppose that Gk # 0 and t; E Gk, then there exists x E Qsuch that y(x) E wk and t; E K. Let us consider the conic body Kt withthe point { as a vertex, which is constructed on the ball B (xk,

z). Since

y(x) E B (xk, z) we have B(y(x), rl) B (xk, z) and KK C KZ C 11. Hence,the set Gk is star-shaped with respect to the ball B (xk, 2 ). Furthermore,

IC - xkl < Iz(x) - xki = Iz(x) -.TI + Ix - y(x)I +- Iy(x) - xkl < h

because Ix - y(x) I = h - rl, therefore Gk C B(xk, h) and diam Gk < 2h.

13 Here and in the sequel k = f, where s E N means k E {1,...,s} and k = 1 oo meanskEN.

14 This is possible because the minimal multiplicity of the covering of R" by balls of thesame radius does not exceed 24.


Let us consider those of the sets Gk which are nonempty. If 0 is bounded,then there is a finite number of these sets - denote this number by s. IfSZ is unbounded, then there is a countable number of these sets (s = oo).

Renumbering them and denoting by Sgt, Q2, ..., we have S1 = U Gk = U Stk.k=1 k=1

Thus, for S2k, k = 1, s, the properties 1) and 2) are satisfied. Finally, 1s

< 1X({S2k}k=1) x({B(xk, h)}k=1) < 2" (1 +n

Ti. O

nRemark 7 In the above proof c6 = r1 and c7:5 2h. Furthermore, dim <lam Bk -4 (1 + r) , k = T ,-s. It is also not difficult to verify that x ({Slk}k=1) < cs6" (1 + h)n.

3.3 Multidimensional Taylor's formulaTheorem 3 Let I C W be a domain star-shaped with respect to the pointxo E 11, l E hl and f E C' (0). Then Vx E 12

f (x) = (D°f)(xo)( - xo)°

C!x

IoI<1

1Q

+l (x --L o )

J(1 - t)'" (D" f)(xo + t(x - xo)) dt (3.36)

a!101=1 o

(here in addition to multi-notation used earlier we mean that xo + t(x - xo) _(x01 + t(xi - x01), ..., xOn + t(xn - xOn)))-

15 We use the inequality

t n(1+

QI

where 0 < Q1 < Q2 < oo. Since for x E R" the number x(x) of the balls B(zk, o) 3 x isequal to the number of the points zk E B(x, Qs), by inequality (2.60) we have

x(x) mesa B(O, Q)) = E mess B(zk. QI)k: B(zh, e)3z

< x((B(zk, Qi))k.1) mesa U B(.., ,p,) 5 x({B(zk, Ql)k=1) measB(x, of + Q,),k: B(a,.p2)3z

and the desired inequality follows.

3.3. MULTIDIMENSIONAL TAYLOR'S FORMULA 101

Idea of the proof. Consider for fixed x and x0 the function V of one variabledefined for 0:5 t < 1 by w(t) = f (xo+t(x-xo)) and apply the one-dimensionalTaylor's formula (3.10) with the remainder in integral form:

i_i(k)

sv(1) _ k(0) + (111)! J(i - dt.

k=0 0

If I = 1, then (3.36) takes the form: V f E C'(S2) and Vx E S2

f (x) = f (xo) + (x, - x0t) f (af ) (xo + t(x - xo)) dt.j=1

°axe

The analogue of this formula for the functions f, which have all the weakderivatives (e )w of the first order on S2 cannot have the same form, even foralmost every x E S2 because it contains the value f (xo) at a fixed point x0.(For, suppose that this formula is valid for some such function f. Then it willnot be valid for any function g, which coincides with f on S2 excluding thepoint x0.)

For this reason we write the above formula in a different way. Suppose, thatS2 C R" is an arbitrary open set and h E R", then it follows that V f E C' (S2)and Vx E S2IhI

f(x+h)= f(x)+Eh3 f1(af )(x+th)dt.i=1 ° ax'

Lemma 5 Let S2 C R;" be an open set and h E R". If f E L10c(11) and foreach j = 1, ..., n the weak derivative (LL )w exists on S2, then for almost everyx E 1111,1

f(x+h)=f(x)+1:hj f 1(8x)w(x+th)dti=1

Ihl 1

=f(x)+Ihl f (f)w(x+ST)r=f(x)+ f0 0

where t; = Th and ( ) w and V w f are the weak derivative in the direction ofthe weak gradient respectively.


Idea of the proof. Apply mollification and pass to the limit. 0Proof. Since A6f E C°°(54) for each 0 < d < y, by Lemma 4 of Section 1.2 wehave: `dx E Qlhl+,

n

h) = (A6f)(x) + Ej=1

f '(A,(f)w)(x+th)dt.oxi

We claim that

f'(A6(.-i_) ) (x + th) dt -4 f (f) (x + th) dt8x j w p OX) w

in L,°C(52,) as 5 -* 0 + . Indeed, for each compact K C St, by Minkowski'sinequality

f ,(4 l-lw)(x+th)dt- f'(8x )w(x+th)dtIIL'(K)

f'11 (A6(8xj)w)(x+th) - (aLxjfL,(K)dt

< IA6(C72)w-( aLx!f

by (1.9)as 5-a0+.Consequently, there exists a sequence 6k > 0 such that bk -+ 0+ as k -4 oo

and

L(A6k(a )w)(x+th)dt- L_)x+th)dt

almost everywhere on Sty. Moreover, by (1.5) (A,, f)(x + h) -+ f (x + h) and(A6k f)(x) -> f (x) almost everywhere on Sty. Thus, passing to the limit ask - oo, we obtain the desired equality for almost every x E Sty and, sincey > 0 was arbitrary, for almost every x E St . 11

We note that one can prove similarly that if f E L10C(1) and `da E Nsatisfying jai = I there exist weak derivatives D° Of on St, then for almost everyx E StjhI

i°f (x + h) _ (Dw°f)(x) h° + I

h I

(1 - t)1-1(DW f)(x + th) dt.I°I<1

aljai=1 0

3.3. MULTIDIMENSIONAL TAYLOR'S FORMULA 103

Corollary 7 Let 0 C R' be an open set, 1 < p < oo, h E R" and f e wp(S2).Then

IIf(x+h) -f(x)IIL,(n,,,l) 5 II IowfIIIL,(n)lhl <_ IIfIIwy(n)Ihl.

Idea of the proof. Apply Lemma 5 and Minkowsli's inequality.Proof. By Lemma 5

IIf (x + h) - f (x)IIL,(nlhi) = I f ((owf)(x + th) . h) dtIIL,(nlhl)

0

I If lI(owf)(x + th) . hlIL,(n,h,) dt < IhI f II Iowf I(x + th)Iln,h,) dt

0 0

= IhI f II Iowf I Ilnih,+th) dt < II owf I IIL,(n) IhI <_ If Ilw,(n) IhI0

since c1h1 + th C St and Iowf I <'E K-20.1.

Next consider for I E N and h E Rn the difference of order I of the functionf with step h:

(Ahf)(x) = (-l) 1-k (k') f (X + kh).k=0

Corollary 8 Let c C Rn be an open set, I E N, 1 < p < oo, h E R" andf E Wy(S2). Then

IlohfIIL,(n(ihi) 5 2'I1fIIL,(n), IlohfllL,(nghl) <- n'-'Ihl'Ilfllwo(n)

Idea of the proof. The first inequality follows by Minkowski's inequality. Toprove the second one apply Corollary 7 and take into account that Q', < n1-1 ifa E N satisfies lal = I.Proof. By induction we get

IIAhfIIL,(n,lh,) = Iloh(oh'f)IILP((n(j-I)Ihl)Ihl)

I-L.r

n \Of`-IhI

j1=1ll(a(axj,

f) / wIIL,(n(t-j)Ihl) hI('...r

ll (ox,, ... axj, )wllLa(n)

= IhI' E IIDwf IIL,(n) <- n'-' IhI' llf Ilwa(n)Iol=1


3.4 Sobolev's integral representationTheorem 4 Let S2 C R" be a domain star-shaped with respect to the ball B =B(xo, r) such that B C S2,

w E LI(R"), suppw CB, fdx=1 , (3.37)R.

I E N and f E C'(/S'2). Then for every x E S2

f (x) = III J (D°f)(Y)(x - Y)°w(Y) dy + J D° f)I(y) w. (x. y) dY,I0I<1 B I°I=1 J(3.38)

where for x,yER",x0y,

w°(x,Y) _ Ial (x-Y)°w(x,Y)a! Ix - Y1101

and

(3.39)

00

rw(x>Y)= f w(x+eiY-xl)f-'do (3.40)

IS-VI

(for x = y E Sl we define w°(x, x) = w(x, x) = 0).

Remark 8 The first summand in right-hand side of (3.38) is a polynomial ofdegree less than or equal to I - 1 while the second one (the remainder) has theform of an integral of potential type.

Both summands in right-hand side of (3.38) consist of integrals containingthe function f and its derivatives and does not involve the values of the functionf and its derivative at particular points - thus, in contrast to Taylor's formula,this is an integral representation of the function f via the function f and itsderivatives up to the order 1.

Let 1=-v1 and k E N. Consider the derivative of the function f in the26direction of C of order k: ( )(y) _ E (k) (D° f)(y){°. Then one may write

I°I=k(3.38) in the following way

1-1

f(x) = f (E Ix

k!y lk () (y)) w(y) dyB k

3.4. SOBOLEV'S INTEGRAL REPRESENTATION 105

+ 1 ?f) (?/w(x, y) dy(l - 1)! f ('9V ) Ix - yI11-1

V.

In particular,

f(x) = f f(y)w(y)dy+ f((Vf)(y) - (x - y))w(x,y) dy.

Ix - yI"B v,

Remark 9 Let us denote for x i4 y by MZ,y the ray, which goes from thepoint x through the point y, and by Lx,y the "subray" of M=,, which goes fromthe point y. As the variable p in (3.40) changes from Ix - yI to infinity, theargument z = x + p of the function w runs along the ray L,,,,y. We notethat p = I z - xI and that (3.40) may be written with the help of a line integral:

w(x, y) =J

w(z)Iz - xI"-t dL.

The ray Lx,y intersects the ball B if, and only if, y E K. For this reasonVxER"

suppy wa(x,,y) = suppy w(x, y) C K. (3.41)

(if w is positive on B, then there is equality in (3.41)). Furthermore, Vx E R"and Vy E K.

suppe w (x + py - x) C B fl Lz,y = [d1, d2].Iy - xI

Here d2 = d2(x, y) is the length of the segment of the ray M=,y contained in K=while d1 = dl (x, y) = max{ Ix - yl, d1 }, where d1 = dt (x, y) is the length of thesegment of the same ray contained in K = \ B. 16

Therefore, actually, the integral in (3.40) is equal to 0 if y ¢ K= and is anintegral over the finite segment [d1, d2] if y E K. We note that

d2 < D, d2 - d1 < d, (3.42)

where D = diam 1 and d = diam B.

16 If Ix - zoI = h and sp is the angle between the vectors xx and 2-1, then 0 < tamp <and d1 and d2 are the minimal root, the maximal respectively, of the quadratic equationd2 - 2dh cos,o + h2 - r2 =0.


Remark 10 If Sl is bounded, then

IIw(x,y)IIC(R"xR") <- II"IIL(R")dz n I <- IIwIIL-(R")D"-1d.

Moreover, Va E 1% satisfying IaI = l

IIw°(x,y)IIC(R"xR") <_ (3.43)

We have taken into account that

I (x - y)° Ixl - Y11 Y, (Ixn - ynl \ on < 1Ix - YI+

I _- Ix - yI J ` Ix - yI J

and that a! > n for IaI = 1.Hence, if w is bounded, then for bounded 11 the functions w and w,, are

bounded on R" x R". If fl is unbounded, then these functions are bounded onK x R" for each compact K.

If w E COO(R" ), then the functions w(x, y) and w°(x, y) have continuousderivatives of all orders Vx, y E R' such that x y. 17

Remark 11 In the one-dimensional case for Sl = (a, b) and B = (x0-r,x0+r)(a, d3), where -oo < a < a < $ y. Furthermore,

Y

w(x, y) = f w(z) dL =f w(u) du,

bf w(u) du,

a < y < x,

x < y:5 b,Y

17 At the points (x, x), where z f I3 they are discontinuous. For n > 1 it follows fromthe fact that for each y E K. \ B lying in some ray going from the point x (for all thesey the vectors have the same value, say, v = (v1, .... v")) the function w(x, y) has

the same value y(x, v) = f w(x + Qv)Q"'1 dQ. Hence, the limit of w(x,y) as y tends to zco0

along this ray is also equal to y(x,v). Respectively for the function w°(x,y) this limit isequal to 4(-v1)°1 ... The discontinuity follows from the fact that theselimits depend on v. For, if the ray defined by the vector v does not intersect the ballB, then y(x, v) = 0. On the other hand, there exists v such that 7(x, v) = 0, otherwise

f w(z) dz = f f w(x + Q v)Q"-1 do) dS = 0, where S is the unit sphere in lit", whichR. S(000contradicts (3.37). For n = 1 the discontinuity follows from the formulas for w and w° givenin Remark 11.


and

W1 (X, y) =1 (x - y)'w(x,

y) _ (sgn (x - y))'-' A(x, y).(l - 1)! Ix - yl' (l - 1)!

Thus, (3.38) takes the form (3.13).

Idea of the proof of Theorem 4. Multiply Taylor's formula (3.36) with xo = yand y E B, by w(y) and integrate with respect to y over Rn. (We assume thatfor y 34,0 w(y)g(y) = 0 even if g(y) is not defined at the point y.) The left-hand side of (3.36) does not change and the first summand in the right-handside coincides with the first summand in the right-hand side of (3.38). As forthe second summand it takes the form of the second summand in (3.38) afterappropriate changes of variables. 0Proof. After multiplying (3.36) with xo = y by w(y) and integrating withrespect to y over R" the second summand of the right-hand side of (3.36) takesthe form

1

I E -i f w(y) (f (1 - t)'-'(D°f)(y+t.(x - y))dt)dyI°1=1 R. 0

= l li f (1 - t)'-' ( f (D°f)(y + t(x - y)) w(y) dy) dt = l 1 J°..

1°I=1 0 fit" 101=1

Changing variables y + t(x - y) = z and taking into account that (x - y)° _

(1 i dy = tl' ", we establish that

J. = f (D°f)(z)(x - z)° (f w I 1 -ttx)(1 dt)n41) dz.

R. 0 \ /

Replacing _ by p, we have

J f (D°f)(z)(-zii ( f w(x+q'-xl)e,,_ldp)dz,

Ix- zz-xIz-aI

which by (3.41) gives (3.38). 13


Remark 12 One can replace the ball B in the assumptions of Theorem 4 bysome other open set G. depending in general on x E 11 such 18 that G= C SZand replace the function w by some function w such that

w.E L1(R"), suppwx C G=, fw. (y) dy = 1.Rn

In this case the same argument as above leads to the integral representation(3.38) in which B, w and the conic body V. are replaced by GwS1 the conicbody (J (x, y) respectively. We shall use this fact in Corollaries 10 -12

yEG.below.

Remark 13 Set w(r) = r"w(xo - rx). Then w(,.) is a kernel of mollification inthe sense of Section 1.1 (in general, a non-smooth one since we have only thatw(r) E L1(R")). The polynomial SS_1(x,xo), the first summand in Sobolev'sintegral representation (3.38), is Taylor's polynomial averaged over the ballB =- B(xo, r) in the following sense:

Sl-1(x, xo) = (A,TI-1(x, ))(xo).

Here T1_ 1(x, y) is Taylor's polynomial of the function f with respect to thepoint y, A8 is the mollifier with the kernel w(r) and the mollification is carriedout with respect to the variable y. For,

(ArPt-1(x, -)) (xo) = f w ( , . )

B(o,1)

1(D°f)(xo - rz)(x - xo + rz)°r"w(xo - rz) dz

I°I<1 B(o,1)

1 f (D°f)(y)(x-y)°w(y)dy=(S1-lf)(x,xo)I°I<1 B(xo,r)

This allows us to characterize Sobolev's integral representation as a "molli-fied (averaged) Taylor's formula with the remainder written in the form of anintegral of potential type" of briefly "averaged Taylor's formula".

18 It is also possible to suppose only that G: C fl replacing the assumption f E C,(fl) byf E C(f1) in the case in which a.-n8fl # 0. See a detailed Remark 1 for the one-dimensionalcase.


Theorem 5 Let Q C R" be a domain star-shaped with respect to the ball B =B(xo, r) such that ,9 C Q,

fWdxWE L,,.(R" ). supp w C B, = 1, (3.44)

R.

I E N and f E (W )t°`(52). Then for almost every x E Q

f (x) = E a! (D' f)(J)(x - iJ)*w(3/) dy +1

D.0 )( twa(x,Y) dy.°I<t B I°I=tt

(3.45)

Idea of the proof. For 0 < 5 < dist (B, on), which is such that - a < IxoI - r <Ixol + r < set 19 X161 = {x E S2 : Yi c 526 fl B (0, 6) } c Q6, write (3.38) for.4y f E C°O(Q161) where 0 < y < b and pass to the limit as y -+ 0+. 0Proof. For each 6 and y such that 0 < y < 6 and Vx E 52161 we have

c, 1 f.__.. .. ._ , . . , f (DoA,f)(y)I°I<t --- I°I=t i ,

n t w°(x,y)dyx-yl1

f /j (A7(Dwf))(y)(l - y)°w(y) dy +(A, (D.* f)) (y)

J Ix - yl"_t wa(x, y) dy.1°I<1 g 101=1 v

We have applied Lemma 4 of Chapter 1 and the fact that the weak derivativesDW f where Ial < l exist (Lemma 6 of Chapter 1).

By (1.5)A, f -+ f a.e. on 5216)

as -y -+ 0+. We shall prove that

R.,-, (x) = f(A(Df))(y)(x - y)°w(y) dyB

(3.46)

'- f (Dwf)(y)(x - y)-w(y) dy = R°(x) on 52161 (3.47)

B

is The necessity of introducing of these more complicated sets than fl6 arises in connectionwith Example 8.


and

Sa,7(x) = J(A7(Dwf))(y)wa(x,y)dy -)

Ix - yl-K.

- f (D' f)(Y)wa(x,y)dy=Sa(x) in L1(Sl[al). (3.48)J Ix - yin-1K,

The relation (3.47) follows from the estimate

IR.,7(x) - R.(x)I < Mlf I (A7(Dwf))(y) - (D.* f)(y)I dy,B

where M1 = II(x - y)allc(n,a,xB)II w(y)IIL (R") < oo and the property (1.9).Set M2 = Ilwa(x,V)IIL (n,d,xR") Then by Remark 10 M2 < oo. Applying

the inclusions V. C 116 = St6 fl B (0, s) for x E nlq and 0[6[ C Q6 we obtain

II Sa,,,(x) - Sa(x)IILi(n,a,) < M2I (A1(Dw I))(y) - tD.0f)(V)I

dy IIL,(ne)

dy= M2 I I A7(Dwf) - Dwf l ( f dxIxyI"-i

n; n;

< M2 II IZI` "IIL,(na-na)IIA_l(Dwf) - D.f IIL,(na)

and 20 (1.9) implies (3.48).From (3.48) it follows that, for some ryk > 0, k E N, such that 'Yk -+ 0 as

k ->oo,wehaveS.,,, (x) -1 Sa(x) a.e. on ft[,q. (3.49)

Now passing to the limit as k -> oo in equality for Al f with 'Yk replacing ry,by (3.46), (3.47) and (3.49) we obtain that (3.45) holds almost everywhere onS2[k[. Since U 11[o[ = f2 we establish that (3.45) is valid almost everywhere on

6>0Q.0

20 In the last inequality in the expression 116 - nj* the sign - denotes vector subtractionof sets inR",i.e.,A-B=(zER":z=x-y where xEA,VEB). Clearly,

flj - n6' C B(o, 8-1) - B(o,6-') = B(o,2a-1).

3.5. COROLLARIES 111

3.5 CorollariesCorollary 9 In Theorems 4, 5 let conditions (3.37), (3.44) respectively, be re-placed by

w E Co (S2), supp w C B, fdx=1. (3.50)

Rn

Then Vf E CI(Q) for every x E 11 and Vf E (Wi)'-(fl) for almost every x E S2

f (x) = f (E(_ aDy

[(x - y)aw(Y)1)f (y) dyaB Ial<1

+Elal=1 V:

("'of '(Y) wa(x, y) dyIx - yl-(3.51)

with D° If replacing DI f in the case in which f E (Wi)!°`(f2)

Idea of the proof. For f E C'(Sl) - integration by parts in the first summand ofthe right-hand side of (3.38). For f E (W1)b0c(Sl) - the same limiting procedureas in the proof of Theorem 5, starting from (3.51) with A, f replacing f.

Corollary 10 In addition to the assumptions of Corollary 9 let Q E l and0 < 1,61 < 1. Then V f E C'(S1) for every x E ) and V f E (WW)!°`(SZ) for almostevery x E Sl

(D1'f)(x)_f(E(-1)1.1+191Dv+'[(x-y)aw(y)])f(y)dy

B Ia1<1-191a!

+(D°f)(y)

Ix - y1n-i+191wQ-9(x> y) dy (3.52)f

Ial=1,a>9 V.

with DO f replacing DO f and D. *f replacing D* f if f E (Wi)'0C(11).

Idea of the proof. For f E C'(S)) write equality (3.38) with Dyf E CHVI(1l)replacing f and with l - 1,61 replacing 1, integrate additionally by parts in thefirst summand of the right-hand side, change the multi-index of summation ato -y -,6 (then E = E ) and write a instead of ry. For f E (Wl)'0c(1))

IaI=1-I0I 171=1,'Y>>0

apply the limiting procedure from the proof of Theorem 4.


Remark 14 For n = 1 equality (3.52) takes the form (3.17). As in the one-dimesional case the first summand in (3.51) and (3.52) can be rewritten in theform

f ( a,"#(x - y)°-1(D°w(y))f(y) dy, (3.53)

where o°,s depends only on a and p.

Corollary 11 Let I E N and SZ C R" be a bounded domain star-shaped withrespect to the ball B with the parameters d, D, i. e., d < diam B < diam SZ <D < oo. Then there exists c9 > 0, depending only on n, 1, d and D, such thatVf E C1(5Z) for every x E St and df E (W)"(0) for almost every x E 0

1(DIf)(x)1 <- cs( f IfIdy+: f(D°f)(y 1dy(3.54)

B I°I=1 v, Ix - yl

where # E Non and 0 < 1,31 < 1, with DW f replacing DI f and Dm f replacingD* f in the case in which f E (W,1)10c(c ). Moreover,

I (Dsf) (x) l 5 clo ((W D)

J If I dye

(D)n-1 E f I(D°f)(y)I )dy (3.55)

+ d Ix - yln-1+Ie1

where c10 > 0 depends only on n and 1.

Idea of the proof. Let p be a fixed kernel of the mollifier defined by (1.1). Inequality (3.52) take w(x) = (s)np(2 xdxo) and apply (3.53) and (3.43) (see alsoCorollary 3).Proof. First of all Vx E f and dy E B

I (x - y)°-a(D°w)(y)15 ()n+°Ix - y1l°I-191I(D°µ)(2(x d xo))

<M1IID°,uIIc(R')(d)10'D-10Id-n

< M2(a)1-1D-I$Id-n,

where M1 and M2 depend only on n and I (since µ is fixed). Hence

( a°,6(x - y)°-B(D°w)(y))15 Ms(a )1-1D-101d-", (3.56)

Ia1<1,0>d


where M3 depends only on n and 1.Secondly from (3.43) it follows that Vx E St and y E V.

D n-1iw°-0(x,y)I < M4()

,

where M4 depends only on n and 1.So (3.53), (3.56) and (3.57) imply (3.55) and hence (3.54).

(3.57)

Remark 15 For n = 1 inequalities (3.54), (3.55) take the form of the firstinequality (3.23), the form of (3.21) respectively.

Moreover, if St = B and diam B = d, then (3.55) implies that

I(D0f)(x)I <_ M.(d-IoI-n Jutdy + f I(D°f)(y)I dy), (3.58)Ix - yIn-1+101B 1°1=1 B

where Mb depends only on n and 1, which is a multidimensional analogue ofthe first inequality (3.22). If, in addition, 1- 1,61 - n > 0, then

I(D0f)(x)I: M5 (d-101-n J If I dy+d!-101-n1 J I(DQf)(y)I dy),B 1°I=1 B

which is a multidimensional analogue of the second inequality (3.22).

Remark 16 If I -IQI - n > 0 (in the one-dimensional case this condition isalways satisfied), then there is no singularity in the integrals of the second sum-mand in the righ-hand side of (3.54). In this case (3.54) implies the inequality

Wf)(x)I <- cll (f IfI dy+ E JI(JYf)(Y)ldv)

rB V

< cll (J If I dy + JI(D°f)(v)Idv), (3.59)B Ia1=! n

where c11 > 0 depends only on n, 1, d and D.Applying the same procedure as in the second proof of Corollary 4 one can

obtain the related inequality with a small parameter: if 1 - IQI - n > 0, then

I(D'f)(x)I 5 c12K(e) Jiii dy + e ft (Df)(y)I dy, (3.60)

B I°I=i B

where now 0 < e < MS d1-191-n,

K(c) = E - 7F (3.61)

and c12 > 0 depends only on n and 1.


Corollary 12 Let 1 E N and f? C Rn be a bounded domain star-shaped withrespect to a ball with the parameters d, D. Suppose that f E C' (0) (or f E(W;)b0c(fz) ). Then there exists a polynomial pl_1(x, f) of degree less than orequal to l - 1 such that V,3 E 1 satisfying 101 < 1 and Vx E 11

aI (Daf)(x) - (D5Pi-1)(x, f)I < c13

(Dd)n-1

J

x(D-

f)-y)IBI dy,1IQI= n

(3.62)

where c13 > 0 depends only on n and 1. (If f E (Wf)t0C(fZ), then (3.62) withDO f and D° f replacing DO f and DI f holds for almost every x E f').

Idea of the proof. Set

P1-1(x,f)=S1-1(x,f) f (E( 1aiIQlDDI(x-y)aw(y)))f(y)dy

B Ial<t

(this is the first summand in (3.51)), where B C fl is such that n is star-shapedwith respect to B and w is the same as in the proof of Corollary 11. Note that

Sl-lal-1(x, D1 f) = (D1 S1-1)(x, f)

and apply (3.51), (3.52) and (3.56). 0

(3.63)

Corollary 13 Let SZ be a bounded convex domain. For x, y E fZ (x t y)denote by d(x, y) the length of the segment of the ray, which goes from thepoint x through the point y, contained in fZ . Then V f E U'(11) and for allx E fZ and Vf E (W11)t°C(0) for almost all x E Cl

1f(x) = measf2 l 1 -t f (Daf)(y)(x-y)adyIoW n

+ 1 1, J(Df)(v)_ a

Ix - yin (d(x, y)n - I x - yI n) dy (3.64)n lol=l a.n

and hence

If(x)I <measfZ at f d(x,y)nI(Daf)(y)I dy

IaI<t n

! f d(xn

+n

W! j Ix - In-1 I(Daf)(y)I dy. (3.65)

I0I=1 n


In particular,

1 DI°IIfx)I <

measS2 E a! f RD°f)(y)I dy, (3.66)Ial<_n n

where D = diam S2. (If f E (W,)'Oc(f ), then D" f must be replaced by DW 'f.)

Idea of the proof. Suppose that in Theorems 4 - 5 supp w C S2 instead ofsuppw C B. Then in (3.38) we have Vn = S2 instead of VZ V1,B - seeRemark 12. Take w(x) = (measS2)-1 for x E 1, then (3.64) follows from (3.38)- (3.40).

Corollary 14 Let S2 C R" be an open set, l E N. Then V f E Co(S2) for everyx E 1 and df E (WW)0(11) for almost every x E S2

_ a

f (x) = o E a! f Ix - yIn(D°f)(y) dy (3.67)

IaI=,

and hence1 1 I I(D° f)(y)l

dIf(x)I S On a!J Ix - yln-i y (3.68)

161= n

with D. Of replacing DI f for f E (W1)0(Q), where vn is the n-dimensionalmeasure of the unit ball in Rn and Un is the surface area of the unit sphere inRn (an = nvn) 21

Idea of the proof. Since supp f is compact in l one can assume, withoutloss of generality, that the function f is defined on R" and f = 0 outside11. Replace in Theorems 4 - 5, keeping in mind Remark 12, B by Gx =B(x, r2) \ B(x, r1), where r1 < r2 are such that supp f C B(x, r1) and w byw.(y) = (measG=)-1, y E G1. Then V1,G = B(x,r2).

Moreover, from (3.40) it follows that Vy E supp f

r2

w(x, y) = nl n) f Pn-1 dp = 1 .vn(r2 - r1) an

rl

Since f = 0 on G. equality (3.67) follows from (3.28). Furthermore, inequality(3.68) follows from (3.67) because a! > ? for IaI = 1.

21 We recall that vn = a+1 where r(u) = f x°-1e_= dx, u > 0, is the gamma-function.0


Remark 17 For n = 1, 1 = 1 and ) = (a, b) equality (3.65) reduces to theobvious equality: V f E Co (a, b) and dx E (a, b)

b

f (x) = 2 f sgn (x - y) f '(y) dy (3.69)

(see (3.7)). Starting from this equality it is possible to give another proof ofequality (3.67).

Remark 18 Consider the following particular case of (3.67)

1

J(Of)(y) . (x -1l) 1 x.

f(x) =an ix - y1" dy - an

lyln * Vf. (3.70)

We note that

Ixi2 = ax e

(in Ix I), n = 2, Ixln n > 3 ,

and for p E (Wi)'-(Rn), -0 E Co(Rn) we haveConsequently, V f E C0 2(R) )

f (x) = - (n -12)anlxi2-n * i f, n2:3 (3.72)

(the Newton potential).

Corollary 15 Let SZ C Rn be an open set satisfying the cone condition withthe parameters r > 0 and h > 0 and K. be the cone of that condition. Supposethat Vx E fl

W. E Co (R"), suppw: C IT., Jiz(v)dY = 1, (3.73)

Rn

I E N, 0 E No and lal < 1. Then V f E C1(Q) for every x E 1 and V f E(W11)(f2) for almost every x E 12

(D"f)(x) =J[(x-y)°w=(y)])f(y)dy

K. io<<-I91


+ r (D°f)(y) .w°-F,:(x, y) dy (3.74)J Ix - yln-1+191

I°I=1, °>9 K.

(with DO f and D. *f replacing DO f and D° f for f E (Wi)1oc(Sl)), where

n +r

lal(x -00'

// y - x n-1

a! Ix - yIn f wylx+ely - xl)y dp. (3.75)

I=-Y1

Remark 19 In contrast to other integral representations the first summandin (3.74) is no more a polynomial.

Idea of the proof. Apply Remark 12 with G. = Kt and wz replacing w. Notethat V:,K= = K. and d2 < h2 + r2 (d2 is defined in Remark 8).

Corollary 16 Let Sn C Rn be an open set satisfying the cone condition withthe parameters r > 0 and h > 0, l E N, 0 E No and 101 < 1. Then there existsc14 > 0, depending only on n, 1, r and h, such that V f E C' (Q) for every x E S2and Vf E (Wl)1ac(0) for almost every x E SZ

D*(D8f)(x)I <_ C14 (f If I dy + f

I

FDye -1191

dy) (3.76)

n I°I=1:z

(with D. Of and D. *f replacing D, f , D' f respectively, for f E (Wl)1Oe(11))

Moreover,

)(Y)I dy),I(Daf)(x)I <- c15 (Ih )-191rn f

Ifl dy+ (rhy-' f I

I(D°in 1+191

K, 101=1 K:

(3.77)where r1 = min{r, h} and c15 > 0 depends only on n and 1.

Remark 20 Compared with (3.54) inequality (3.76) is valid for a wider class ofopen sets satisfying the cone condition. On the other hand, (3.54) is a sharperversion of (3.76) (for in the right-hand side of (3.54) f If I dx replaces f if I dx)

B afor a narrower class of domains star-shaped with respect to the ball B.

Idea of the proof. Let K = K(r, h) if h > r and K = K(h, h) if h < r, and letB(xo, r2) be the ball inscribed into the cone K. Here r2 = rh( r + h +r)-1 >-


r(l+f)-' if h > r and r2 = h(1+f)-' if h < r. Hence B(xo, rl(l+f)-') CK. Moreover, let w be a fixed function defined by (3.50). Suppose that in (3.74)wx is defined by: Wx(y) = (i+ 22)-1(y-£y)), where £_ = A1(xo) andA. is a linear transformation such that Kx = A.(K). Following the proof ofestimates (3.56) and (3.57), establish that b'x E Sl and Vy E K.

1 IQH Ifll(- )' D,+R[(x - y)awx(y)]I < Ml(h r1)'-ih-101r1

(3.78)Ial<t-Idl

andh n-'

Iwa-#,.(x, y)I <- M2 (rl) , (3.79)

where Ml and M2 depend only on n and 1. Estimates (3.77) and hence (3.76)follow from (3.74), (3.78) and (3.79). 0

Remark 21 From (3.77) it also follows that in (3.76) f If I dx can be replacedn

by If IIL,(n) for any p E [1, oo].

Chapter 4

Embedding theorems

The main aim of this chapter is to prove various inequalities related to thoseof the form

II1 fIIL,(n) < lIfIIW ini,

where a E K, lal < l (D,°, f a f) and cl > 0 does not depend on f.These inequalities may be also presented in an equivalent form as the so-

called embedding theorems.

4.1 . Embeddings and inequalitiesWe start with the consideration of the notion of a continuous embedding as itrelates to the general theory of function spaces, which are, in the framework ofthis book, normed or semi-normed vector spaces.

First let Z1 and Z2 be normed vector spaces. We say that Zl is embeddedin Z2 if

Z1 C Z2. (4.1)

The identity operator I, considered as an operator acting from Z1 in Z2:

Vf E Z1 If = f, I : Zl -+ Z2, (4.2)

which is possible because of (4.1), will be called the embedding operator corre-sponding to the embedding (4.1).

Definition 1 Let Z1 and Z2 be normed vector spaces. We say that Z1 is con-tinuously embedded in Z2 and write

Z1 C:; Z2 (4.3)

120 CHAPTER 4. EMBEDDING THEOREMS

if, in addition to (4.1), the corresponding embedding operator is continuous,i.e., there exists c2 > 0 such that V f E Z1

illiIZ, <_ c llfliz, (4.4)

In the cases we are interested in relations (4.1) and (4.3) are equivalent,

which follows from the next statement.

Lemma 1 Let Z1 and Z2 be Banach spaces such that Z1 C Z2. Suppose thatthe corresponding embedding operator is closed, i.e., for any fk E Z1 wherekEN,g,EZ1andg2EZ2

lion fk = 91 in Z1, Urn fk = 92 in Z2 = 91 = 92. (4.5)k-+oo klao

Then (4.4) is satisfied and, hence, Z1 Ci Z2.

Idea of the proof. Since the embedding operator I : Z1 -a Z2 is defined on thewhole Z1 and is closed, by the Banach closed graph theorem the operator I isbounded.

Remark 1 Let us introduce for Banach spaces Z1 and Z2 such that Z1 C Z2one more norm on Z1, namely, df E Z1

U 11z" = ill liz, + Iif lIz,.

It is a norm on Z1 considered as the intersection of Z1 and Z2. Condition (4.5)is equivalent to the following: Z1 is complete with respect to the norm II' lizIndeed, if { fk}kEN is a Cauchy sequence with respect to II liz,,, then

lim IN - fmiiz, = urn Ilfk - fmiiz, = 0.k,m-oo k,m-400

Consequently, by the completeness of Z1 and Z2 elements g1 E Z1 and 92 E Z2exist such that limk,o fk = 91 in Z, and limk-wo fk = g2 in Z2. By (4.5)91 = 92 and, hence, slim li fk - g1 li z = 0. Conversely, let the above relations be

satisfied. Then { fk}kEN is a Cauchy sequence with respect to the norm II ' lizBy the completeness of Z12 an element g E Zl exists such that

lirnkoo(Iifk - 9112, + 11A - 9ii2,) = 0.

From the uniqueness of limits in Z1 and Z2 it follows that gl = 92(= g).We note also that the closedness of the embedding operator I is a necessary

and sufficient condition for the equivalence of (4.1) and (4.3). The sufficiencyis proved in Lemma 1, while the necessity is obvious, since the boundness of Iimplies its closedness.

4.1. EMBEDDINGS AND INEQUALITIES 121

Now let Z, and Z2 be semi-normed vector spaces and

Bk= If E Zk: IIf IIz. =0}, k = 1,2.

Definition 2 Let Z, and Z2 be semi-nonmed vector spaces, which are subsetsof a linear space Z. We say that Z1 is continuously embedded in Z2 and write

Z, C=y Z2 (4.6)

if'Z, C Z2 + 81,

and the corresponding embedding operator I : Z1 -+ Z2, defined by If = g, isbounded. 2

Remark 2 This means that V f E Z, there exists g E Z2 such that g is equiv-alent to f in Z1, i.e., g - f E 81, and there exists c3 > 0 such that Vf E Z,

II9IIz, <- c31IfiIz,. (4.7)

Remark 3 If 8, C 82 (in particular, if Z1 and Z2 are normed vector spaces),then Z2 + 01 = Z2, 1I9I1z, = Ill IIz, and Definition 2 has the same form asDefinition 1.

Remark 4 Assume that the semi-normed vector spaces Z, and Z2 possess thefollowing property:

the semi-norm IIf IIz, makes sense for each f E Z, with IIf IIz, < oo orIIf IIz, = oo and V f E Z, there exists g E Z2 such that

inf IIf - hllz, = II9IIz2-hEel

In this case Definition 2 is equivalent to:there exists c3 > 0 such that Vf E Z1

inf IIf - hllz, 5 csllf Ilzi.heo,

' The sign + denotes the vector sum of sets.2In this case, in general, the embedding operator is not unique. However, one may easily

verify that for different embedding operators, say I, and 12, Vf E Z, we have IIIiflIz, _1112flIs,-


Lemma 2 Let Z1 and Z2 be semi-Banach spaces such that Z1 C Z2. Supposethat for any fk E Z1, k E N, g1 E Z1 and g2 E Z2

lim fk = 91 in Z1i lim fk = g2 in Z2 91 - 92 E 01. (4.10)k-+oo k-+oo

Then (4.9) is satisfied.

Idea of the proof. Apply the Banach closed graph theorem to the factor spacesZl = Z1/81 and Z2 = Z_2/81 and 3 the embedding operator I : Z1 -> Z2.Proof. We recall that Z1 is a Banach space and b'/ E Z1

If 112, = 11f 11z" (4.11)

where f is an arbitrary element in f. (If fl, f2 E f, then 11f, 11 = 11f211.) As forZ2 it is, in general, a semi-Banach space and

IIJ II2, = inf If - hllz2h0i(4.12)

where f E j. (The right-hand side does not depend on the choice of f E f.)From Z1 C Z2 it follows that Z1 C Z2 and by (4.10) the corresponding

embedding operator I is closed. For, let 1k E ZZ1, k E N, 91 E Z1, 92 E Z2 and

k Ik=91 in Z1, k Ifk=kl Ik=g2 in Z2.

Suppose that fk E fk, 91 E 91 and g2 E 92 Then fk - 91 E It - 91, fk - 92 Efk - 92 and

limllft -91IIz, = 0, lim (hnf I1fk-92-hIlz,) = 0.

Therefore, Vk E N there exists hk E 81 such that 1lim Ilfk-92-hkllz, = 0. Thus

fk - hk -+ 92 in Z2 ask -+ oo. Moreover, since Ilft - hk - 91IIz, = lift - 9111zwe also have that fk - hk --1 91 in Z, as k -3' oo. By (4.10) 91 - g2 E 81 and,hence, g1 = 92

Now by the Banach closed graph theorem the operator I is bounded: forsome c4 > 0 we have b'/ E Zl

111112,=IIIJII2,:5 04II!II2,.

Consequently, by (4.10) and (4.12) the desired inequality (4.9) follows.

3 The spaces Z'1 and Z2 consist of nonintersecting classes f C Z1, f C Z2 respectively,such that fl, f2 E 1 . = fl - f2 E 01, fl - fs E B2 respectively.


Corollary 1 If, in addition to the assumptions of Lemma 2, (4.8) is satisfied,then Z, C Z2 is equivalent to Z, C:; Z2.

Idea of the proof. Apply Remark 4.

Corollary 2 In addition to the assumptions of Lemma 2, let

01 C 02. (4.13)

Then there exists c5 > 0 such that V f E Z1

IIfIIz2 <- Cs IIfIIz, (4.14)

Idea of the proof. Since 01 C 02, we have IIhIIz2 = 0 for each h E 01. Furthermore,Vf E Z, we also have IIf - hIIz2 = IIf 1122 and (4.9) coincides with (4.14).

Corollary 3 Let Z be a semi-normed vector space, equipped with two semi-norms II III and II . 112 and complete with respect to both of them. Moreover,suppose that for any fk E Z, k E N, g1, g2 E Z

kHM MIIfk-91111=0, llfk-92112=0 =* g1- 92E91n82. (4.15)

Then the semi-norms II 'III and II.112 are equivalent 4 if, and only if,

01 = 82. (4.17)

Suppose, in particular, that Z is a normed vector space, equipped with twonorms 11.11, and II.112 and complete with respect to both of them. If for anyfkEZ, kEN, 9,,92EZ

kIIfk-91111=0, k IIl-92112=0

then the norms 11 'III and II 112 are equivalent.

4Le., there exist ce, or > 0 such that V f E Z

(4.18)

CO IIf112 <- IIfI11 <_ CT IIf112. (4.16)


Idea of the proof. Necessity of (4.17) follows directly from (4.16). To provesufficiency apply Corollary 1 to the semi-Banach spaces Z, and Z2, which arethe same set Z, equipped with the semi-norms II - IIl and II.112 .

Now we pass to the case of function spaces. Let SZ C Ri" be an open set.Moreover, suppose that Z(ft) is a semi-nonmed vector space of functions definedon Q. Denote

ez(n) = If E Z(Q) : IIf1Iz(n) = 0}

and

9(0) = If : f (x) = 0 for almost every x E 0}.

All function spaces Z(0), which are considered in this book, possess thefollowing property:

Z(n) Ci L' -(Q), (4.19)

i.e., Z(S2) C L'-(O) and for each compact K C 0 there exists ca(K) > 0 suchthat Vf E Z(S2)

hEin ezf IIf - hIIL,(x) <- c8(K) IIf 112(n).

For many of the function spaces considered

8z(a) = t)(ft).

If this property is satisfied, then (4.20) takes the form

(4.20)

(4.21)

IIf IIL,(x) <- c8(K) IIf 112(n). (4.22)

Remark 5 If two semi-normed vector spaces Zl(f1) and Z2(SZ) satisfy (4.19)and 0z,(n),9z,(n) C 0(ft), then f o r a n y f k E Z I (SZ) n Z2(S2), k E N, gl E Z1(())and g2 E Z2(ft)

lim fk = gi in Zl (ft), k m fk = 92 in Z2(1) 9i - 92 on S2,

i.e., 91 - 92 E 9(S2).

Lemma 3 Let Sl C R" be an open set and let Zl(f2) and Z2 (fl) be semi-Banachfunction spaces satisfying (4.19) and (4.21). If

Zi(p) C Z2(1Z) (4.23)

then there exists cs > 0 such that 'If E Z1(f2)

IIfIIzs(n) < cs II/IIz,(n) (4.24)

and, hence, (4.23) is equivalent to

ZA(0) C4 Z2(11). (4.25)


Idea of the proof. Apply Corollary 2.Proof. If fk E Z1(12), k E N, gl E ZI(f ), 92 E Z2(S2), and

urn fk = gi in Zi(I), k fk = 92 in Z2(Q), (4.26)

then by (4.22)

lim fk = gl in L"(9), lim fk = 92 in L1°`(SI) (4.27)k-+oo kloo

and gi - g2 E 9(e) = 9z,(n). Hence, (4.24) follows from (4.13)From (4.21) it follows that Z2(S2) + Bz,(n) = Z2(St). Moreover, for each

g E Z2(11), for which g- f E Oz,(n) we have II9IIz2(n) = IIf1Iz2(n) Consequently,(4.23) coincides with (4.6), (4.24) coincides with (4.7) and, hence, (4.23) isequivalent to (4.25).

Corollary 4 Let 9 C R" be an open set and Z(i2) a semi-normed vectorspace equipped with two semi-norms and complete with respect to both of them.Moreover, suppose that conditions (4.19) and (4.21) are satisfied. Then thesesemi-norms are equivalent.

Idea of the proof. Apply Lemma 3 to the semi-normed vector spaces Z1(il) andZ2(0), which are the same set Z(1l), equipped with the given semi-norms.

Finally, we collect together all the statements about equivalence of inequal-ities, embeddings and continuous embeddings in the case of Sobolev spaces.

Theorem l Let l E N, m E No, m <1,1 < p, q < oo and let Il C R" be anopen set.

1. The continuous embedding

WW ((1) Ci WQ (0), (4.28)

i.e., the inequalityIll II Wr(o) < CIO llf llww(n) , (4.29)

where cto > 0 does not depend on f, is equivalent to the embedding

Wp(11) C WQ (11).

2. The continuous embedding

(4.30)

W,(11)C:; Cb (c), (4.31)


i.e., the statement: Vf E WP(SZ) there exists a function g E Cb (S1) such thatg - f on SZ and

Ilgllcm(n) <_ cil Of Ilwp(n), (4.32)

where c11 > 0 does not depend on f, is equivalent to inequality (4.29) andembedding (4.30), where q = oo, and in (4.29) c1o1q = cil

3. If inequality (4.29) holds for all f E WW(I) fl C°°(fl), then it holds forall f E W,(SZ).

Idea of the proof. Apply Definition 2 and Lemmas 2 and 3. To prove theequivalence of (4.32) and (4.29) when q = oo apply Theorem 1 of Chapter 2and the completeness of the spaces under consideration. If m > 0 apply alsothe closedness of the differentiation operator D° where Iai = m in Cb(Q). Toprove the third statement of the theorem again apply Theorem 1 of Chapter2, the completeness of W, ,'(Q) and, for m > 0, the closedness of the weakdifferentiation operator DW where Jai = m in Lq(Sl). 0Proof. 1. Clearly, (4.30) follows from (4.28). As for the converse, it is a directcorollary of Lemma 3, because 8wp(n) = ow-(A) = 0(g).

2. Furthermore, (4.31) implies (4.29) with q = oo, which, by the firststatement of the theorem, is equivalent to (4.30) with q = oo.

Let us prove that (4.29) with q = oo implies (4.32) where c11 = c1019=00.First suppose that m = 0 and 1 < p < oo. Then V f E W,1(0) there existfk E C°°(Q) n WP(SZ), k E N, such that fk -4 f in Wp(SZ) - see Theorem 1 inChapter 2. By (4.29) with q = oo, fit E Cb(SZ), k E N, and'dk, s E N

Ilfk - fallC(n) = Ilfk - 5 C11 Ilfk - fsIIW (n)

(see footnote 4 on page 12). Hence, { fk}kEN is a Cauchy sequence in Cb(SZ).Since Cb(fl) is complete, there exists g E Cb(11) such that fk -+ g in Cb(SZ) ask -> oo. Since both WP(1) and Cb(SZ) are continuosly embedded into L10Q(0),it follows that g - f on SZ - see Remark 5.

If m = 0 and p = oo, then 'If E W00(SZ) there exist fk E C00(11) nW,10(SZ), k E N, such that fk -+ f in WO0(f) for r = 1, ...,1 - 1 andIIfkIIw., (n) - Ilf Ilw;,(n) as k -> oo (see Theorem I in Chapter 2). Sincellfk - fallC(n) = Ilfk - fell L (n), {fk}kEN is again a Cauchy sequence in C6(SZ).The rest is the same as for the case in which 1 0 and 1 < p < oo, then the same argument as above shows thatthere exist h E Cb(S2) and h° E Cb(fl) where Ial = m such that fk -+ h in Cb(SZ)and D° fk -+ h° in Cb(SZ). Since the differentiation operator D° is closed inCb(11), it follows that h° = D°h. Hence h e Cr(fZ) and fk -+g in Cr(fZ).


Finally, for all m > 0 and 1 < p < oo, we take f = fk in (4.29) whereq = oo and passing to the limit as k -+ oo we get (4.32) since

IIgIICm(n) = kliM IIfkIICm(11) = kliM Ilfkllw(n)-+00 oc

< CIO lim IIfkIIwn1(s1) = C10IIf IIw;(n)k-4oo

3. The proof of the third statement of the theorem is analogous.

4.2 The one-dimensional caseWe start with inequalities for intermediate derivatives.

Theorem 2 Let -oo < a 0 depends only on I and b - a.2. If -oo<a<a<Q<b<oo, then form=l,...,1-1

IIf,S,)IIL,(a,b) <- C13 (IIfIIL,(a,A) + Ilf,(`'IIL,(ab)),

(4.33)

(4.34)

where c13 > 0 depends only on 1, b - a, fi - a and f is such that the right-handside is finite.

Remark 6 If b - a = oo and B - a < oo, then inequality (4.34) does not hold.This follows by setting f (x) = xk where m < k < 1.

Idea of the proof. Apply inequality (3.21) and Remark 5 of Chapter 3.Proof. Let -oo < a < b < oo. From (3.21), by Holder's inequality and Remark5 of Chapter 3, it follows that

Ml (b -a)'_m (0 - a)-'IIfIIL,(a,O) + (b - a)i' Ilfw')IIL,(a,b)),

(4.35)where M1 depends only on n. This inequality implies (4.34).

Now let b - a = oo. Say, for example, -oo < a < oo and b = oo. If1 < p < coo, then by (4.33)

ao 1

=(E Ilf-(_)IIL,(a+k-1,a+k))

k=1


M2 ( E(IIf IIL,(a+k-1,a+k) + Ilfwl) IIL,(a+k-1,a+k)))k=1

= M2(IIfIIL,(a,-) + Ilfw`)IIL,(a,.))° M2 (IIfIIL,(a,aa) + Ilfw`)IIL,(a.ao))

Here M2 is the constant c12 in (4.33) for the case in which b - a = 1.If p = oo, then

Ilfwm'IILoo(a,ao) = Sup I1fwm)IIL.(a+k-1,a+k)kEN

M2 SUP(IIf llfw`)IIL,o(a+k-1,.+k))

kEN

< M2 (IllIILW(a.00) +

Corollary 5 Let 1 < p < oo. The norm

1

F, Iifw)IIL,(a,b) (4.36)M=0

is equivalent to If IIw;(a,b) for any interval (a, b) C It. The norm

IIf IIL,(a,0) + Ilf, lIL,(ab)

is equivalent to Ill II W (a,b) if -oo < a < a <,8:5 b < oo.

(4.37)

Idea of the proof. Apply inequality (4.33) and inequality (4.34) with m = 0.

Corollary 6 Let -oo < a < b < oo, t E N and 1 < p < oo. Then

w4(a, b) = WW(a, b). (4.38)

( If b - a = oo, then 1 E w,(a, b) \ W, (a, b). Thus, the embedding Wp(a, b) Cwp(a, b) is strict.)

Idea of the proof. If f E wp(a, b), then f E L11-(a, b) , and Corollary 5 implies(4.38).0

Remark 7 Equality (4.38) is an equality of sets of functions. Since B,,,D(a,b)Bwo(a,b), the semi-norms II - IIw;(a,b) and II - IIw;(a,b) are not equivalent. (SeeCorollary 3 of Section 4.1.)


Corollary 7 Let 1, m E N, m < l and 1 0 independent off is equivalent to the validity of

Ilfwm)IILP(a,b) C14 ((b - a)-mIIfIIL,(a,b) + (b - a)'-mlIfw')IIL,,(a,b)/1, (4.39)

156 >= IIfIIL,(a,b) + IF IIff')IIL,(a,b) (4.40)C156-"7-

for 5 0 < e < c14(b - a)'-' and

IIfwm)IIL,,(a,b) <- C16IIfIIL,,(a,b) IIfIIW,(a,b) (4.41)

with some c14, c15 > 0 independent of f, a and b and some c16 > 0 independentof f.

2. If b - a = oo, then inequality (4.33) is equivalent to

ii;(m)1LP(a,b) <_ C12 e ' m IIfIILP(a,b) +e IIfW')IILP(a,b) (4.42)

for0<e<00,

II c(m)IIL,ca,b) <_ C12 t (1 l / 1 T -IIIfIILP(a,b) Ilf,;`'IIL,ca,br (4.43)

b

rllm

JIf-)IPd2 <'12(( [( ) (1

a

if1<p<ooand6

//t)1-T11-P(4.44)

m e-'-f IfIPdx+e f Ifw')IPdxfIf(m)IPdx<crl l(!

)

T(1 t /1 1-'T] '

'

a a a

for0<e<oo if l <p<oo.(4.45)

b One may consider 0 < e < eo, where co is an arbitrary positive number. In this casecls > 0 depends on eo as well. It also follows that Ve > 0 there exists C(e) such that

Ilfwm)IIL,(a,b) < C(e) IIfIIL,(a,b) +f Ilf, IIL,(a,b)

Note that one cannot replace here C(e) by a and a by C(e). (If it were so, then takingf (z) = xm and passing to the limit as a - 0+ would give a contradiction.)

From (4.40) it also follows that

Ilfwm)IIL.(a,b) < M((b - a) m + 1) II/II W,(a,b),

where M > 0 is independent of f,a and b.

b

f (IfIP+lf.(')IP)dxa

b b


Idea of the proof. 1. Changing variables reduce the case of an arbitrary interval(a, b) to the case of the interval (0, 1) and deduce (4.39) from (4.33). For0<b<b-asetN=[b6a] +1,b1= N°,ak=a+61(k-1),k=1,...,N+1.Apply the equality 6

j

N

IIf II L,(a,o) = IIf IILp(ak,ak+,))

p

k=1

(4.46)

and the inequality s < 61 < b to deduce (4.40) from (4.39). Minimize theright-hand side of (4.40) with respect to e in order to prove (4.41). Finally,note that inequality (4.39) follows from (4.41) and the inequality

x°y1-Q < a°(1 - a)'-* (x + y), (4.47)

where x,y > 0, 0 < a < 1.2. Apply (4.33) to f (a + 5(x - a)) if b = oo, to f (b - b(b - x)) if a = -oo

or to f (bx) if a = -oo, b = oo where b > 0, and deduce (4.42). Minimize theright-hand side of (4.42) to get (4.43). Note that (4.33) follows from (4.43)and (4.47). Raise (4.43) to the power p and apply (4.47) to establish (4.44).Deduce, by applying dilations once more, (4.45) from (4.44). Minimizing theright-hand side of (4.45), verify that (4.45) implies (4.43). 0Proof. 1. Setting y = e-a we get

Ilfwm)IIL,(a,b) _ (b - a)',-mII(f (a + y(b -

a) p-m(IIf(a + y(b - a))IILp(o,1) + 11(f (a + y(b - a)))(') jILp(o,1))

= M1((b - a)-'11f IIL,(a,b) + (b -

where M1 is the constant c12 in (4.33) for the case in which (a, b) = (0, 1).Moreover, for 0 < a < b - a and 1 < p < oo from (4.46), (4.39) and

Minkowski's inequality it follows that

Nt

,

Ilfmm)IILp(a,b) mIIJ IILp(aiAk+1) + o1 mIIfwj)IIL,(ak,ak+t))Pj pk=1

C14 (al m (E IIf IILp(ak,ak+i)) al-m (E II fw1) IIPLP(ak,ak+i))p

)k=1 k=1

'If p= oo this means that IIIIIL,.(o,b) = kmaxN UfHL..(ok,ak+,)


< C14(2'6-m IIIIILp(a,b) + b'-' 11f(')IILp(a,b))-

Setting e = c1461-' we establish (4.40). The minimum of the right-hand sideof (4.40) is equal to

C15 T((';) (1- -1)1- )-IIIf IILp(ab) Ilfwl)IILp(a,b)

and is achieved for e = e1, where

e1= (1 C15 U114(0) IVw`'IIL,(a,b))1

If el < (b - a)l-', then, setting e = el in (4.40) we get (4.41). Now letel > (b - a)"-'". This is equivalent to

Iifw')IILp(a,b) <- ,, Cl5 (b - a)-'IIfIILp(a,b).

Since

_T TIllIILp(a,b) 5 IIfIILpa,b) IIlIIw (a,b)' Ilfw''IILp(aa) <_ Ilfw`'IILpa,b) IIfIIWy(a,b)'

inequality (4.39) with e = c14(b - a)I-' implies that

Iifw')IILp(a,b) 5 M2((b -a)-m + 1) IIfIILp(,b) IIfIIW;(a,b)'

where M2 depends only on 1, and (4.41) follows. In its turn (4.41) and (4.47)imply (4.33).

2. Let b - a = oo, say a = -oo, b = oo. Given a function f E WP'(-00' oo)and b > 0, by (4.33) we have

Ilfw''IIL,(-aa,aa) - b-m+;II(f(bX)),am)IILp

< C12 b-'+= IIf (bx)IILp(-00,00) + II(f(bx))w')IILp(-oo,oo))

= C12 (b 'IIfIILp(-00,00) +8' ' Ilfw')IILp(-.,.)

Setting c1261-m = e, we get (4.42).The rest of the proof is as in step 1. 0

Corollary 8 Let 1, m E N, m < 1. Then

Ilfwm)IILa(-ao,oo) IIfIIL2(- ,oo) Il1w')IIL (-00,00) (4.48)

and the constant 1 in this inequality is sharp.


Idea of the proof. Prove (4.44) when p = 2, a = -oo, b = oo by using Fouriertransforms and Parseval's inequality, and apply Corollary 7.Proof. By Parseval's equality and inequality (4.47)

00

Ilfwm)IIL3 (-00,00) - IIF(fwm))IIL2 (-00,00) = J

00

-00

< ('!')m (1 - m)1-T f(i-00

_ (';)T(1 - m)1 T (IIFf11L2(-00,00) + IIF(f.())IIL2

2(-o,0))

_ (m)'(1 - it)1 T (IIfIIL2 22(-00,00) + I1fw!)IIL2(-00,00)).

Since Z2in = ('")T(1 - if, and only if, ICI = (1 m) - Co we set

f = f, where ff(x) = (F-1(x(Eo-,Fo+E)))(x) = A stti zz a-'{0y. Passing to the

limit as e -+ 0+ we obtain that (m)9 (1 - m)1-T is a sharp constant.

Remark 8 Let 1, m E N, m < l and 1 < p < oo. The value of the sharpconstant cm,,,P in the inequality

IlAm)II4(-00,00) m,l,p IIfwIIL,(_oo,oo) Ilfwt)IILp(-00,00) (4.49)

is also known in the cases p = oo and p = 1:

KK-mcm,1,1 = Cm,l,oo = K-m ,

t

where for j E N

400

1 4X00` (-1)'K2,i-1 = r (2i + 1)2j K21

_ ±` (2a + 1)2j+1

i=O =o

The following inequality holds

1 = Cm,1,2 < Cm4,P < C ij,l = Cd,00 <

Thus, df E Wp(-oo, oo) for each 1 < p < 00

Ilfwm)IILp(-00,00) 52 IIf (-oo,oo)' (4.50)


Remark 9 We note a simple particular case of (4.49):

IIfwIILoe(-oo,oo) :5 1/2 If (4.51)

where f is a sharp constant. One can easily prove this inequality by applyingthe integral representation (3.27) with a = x + e, b = x - e, e > 0. It followsthat for each function f whose derivative f' is locally absolutely continuous

If'(x)I <-E 11AL,,(-oa,o) + 2

We get the desired inequality by minimizing with respect to e > 0 and applyingDefinition 4 of Chapter 1.

In the sequel we shall need a more general inequality: for 1 < p:5 oc

1(1-1) 11+1IIfwIIL,(-oo,oo) (4.52)

To prove it we apply Holder's inequality to the integral representation (3.24)and get

If'(x)I <- IIw IIL,,(a,b)IIfIILp(a,b) + IIA(x,')IIL,,(a,b)IIfwIILp(a,b)

almost everywhere on (a, b). Choosing w in such a way that IIw'IIL,,(a,b) is min-imal, we establish that'

W(X)+1)(b-a)p(6 I2x -(a+b)I° +pb 1 a

and

Moreover,

IIw IIL,,(a,b) = 2 (p + 1)0.

b

f w(u) du < (1 + - )',IA(a

2b,y)I b

y

f w(u) du < (1 + p b-aY

for a < y < °ab

for Sib- <y<b,

b b

7Euler's equation for the extremal problem f lw'(x)IP' dx - min, f w(x) dx = 1 wherea

1 < p < oo has the form (Iw'(x)I" sgnw'(x))' = A. So, m'(x) = Jalx + A2IP 'sgn (AIx + A2)Here A, A1, A2 are some constants.


and

Taking a=x+s,b=x-e, we get

If'(x)I

°IIf lIL,(_,,,oo)

almost everywhere on (-oo, oo). By minimizing with respect to e > 0 andapplying Definition 4 of Chapter 1 we have

IIf AP IllIILp(-00,00) IIf wIILp(±oo,00)'

where 8

AP-p'+1\p'+1Ja)- " < 2e< <2.

Remark 10 By (4.52) and (4.50) it follows that VI E N, I > 2,

Ilf<_ Ilf(`)III('+;)(1-00,00) w La(-00,00)

< 2 1 2 IIf IIL,(-oo,oo) Ilfwr)IILp 00,00)) ( a)lfw+)IILa(±.+.)

- 27r IIf 11Lp(_oo,00) Il '1' IIL; 00,00)'

Remark 11 Let I, M E N, m < I and 1 < p:5 oo. Then `df E Wp(0, oo)

(4.53)

IIf (')IIL°(O,ao) :52

& IllIILp(00)Ilf 2IIL(0,-). (4.54)

This can be proved with the help of the extension operator T2, constructed inSection 6.1 of Chapter 6 (see Remark 1):

IIfwm)IILp(0,C0) C II(T2f)wm)IILp(-.1.) <-

2

IIT2f IILa(-oo,w) lI(T2f)((a')IILa(-OO,oo)

8This inequality is equivalent to 2p-1 < Qv 'T)P(1 - a)P, where v = which isclear since for p > 1

/f

lP-1<2<2et<ei11 .az)P(1-2p/a.


IIfIILp O,oo) IIfw"11ILp(O,oo)'2

It can be proved that, in contrast to the case of the whole line, the best constantin the case of the half-line for fixed m tends to oo as I -+ oo. To verify this onemay consider the function f defined by f (x) x)' for 0 < x < Y l-!, andf(x)=0forx> '1!.

Corollary 9 Let -oo < a < b < oo, I, m E N, m < I and 1 < p:5 oo . If asequence { fk}kEN is bounded in WP (a, b) and converges in Lp(a, b) to a functionf, then it converges to f in Wpm (a, b).

Idea of the proof. Apply inequalities with a parameter (4.40) and (4.42) ormultiplicative inequalities (4.41) and (4.43). 0Proof. Let IIfkIIWP(a,b) < K for each k E N and fk -+ f in Lp(a,b) as k -+ oc.

1. By footnote 5 it follows that Ve > 0 there exists C(c) such that

5 C(c) IllIILp(a,b) + e

for each f E W, (a, b). Consequently, Vk, s E N

II(fk)y,'") - (fs);p"`)IILp(a,b) <- C(e) Ilfk - fall Lp(a,b) + 2eK.

Given 6 > 0 we choose a in such a way that 2eK < 6. Since fk is a Cauchysequence in Lp(a, b), there exists N E N such that C(e) Ilfk - fsIILp(a,b) < sif k,s > N. Hence, bk,s > N we have II(fk)W'") - (fs)w'")IILp(a,b) < 6, i.e., thesequence (fk)(W'") is Cauchy in Lp(a, b). Because of the completeness of Lp(a, b)there exists g E Lp(a, b) such that (fk)(W") -+ gas k -+ oo in Lp(a, b). Since theweak differentiation operator is closed (see Section 1.2), g is a weak derivativeof order m off on (a, b). Consequently, fk -4 f in Wpm (a, b) as k -+ oo.

2. By (4.41) and (4.43) it follows that dk, s E N

II(fk)y,'") - (fs)w'' llL; (a,b) <- M Ilfk - fall cp',b) Ilfk - Jsllwp(a,b)

< M(2K)'` llfk - fsllip(a,b) ,

where M depends only on 1. Consequently, we can again state that (fk)W('") isa Cauchy sequence in Lp(a, b). The rest is the same as in step 1. 0

Theorems Let -oo < a < b < 00,1 E N, m E N, m < I and l p : 5Then the embedding

W, (a, b) c Wp (a, b)


is compact, i.e., the embedding operator I : W,(a, b) -+ Wp (a, b) is compact. 9

Idea of the proof. Let S be an arbitrary bounded set in W,(a, b). In the casem = 0 consider any bounded extension operator T : WW (a, b) -i Wp(-oo, oo).(See Section 6.1.) For d > 0 set f j = Aa(T f ), where A6 is a mollifier witha nonnegative kernel. Prove that there exists All > 0 such that Vf E S and`db>0

if - faiIL,(a,b) <- M15. (4.55)

Moreover, prove that there exists M2(5) > 0 such that b' f E S and Vx, y E [a, b]

I fa(x)15 M2(6), Ifa(x) - f6(y)15 M2(5) Ix - yI. (4.56)

Finally, apply the criterion of compactness in terms of c-nets and Arzela'stheorem. 10 In the case m > 0 apply Corollary 9.Proof. By (1.8) we have

IIf - f3II L,(a,b) 5 II Ab (Tf) - T f I I L,(-oo,aa)

< sup I I (T f) (x + h) - (T f) (x) II L,(-aa..).lnl<s

By Corollary 7 of Chapter 3 and inequality (4.33)

II(Tf)(x+h) - (Tf)(x)IIL,(-,,oa) <- IhI II(Tf)',IIL,(-.,.)

< M3 IhI IITfIIw,,(-oo,.) <- M4 IhI IIfIIw,,(a,b),

where 1W3 and Af4 are independent of f.Since S is bounded in WW (a, b), say 11f IIW;(a,b) < K for each f E S, inequal-

ity (4.55) follows. Furthermore, by Holder's inequality Vx E [a, b]

x+6

If6(x)I 5 fw(_-)I(Tf)(y)I dy 5 ds (26)' IITfIIL,(R)

X-4

< M6(6) IIf IIW (a,b) S K M6(b)

9 This means that each set bounded in W,(a, b) is compact in W, (a, b) (e precompact),i.e., each of its infinite subsets contains a sequence convergent in Wy (a, b).

10 Let it C R" be a compact. A set S C C(f)) is compact in C(fl) (= precompact) if, andonly if, S is bounded and equicontinuous , i.e., Ve > 0 36 > 0 such that Vf E S and Vx, y e flsatisfying Ix - yj < 6 the inequality if (x) - f (y)l < e holds.


and the first inequality (4.56) follows.

independent of f.Moreover, Vx, y E [a, b]

Here M5 = max Iw(z)I and M6(b) is

If6(x) -f6(y)I =b I f ( (x 8 ) -w(y d u)) (Tf)(u)du

R

<6 f Iw(xb!)-6u)

I(Tf)(u)Idu(x-6,x+6)U(y-6,y+6) 1

-yI KM8(6)Ix-yi<M'I662

and the second inequality (4.56) follows. Here M7 = maxlw'(z)I and M8(6) is

independent of f .Given e > 0 by (4.55) there exists 6 > 0 such that if - f6IIL,(a,b) < z for

each f E S. Then an z-net for the set S6 = { f6 : f E S} will be an a-net for S.If we now establish the compactness of S6 in L,,(a, b) it will imply the existenceof a finite i-net for S6. This means that we may construct finite c-nets for Sfor an arbitrary e > 0, which implies that S is compact in Lp(a, b).

Finally, it is enough to note that from (4.56) it follows that the set S6is bounded and equicontinuous in C[a, b]. Hence, by Arzela's theorem S6 iscompact in C[a, b] and consequently in Lp[a, b] since convergence in C[a, b]implies convergence in Lp[a, b].

2. Let m > 0. By step 1 each infinite subset of S contains a sequence{fk}kEN convergent to a function f in Lp(a, b). By Corollary 9 f E Wp (a, b)and fk -+ f ask -*oo inWp (a,b).0

Example 1 If b - a = oo, then Theorem 3 does not hold. Let, for example,(a, b) = (0, oo). Suppose, that cp E Co (-oo, oo) is such that supp cp C [0, 1]and W ; 0. Then the set S = {,p(x - k)}kEN is bounded in Wp(0, oo) sinceII P(x - k)IIw;(o,w) = IIwIIwo(o,0o) However, it is not compact in W(0, oo)because for each k, m E N, k m

IIw(x - k) - p(x - m)Ii w; (o,oo) >- IIw(x - k) - W(x - m)IIL,(o,co) = 20.

(Consequently, any sequence in S, i.e., {w(x - k,)}$EN, is not convergent inwpm (0, 00))


Next we pass to the embedding theorems in the simplest case of Sobolevspaces Wp (a, b). In this case it is possible to evaluate sharp constants in manyof the relevant inequalities.

Theorem 4 Let -oo < a < b < oo and 1 < p < oo. Then each functionf E Wp (a, b) is equivalent to a function h E C(a, b). Moreover,

1) if-oo<a<b<oo,then

If IIL (a.b) <- (b - a) "IIfIIL,(a.b) + (p' + 1)-;"(b - a) IIf' .IIL,(a,b) (4.57)

and, consequently,

II!IIL-(..b) C17IIfIIW, (a,b),

whereC17 = max{(b - a)-D, (1/+1) '(b -

2) if-oo<a<b<oo,thenb

IIf (x) - b 1 a f f (y) dyIIL(a,b)< (p' +

1)-'(b - a) ° IIf,IIL,(a,b);

a

(4.58)

(4.59)

3) if -oo < a < b = oo, then lim h(x) = 0 and

IIfIIL (a,oo) <- (p')-'IIfIIwo(a,aa); (4.60)

4) if (a, b) = (-oo,oo), then Zlimah(x) = 0 and

Ill <IILm(-oo,oo) 2 v (Y') Of II Wo(-oo,oo). (4.61)

All the constants in the inequalities (4.57), (4.59) - (4.61) are sharp. Theconstant c17 in inequality (4.58) is sharp if b - a < (p' + 1) '.

Remark 12 For p = 1 inequality (4.57) takes the form

Ill (b - a)-'IIf IIL,(a,b) + IIf,,IIL,(a,b). (4.62)

We also note that for p = 1 inequalities (4.60) and (4.61) are equivalent to

IIfIIL (a,.) <- IIfwIIL,(a,oo) (4.63)

and

IIlIIL ,(-oo,oo) <- z IIfwIIL,(-aa.aa) (4.64)

respectively.


Idea of the proof. Apply Definition 4 and Remark 6 of Chapter I. In order toprove inequality (4.59) apply the integral representation (3.6). Deduce (4.57)and (4.58) from (4.59). Inequality (4.63) follows from (4.62) and implies in-equality (4.64). Apply (4.63) and (4.64) to If Ip and deduce (4.60) and, respec-tively, (4.61). Set f a 1 to prove sharpness of c17 in (4.58) and of the constantmultiplying IIfIIL,(a,b) in (4.57). Set f (x) = (x - a)"' to prove sharpness of theconstant in (4.59). Set f (x) = f, (x) = 1 + e(x - a)"' and pass to the limit ase -a 0+ to prove sharpness of the constant multiplying IIfti,IIL,(a,b) in (4.57).Set f (x) = e-,,(z-a) in (4.60), f (x) = e-µI=I in (4.61) respectively, and choosean appropriate µ to prove sharpness of the constants in those inequalities. 0Proof. 1. Let f E Wp (a, b). By Definition 4 of Chapter 1 there is a functionh equivalent to f on (a, b), which is locally absolutely continuous on (a, b).Moreover, if a > -oo or b < oo, then the limits lim h(x) and lim h(x) exist.

x_+a+ x_+b-If a = -oo or b = oo, then as will be proved in steps 3-4 lim h(x) = 0,

x_+-00lim h(x) = 0 respectively. Hence h is bounded and uniformly continuous on

2_+00

(a, b), i.e., h E V(a, b) for all -oo < a < b < oo.2. By applying Holder's inequality to (3.6) we obtain that for almost every

xE(a,b)

If (X) - b a f f(y)dyI0

< (b - a)-' (f (y - a)"' dy + (b - y)'' dy)' IIf,,IIL,(a,b)

n

_ (p' + 1)-a (b - a)-'[(x - a)"'+' + (b - (4.65)

Since max [(x - a)"'+' + (b - x)"'+'] (b - a)'+a we have established (4.59).a<x<b

Inequality (4.57) and, hence, (4.58) follow sinceb b

If (x)I5Ibla f f(y)dyl+If(x)-blaf f(y)dyla a

b

< (b - a) = IIf IIL,(a,b) + I1(x) -b

1 a f f (y) dyl.a

3. By letting b -+ +oo in (4.62) we obtain (4.63). Moreover, Vx E (a, oo)

Ih(x)I <- IIhItC(x,oo) = IIfIIL (x,.) <- ZOO If,. (y)Idyx


and it follows that lim h(x) = 0.x-+oo

4. If f E W1(-oo, +oo), then

lh(x)I <- f Ifaldy-00

and lim h(x) = 0 as well. Adding the last inequality and the previous one,W-00

we getIh(x)I IIfwIIL,(-oo,oo)

and (4.64) follows since IIhIIc(-oo,oo)5. If p > 1, then by (4.63) and Holder's inequality

II IJ II(IfIP)wlli,(a,oo) =PPII IfIP-1fwlli,(a,oo)

s PpIl/IILp(a,oo)IIfwIIL,(a,oo)*

(4.66)

We establish (4.60) by applying inequality (4.47). Inequality (4.61) is provedin a similar way.

6. Setting f = 1 we obtain that the constant (b-a)-P multiplying IIf IIL,(a,b)

in (4.57) and the constant c1. in (4.58), if b - a < (p' + 1)', cannot be dimin-ished. If f (x) = (x - a)P', then one can easily verify that there is equality in(4.59).

Now let us consider the inequality

IllIILo(a,b) 5 (b - a) o IIf IIL,(a,b) + AIIf,IIL,(a,b)

We prove that A >(p'+1)-P(b-

a) ', which means that the constant multiply-ing IIf,IlL,(a,b) in (4.57) is sharp. Indeed, set f (x) = f,(x) = 1+e(x-a)"', where

e > 0, then IIfAIIL (a,b) = l+e(b-a)P',II(fe)'IIL,(a,b) =ep'(p'+1)-P(b-a)"'-and II fc 1I (b-a) P (1+e (p'+1) -'(b- a)P' +o(e)) as a -+ 0+. Consequently,

A > lim llfsl$L (a,b) - (b - a) PIIfaIIL,(a,b) + 1)"a (b - a)a.-o+ II(fe)'IIL,(a,b)

Finally, for f (x) = e-µ( 2-a) inequality (4.60) with 1 < p < oo is equivalent tothe inequality 1 < (p')-gyp-o (,u P + pt ). For p = P11 the quantity p o + µ

is minimal and this inequality becomes an equality. Hence, for f (x) = e-


there is equality in (4.60). Analogously for f (x) =e-Pu there is equality in

(4.61).If p = 1, then equality in (4.60) and (4.61) holds if, and only if, f is equiv-

alent to 0. This follows from inequalities (4.63), (4.64) respectively. However,the constants 1 in (4.60) and z in (4.61) are sharp, which easily follows bysetting f (x) = e-a(x-a), f (x) = e-lIx[ respectively, and passing to the limit asµ -4 +00. 0

Remark 13 We note that for a function It, which is equivalent to f on (a. b)and which is absolutely continuous on [a, b], inequality (4.63) may be rewrittenas

max Ih(x)I < Var h.a<x<oo [a,oo)

The maximum exists since h(x) -r 0 as x -- +oo. The inequality is clear sincefor each function h of bounded variation Vx E [a, oo) we have

Ih(x)I = limy Ih(x) - h(y)I < Var It.Y [a.00)

It is also clear that for f E Wp (a, oo) equality is achieved in (4.63) if,and only if, f is equivalent to a nonnegative and nonincreasing function or anonpositive and nondecreasing one on [a, oo). Similarly for f E Wp (-oc, oo)equality is achieved in (4.64) if, and only if, f is equivalent to a function,which is nonnegative, nondecreasing on (-oo, xo) and nonincreasing on [xo, oo)for some xo or nonpositive, nonincreasing on (-oo, xo) and nondecreasing on[xo, oo).

Remark 14 Analysis of the cases, in which there is equality in Holder's in-equality, 11 suggests the choice of test-functions, which allows one to state thesharpness of the constants. In the case of inequality (4.59) we take x = b in(4.65). If (f')" = Ml(x - a)9' on (a, b) for some M1 > 0, and, in particular,f (x) = (x - a) e, then there is equality in inequality (4.65) and, consequently,in (4.59). Let f > 0 and f < 0 on (a, oo) in the case of inequality (4.60). Thenby Remark 13 there is equality in the first inequality (4.66). Furthermore, if(- f')P = M2(f

n_1)p'on (a, oo), M2 > 0, then there is equality in the second

ii Let f and g be measurable on (a, b) and 1 < p < oo. The equality

IIf9IIL,(a,b) = IIfIIL,(a,b)II9IIL,,(a,b)

holds if, and only if, Alf I° = B191° almost everywhere on (a, b) for some nonnegative A andB.


inequality (4.66). All solutions f E Lp(a, oo) of this equation have the fromf (x) = e-"(x-a) for some a > 0.

A more sophisticated argument of similar type explains the choice of test-functions f (x) = 1 + E(y - a)" in the case of inequality (4.57).

Corollary 10 (inequalities with a small parameter multiplying the norm of aderivative) Let -oo < a < b < oo,1 < p < oo.

1) If -oo < a < b < coo, then Ve E (0, Eo), where Eo = ((p' + 1) (b - a)) ',

If IIL (a,b) 5 W+ 1) °E IIfIIL°(n,b) +EIIfwIIL°(a,b)

2) If -oo < a < b = oo, then VE E (0, oo)

If (a, b) _ (-oo, oo), then Ve E (0, oo)

(4.67)

(4.68)

(p')-'(2E)-

IIfIIL°(-aa,aa)+6IIfwIIL°(-aa,aa) (4.69)

The constants in inequalities (4.68) and (4.69) are sharp.

Idea of the proof. Apply the proofs of inequalities (4.40) and (4.42). Verify

that there is equality in (4.68) for f (x) =exp(-`-dr

-6) and in (4.69) for

f (x) = exp (- Qa

r II) . See also Remarks 15 -16 and 18 below. C3

Corollary 11 (multiplicative inequalities) Let -oo < a < b:5 oo, 11) If -oo <a<b<oo, then

II/IIL,°(a,b) 5 C38IIfIIL (a,b) IllIIW;(a,b)'

where cls = (b - a) a 1)-' < (b- a)-'p + 2.2) If -oo < a < b = oo, then

IIf II t (a,oo) 11fw11 i°(a,aa)

3) If (a, b) = (-oo, oo), then

5 (Z)° IIfII (-,a,,,) IIfwIIL°(a,b)

The constants in inequalities (4.71) and (4.72) are sharp.

<p<oo.

(4.70)

(4.71)

(4.72)


Idea of the Proof. Inequalities (4.71) and (4.72) have already been establishedin the proof of Theorem 4. If f (x) = exp (-(x - a)) or f (x) = exp (-IxI), theninequalities (4.71) and (4.72) become equalities. Apply the proof of inequality(4.41) to prove (4.70). See also Remarks 15-16 and 18 below. 0

Remark 15 We note that for 1 < p < oo the additive inequality (4.60), theinequality with a parameter (4.61) and the multiplicative inequality (4.71) areequivalent. Indeed, (4.68) was derived from (4.60) with the help of dilationsand (4.60) was derived from (4.71) with the help of inequality (4.47). Finally(4.68) implies (4.71) by minimizing the right-hand side of (4.68) with respectto a parameter.

These inequalities are also equivalent to the following ones:

00

IIfIILW(a.aa) < (P- 1) 7(f(IfIP+ Ifv,I°)dx)= (4.73)

a

and, Ve > 0,

I fI dx+e I IfwIpdx)P. (4.74)a a

For, (4.73) follows from inequality (4.71) raised to the power p and (4.47),(4.74) follows from (4.73) with the help of dilations and (4.71) follows from(4.74) by minimizing its right-hand side.

For the same reasons inequalities (4.61), (4.69), (4.72) and the inequalities

00

2- i (p-1)--' ( f (IfIPdx+If'IP) (4.75)

-00

and00

1

F-IT+e f If1IPdx)° (4.76)

-CO

with an arbitrary e > 0, are equivalent as well. Equalities in (4.73) - (4.74),(4.75) - (4.76) respectively, hold for f (x) = exp (-µ(x-a)), f (x) = exp (-pIxI)respectively, with appropriate choice of A. For example, in the case of inequality(4.75) p = (p - 1)=.

Moreover, the listed inequalities for the halfiine and for the whole line arealso equivalent. This follows from the equivalence of (4.73) and (4.75). For, if


(4.73) holds, then, replacing x by 2a - x, we have also that

a

IIfIILoo(-oo,a) < (p- 1)7 (f (Iflp+ If' I°) )dx)o.-00

Consequently,

00

z (IIfIIL (-oo,a) + IIfIIL (a,0a)) _(y

2 f (Iflp+ Ifwlp)_00

and (4.75) follows. Conversely, if (4.75) holds and f E Wp (a, oo) we apply(4.75) to the even extension F of f: F(x) = f(x), if x > a, and F(x) _f (2a - x), if x < a. Then

00

P

IIfIILo (0,00) = IIFIIL-(-00,00) <- 2(p- 1);- (f (IFI°+IF' IP)dx) I-00

00

=(p-1)T (f (Ifly+IfwI")dx)°a

Remark 16 For p = 2 all the inequalities discussed in Remark 15 may bededuced by taking Fourier transforms since by Parseval's equality

IIf II F-'Ff II L,,(-,,..)

00

= 2 II f eu{(Ff)( ) d-00

II(1+t2)-#(1

+t2)j(Ff)(t) IIL,(-00,00)27r

IIFfIIL,(-00,00) =r2-,,

00 00

1727 (f (1+e2)-'4)}'(f-00 -00

00 00

_ f (IFfI2+IFffI2)dS)3 = f (If l2+Ifwl2)dx)'-00 -00

Thus we obtain (4.75) for p = 2.


There is an alternative way of using Fourier transforms, which leads to twoother inequalities 12

001

If ±f ,I2dx)5,-00

which hold for each 13 f E W2 (-oo, oo). The constant 7 is sharp. Indeed,by the properties of Fourier transforms and by the Cauchy-Schwartz inequalitywe have

,Of IILm(-oo,ao) =IIF-1

1

F(f ± ft)(11

^t ) IILm(-oo,00)

27IIF-1(1 t g) * (f f

-00

27r

IIF-1(1

fg)IIL,(-oo,ao)

00

F' 1

(x - y) (f (y) f fW(U)) d1/27r II ( (1 f

II 2(-oo,oo)

27r II 1 ± t IIL,(-oo,oo)Of ± fwllLz(-oo,ao)

and the desired inequality follows. If f (x) = e-1=t, then all three inequalitiesunder consideration become equalities.

The second approach is applicable to the case 1 < p:5 oo as well and leadsto the inequalities

/ ,,00 a

If ±fwlpdx)v-00

for each f E WP (-oo, oo) since, for example,

IIF(-1) (1 I 9) IILd(-oo,oo) = 27rIIexIIL,,(o,oo) =

If 1 0 and f (x) = e9 T for x < 0, then theseinequalities become equalities.

12 If we square and add them, we obtain the previous inequality.Is We note that this inequality does not hold for each function f, which is such that the

right-hand side is finite. (It does not hold, say, for f (x) = e*=.)


R.emark 17 We note two corollaries of (4.65) under the supposition that f isabsolutely continuous on [a, b]:

If(a)I S (b - a)° IIfIIL,(e,b)+(p +1)-o(b-a)tIIf'IIL,(a,b) (4.77)

(for p = 1 (4.77) coincides with (3.8)) and

If(a

2 b) I(b- a) 'IIfIIL,(a,b)+2 (p'+1)-o(b-a) IIf'IIL,(a,b). (4.78)

Both constants in (4.77) are the same as in (4.57) and are sharp. This is provedby using the same test-functions as in the case of inequality (4.57).

In inequality (4.78) both constants are also sharp. The test-function f =_ 1shows that the constant multiplying If IIL,(a,b) is sharp. Moreover, the test-functions f = fE, where e > 0 and fE is defined by ff(x) = 1 + e(x - a)", ifa < x < a2, and fE(x) = 1 +e(b- a)"', if azb < x < b, show by passing to thelimit as a -4 0+ that the constant multiplying IIf'IIL,(a.b) is sharp.

Corollary 12 Let I E N,1 0 is independent of f, i.e., W, (a, b) C; V'4 (a, b).If a = -oo, then lim h(m)(x) = 0, if b = oo, then lim h(m)(x) = 0, where

x-r-oo x-ioom=0,...,1-1.

Idea of the proof. Apply Remark 6 of Section 1.3 and Theorems 4 and 2.

Theorem 5 Let lEN,mENo,m<1,1<p,q<oo and -oo< a <b<00.Then the embedding

W, (a, b) C; WQ (a, b) (4.80)

holds if, and only if, b - a < oo, or b - a = oo and p < q. Moreover,this embedding is compact if, and only if, b - a < oo and the equalities m =I - 1, p = 1 and q = oo are not satisfied simultaneously.

Remark 18 As in the simplest case discussed in Corollary 7, from the inequal-ity, accompanying embedding (4.80),

Ilfy,m)IILa(a,b) < M IIf IIWW(a,b), (4.81)


where M > 0 is independent of f, it follows that, for b - a < oo,

Ilf,m)IIL,(a,b) 5 c20 (b -a) 9 0 ((b - a)--Ilf IIL,(a,b) + (b -(4.82)

where c20 > 0 is independent of f, a and b.If q > p, then, excluding the case in which m = l - 1, p = 1 and q = oo, it

also follows that

IIL,(a,b) <- C21 E I1f IIL,(a,b) + EIIf,S,) IIL,(a,b) , (4.83)

where 0 < s < c22(b -a)1_m_ o+Q and

c21, c22 > 0 are independent of f, a andb, and (I-1- ykm-l

(Q)+, V)

11fIIWp(a,b)+°',(4.84)11fwm)IIL,(a,b) <- C2311fII1,

where c23 > 0 is independent of f . Moreover, inequalities (4.81) - (4.84) areequivalent.

The proof is similar to the proof of Corollary 7. One should notice, inaddition, that since q > p, by Jensen's inequality

II911L,(ak ak+,))V

< ( II9IIL,(ak,ak+.))a

= II9IILp(a.b).k=1 k=1

If b - a = oo, then inequality (4.83) holds Ve > 0 and in inequality (4.84)IIfIIW,(a,b) can be replaced by Ilfw')IIL,(a,b)

Idea of the proof. To prove (4.81) apply Corollary 12 and Holder's inequality ifb - a < oo and the inequality

IIfIIL,(a,b) s IIfIIL,(a,b)IIf11L1 (a,b),

whereifb-a=ooandp<q. Ifb-a=ooandq<p,set f(x)=(1+x2)-a,or

f(x) _ IkI Qw(x - k), (4.85)

kEZ:(k,k+1)C(a,b)

where W E Ca (R), W 0 0 and supp V C [0,1], to verify that (4.81) does nothold.

To prove the compactness apply Theorem 3 and inequality (4.83) or (4.84).If b - a < oo, m = ! - 1, p = 1 and q = oo, consider the sequence

fk(x)=kl'iq(a26+k(x-a2))' (4.86)


where k E N, q E Co (-oo, oo), supp q C (a, b) and (°-2) = 1. Finally, ifb - a = oo, apply Example 1.Proof. The proof of the statements concerning embedding (4.80) being clear,we pass to the proof of the statements concerning the compactness.

1. Let b - a < oo and fk, k E N, be a sequence bounded in W,(a, b). Thenby Theorem 3 there exists a subsequence ft., s E N, and a function f E L,(a, b)such that fk. -* f in LD(a, b). If I - m - v + o > 0, then from (4.83) or (4.84)it follows that fk, -i f in WW (a, b).

2. Ifb-a<ooandl-m-v+9=0,i.e.,m=l-1,p=Iandq=oo,then for the functions ft defined by (4.86) we have

IIfkIIW;(a,b) = k-' II 7 L,(a,b) + II?7(')IIL,(a,b) - IInIIW;(a,b)

and k m f(k'-')(x) = h(x), where h(0) = 1 and h(x) = 0 for x 96 0. Con-

sequently, the sequence fit, k E N, is bounded in Wi (a, b), but none of itssubsequences fk., s E N, converges in Laa(a, b). Otherwise, for some subse-quence fk., lim 11A. - fkeIICIa,b) = m 11A. - fk,IIL (.,b) = 0. Hence, fk.a,o-wo ao

h-00

convergers uniformly on [a, 6) to h, which contradicts the discontinuity of thefunction h.

3. If b - a = oo, then Example 1 shows that embedding (4.80) is notcompact for any admissible values of the parameters.

4.3 Open sets with quasi-resolvable, quasi-continuous, smooth and Lipschitz bound-aries

We say that a domain ) C R" is a bounded elementary domain with a resolvedboundary with the parameters d, D, satisfying 0 < d:5 D < oo, if

11={xER":an<xn<tp(x), 2EW}, (4.87)

where14 diam H < D, 2 = (xi, ..., xn_i), W = {2 E R"-1 : ai < xi < b;, i =1, ...n - 1}, - oo < aj < b, < oo, and

an+d<<p(2), 2EW. (4.88)

1a Since fl is a domain, hence measurable, by Fubini's theorem the function v is measurableon W and mess fl = f d2.

w

4.3. CLASSES OF OPEN SETS 149

If, in addition, w E C(W) or w E C'(W) for some l E N and IID°wllc(w) 5M if 1 < lal < I where 0 < M < oo or w satisfies the Lipschitz conditon

I w(i) - w(y)I 5 M la - yl, x, y E W, (4.89)

then we say that Sl is a bounded elementary domain with a continuous boundarywith the parameters d, D, with a C'-boundary with the parameters d, D, M,or with a Lipschitz boundary with the parameters d, D, M respectively.

Moreover, we say that an open set Sl C R" has a resolved boundary withthe parameters d, 0 < d < coo, D, 0 < D < oo and x E N if there existopen parallelepipeds Vj, j = !-,s, where s E N for bounded Sl and s = oo forunbounded Sl such that

1) (Vj)d n Sl i4 0 and diamV, < D,

2) Sl C U (V )d,j=1

3) the multiplicity of the covering {Vj}'=1 does not exceed x,4) there exist maps )1j, j = 1, s, which are compositions of rotations, reflec-

tions and translations and are such that

Aj(Vj)={xER": aij<xi<bij,i=1,..... n}and

Aj(f2 n vj) = {x E R: a"j < x" < cpj(-*), 2 E Wj}, (4.90)

where 2 _ (x1, ..., x"-0, Wj = {i E R"-1 aij < xi < bij, i = 1, ..., n - 11,and

a,,,+d<cpj(±)<b"j-d, XEWj, (4.91)

if vi n on 0. If Vj C ft, then wj(:i) _- b"j. (The left inequality (4.91) issatisfied autimatically since by 1) a"j > 2 d.)

W e note that Aj(SlnVj) and, if Vjnofl 34 0, also ) ((`N)nVj) are boundedelementary domains with a resolved /boundary with the parameters d, D, whereA (x) = (41(x), ..., Am-1W , -,\j,.W )-

Since by 1) by -ai,j > 2d, i = I,-, n-1, by 4) it follows that meas (Stnvj) >d", j = 1, s. So by 3), for unbounded 0, mess Sl = oo, because by (2.60)0E meas (Stn vj) < x mess Q.=1

If an open set Sl C R" has a resolved boundary with the parameters d, D, xand, in addition, for some I E N all functions cpj E C'(Wj) and IID°'pjilc(wt) <_M if 1 < Ial < I where 0 < M < oo and is independent of j or all functions wjsatisfy the Lipschitz condition

Iwj(x)-wj(v)I :5 M12-9I, 1,9 EWj, (4.92)


where M is independent of x, 9 and j, then we say that Q has a C'-boundary(briefly O E C1) with the parameters d, D, x, M, or a Lipschitz boundary(briefly 8S2 E Lipl) with the parameters d, D, x, M respectively.

If all functions wj are continuous on W we say that 11 has a continuousboundary with the parameters d, D, x.

Furthermore, an open set S1 C R" has a quasi-resolved (quasi-continuous)a

boundary with the parameters d, D, x if 11 = U Ilk, where s E N or s = oo,k=1

and 51k, k = 1-.s , are open sets, which have a resolved (continuous) boundarywith the parameters d, D, x, and the multiplicity of the covering {Qk}k=1 doesnot exceed x. (We note that if 0 is bounded, then s E N.)

Finally, we say that an open set S2 C R" has a resolved (quasi-resolved, con-tinuous, quasi-continuous) boundary if for some d, D, x, satisfying 0 < d <D < oo and x E N, it has a resolved (quasi-resolved, continuous, quasi-continuous) boundary with the parameters d, D, x). Respectively an openset Sl C W has a C'- (Lipschitz) boundary if for some d, D, x, M, satisfy-ing 0 < d < D < oo, x E N and 0 < M < oo, it has a Cl- (Lipschitz) boundarywith the parameters d, D, x, M.

Example 2 Suppose that S1 = 1(X1, X2) E R2 : -1 < x2 < 1 if -1 < x1 <0, -1 < x2 < xi if 0 < x1 < 1} where 0 < y < 1. Then 11 is a boundedelementary domain with a resolved boundary, which is not a quasi-continuousboundary.

Example 3 Let 11 = {(x1i x2) E R2 : 0 < x1 < 1, xj < x2 < 2 xi } where0 < y < oo, -y 34 1. Then 811 is not a quasi-resolved boundary while INsatisfies the cone condition.

Example 4 For the elementary domain 11 defined by (4.87) the Lipschitz con-dition (4.89) means geometrically that Vx E 811 the cones

Ki = {y E R" : yn < W(2)-MIa-yI}, Ky = {y E R" : V(2)+MI2-91 < yn}

are such thatKz n W C 1, K= n W C `S1, (4.93)

where W=Ix ER":2EW,a"<xn<oo}.For, if (4.89) is satisfied and y E K= nW, then y" < W(2) - MI2 - y'j < V(9)

and y E Q. Similarly, K; n W C °N. Suppose that (4.93) is satisfied. Since(y,,p(y)) V f1 the inclusion K; n W C 0 implies that w(9) > W(2) - MI! - 91.


(For, if p(y') < gyp(.t) - Ml - 91, then (9, sp(y)) E 0.) Similarly, Kz n W C` Qimplies that cp(y) < ap(t) + MI± - 91 and (4.89) follows.

We note also that the tangent of the angle at the common vertex of bothcones K= and K= is equal to may.

Example 5 Let 52 = {(x1,x2) E R2 : x2 < cp(xl)}, where tp(x1) _ Ix, 17

if x1 < 0, cp(x1) = xi if x1 > 0 and 'y > 0. Then the function cp satisfies aLipschitz condition on R if, and only if, y < 1, while 0 has a Lipschitz boundaryin the sense of the above definition for each y > 0.

Example 6 Let y > 0. Both the domain 521 = { x E R" : 1xI" < x,, < 1, Ixl <1 } and the domain 522 = { x E R" : -1 < x,, < 1x1 < 1 } have a Lipschitzboundary if, and only if, y > 1. (Compare with Examples 6 and 7 of Chapter3.)

Lemma 4 If an open set 52 C R" has a Lipschitz boundary with the param-eters d, D, x and M, then both 52 and `S2 satisfy the cone condition with theparameters r, h depending only on d, Al and n.

Idea of the proof. Let z E Vj n 852 and x = ad(z). Consider the cones K= andK= defined in Example 4, where 1P is replaced by tp,.Proof. By Example 4 we have

K= n A, (V;) c.\;(V; n 52), K= n a; (V,) c A, (V; n' 52).

By 1) bid - aid > 2d and (4.90) implies that there exist r, h > 0 depending onlyon d, M and n such that ) (V, n c) and \j(Vj n` S2) satisfy the cone conditionwith the parameters r and h. (The cone condition is satisfied for the largestcone with vertex the origin, which is contained in the intersection of the coneK(d,

M)defined by (3.34) and the infinite rectangular block x1, ..., x"_1 > 0).

Since aj is a composition of rotations, reflections and translations, the setsVj n 52, Vj n` S2 and, hence, the sets 52 and `S2 also satisfy the cone conditionwith the parameters r and h.

Example 7 Let 52 = Q1 U Q2, where Qt and Q2 are open cubes such that theintersection iU1 n Z72 consists of just one point. Then both i and `S2 satisfythe cone condition, but the boundary of S2 is not Lipschitz. (It is not evenresolvable.)

Lemma 5 A bounded domain i C R" star-shaped with respect to the ballB C Cl has a Lipschitz boundary with the parameters depending only ondiam B, diam 52 and n.


Idea of the proof. Apply the proof of Lemma 2 of Chapter 3 and Example 1. 0Proof. Let SI be star-shaped with respect to the ball B(xo, r) and z E BSI. Weconsider the conic body V. = U (y, z) and the supplementary infinite cone

yEB(xo,r)

V= _ U (y, z) , where (y, z)" _ {z + Q(z - y) : 0< @< oo} is an open rayyE B(xo,r)

that goes from the point z in the direction of the vector yd. Then V2 CSI andby the proof of Lemma 2 of Chapter 3 V2 C` N.

Without loss of generality we assume that the vector x is parallel to theaxis Ox, hence, z = (xo, zn), where to = (xo,1, ...) xo,n-1) and z,+ > xo,,,, andconsider the parallelepiped U2 = {y E R" : xo,n < y,, < 2zn - xo,n, y E U,},where U= = {y E Rn-1

: jy; - xo,;j < Z, i = 1, ..., n - 1}. Then Vp E U, the raythat goes from the point (y, xo,n) in the direction of the vector x intersectsthe boundary 8) at a single15 point, which we denote by y = (y, p(p)). Inparticular, p(I) = zn.

Since the tangent of the angle at the common vertex of Vs and V2 is greaterthan or equal to where R2 = max Ixo - yl, it follows (see Example 1) that

VEOn

y E U=. We note that if 9 E U,, then the conic bodyV, contains the cone Ky with the point y as a vertex, whose axis is parallel toOx,, and which is congruent to the cone defined by (3.34) with the parameters

xo,n. Moreover, V. contains the supplementary infinite cone Ky. Thetangent of the angle at the common vertex of these cones is equal to

r2 iP OT x0.n 2(p(s)-xo,n+ ? Ir-flU 4R,

Consequently (see Example 1),

1'P (.t) - V(9) x - U.4

Moreover, since V, C a and Vs C` SI, we have x0,, + z < V(2) < 2zn - xo,n -z 2 E Us . We note also that

B(z, E) C Us C B(z, (Rl + (n - 1)2(x)2){). (4.94)

Finally, we consider a minimal covering of R" by open balls of radius e. (Itsmultiplicity is less than or equal to 2".) Denote by B1, ..., B, a collection ofthose of them, which covers the 6-neigbourhood of the boundary BSI. Each of

'5 Suppose that q E 8fl, rr # y and Q = P. If nn > yn, then y E V,r C fl. If nn < yn, theny E Vs C° 3f. In both cases we arrive at a contadiction since y E M.


these balls is contained in a ball of the radius z centered at a point of 852. SinceVz E 8Sl we have U= D B(z, a), we can choose Us ..., Us,, where zk E 8i2 insuch a way that UEk D Bk. Consequently, the parallelepipeds UZ, , ..., U, coverthe B-neigbourhoods of 852. From (4.94) it follows that the multiplicity of this

i ncovering does not exceed x = 2' (1 + r + (n - 1)2

(2)2)

) . (See footnote

15 of Chapter 3.)Thus, 0 has a Lipschitz boundary with the parameters d = e, D =

diam il, M = 4f < 4D and x.

Lemma 6 1. A bounded open set it C R" satisfies the cone condition if, andonly if, there exist s E N and elementary bounded domains 12k, k = 1, ..., s, with

Lipschitz boundaries with the same parameters such that S2 = U Stk.k=1

2. An unbounded open set 52 C R" satisfies the cone condition if, and only if,there exist elementary bounded domains ilk, k E N, with Lipschitz boundarieswith the same parameters such that

00

1) iz = U Qk,k=1

and2) the multiplicity of the covering x({ilk}k 1) is finite.

Idea of the proof. To prove the necessity combine Lemma 4 of Chapter 3 andLemma 5. Note that if the boundaries of the elementary domains ilk, k = !-,sare Lipschitz with the parameters dk, Dk and Mk then they are Lipschitz withthe parameters d = inf dk, D = sup Dk, M = sup Mk as well if d > 0, D < 00

k=1s k=1,-s k=Tiand M < oo. To prove the sufficiency apply Lemma 4 and Example 5 of Chapter3.0

Remark 19 If in Lemma 6 ilk are elementary bounded domains with Lips-chitz boundaries with the same parameters d, D, M, then fl satisfies the conecondition with the parameters r, h depending only on d and M.

Remark 20 If we introduce the notion of an open set with a quasi-Lipschitzboundary in the same manner as in the case of a quasi-continuous boundary,then by Lemma 6 this notion coincides with the notion of an open set satisfyingthe cone condition. If we define an open set satisfying the quasi-cone condition,then this notion again coincides with the notion of an open set satisfying thecone condition.


Lemma 4 of Chapter 3 and Lemma 6 allow us to reduce the proofs ofembedding theorems for open sets satisfying the cone condition to the case ofbounded domains star-shaped with respect to a ball or to the case of elementarybounded domains having Lipschitz boundaries. To do this we need the followinglemmas about addition of inequalities for the norms of functions.

Lemma 7 Let mo E N, 1 < p1, ...pm, q < oo and let f? = 6 Slk, where Slk Ck=1

R" are measurable sets, s E N for q < max pm ands E N or s = com=1,...,mo

otherwise. Moreover, ifs = oo and q < oo, suppose that the multiplicity of thecovering x e x({Qk }k=1) is finite. Furthermore, let fm, m = 1, ..., mo, and gbe functions measurable on Q.

Suppose that for some am > 0, m = 1, ..., mo, for each k

mo

II9IIL,(nk) <_ E am llfmIILp,,,(nk) (4.95)m=1

Thenmo

IIgIIL,(n) < AaM=1

when A = s if q < max pm and A = x otherwise.m=1,...,mo

(4.96)

Idea of the proof. If pi = ... = pm = q = 1 add inequalities (4.95) and applyinequality (2.59). In the general case apply Minkowski's or Holder's inequalitiesfor sums (for q > pm, for q < pm respectively) and inequality (2.59). 0Proof. Let q < oo. 16 By (4.95) and Minkowski's inequality it follows, that

a 1 a mpV

IIgIIL,(n) < ( II9II9Lq(nk))a

-`k=1 k=1 m=1

S fmllLp(f1k))Q) a = Qm ( IIfmIILp,,,(nk))m=1 k=1 m=1 k=1

le The case q = oo is trivial and the statement holds for f2 = U f4, where I is an arbitraryif f

set of indices:Me

11911L- (n) = SUP F, QmhIfmHIL,.,(n).M=1


If q > p,,,, denote by Xk the characteristic function of Stk. Since E Xk(x) < x,k=1

by Minkowski's inequality we have

s i a

IIfmIILvm(flk)) Xk=1 k=1

_ ((u(fk=1Ifm(x)I°mXk(x) dx)

) q) Pm

(J ( I fm(x) I Q7Ck(x) 1 ' dx) n

k=1

_ (f Ifm(x)IP- (:L Xk(x)) " dx) "- < x4IIfmIILvm(n).0 k=1

If q < p,,, < oc, then by Holder's inequality with the exponent > 1 and(2.59) ( sa am (E IIfmIILvm(nk))

oM

k=1 k=1

<s xD IIfIIL,,,,(n) <_ 3aIIfIIL,,(n)

and inequality (4.96) follows.The case in which some pm = oo is treated in a similar way with suprema

replacing sums. 0

Corollary 13 Let I E N, Q E 1 satisfy 101 < l,1 < po, p, q < oo, f = U S2k,k=1

where11kCW' are open sets,sENifq<poorq<pandsENors=coifq > po, p. Moreover, ifs = oo and q < oo, suppose that the multiplicity of thecovering x = is finite.

Suppose that f E Lpo(Il) f1 w,(11), c25, c26 > 0 and

II Dwf II Lq(nk) s C25IIIIIL,o(nk) + ccIIf Iiwi(nk), k = 7s- (4.97)

Then

IIDwOfIILq(n) <_ Aa (c IfIIL,a(n)+02611f11wp(n)). (4.98)

Idea of the proof. Direct application of Lemma 7. O


Lemma8 Let lEN,m ENo,m<l,1 <p,q<oo,fl = Uf1k, wheresENk=1

and Slk C R" are open sets such that

Wp(11k) CZ; Wq (f1k), k = 1, ..., s. (4.99)

Then

W, (1l) C.; WQ (Sl). (4.100)

Moreover, if embeddings (4.99) are compact, then embedding (4.100) is alsocompact.

Idea of the proof. Apply Theorem 1 and Lemma 7 to prove embedding (4.100).To prove its compactness consider a sequence of functions bounded in Wn(Sl)and, applying successively the compactness of embeddings (4.99), get a subse-quence convergent in W91401).Proof. 1. By Theorem 1 (4.99) is equivalent to the inequality

Ill IIW (n4) <- Mk IlfIIW (n,k),

where k = 1, ..., s and Mk are independent of f. By Lemma 7 it follows that

1If11w, (n) < Mo kmax.Mk IIfIIW;(n),

where Mo depends only on n, m, p, q, and (4.100) follows.2. Let M > 0 and II f fl w (n) < M for each i E N. Then , in particular,

11f II wP(n1) < M. Consequently, there exist a function gl E W. 'n(111) and asubsequence f,w -+ 91 in W9 n(111) as j - oo. Furthermore, lI IIw,(n,) 5 Mand, hence, there exist g2 E W9 (S12) and a subsequence f,(,) of f;(1) such

j ,that f1(2) -i g2 in WQ (S)2). Moreover, f(2) -+ g1 in W."'(111). Repeating

this procedure s - 2 times, we get functions gk E W9 (Slk), k = 1, ..., s and asubsequence f,, such that fj -a gk in WQ (Slk) as j -4 oo. We note that gk isequivalent to go on Slk f1 Sl,. Hence, there exists a function g, defined on Sl,such that g - gk on Slk, k = 1, ..., s. By the properties of weak derivatives (seeSection 1.2) g E W, 'n(11) and

IIA - 9IIw; (n) < E 11fii - 9kIIW (nr) -+ 0k=1

as jioo.


Lemma 9 Let I E Id, m E No, m < 1 and 1 < p, q < oo. Suppose thatfor each bounded elementary domain G C Rn with a resolved (continuous)boundary there exists c26 > 0 such that for each Q E R satisfying IQI = m andVf E WW(G)

IID!fIIL,(G) :5 C26IIf1IWD(c) (4.101)

Then for each bounded open set 11 C Rn having a quasi-resolved(respectively, quasi-continuous) boundary there exists c27 > 0 such that

II D,'f IIL,(n) <- C27 Ill IIw;(n) (4.102)

for each 13 E I satisfying 1#1 = m and V f E Wp(0).If p:5 q and c26 depends only on n, 1, m, p, q and the parameters d and D

of a bounded elementary domain with a resolved (continuous) boundary, thenfor each unbounded open set Sl C Rn having a quasi-resolved (quasi-continuous)boundary there exists C27 > 0 such that inequality (4.102) holds.

Idea of the proof. Apply (4.101), where G, f are replaced by \,(n fl V,), f, _f (A,) respectively, and the parallelepipeds V, and the maps a, are as in thedefinition of a resolved (continuous) boundary. Change the variables, settingy = and obtain (4.101) where G = SZ f1 V j. Apply Corollary 13 twiceto prove (4.102) succesively for open sets 0 with a resolved (continuous) andquasi-resolved (quasi-continuous) boundary.Proof. First suppose that St has a resolved boundary. We notice that A,(x) =

A,x + b,, A71x - A,1b,, where b, E Rn, A, _ A-1 =ik=1

(bV))n , Ia k)I, Ibb)I < 1 and Idet A)I = Idet A; 1I = 1. Consequently, we haveik=1

(D!f)(x)I = I Dw(f;(at ')(x)))I = I (aa;,)w... (a ,,)w(f,(A(_1)(x)))I

n=I

In!

k,,...,km=1

< I7 y! I (Dwfj)(,\(7') (x))

I < n'"17

I (Dwfj) 1)(x))

Setting y = Af-1)(x) we establish that

IIDwf IIL,(nnvj) <_ n- II D.7fj II L,(a;(nnv,))-I,I=m


Similarly, for a E Nt) satisfying j al = 1,

IIDwfjIIL,(A,(invtn 5 n` IID,fjIIL,(1nvj).Iti1=1

Hence, inequality (4.101) implies that

IIDwfIIL,(S2nv,) <nm+lc2s(Aj(slnV1))IIf1Iwp(nnvt)

If s2 is bounded, then the number of parallelepipeds Vj is finite, say, s.Hence, by Corollary 13,

IIDwfIIL,(n) < nm+l sv max c26(Aj(1l n Vj)) IIf IIw;(a)

Let SZ be unbounded, then the set of parallelepipeds Vj is denumerable.Suppose that n, 1, m, p, q are fixed (p< q). Then in (4.101) c26(G) = c (d, D).Hence Vj E N

IIDwfIIL,(nnvj)<nm+lc (d, D)IIfIIwa(nnv,)

By Corollary 13 it follows that

IIDwf IIL,(n) <_ nm+l x+ c ;6(d, D) IIf Ilwo(n) = c;(d, D, x)II f Ilwo(n).

Thus, (4.102) is proved for an SZ with a resolved boundary. If sZ has a quasi-resolved boundary one needs to apply Corollary 13 once more, in a similar way.The case of 11 having a quasi-continuous boundary is similar. O

Lemma 10 Let I E N, m E 1%, m < 1 and 1 < p, q < oo. Suppose that for eachbounded domain !Q C R" star-shaped with respect to a ball there exists c26 > 0such that Vf E WI,(G) inequality (4.101) holds.

Then for each open set 11 C W' satisfying the cone condition there existsC27 > 0 such that Vf E WW(ft) inequality (4.102) holds.

If p < q and c26 depends only on n, 1, m, p, q and the parameters d and Dof a domain star-shaped with respect to a ball, then for each unbounded open sets2 C R" satisfying the cone condition there exists c2? > 0 such that inequality(4.102) holds.

Idea of the proof. Apply Lemma 4 and, if SZ is unbounded, Remark 7 of Chapter3 and Corollary 13. 0


Proof. Let 1 satisfy the cone condition with the parameters r, h. By Lemma 4

and Remark 7 of Chapter 3, St = U 1k, where s E N for bounded 1, s = 00k=1

for unbounded Sl, and S1k are bounded domains star-shaped with respect to theballs Bk C Wk C Stk. Moreover, 0 < M1 < diam Bk < diam Slk < M2 < oo andx({Slk}k=1) < M3 < oo, where MI, M2 and M3 depend only on n, r and h.

If Sl is bounded, then

IIDwfIILq(nk) <_ G26(Qk) IIfllW;(nk), k = 1,...,s.

Hence, by Corollary 13,

1IID°fIIL,,(n) <_ 89 kmax C26(Qk) IIlIIW"(n)

(4.103)

(4.104)

Suppose that Sl is unbounded. Denote by A(d, D) the set of all domains,whose diameters do not exceed D and which are star-shaped with respectto balls whose diameters are greater than or equal to d and set C& (d, D) =

sup c26(G). Clearly A(d, D) C A(d1, D1) if 0 < d1 < d < D < D1 < oo.GE A(d,D)

Then bk E N

II Dwf IILq(nk) < c26(dk, Dk) IIf II W (nk) <_ c26(Ml, M2) Ill IIw;(nk)

and, by Corollary 13,

IIDWf IIL,(n) <_ M3 c26(Ml, M2) 11f Ilw;(n)

Lemma 11 Let I E N, m E No, m < l and 1 < p, q < oo. Suppose thatfor each bounded elementary domain G C R' with a Lipschitz boundary thereexists c26 > 0 such that for each 0 E No satisfying 1/31 = m and V f E Wp(G)inequality (4.101) holds.

Then for each bounded open set St C R satisfying the cone condition thereexists c27 > 0 such that for each 03 E I satisfying 101 = m and V f E Wp(sl)inequality (4.102) holds.

If p:5 q and c26 depends only on n, 1, m, p, q and the parameters d, D andM of a bounded elementary domain with a Lipschitz boundary, then for eachunbounded open set St C R" satisfying the cone condition there exists c27 > 0such that inequality (4.102) holds.

Idea of the proof. Apply Lemma 6, Remark 19 and the proof of Lemma 9.


Proof. Let Q satisfy the cone condition with the parameters r, h. By Lemmaa

6 and Remark 19, St = U IZk, where s E N for bounded Q and s = ook=1

for unbounded St. Here Qt are bounded elementary domains with Lipschitzboundaries with the same parameters d, D, M depending only on n, r and h.Moreover, x({SZk}k=1) < Mg, where M3 also depends only on n, r and h. If SZis bounded , then as in the proof of Lemma 10 we have inequalities (4.103) and(4.104). Let fl be unbounded. Suppose that n, 1, m, p, q are fixed (p < q).Then in (4.101) c26(G) = c28(d, D, M). Hence, Vk E N

IIDWf IILq(nk) < c26(d, D, M) If II W;(nk)


IID.'f IILq(nk) 5 M c28(d, D, M) IIf Ilwgn) = cze(r, h)11f11W;(n)I

4.4 Estimates for intermediate derivativesTheorem 6 Let I E N, 8 E N satisfy 1(31 < l and let 1:5 p:5 oo.

1. If 11 C R" is an open set having a quasi-resolved boundary, then Vf EWW (n)

IIDwfIIL,(n) 5 cas Ilfllwl(n), (4.105)

where c28 > 0 is independent of f.2. If SZ C R" is a bounded domain having a quasi-resolved boundary and the

ball B C Sl, then df E wP(S2)

IIDmfIIL,(n) <_ 2a (II/IIL,(B)+IlfIlwa(n)), (4.106)

where c29 > 0 is independent of f.3. If 11 C R" is a bounded open set having a quasi-continuous boundary,

then be > 0 there exists cgo(e) > 0 such that df E WW(S1)

(n) <_ cso(e) Ill 11n> +e llfllw;(n) (4.107)

Idea of the proof. Apply successively the one-dimensional Theorem 2 to prove(4.105) and (4.106) for an elementary bounded domain Q with a resolvedboundary. In the general case apply Lemma 9 and the proof of Lemma 7.Deduce inequality (4.107) from Theorem 8 and Lemma 13 below.

4.4. ESTIMATES FOR INTERMEDIATE DERIVATIVES 161

Proof. 1. Suppose that S2 is a bounded elementary domain (4.87) with theparameters d, D. By inequality (4.35) it follows that VQ E N satisfying IQI < 1and V2 E W

MI \ De, x, ' s ) + II \a \D x' II J'll( f)( 8xn w w.f ( )

where Q = (.8 1, 0 < 6 < 2 and MI depends only on 1, p, b and D. (Werecall that +p(2) - an < D.)

By the theorem on the measurability of integrals depending on aparameter1 ' both sides of this inequality are functions measurable on W.Therefore, taking Lp-norms with respect to ± over W and applying Minkowski'sinequality for sums and integrals, we have

IID ,f IIL,(n) <- MI (11 11 (Dwf) (x+ xn) II L,,,n -6,w,+ d+6) II L,s(W )

+1I\ \a2n /"I Dwf

< MI (II IIfIIwp(n)).

Let a = (Q, an), where an = an + Z, 8 E W5, and Q = ((i, Q =(QI, ..., Qn-a). We consider the cube Q(a, 6) _ {x E R" : Ixj - ail < 6, j =1, ..., n} and set U = {x = (xi, ..., xn_z) E R"-I : aj < xj < bj, j = 1, ..., n- 2).Applying the same procedure as above, we have

II II(Dwf)(x,xn)IIL,.i(w)IIL,(a + -a a++ +a)

S MI(II it

+Q Ilflu g,' (W) IIL,,=n(?n-6,on+s))

Substituting from this inequalty into the previous one and applying Holder'sinequality, we get

Il D.f II L,(n)

17 We mean the following statement.Let E C R"' and F C R" be measurable sets.Suppose that the function f is measurable on E x F and for almost all y e F the functionf Y) is integrable on E. Then the function f f (x, ) dx is measurable on F.

E


M2(1111 ,,+6)

+IlfIIwi(n)),

where M2 depends only on 1, p, b and D.Repeating the procedure n - 2 times, we establish that

IIDwfIIL,(n) <- M3(IIf0IL1(Q(a,d))+IIfIIi4(n)), (4.108)

where M3 depends only on n, 1, p, 6 and D.2. Taking b = 4 and applying Holder's inequality, we establish that inequal-

ity (4.105) holds for all Q E N satisfying 101 < 1 for each bounded elementarydomain with a resolved boundary and c2s depends only on n, 1, p, d and D.Hence, by Lemma 9, the first statement of Theorem 6 follows.

3. Suppose that B - B(xo, r) C fl, where 12 is an elementary domainconsidered in step 1. Without loss of generality we assume that xo,n > an andset or = (x0,1, ..., xo,n_1 i an), 6 = min{ , f). Then Q(xo, 6) C B(xo, r) and theparallelepiped G = {x E Rn : 1 xj - xoi 1 < 6, an - 6 < xn < xo,n + b} C S2.Applying inequality (4.108) in the case, in which ,0 = 0, p = 1 and SZ, Q(o, 6)are replaced by G, Q(xo, 6) respectively, we get

IllIIL,(c) -< M4 (11f11L,(Qt=o,o)) + Ilf ll.,(c))

5 M5(IIIIIL,(B)+IIf114(c)),

where M4 and M5 are independent of f. Hence, from (4.108) it follows that(4.105) holds for each Q E No' satisfying 1131 < 1.

4. From step 3 and the proofs of Lemmas 7 and 9 it follows that for eachbounded domain 1 having a resolved boundary

s

IIDwflIL,(n) 5 Ms (t 11f11 L,(8;) + I1f111,(n)),j=1

(4.109)

where s E N and Bj are arbitary balls in n fl Vj. (Vj ands are as in thedefinition of SZ having a resolved bondary.)

Let the ball B c fl. We choose m E N and the ball B0 in such a way thatB0 C B fl Cl fl V,n. By step 3 and Holder's inequality it follows that

II!IIL,(Bm) <- M7 (Ilf IIL,(Bo) + Ilf IIW,(nnvm)) <- Ms (11/ IIL,(B) + Ilf 11t;(n))


Let j 96 m. We choose a chain of parallelepipeds U1, ..., U, of the coveringwhich are such that U1 = V3, Uk n Uk+1 54 0, k = 1, ..., or - 1, and

U, = Vm. Next we consider balls Bk c Uk n Uk+1, k 1, and setBo Bj, B, := Bm. Then by step 3

-1.IIf II L,(Bk) < M9 (IIf IIL,(Bk+,) + IIf IIW;(nM/k)) I k = 01 ..., or

Consequently, for each j = 1, ..., s

IIfIIL,(B,) 5 M1o (IIIIIL,(B) + IIlIIw,(n)),

and inequality (4.106) for bounded domains having a resolved boundary followsfrom (4.109). (We note that M6, ..., Mlo are independent of f .)

The argument for a bounded domain Sl having a quasi-resolved boundaryis similar.

5. By Theorem 8 embedding (4.118) is compact. hence, by Lemma 13inequality (4.121) holds where q = p and inequality (4.117) follows.

Next we give some examples showing that assumptions on St in Theorem 6are essential. The first two examples show that for open sets 0, which do nothave a resolved boundary inequality (4.105) does not always hold.

00

Example 8 Let l E N, 0 E K, 0 < 10 1 < 1, 1 < p:5 oo and 0 = U B(xk, rk),k=1

where rk > 0 and B(xk, rk) are disjoint balls. Suppose that rk - 0 as k -+ 00

and rklpi- )° < oo if p < oo. We set f (x) := rk ° (x - xk)" on Qk, k E N.00k=1

Then f E W,(1) but D' Of ¢ Lp(St).

Example 9 Let 1 E N, (01, 02) E No, 01 96 0, /31 + 02 < 1, 1 < p < oo and let0 be the domain considered in Example 3. We set f (x1 i X2) 4' x22 wherea2 0 No. Then, for 1 M3 ( f xi((02-02)n+1) dX1) y,

0


where M1, M2, M3 > 0 are constants. Suppose that 0 < y < 1, 1 < +,132 and

p(1+1.'Y

'Y

)-@L<a2< 02-p(1+1.).Then f EW,(f)butD0f Lp(S2).

(The case p = oo is similar: if l - 7 < a2 < 132i then f E W,',(S2) butD# f 0 L,o(Sl).) If 7 > 1 an analogous counter-example may be constructed bysetting P X1, x2) := xl' 429' where a1 0 No.

Example 10 For any open set n C R", which has infinite measure or is dis-connected, inequality (4.106) does not hold. In the first case we arrive at acontradiction by setting f(x) = xO. In the second case let G be any con-nected component of S2 containing the ball B. Inequality (4.106) does not holdif f (x) = 0 on G and f (x) = xs on S2 \ G. We note that if 92 has a re-solved boundary and is unbounded, then mess 0 = oo because in this case

s = oo, meas (S2 fl Vj) > b", j E N and F, mess (1 fl V;) < x meal 0.001=1

The last example shows that for bounded open sets having a quasi-resolvedboundary inequality (4.107) does not necessarily hold.

00Example 11 Let 1 < p:5 oo, w = U (2-(2a+1), 2-2s) and fl = { (xl, x2) E R2 :s=0

xiEwif05X2<1and0<xl<1if-1<x2< 0). Suppose that Ye >0there exists M1 (e) such that V f E W, (S2)

IIMi(e)IIfIIL,(n) +611fIIw (n)

Let f (XI, x2) = g(xl)h(x2), where g, g', g" E L,(w) and h(x2) = xz if O < x2 <1, h(x2) = 0 if -1 < x2 < 0. Then

119'IIL,(W) IIhIIL,(o,l) <- M1(e) IIgIIL,(.) IIhIILp(o,1)

+e(II9"IIL,(w) IIhIIL,(o,1) + II9'IIL,(W) IIh'IIL,(o,l) + 11911L,(w) IIh"IILp(o,l)).

Choosing sufficiently small e, we establish that there exists M2 > 0 such that

119'I1Lp(u) <- M2 (11911Lp(W) + 119"IILp(w))

for all the functions g. In fact, we have arrived at a contradiction. To verifythis we set gk(xl) = xl - 2-(2k+1) if xl E (2-(2k+1), 2-2k) and gk(x) = 0 for allother xl E w and pass to the limit as k --> oo.


Corollary 14 Let I E N and 1 < p < oo.1. If Sl C R" is an open set having a quasi-resolved boundary, then the

normOf II`w;(n) _ E 11D' f IIL,(n) (4.110)

IaI<1

is equivalent to If IIw;(n)2. If 11 C R" is a bounded domain having a quasi-resolved boundary and

the ball B C St, then the norm

IIf IIL,(B) + IIDwf IIL,(n)lal=

is equivalent to IIf Ilw,,(n)

Idea of the proof. Apply inequality (4.105) and inequality (4.106).

Corollary 15 Let I E N,1 < p < oo and let Q C R' be an open set havinga quasi-resolved boundary. If 0 is bounded, then w,(SZ) = W,(Sl). If Q isunbounded, then the inclusion W, "(Q) C r 4(Sl) is strict.

Idea of the proof. If Sl is bounded, apply inequality (4.106) to each connectedcomponent of 11. If Sl is unbounded, apply Example 10.

Remark 21 Since 9w, (n) # 9wo(n), the semi-norms II II,,,(n) and II ' (Iw;(n), arenot equivalent. (See d'orollary 3 and Remark 9.)

Generalizations of the one-dimensional inequalities (4.39), (4.40) and (4.41)hold under stronger assumptions on fl than in Theorem 6.

Theorem 7 Let 1 E N, 0 E N satisfy 0 < IQI < 1 and let 1 < p, q < oo.1. If Cl C R" is a domain star-shaped with respect to the ball B, then

dfEW(C)

IIDwfIIL,(i) <c31((D)+-'D-lyl IIfIIL,(B) +(D)'-'D,-1,91 IIfIIw;(n) )(messSl) a d (meal B, d (mess Sl)

(4.112)where d = diam B, D = diam Sl and c31 > 0 depends only on n and 1.

2. If Cl C R" is an open set satisfying the cone condition and co > 0, thenV f E WI (fl) and de E (0, co]

IIDmf IIL,(n) _< c32e-T

IIf IIL,(n) + 6 IIf Iiw,(n), (4.113)


where c32 > 0 is independent off and e.Moreover, Vf E Wp(S2)

101

IIDwfIILp(n) 5 C33IIfIIL,(n) IIfIIW(n),

where c33 > 0 is independent of f.

(4.114)

Idea of the proof. Starting with inequality (3.57) apply Young's inequality forconvolutions 's to prove (4.112). If Sl satisfies the cone condition with theparameters r, h > 0 apply, in addition, Lemma 3 and Remark 7 of Chapter3 and Corollary 13. Replacing r and h by rb and hb, where 0 < 6 < 1,deduce (4.113). Verify that (4.113) implies (4.114) as in the one-dimensionalcase considered in Section 4.2.Proof. 1. Let 0 be a domain star-shaped with respect to the ball B C Q. ByCorollary 10 of Chapter 3 and inequality (3.57), in particular by (4.115), wehave

IIDwfIIL,(n) 5 M1((messSl)D(d)`-'D-161d-n

f IfIdya

+(D)n-1 E II fI(Dwf)(y)I dyII)

I2 - yIn-1+161 L,(st)IaI=( n

< M2 ((meas SZ)ao(Dd)(-1D-161(meas

B)-,fIlIdY

a

+(d)n IIIxI-n+(-161IILt(n-n) IIfIIwp(n)),

's We mean its following variant: if 1 < p < oo, G, fl C IRn are measurable sets, g ELD(G), f E L1(fl - G), where fl - G is the vector difference of fl and G, then

II f f(x-v)9(v)dvll -IIfIIL,(n-c)I1911L,(c)c LAO)

In the sequel we shall also need the general case:

II f f(z - v)9(v)dvll :5 1IfIIL.(n-o)119IIL,(o), (4.115)Q L.(n)

where1 <p,r<q<oo, r =1-p+Q.


where M, and M2 depend only on n and 1. Since 19 S1- 9 C B(0, 2D),

zd

IIIZI-n+1-191IIL1(fl-n) < On f d-191-' do < 21 an D'-191

0

and, by Holder's inequality, (4.112) follows.2. Next let 0 be an open set satisfying the cone condition with the param-

eters r, h > 0. By Lemma 3 and Remark 7 of Chapter 3 0 = U S1k, where eachk

S1k is a domain star-shaped with respect to a ball of radius rl whose diame-ter does not exceed 2h, and the multiplicity of the covering does not exceed6"(1 + p)n. By (4.112) and Holder's inequality it follows that for all k

IIDwfIILp(flk) <- M,((h )'-1+"

IhI-191 IlllILp(flk) +(h) n-1

hl-191 11f IIwD(flk),)7'1 rl

where Ml depends only on n and 1. By Corollary 13

II D1Wf Iltp(n) M2 1 +hl p (( '

1`-l+p IhI-191IIf IILp(n)r Jl rl

+( h ) n-1 h'-191IIflIwo(fl)rl

where M2 depends only on n and 1. We note that 11 satisfies the cone conditionalso with the parameters rb and hb where 0 < 6 < 1 and replace r and h by rband U in this inequality. Setting e = M2(1 + ;)$(r )n-lh'-19161-191, we obtaininequality (4.113) for 0 < e < eo = M2(1 + T) a (

Suppose that e > eo and ea < E < co. Let c32 = Then

IIDwf lILp(n) s X32 (e0)-) IllIILp(fl) +EO IllIlw;(fl),

_< X32 \(E

E*/' E-' If IILp(n) + E 11f 11."(n),Q

and (4.113) again follows.Finally, inequality (4.114) follows from (4.113) in the same way as inequality

(4.41) follows from (4.40).

19 We apply the formula

f 9(IxI) dx = on f 9(e)Qn-1 d9, (4.116)

B(O,r) 0

where on is the surface area of the unit sphere in R.


Corollary 16 Let 1 E N, Q E N satisfy 0 < IQI < 1 and let 1 0andbf EW,(B,)

II Dwf II L,(B,) <_ c31(r-Ial IIf II L,(B.) + ri-ISI IIf II.p(B.)) , (4.117)

where c31 > 0 is independent off and r.

Idea of the proof. Apply (4.112) where B = SZ = B?.

R.emark 22 The statement about equivalence of this inequality, the relevantinequality with a parameter and the multiplicative inequality, analogous to theone-dimensional Corollary 7, also holds.

Remark 23 Let I E N, m E No, m < 1. By Section 4.1 inequality (4.105) forall ,Q E No satisfying 1,31 = m is equivalent to the embedding

WW (1l) Ci W742) . (4.118)

Next we pass to the problem of compactness of this embedding and startby recalling the well-known criterion of the precompactness of a set in Ln(f2),where SZ C R" is a measurable set and 1 < p < oo. We shall write fo for theextension by 0 of the function f to R": fo(x) = f (x) if x E 0 and fo(x) = 0 ifx 0 f2. The set S is precompact if, and only if,

i) S is bounded in L,(f)),ii) S is equicontinuous with respect to translation in Ly(ft), i.e.,

lim sup I)fo(x + h) - f (x) IIL,(n) = 0h-0 fES

andiii) in the case of unbounded fl, in addition,

l iM )ESIIf II L,(n\Br) = 0

Lemma 12 Let I E N0, 1 < p < 00. Moreover, let fl C R" be an open set andS C W,1(0). Suppose that

1) S is bounded in Wa(ft),2) all suupp IIf IIw,,(n\n,) = 0,

3) aim sup IIf (x + h) - f (xIIw:(nihi) = 0fES

and4) in the case of unbounded Cl, in addition, lim sups IIf II wa(n\B.) = 0.

r-+00 fe

Then the set S is precompact in WP(C).



Ilfo(x + h) - f (x)IIL,(sz) <- 2IIf IIL,(B\n,1,,,) + IIf (x + h) - f (x) IIL,(ni,,i) (4.119)

and the closedness of weak differentiation.Proof. Inequality (4.119) clearly follows from the inequality

Ilfo(x + h) - f (x)IIL,(n) <- Ilfo(x + h)IIL,(n\ni,,i)

+IIf(x)IIL,(n\n1,1) + IIf(x+ h) - f(x)IIL,(RIhI)

If I = 0 then condition ii) follows from (4.119) and conditions 2), 3). HenceS is precompact in Lp(fl).

Next let I > 1. From 1) - 4) it follows that the set S and the setsSa = {Dwf, f E S} where a E N and lal = I are precompact in Lp((2).Consequently, each infinite subset of S contains a sequence fk, k E N, suchthat fk - f and Dwf - ga in Lp(1). Since the weak differentiation operatorDw is closed in Lp(SZ) (see Section 1.2), 9a = Dwf on Sl, f E Wp(1) and fk -+ fin Ww(sl).

Theorem 8 Let I E N, m E No, m < l , 1 < p< oo and let 1 C W be abounded open set having a quasi-continuous boundary. Then embedding (4.118)is compact.

Idea of the proof. If S2 is a bounded elementary domain with a continuousboundary, given a set S bounded in WW(SZ), apply Corollary 12 of the one-dimensional embedding Theorem 4 and Theorem 6 to prove property 2). Fur-thermore, apply Corollary 7 of Chapter 3 and Theorem 6 to prove property 3)with m replacing I. In the general case apply Lemma 8.Proof. By Lemma 8 it is enough to consider the case of a bounded elementarydomain with a continuous boundary (t defined by (4.87). Let Mz > 0 andS = (f e Wp(12) : IIf Ilw;(n) < Ml}. By inequality (4.79) for almost all x E Wand 0 < y < d

7°II1(x,

< M,-y1 II \).(I,a )IILv(m(s)-d,w(s))

where M2 is independent of f and y.By the theorem on the measurability of integrals depending on a parameter

(see footnote 17) both sides of this inequality are functions measurable on W.


Therefore, taking LP -norms with respect to x over W and applying Minkowski'sinequality, we have V f E S

16

IIfIIL,(:z\(n-rya.)) M27" IIlIIWW(n) <_ M37° ,

where M3=Ml M2 and en = (0,..., 0, 1).If f is replaced by Dw f , where,6 E l satisfies 101 = m, then by Theorem

6 we get

1-mIIDwflit,(n\(n-Yen)) < M47n (IIDwfIIL,(n) + II (

aax. W) D"'f IILn(n))

< Ms'y Ilf llw;(n) <- Ms 7p ,

where M4, M5 and M6 are independent off E S and 7.2. Since V is continuous on W, the sets r = {(x, 0(1)), x E W} and r -Ten

are compact and disjoint. Consequently, p(7) := dist (I', r - Ten) > 0 andre(") n Il c S2 \ (Q - 7e") . Hence, given s > 0, there exists po such thatV f E S IIf IIw; (cn,) < c2-", where Gnl = r°° n S2.

Next let F1o = {x E IR" : xi = ai; ak < xk < bk, k = I,-, n - 1, ki; an < x" < ,p(x) - ffi}, i = I,-, n - 1, and let Fil be defined similarly withxi = bi replacing xi = ai. Moreover, let tno = {x E R" : x" = a.; ak <xk < bk, k = 1, ..., n - 1). Since these sets are compact and do not intersect t,

for sufficiently small p E (0, go] we have Gij C ci and n \ 11g e U ( U Gi,).1=1 t=o

Here, for i = I,-, n - 1, Gio = {x E lR" : ai < xi < ai + p; ak < xk <bk, k = 1, ..., n - 1, k 54 i; an < x" < V(x) -

a}, Gil is defined similarly with

bi - p < xi < bi replacing ai < xi < ai + p. Finally, G"o = {x E R" : an <xn < an + p; ak < xk < bk, k = 1, ..., n - 1) .

The same argument as above shows that for sufficiently small p

IIfIIw;(c,;) <e2-", i=1,...,n, j=0,1.

Hencen 1

IIfIIwa(n\n,)<_E flfIIw;(c,i)<e,i=1 7=0

and property 2) follows with m replacing 1.3. By Corollary 7 of Chapter 3 and Theorem 6

IIf (x + h) - f (x)llwp (nIAI) = IIf (x + h) - f (x)IIL,(n,,,i)


+ E II(Dwf)(x+h) - (Dwf)(x)IIL,(n1,,1)

IfI=m

< M7 Ihl (Ilf IIwD(n) + II Dwf IIWD(n)) < Ms IhI IIf Ilwp(n) <_ Mg IhI,IBI=m

where M7, M8 and M9 are independent of f E S and h, and the property 3)follows.

By Lemma 12 the set S is precompact in Wp ()) and, hence, embedding(4.118) is compact.

If m = 0, then embedding (4.118) always holds, but it can be non-compactas the following simple examples show.

Example 12 If the unbounded set Q contains a denumerable set of disjointballs B(xk, r) of the same radius, then embedding (4.118) for each m =0,1, ..., 1 -1 is not compact. To verify this it is enough to set fk (x) = 9(x - xk),where v E Co (B(0,r)), V = 0, k E N. In that case Ilfkllwy(n) = II97IIwo(e(O,r))

and Ilfk - fallwp(n) = 2= Ilcvllwp(a(O,r)), k # s. Hence, any subsequence of{fk}kEN is divergent in WP '4(Q).

Example 13 If 11 is a bounded or unbounded open set, which is such that00

f, = U Ilk, where S2k are disjoint domains, then embedding (4.118) for eachk=1

m = 0,1...,1-1 is not compact. To verify this it is enough to consider functionsfk, which are such that fk = 0 on n \ fk, 1 and Ilfkllw,,(n,k) S M,where M is independent of k E N. The sequence { fk}kEN is bounded in Wp(Q),

but Ilfk - fallwp (n) > llfk - f3IIL,(n) > 2P. Hence, again every subsequence of{fk}kEN is divergent. If measf1k < oo, one may just set fk = (messfIk) onStk. If mess Qk = oo, let fk(x) = n(4) on cck, where r) E Co (R" ),'7(x) = 1 iflxl < 1, and rk > 1 are chosen in such a way that mess (f k f1 B(0, rk)) > 1 .

A more sophisticated example shows that embedding (4.118) for bounded do-mains having a guasi-resolved boundary can also be non-compact.

Example 14 Let 1 < p < oo and 0 be the domain in Example 11. Then theembedding WP (Q) C- Lp(S2) is not compact. For, let fk(x1, x2) = 2

azk+lx2 if

2-(2k+1) < x1 < 2-2k and fk(x1, x2) = 0 for all other (x1, x2) E fI . Then thesequence { fk}kEN is bounded in Wp (1): IIfkIIW;(n) = 1 + (p + 1)-P. However,it does not contain a subsequence convergent in Lp(S2) since Ilfk - fmllt,(n) _2P (p+1)-P ifkAm.


Lemma 13 Let l E N, m E No, m < 1, 1 < p, q < oo and let Sl C R" be anopen set.

1. If the embeddingW,(0) C.; WQ (I) (4.120)

is compact , then de > 0 there exists c34(e) > 0 such that Vf E Wp(sl)

Ill Ilwa (n) <- c34(e) II/IIL,(n) +ellfllw;(n) (4.121)

2. If e > 0 (4.121) holds and the embedding W,()) C; Lp(Q) is compact,then embedding (4.120) is also compact.

Idea of the proof. 1. Suppose that inequality (4.121) does not hold for alle > 0, i.e., there exist co > 0 and functions fk E Wp(sl), k E N, such thatIlfkllw;(n) =1 and

IIfkIIW; (n) > kIIfkIIL,(n)+e0llfkIIw(n) (4.122)

Obtain a contradiction by proving that lim = 0 and, consequently,

liminfllfkllw, (n) 2! co.

2. Given a bounded set in Wp(1), it follows that it contains a sequence{ fk}kEN convergent in L,,()). Applying inequality (4.121) to fit - f, , provethat

kiim Ilk - fs II W; (n) = 0. 13

Proof. 1. Since IIfkIIw,.(n) = 1, by (4.120) it follows that IIfkIIw, (n) <- M1, whereM1 is independent of k. Consequently, by (4.122) we have IIfkIIL,(n) < M1 k-1.Thus, lim IIfkIIL,(n) = 0 and lim IIfkllti,i(n) = 1 . Hence by (4.122)

k-4o0 k- oo

lim inf Ilfkllwam(n) >- co lim inf IIfkllVl (n) = CO.k-+00 k-.o0 P

Since embedding (4.120) is compact, there exists a subsequence fk, convergingto a function f in W, ,'n(11). The function f is equivalent to 0 since fk, -r 0 inLp(O). 20 This contradicts the inequality

II!lIW; (n) = lim IIfkIIW (n) >- co

2. Let M2 > 0 and S = { f E Wp(fl) : Ilf IIw,(n) <- M2}. Since the embeddingW,(0) C, L,()) is compact, there exists a sequence fk E S, k E N, which isCauchy in Lp(SZ). Furthermore, by (4.121)

Ilfk - fall w; (n) 5 C34(e) Ilfk - fsII L,(n) + e Ilfk - fsll,o,(n)

20 If W. - ail in LP(fl) as a -i oo, then there exists a subsequence ,p,, converging to 4)xalmost everywhere on Cl. Hence, if also W, -+ 02 in L5((l), then 01 is equivalent to It on fl.


< c34 (E) I I fk - fa ll L,(n) + 2c M2.

Given b > 0, take c = Since fk is Cauchy there exists N E N such

that Vk,s > N we have Ilk - faII L,(n) < 2 (c34 (.L)) Thus, Vk,s > NIlk - fall w, (n) < b, i.e., the sequence fk is Cauchy in Wq (SZ).

By the completeness of WQ (SZ) there exists a function f E W, '(Q) suchthat fk -> f in W."' (SZ) as k -a oo.

Corollary 17 Let1EN,mENo,m<l,1 <p,q<ooand let SZCRn beabounded open set having a quasi-continuous boundary. Then the compactnessof embedding (4.120) is equivalent to the validity of inequality (4.121) for alle>0.

Idea of the proof. Apply Lemma 13 and Theorem 8.

Lemma 14 Let 1 < q < p < oo and let SZ C Rn be an open set such thatmeas SZ < oo. Then the embedding

Wp (SZ) C; Lq(1) (4.123)

is compact.

Idea of the proof. Given a bounded set S C Wp (Il) apply Holder's inequalityand Corollary 7 of Chapter 3 to prove that conditions 1) -4) of Lemma 12 aresatisfied with Lq(SZ) replacing WP '(0).Proof. Let M > 0 and S = (f E Wp (I) : IIf IIw;(n) < M}. By Holder'sinequality V f E S

IllIIL,(n) <- (meas 9)1-1 If IIL,(n) <- M (meal SZ)9D

and

Of IIL,(n\n,) <- (mess (Q \ SZ6))Q ° IIf IIL,(n\n,) 5 M(meas (Q \ 116))a P.

Since mess SZ < oo, we have dli meas (SZ \ SZa) = 0 and slim 'ES If IIL,(n\n,) _

0. Thus, properties 1) and 2) are satisfied. Moreover, by Corollary 7 of Chapter3 it follows that V f E S

IIf (x + h) - f (x)IIL,(nl,,l) S IhI If IIwt(n)

< IhI (mews SZ)a s Of Ilw,.(n) <- M (messSZ)o_D

IhI.


Hence, property 3) is satisfied (with Lq(I) replacing WW(SZ)). If f is un-bounded, then again by Holder's inequality

IIfIIL,(n\Br) 5 M meas(1 \ B,.)9

and property 4) follows.

We conclude this section with several statements, which are based essen-tially on the estimates for intermediate derivatives given in Theorems 6-7.

Lemma 15 Let I E N and let 1 < p, p1, p2 < oo satisfy 1 = aL + i H. Supposethat SZ C W' is an open set having a quasi-resolved boundary. ien Vf1 EWW, (11) and `d f2 E WD, (Q)

IIf1 f211w;(n) _< C35 IIf1IIwp,(n) IIf2IIww,(n), (4.124)

where c35 > 0 depends only on n and 1.

Idea of the proof. Apply the Leibnitz formula, Holder's inequality and Theorem6. E3

Proof. If fk E c- (n) fl Wpk (Q), k = 1, 2, then starting from

D°(f1 f2) = (C') D"f1 D°-0f2,0<0<0 0

we have

Ilfi f211ww(n) <_ IIf1 f2IIL,(n) + n' II D'sf1 D°-l f2, IIL,(n)I°I=t o<fl<°

<_ IIf1IILa,(n) IIf2lIL, (n) +n' E E IID°fI11L,,(n)I°I=10<,l<°

< M1I01:51 IAI<1

< M2 IIf1IIw;,(n) IIf2llwp,(n),

where M1 and M2 depend only on n and 1.If fk E Wak (f2), k = 1, 2, then (4.124) follows by applying, in addition,

Theorem 1 of Chapter 2.


Corollary 18 Let I E N, 1 < p < oo and let 52 C R" be an open set having aquasi-resolved boundary. Suppose that W E Cb(fl). Then Vf E W,(Sl)

IIf'PIIWW(n) 5 C36 IIfIIWW(supp ana) _(4.125)


Idea of the proof. Direct application of the proof of Lemma 15.

Lemma 16 Let I E N, 1 < p < oo and let SZ C 1R" be an open set with a quasi-resolved boundary. Moreover, let g = (91,...,g") ft -+ R", gk E C'(52), k =1, ..., n. Suppose that Va E Z satisfying 1 < IaI 0. Furthermore,

let g(fl) be also an open set with a quasi-resolved boundary.Then V f E Wp(1)

C37IIfIIww(9(n)) <_ IIf(9)I1ww(n) 5 C38IIfIIWo(9(c)), (4.126)

where c37, c38 > 0 are independent off and p.

Remark 24 By the assumptions of the lemma on g it follows that there existsthe unique inverse transform g('1) = (9(1-1), ..., 9n(-')) : g(f2) - Sl such thatgk-1) E C'(g(cI)), k = 1,...,n. Moreover, Va E K satisfying 1 < IaI 0.

YE9(n)

Idea of the proof Apply the formula for derivatives of f (g), keeping in mindthat for weak derivatives, under the assumptions of Lemma 16 on g, it has thesame form as for ordinary derivatives, i.e.,

Dw(f(9)) = E (Dmf)(9) D"g...D71o19, (4.127)

0<_a,101>1

where ryk E NJ and cp,.,......,., are some nonnegative integers. Apply also The-orem 6 of Chapter 4.Proof. Let a E 1Vo and IaI = 1. By (4.127), Minkowski's inequality and Theorem6 it follows that

IID.*(f(9))IIL,(n) < M1 E II (Dwf)(9)IIL,(n) = (y = 9W)0:50,101>1


M, II(Dff)(y)IDx(g-1(y))I

IILp(9(n))d<_a,181>1

< M2 > IIDWfIILp(9(n)) :5 M3 II f Ilwa(9(n))95x,181>_1

where Ml, M2i M3 > 0 are independent off and p. Hence, the second inequality(4.126) is proved in a similar way. 0

Remark 25 From the above proof it follows that

C38 C39 (inf I Dx (x) I)p

1maa <fIIDagIIc(n),

(4.128)

where c39 depends only on n and 1.

Theorem 9 Let I E N, 1 > 1, 1 < p < oo and let 0 C R" be an open setsatisfying the cone condition. Suppose that f E Lp(S2), the weak derivatives(e ) w, j = 1, ..., n, exist on I and are in Lp(SZ). Then '13 E N satisfying

(aI = l the weak derivatives L), Of also exist on 0 and

II Dwf IIL,(n) <_ C40 (iii IILp(n) + E II (8`;f) II ) ' (4.129)1=1 w LP(n)


Remark 26 For an open set 12 C R" consider the space of all func-

tions f E Lp(fl) whose weak derivatives (1) exist on sz and

n

IIfIIW;

.

(n) =IIfIILp(n) }'EII(a )WIIL,(fl) <o

By Theorem 9 Wp--,'(0) = W,(fl) if 1 < p < oo and n satisfies the conecondition and the norms II IIw; (n) and II - IIw;(n) are equivalent.

Idea of the proof. By Lemma 11 it is enough to consider the case of opensets c2 with a Lipschitz boundary. Applying the extension theorem for thespaces Wp...1(SZ) (see Remark 18 of Chapter 6) and the density of C0 (R") inWp which is proved as in Lemma 2 of Chapter 2, it is enough to prove(4.129) for ft = R" and f e Co (R"). For p = 2 (4.129) easily follows by

4.5. HARDY-LITTLEWOOD-SOBOLEV INEQUALITY 177

taking Fourier transforms. If 1 < p < oo one may apply the Marcinkiewiczmultiplicator theorem: 21

IID"f IIL,(R") = II E F-' { (i)0(-i sgn )1(E ICkll) -1F(a f) } IIL,(R.)j=1 k=1 3

tt

< M1II af IIL,(R")

M11=1 J

where M1 depends only on n, I and p. 0

Example 15 Let p = oo, I E N, 8 E K, 101 = l and let f be a function definedby f (x) = xs In I xI n(x) if x # 0 (f (0) = 0), where r) E Co (R") and vl = 1 in aneighbourhood of the origin. Then f E but Dwf ¢ La,(IIt").

Thus Theorem 9 does not hold for p = oo. One can also prove that it doesnot hold for p = 1.

4.5 Hardy-Littlewood-Sobolev inequality forintegral of potential type

Let f E Li0c(R"). The convolution

IzI_'*f -J Ix(Jy,Ady, A<n, (4.131)

R.

is called an integral of potential type.

Remark 27 One may verify that1) if A > n, f is measurable on R" and f is not equivalent to 0, then IxI-a * f

does not exist on a set of positive measure,2) if A < n, f is measurable on R" and f ¢ then the convolution

IxI-a * f does not exist for almost all x E R".3) if A < n, f E Lr(R") and IxI''` * f exists for almost all x E R", then the

function IxI-a * f is measurable on R".

2i Let 1 < p < oo. Suppose that, for Va E 1%' satisfying 0 < a < 1 (i.e., a, = 0 or1, j = 1,...,n), the function p E L. has the derivatives D°p on the set R = {x E R"xl ... x" # 0}. If jx°(D°p)(x)j < K, x E R; , then

11F-'(pFf)IJL,(R-) <

M2 depends only on n and p.


Integrals of potential type are contained in the inequalities deduced fromthe integral representations of Chapter 3, namely (3.54), (3.58), (3.65), (3.66),(3.69) and (3.76). For this reason we are interested in conditions on f implyingthat IxI'a * f E LQ(R").

Theorem 10 (the Hardy-Littlewood-Sobolev inequality). Let n E N,

1<p<q<oo (4.132)

and

A = n(-L + 1). (4.133)

Then Vf E Lp(R") the convolution IxI-' * f exists for almost all x E R"and

II IxI-' * f IIL,(R") <- C41IIfIILp(R"), (4.134)

where c91 > 0 depends only on n, p and q.

Remark 28 By applying inequality (4.134) to f (ex), where f E Lp(R" ), f *0, is fixed and 0 < e < oo, one may verify that if A n(-L + 1), then inequality(4.134) does not hold for any choice of c41.

We give a sketch of the proof of Theorem 10 based on the properties of

maximal functions. 22

Lemma 17 Let n E N, p < n. Then for all functions f measurable on RnVxER" andVr>0

f Ix - yl-"If (y)I dy C43 r"-µ(M f)(x), (4.136)

B(z,r)

where c43 > 0 depends only on n and p.

For f E LiOO(R") the maximal function Mf is defined by

(M j) (x) =r >o mess B(x, r) JB(z,r) If d11, x E R".

For almost all x E R" If (x)I < (Mf)(z) < oo. Moreover, M f is measurable on R" and for1 0 such that V f E La (R" )

IIMf IIL,(R,) < C42 Ilf IIL,(R-). (4.135)

(If p = 1, this inequality does not hold.)

4.5. HARDY-LITTLEWOOD-SOBOLEV INEQUALITY 179

Idea of the proof. Split the ball B(x, r) into a union of spherical layersS(x, r2-1) = B(x, r2-k) \ B(x, r2-k-1), k E N0, and estimate f If I dy

S(x,r2-k)via the maximal function MI.

Lemma 18 Let n E N, 1 0 depends only on n, p and µ.

Idea of the proof. Split the integrals defining IxI-µ * f into an integral overB(x, r) and an integral over `B(x, r). Applying inequality (4.136) to the firstintegral and using Holder's inequality to estimate the second one via IIf IIL,(R^),establish that

IxI-µ * f I < Ml r"-µ(Mf)(x) + M2 r-(µ-P)IIf IIL,(a ),

where M1, M2 depend only on n, p and µ. Finally, minimize with respect tor.Idea of the proof of Theorem 10. Apply inequalities (4.137) and (4.135).Proof. Since (Mf)(x) < oo for almost all x E R", the convolution IxI-A * fexists, by (4.137), almost everywhere on R" and, by Remark 27, is measurableon R". Since Mf is also measurable on R", taking Lq-norms in inequality(4.137) and taking into account (4.135), we get

II IxI-a

* fIIL9(Rn) s C44IIf'IL,(Rn) IIMfIIL,(Rn) <_ C44 C 2IIfIIL,(Rn).

Remark 29 One may verify that from the above proof it follows that

C41 = (1 + o(1))(! ) , as q -+ oc. (4.138)

Remark 30 Let (P f) (x) = IxI --k * f. Theorem 10 states that for 1 < p <q < oo the operator P is a bounded operator mapping the space Lp(R") intothe space Lq(R ). There is one more, trivial, case in which the operator Pis bounded: p = 1 and q = oo. In all other admissible cases the operatorP is unbounded, thus, inequality (4.134) does not hold for any c41 > 0. Ifp = 1 or q = oo, it follows from the explicit formulae for the norms of integral


operators, 23 by which II IxI ' IIL,(R-) = oo for 1 < q < 00

and IIPIIL,(Rn).L (R') = II Ixl-AIIL (Rn) = oo for 1 < p < oo. If, finally,1 < p = q < 00, it follows by Remark 27.

Next we discuss the case q = oo in Theorem 10, i.e., behaviour of theconvolution IxI - A * f for f E Lp(R" ). The cases p = 1 and p = oo are trivial.(If p = 1, then this convolution is just a constant; if p = oo, see Remark 27.)If 1 e. Then f E LL(R"), but IxI_A

* f = oo for each x E R.

For this reason we consider the case in which 1 < p < oo for functions inLp(R") with compact supports.

Theorem 11 Let 1 < p < oo, f E Lp(R"), f x 0. If Q < -L, then for eachcompact SI C R"

lfQIXl*f I /I Ill IIL

,(R"d--

< oo.(4.141)

Idea of the proof (in the case j3 < ). Suppose that if IIL,(R*) = 1, the case inVft

which IIf IIL,(R^) # 1 being similar. Following the proof of Lemma 18, establishthe inequality

I IxI-A * f 15 M1 rp (Mf)(x) + (Q" I lnri)' +M2i

where 0 < r < 1 and M1, M2 depend only on n, p. Take r = (X))P)- In

and apply inequality (4.47). 0

25 Let E, F C II" be measurable sets, k be a function measurable on E x F and (K f)(y) _f k(x, y) f (y) dy. Then for 1 < q < 00E

IIKIIL,(F)_,L,(E) = II IIk(z,y) IIL,,.(E) IIL,,. (F) (4.139)

and for 1:5 p:5 00

IIKIIL,(F)NL,(E) = IIIIk(x,Y)IIL,,,,(E) IIL..,,(F) (4.140)

4.6. EMBEDDINGS INTO THE SPACE C

Corollary 19 If 0 < µ

fexp

< p', then b'Q > 0

(t lxr_a * fI") dx < oo.

A

181

Idea of the proof. Apply the elementary inequality aµ _< 6- + BaP', wherea > 0, 6 > 0, which follows from (4.47).

Remark 31 If 33 < e1 , there is a simpler and more straightforward way ofproving inequality (4.141), based on expanding the exponent and applicationof Young's inequality for convolutions (4.116).

Example 17 Let 1 2. Then f E LP(W') and

M, I In IxIl < jxI -d * f < M21 In IxII", 0 < jxj i ,

where Ml, M2 > 0 are independent of x.Idea of the proof. To obtain the lower estimate it is convenient to estimatejxj- * f from below via the integral over B(0,1) \ B(0, Y). To get the upperestimate one needs to split the integral defining 1x1-A * f into integrals overB(0,

2)\ B(0, 21x1), B(0, 21x1) \ B(x, ll) and B(x,

z)and to estimate them

separately.

Remark 32 This example shows that the exponent p' in inequality (4.141)is sharp. Indeed, if µ > p', then for < ry 0 and for each compact fl C R".

A more sophisticated example can be constructed showing that for 6 > nTheorem 11 does not hold.

4.6 Embeddings into the space of continuousfunctions

Theorem 12 Let I E N, 1 p for 1<p<oo, l>n for p=1, (4.142)


then each function f E Wp(SZ) is equivalent to a function g E Cb()) and

tIglIC(n) <_ C45 If II W, (n), (4.143)

where c45 > 0 is independent of f, i.e., Wp(S1) C; Cb(S)).If Il is unbounded, then

lim g(x) = 0. (4.144)x-4co, ZEn

Idea of the proof. By Theorem 1 (4.143) is equivalent to the inequality

IIf IIL (n) = IIgIIC(n) <_ C45 IIf IIwp(n)

for all f E W,(SZ) fl C°°(S2). Since IIf IIc(n) = sup IIf IIc(K.), where K. are the'ten

cones of the cone contition, which are congruent to the cone K defined by(3.34), it is enough to prove that

II f II c(K) <_ C45 Ilf llwy(K) (4.145)

To prove (4.145) apply inequality (3.76). In the case of unbounded open sets11 apply inequality (3.77) to prove (4.144).Proof. By (3.76) where Q = 0 for Vf E Wp()) fl C' (11) and Vx E K

If(x)I 5 M1 (f IfI dx+E f I(D°f)(y)I dy)Ix - yl"-'K IQ1_,K

where M1 is independent of f and x. Hence, by Holder's inequality,

If W1 <M1((measK)pIIfIIL,(K)+EIIIx-yI` "IIL,,.,,(K)OD°fIIL,(K)).101=1

Let D be the diameter of K (D = h2 +r2). If 1 P, thenapplying (4.116), we have Vx E K

II Ix - YI` "IIL,,.,(K) < II IZI' "IIL,,(B(O,D))

D

_ an f e(!-n)p'+n-1 dLo i= ( a" Df- n

J Alp)0

4.6. EMBEDDINGS INTO THE SPACE C 183

If p = 1 and 1 > n, then

II Ix - yI` "IIL-.r(K) <- D'-".

Consequently,If(x)Is M211f1IW;(K),

where M2 is independent off and x, hence, (4.145) and (4.143) follow.If S2 is unbounded and f E WP(S2), then, applying inequality (3.77) where

0 = 0 to the function g in (4.143), we get that Vx E S2

19(x)1 <- M2(f If I dy+ E f I(Dwf)(y)I dy)K, iai=1K.

IX - yI"-`

< M3 IIf II W;(K.) <- M311111 W/(n\B(O,IzI-D))

if (xI > D, where M2, M3 are independent of f and x . Therefore, lim g(x) _X400

0.0

Corollary 20 If S2 satisfies a Lipschitz condition, then the function g E C(S2).

Idea of the proof. For 11 = R" apply (4.143) to f - fk, where the functionsft E CO (R") converge to the function f in Wp(R ). If S2 satisfies a Lipschitzcondition, apply the extension Theorem 3 of Chapter 6.Proof. If 52 = R", then from (4.143) it follows that IIfk-9IIc(R) -+ 0 ask -r oo.Hence, g E ?7(R"). Let 11 satisfy a Lipschitz condition and T be an extensionoperator in Theorem 3 of Chapter 6. For f E WP (S2) consider a sequence offunctions hk E CO (R") converging to Tf E Wp(W°). Then hk -+ f in Wy(S2)and by (4.143) IIg - hkIIc(n) -+ 0 as k -* oo. Hence, again g E C(f ).

Corollary 21 Let 1, m E N, 1 m+n for 1<p<oo, 1>m+n for p=1, (4.146)

then each function f E WP(Q) is equivalent to a function g E Cb (S1) and for0 E N satisfying 101 < m

IID19IIc(n) <- c46IlfIIww(n), (4.147)

where c46 > 0 is independent of f, i.e., WW(Q) C:; Cb (S2).If S2 is unbounded, then

lim (DO9)(x) = 0, ,6 E N , IQ1 < M.X-+00' ZC-n


Idea of the proof. It is enough to apply Theorem 12 to Dw f , where IQI < m,since by Theorem 6 DO f E W,'_1"1(Q) and inequality (4.105) holds.

We note that conditions (4.142) and (4.146) are also necessary for the va-lidity of (4.143) , (4.147) respectively. (See the proof of Theorem 14 below.)

Corollary 22 Let 1 0 depending only on n, I and p such that forconvex domains 1 satisfying D = diam S2 < do

IIgIIc(n) S (meas Q) p Ilf ll-wa(n)

where IIfIIi (n) is the norm defined by (4.110), equivalent to IIfIIwi(n) (coin-

ciding if l = 1). The constant (meas1l)-p is sharp since for f =_ 1 equalityholds.

This inequality follows from the proof of Theorem 12 if to start from theintegral representation (3.65). Let, for a E S"-1, where S"-1 is the unit spherein R", r(x, a) be the length of the segment of the ray {z E R" : z = x+ pa, 0 <2 < oo} contained in Q. Then, for d(x, y) defined in Corollary 13 of Chapter3, we have d(x, x + @a) = r(x, a). Hence

r(z,o),

" 'II d (,y)ìi_ ( \

(d(x,x Qa))p on-1de)da)Ix

s1 1Q"-1

(P

_ (p'(l - f (r(x,a))1p'+nda)°' < (p'(l -sn-1 J

since

n f (r(x, a))" do. = mess Q.

sn-1

Thus, by (3.65) and Holder's inequality,

I°IIIglIc(n) <_ (meal il)

ia D; IIDwf IIL,(n)"°I<I

4.6. EMBEDDINGS INTO THE SPACE C 185

ICI

+I(p'(1 -pn))-7' Ea! IIDWfI1L,(n)),

IQI=1

and the desired statement follows. In the simplest case p = 1, 1 = n thisinequality takes the form (3.66) and, hence, one can take do = 1.

Remark 34 There is one more case, in which the sharp constant in the in-equality of the type (4.143) can be computed explicitly, namely I = R", p =2. 1 > 2. In this case

7F. I

I1911C(R-) - (21r)-'v,,l1 )'IIfIIw;(Rn)

21

(See Remark 8 of Chapter 1.) Equality holds if, and only if, for some A E C

f(x) = A(F-'((1 + 1e121)-'))(x)(if ! = n = 1, then f (x) = B exp (- Ix l)) for almost all x E W. If ! = n = 1, thisinequality coincides with (4.75) where p = 2. This follows since V f E WW(R)

IIfIIL (R-) = IIF-'Ffi1L_(R") =

(27r)-!2' 11 1R"

< (2ir)-111FfIIL,(R^) = (2r)- 11(1 +ICI21)-j'

(1 +

(27r) II(1 + II(1 + IL7W,,

_ (27r)-= (f (1 + I2 + IVhfl2) dx) aR" R^

The desired inequality follows since by (4.116)00

f(1 +IfI2)-tdC=a" f(1 +e j')-1 e-1 do I(1+®'t)-'=t

R" 0

1r

l= 21J

V§ (1-t)N-1dt=21 B(I 2!'2I1

0

- v"2Ti r(2!) rl1 - 2!) "" 2!(sin

2! )In the second inequality equality holds, if, and only if, for some A E C wehave (Ff)(C) = A(1 + (0121)-1 for almost all C E R". (See footnote 11.) SinceI f (Ff)(C) dCI = 11Ff IIL,(R"), equality holds also in the first inequality.R^


4.7 Embeddings into the space LqTheorem 13 Let I E N, 1 < p < oo, 1 < R and let Sl C R" be an open setsatisfying the cone condition. Moreover, 24 let q. be defined by

l = n(D - - ). (4.148)

Then for each function f E Wa(0)

Ilf IILq.(n) <- C47 II!IIw;(n) (4.149)

where c47 > 0 is independent of f , i.e, W,(SZ) C; L9. (c ).

Idea of the proof. By Lemma 10 it is enough to prove (4.149) for boundeddomains 11 with star-shaped with respect to a ball. Apply inequality (3.54)and Theorem 10 to prove (4.149) for such S1. 0First proof (p > 1). Let S2 be a bounded domain star-shaped with respect tothe ball B = B(xo,

z)and let ddiam) = D. By (3.54) where Q = 0

If(x)I <- M1(J IfIdy+E f I(D°f)(3l)IIx - yl°-1

dy) (4.150)

B lnI=l V,

for almost all x E 11, where Ml depends only on n, 1, d and D. By Holder'sinequality f If I dy < (mess B) I I ! II L,(n) Hence

B

IIfIILq.(n) <- M2 (11f IIL,(n) + II fda(y)

x - yl^(i+a) dyll[,o.(R"))'

R" I

where Da(y) = I(Dwf)(y)I if y E 11 and 6.(y) = 0 if y ¢ fl and M2 dependsonly on n, 1, p, d and D. By Theorem 10

fix dylIL(R) < MII'aIIL,(R") = MsIIDf IIL,(),q.- yl

where M3 depends only on n, l and p. Thus (4.149) follows, where c47 dependsonly on n, 1, p, d and D. Hence, by Lemma 10, the statement of Theorem 13follows. 0

24 Often q. is called "the limiting exponent."

4.7. EMBEDDINGS INTO THE SPACE LQ 187

Remark 35 It is also possible to start, without using Lemma 10, from in-equality (3.76) and argue in a similar way. (If 0 is unbounded, one shouldtake into account that, by Remark 21 of Chapter 3, in (3.76) IllIIL,(R-) can bereplaced by Ilf

Remark 36 By Holder's inequality and by the interpolation inequality

1-0IIfIIL,(n) <- IlfIILD(n)IIfIIL, (n)+ (4.151)

where p < q < q. and 1 = v + 19-01 it follows that inequality (4.149) holds foropen sets 12 satisfying t?ie cone condition if q, is replaced by q, where 1 < q < q.for bounded sets S2, p < q < q. for unbounded sets S2 respectively.

This statement may be proved, including the case p = 1, by simpler means -just by applying Young's inequality. By Lemma 10 it is enough to consider thecase of bounded domains star-shaped with respect to a ball. Starting startingfrom (4.150) and (4.155), it is sufficient to note that

II fI(wf)'Y) dyII

L,(n)<- II Ixl1-nllLrro-n) IIDwfIIL,(n)Ix - yI -

and by (4.116)

2D

II Ixl'-'IIL,(n-n) <- II I ri1-nIILr(B(0,2D)) = (an f 9(l-n)r+n-1 de)

0

-)1 ° (2D )!-"(n a < 00.l-n(y-Q

Remark 37 For p = 1 inequality (4.149) cannot be proved by applying Theo-rem 12, which does not hold for p = 1. Moreover, in this case inequality (4.149)does not follow from (4.150). More than that, inequality (4.149) can not beproved by estimating separately the Lq.-norms of each summand in the remain-der of Sobolev's integral representation (3.51) and not taking into account thatD. *f are not arbitrary functions in Lp(1l) , but are the weak derivatives of afunction f E Lp(SZ). For, let

(Kaco)(y) =f we(x, y)

W(y) dy.

V.


Then by (4.141)

Wa(x, Y)X' (y)IIKaIIL,(n,-,L,.(n, = II

II a II IIIx_YI La..:(n) Loo.v(n)

and one can prove that IIKaIIL,(n)_*L..(n) = 00.

Idea of the second proof of Theorem 13. 1. Verify that by Lemma 11 andTheorem 3 of Chapter 6, it is enough to prove inequality (4.149) for SZ = W'.

2. Let S2 = R", n > 1. First suppose that l = 1 and p = 1. Then q. = n"1Starting from the inequality

n n

Ifl n-I = [I Ifl"=' <_ II IIfIIim;=,(R)IM=1 m=1

(4.152)

which holds almost everywhere on R, apply the one-dimensional embeddinginequality (4.64) and the following variant of Holder's inequality for the productof functions gn, - g(xl, ..., xm-1)xm+1, ..., xn), which are independent of m-thvariable: 25

n n1,

ri 1IM=1 M=1

(4.153)

Here ft; 1 is a space of (xl, ..., xm-1, xm+i, ...,xn) where x1 E R Obtain forf E Wp (Rn) the inequality

n

( Of lb

IIfIIL.(R") G 2 II \axm/wllt,(R")(4.154)

2s This inequality can be easily proved by induction. On the other hand, it is a particularcase of Holder's inequality for mixed Lp-norms, where p = (pl,...,pn) and

IIfIIL,(R") =

which has the formk k

II II fm1IL'(R") < H IIfrIIL,.,(R"),m=1 m=1

I., nwhere An = (plm,..., pnm) are such that 1 < pjm < oo and E PJI- = 1,m = 1,...,k. It Is

proved by successive application of the one-dimensional Holder's inequality. If pjm = n - 1for j # m and pmm = co, we obtain (4.153).


3. If 1 = 1 and p > 1, apply (4.154) to If I{ with appropriate > 0 andprove that for f E Co (R" )

2(n'p

"

ll ax- L..(R"1'(4.155 )nafll"

4. If l > 1, apply induction and, finally, Lemma 2 of Chapter 2.Second proof (p > 1). 1. First let SZ be an open elementary domain with aLipschitz boundary with the parameters d, D and M. (See Section 4.3.) More-over, let T be an extension operator constructed in the proof in Theorem 3 ofChapter 6. By inequality (4.149) where ) = R" we get

IllllL,.(n) <_ IITfIIL,.(R") < M1IITfIIw,(R") Mi IITI %2IlfIIwp(n).

Here Ml depends only on n, 1, p and M2 depends only on n, 1, p, d, D andM. Since q. > p, by Lemma 11 inequality (4.149) holds for each open set Ilsatisfying the cone condition.

2. Now let Q = R". First suppose that I = 1, p = 1 and let f E Wi (Ill"`).By (4.152), (4.64) and (4.153) we have

s=d n t

"nt

IIfIIL(R") = II Ifi"=fIIL,(R") II rj

II I \(7x ).11L,.m(R) IILi(R")

II (8x )tvIIL,(R").

2 n II II (ate )wDLt.sm(R) IILt(Rm t)2

M=111

3. Let I = 1 and 1 0 1( )w1 = t If It-I Imo I , applying (4.154) to If It,

where An' q., we have

IIfIIL,.(R") =11IfI{IIL2t(R") = II

n of(z) ii II(alf')w1IL(R") = (2) n II Ift Ox

L.(R").

By Holder's inequality

IIIfIE-lax IIL,(R') < 11 lfl,OX IIL,(R")


IIfII .(R")IlafIIL,(R")

Hence

IIf IIL,.(R") < (2) ` IIf IILq(R") r, II t9f IIL (R")*m=1 M a

Since 0 < IIf IIL,.(R") < oo and 1 - I'- = 1, we obtain (4.155).4. Next suppose that p > 1,1 < 1 < a and f E Co (R"). We define the

exponents q1, ..., qq-1 by 1 = n(1 - 1), 1 = n(1 - i ), ..., 1 = n(1 - i ).9j 9

Applying (4.155) successively, we get0 9, D q,_,

nof

...IIfIIL,.(R")_<M1II 8xm,IL,I(R")<

1m,=1

n n0 f

< M1 mE1...

mmr

IIC7x, ...l7xm, IILy(R")

M1+1 E II D°f IIL,(Rn) = M1+1 IIf IIt4(Ra) ,

1°i=1

where M1, ..., M1+1 depend only on n, I and p. Finally, taking into considerationLemma 2 of Chapter 2 and passing to the limit, it follows that this inequalityholds Vf E WW(R"). O

Remark 38 Inequality (4.149) for S1 = R", n > 1, 1 = 1, 1 <_ p < n (steps 2and 3 in the second proof of Theorem 13) can also be proved with the help of thespherically symmetric rearrangements f' of functions If I defined by f'(x) =sup{t µ(t) > V. IxI"}, where u(t) = mess{x E IV : If (x)I > t}. Clearlyf' (x) = g(Ix1). The following properties of f' are essential:

IIf'IIL,(R") = IIf IIL,(R") , 15 P:5 00 (4.156)

and

Ilowf'IIL,(R") <_ IIVwfllL,(R"), 1 , f is locally absolutely continuous on (0, oo)and lim f (x) =0. W e note that f o r q = q. = p, 1 <p < n and a = nP'inequality (4.158) takes the form

00

1

0

(f(J If(x)I°'xn-1dx)9 <C49( f If'(x)IPxn-1dx)°. (4.159)

0 0

Applying (4.156), (4.116) and (4.159), we have

00

IIfIILq.(R^) = IIf*IIL..(R") = (an f I9(e)IQ- en-1 de)

0

00t

< an v. p c49(an f I9,(e)IP en-' de) p = an ^ CO IIVwfÌILp(R^)0

Hence by (4.157)

IIf IILg.(R") <- C50 IIVwf IILp(Rn) , (4.160)

where c5o = an ^ c49 and (4.149) follows.Moreover, it is also possible to prove that the minimal value of c5o in (4.160)

is equal to7r- n-o (p - 111-0 r(z + 1)r(n)

n-p r(p)r(1+y)J(If p = 1, one must pass to the limit as p - 1+.) In the case p > 1 equality in(4.160) holds if, and only if, f o r some a, b > 0 If (x) I = ( a + -; almosteverywhere on R".

Remark 39 As in Remark 33 it can be proved that there exists d0 dependingonly on n, 1, p and q satisfying 1 < q < q. such that for convex domains fZsatisfying D diam SZ < do

IIfIIL.(n) 5 (meas0)9-D IIfIIwa(n),

where the constant (mess fl) 9 - a is sharp.To obtain this inequality one should apply the inequality

(f I f k(x, y) f(y) dyI° dx)1 < A IIf IILp(n),

G A


where 1-1 1

A =sup ,(n)sup Ilk(., 01111,(G)MEG yEc

Sl C R' and G C R" are measurable sets, k is a measurable function on G x SE,1 < p < q < oo and + q. (The proof is similar to the standard proof ofYoung's inequality (4.115).)

As in Remark 33 one can prove that for convex domains St and k(x, y) _d(z,y "Iz-yl^_

Hence by (3.65)

A < (r(l - n(p -Q)))

!

_i D1alIIfIIL,(n) - (measSt)1 o(E

a( IIDwfIIL,(n)lal<l

+,(-,(r(l-n(p-q))) 3 i IIDwfIIL(n)lal=(

and the desired statement follows.

Corollary 23 Let I E N, m E 1%, 1 < p, q < oo, m < 1 < m + -y and let0 C 1[i" be an open set satisfying the cone condition. Moreover, let q. bedefined by

1=m+n(p-q), (4.161)

1 < q < q. if Cl is bounded and p < q < q. if Sl is unbounded.Then V f E Wp(Q) for ,B E N satisfying 1131 = m

IIDwfIIL,(n) <- C51IIfIIW,(n), (4.162)

where csl > 0 is independent of f, i.e., WW (11) c Wy (SZ).

Idea of the proof. Apply Theorem 6 and 13 to D.Of, Holder's inequality if Cl isbounded, and the interpolation inequality (4.151) if SZ is unbounded.

Corollary 24 Letl E N, 1 <p<q<oo,l>m+n(n-1),eo>0 and letfl C W' be an open set satisfying the cone condition. Then V f E WI(fl) forQ E N satisfying I#I = m

ru+e( H)

II Dwf IIL,(n) 5 cat 6 `-m-"(-) Ill IIL,(n) + e II f IIW;(n) , (4.163)


where 0 < e < co and c52 > 0 is independent off and E. Furthermore,

3(i-m-n(n-y)) 7(m+n(p-y))IIDwf IIL,(n) <- c53 IIf IIL,(n) IIf IIw, (n) (4.164)

where c53 > 0 is independent of f.If 0 = R" or, more generally, Q is an arbitrary infinite cone defined by

Q={xEW:x=pv,0<e<oo,crES}, (4.165)

where S is an arbitrary open (with respect to Sn-1) subset of S"-', then in-equality (4.163) holds for an arbitrary e > 0 and in inequality (4.164) IIf Ilwp(n)can be replaced by IIfIIw;(n)

Idea of the proof. If 1 has the the form (4.165), then inequality (4.163) may beobtained by applying (4.162) to f (ex), e > 0, since cQ = Q. Inequality (4.164)follows from (4.163) by minimization with respect to e. To obtain (4.163) foran 11 having a Lipschitz boundary, apply the extension theorem of Chapter 6(Theorem 3 and Remark 16) and (4.163) for l = R". If Q satisfies the conecondition, apply, in addition, Lemma 6 and Corollary 13. Inequality (4.164) isderived from (4.163) as in the proof of the one-dimensional inequality (4.43). 0Proof. If S2 has a Lipschitz boundary, then by Theorem 3 and Remark 16 ofChapter 6, for ally > 0,

IIDwfIIL,(n) 5 IIDwTfIIL,(R^) <-MIy-6IITfIIL,(R-)+7IITfIIw,(Rw)

< M1y-'IITIIoIlfIIL,(n) +yIITIIjIIfIIw,(n)

_(Mly-6

IITIIo + y IITII1) If IIL,(n) +,y IITIIh Ilf 11.,(n)

Here 6 = (m + n(I - !))(l - m - n(I - q))-', T is the extension operatorconstructed in Theorem 3 of Chapter 6, IITIIo - its norm as an operator actingfrom Lp(1) in Lp(R") and IITII: - its norm as an operator acting from Wp(f1) toWW(R"). Both IITIIo and IITIIh depend only on n, 1, p and the parameters of theLipschitz boundary. Setting 7IIT II+ = e and noticing that Miry 'IITIIo+7IITII: 5M2e-6 if 0 < e < co, we get

IIDwfIIL,(n) 5 M2e-6IIfIIL,(n)+eIlfIlwo(n) ,

where M2 depends only on n, 1, p, q, eo and the parameters of the Lipschitzboundary.


Next suppose that Q satisfies the cone condition. By Lemma 6 there existelementary domains Ilk, k = 7s, such that Sl = U Ilk, where s E N for bounded

kSt and s = oo for unbounded Q. They have Lipschitz boundaries with the sameparameters, and the multiplicity of the covering x = x({Slk} k) is finite if.lSt is unbounded. Consequently for each k = 1, s

II D!f IIL,(n,,) 55 M2E_6

IIf IIL,(nk) + E IIf IIv,(n.)


II Dwf IIL,(n) <- xo (M2E_6

IIf IIL,(n) + E IIf Ilwp(n))

hence, (4.163) follows.

To prove (4.164) we set E. _ Of IIL,(n) If IIp(n))£° where 1; = i (l - m -

n (I - q)) . If e. < E0, then (4.164) follows from (4.163) directly. If E, > ED,

then IIf Ilw;(n) 5 Eo c IIf IIL,(n) and by (4.163)

IIDWfIIL,(n) <- Ma IIfIIL,(n) <- Ma IIf IIi,(n) IIf Ilwp(n)

where M4 is independent of f. Hence (4.164) follows.

Corollary 25 Let I E N, 1 0 work out that

II D°wf IIL,(n) <- C51 (E-' If IIL,(n) + Ilf IIi,(n))

and pass to the limit as a - oo.

Theorem 14 Let I E N, m E N0i 1 < p, q < oo and let 11 be an open setsatisfying the cone condition. Then the embedding

W,(Q) C; WV- (0) (4.166)

in the case of bounded Sl holds if, and only if,

l > m + p for q = oo, 1 m+n(p-q) for q=oo,p=1 or q<oo,1<p<oo. (4.168)

In the case of unbounded SI if, and only if, in addition, q > p.Moreover, embedding (4.166) is compact if, and only if, SI is bounded and

l > m + n(p - Q). (4.169)

Idea of the proof. Apply Corollaries 20 and 21, Example 8 of Chapter 1 and, forq < p, modify the function defined by (4.85). As for compactness, apply Corol-laries 17 and 24 and modify the sequences defined by (4.86) and in Example1.0Proof. 1. If conditions (4.167) or (4.168) are satisfied, then embedding (4.166)follows from Corollaries 20 and 21.

Let us assume without loss of generality that 0 E Q. Suppose that I <m + n (1-1), then 26 there exists p satisfying I - o < p < n - a , whichis not a nonnegative integer. By Example 8 of Chapter 1 JxJ" E W,(Q) butJxV' V Wq (SI), and it follows that embedding (4.166) does not hold. Nextsuppose that l = m + p, q = oo and 1 < p < oo. Let 0 < v < 1 - 1. ByExample 8 of Chapter 1 xm(I In JxJ I)" E WP(Q) but clearly this function doesnot belong to WW(SI). Hence again embedding (4.166) does not hold.

Let q 0 and disjoint balls B(xk, p) C 1, k E N. We set f (x) _00

E k-a77(l), where 77 E Co (RI), suppr C B(0,1) and p $ 0. Then, as ink=l

the proof of Theorem 5, f E WW'(SI) but f 0 WQ (SI), hence embedding (4.166)does not hold.

2. If condition (4.169) is satisfied, then the compactness of embedding(4.166) follows from Corollaries 17 and 24.

If SI is bounded and I < m + n ( P - 4) , consider the sequence fk (x) _

ko-1r/(kx) where k E N. Then IIfktIwl(n) <- lI77IIwl(R"). Suppose that, forsome g E W.'(11) and some subsequence fk, , fk. -* g in WQ (SI). Sincefk,(x) -+ 0 as s -+ oo for all x 0 0, it follows that g - 0. On the other hand,

Ilfk.llw, (n) >-k'"+(a-a)-IIIB

I)

. Hence fk, -o+ 0 in WQ (SI).1 a(R")

26 We note that the necessity of the inequality ! > m + n (D - a) also follows for 1 <p,q < oo and f1 = R" by comparison of the differential dimensions of spaces WW(R") andWa (R"). See footnote 14 of Chapter 1. With slight modifications a similar argument worksfor open sets f1 34 R".


If SZ is unbounded, consider, as in step 1, the disjoint balls B(xk, p) andset ft(x) = (!). As in Example 1, fk does not contain a subsequenceconvergent in Wy (S2).

Next let us consider in more detail the case I = p. By Theorem 14, for an

open set 0 satisfying the cone condition, it follows that Wp (S2) C L,(S2) fora

each p < q < oo. However, Wp (0) ¢ L,,. This statement may be improvedin the following way.

Theorem 15 Let 1 0 depending on n, p and theparameters r, h > 0 of the cone condition such that V f E Wp (S2), f x 0,

Jexp (c541

IIf II f aJP) dx < oo. (4.170)

wo (n)

Idea of the proof. Apply inequality (3.76), Remark 21 of Chapter 3 and Theorem11.13Proof. By (3.76) and Remark 21 of Chapter 3 for almost every x E SZ

If(x)I <_ Mi(IIfIIL,(n)+I0 *W),

where w(x) _ E I (Dw f) (x) I for x e SZ and cp(x) = 0 for x 0 0. SinceIal=;

IIWIIL,(R) <_ IIf IIwD (n), we have

If(x)I )P <Mz 1+ IxI3*(Ill II a (

wi (n)

Here M1, M2 > 0 depend only on n, p and the parameters r, h of the conecondition. Hence inequality (4.170) where ccd = M2 v;1 follows from (4.141).

Remark 40 The cone condition in Theorems 12-15 is not necessary but issufficiently sharp, because for the domain considered in Example 6 of Chapter3 these theorems do not hold for any ry E (0,1). See Remark 19 and Example1 of Chapter 6.

Chapter 5

Trace theorems

5.1 Notion of the trace of a functionLet f E Ll0c(R") where n > 1. We would like to define the trace tr f =_ tr,m f

f lam of the function f on Wn where 1 < m < n.

We shall represent each point x E Rn as a pair x = (u, v) where u =(xl, ..., x,n), v = (xm+i, ..., xn) and suppose that RI(v) is the tn-dimensionalsubspace of points (u, v), where v is fixed and u runs through all possible values.We shall also write Rm for R'" (0) if this will not cause ambiguity.

If f is continuous, it is natural to define the trace tr f as a restriction ofthe function f : (tr f)(u) = f (u, 0), u E R"'. However, this way of defining thetrace does not make sense for an arbitrary function f E Li°C(Rn), since actuallyit is defined only up to a set of n-dimensional measure zero. In fact, one caneasily construct two functions f, h E Ll°C(R" ), which are equivalent on R", butf (u, 0) 96 h(u, 0) for all u E R'". Finally, it is natural to define the tracesthemselves up to a set of m-dimensional measure zero.

The above is a motivation for the following requirements for the notion ofthe trace on R" of a function f E Lll°C(R"):

1) a trace g E Li"(Rn),2) if g E L'i C(R'n) is a trace of f, then 0 E L'i C(Rm) is also a trace of f, if

and only if, t/' is equivalent to g on R'n,3) if g is a trace of f and his equivalent to f on R", then g is also a trace

of h,4) if f is continious , then f (u, 0) is a trace of f .

Definition 1 Let f E Lj-(R") and g E Li°C(Rm). The function g is said tobe a trace of the function f if there exists a function h equivalent to f on Rn,

198 CHAPTER 5. TRACE THEOREMS

which is such that 1

h(-, v) -r in LiOC(R'") as v -+ 0. (5.1)

Clearly the requirements 1)-4) are satisfied. In fact, if g is a trace of f andis equivalent to g, then (5.1) implies v) in L oc(Rm) and tP is also

a trace of f . Next suppose that both g and tp are traces of f , then we have(5.1) and also v) -4 0 for some H - f on R". Wenote that for each compact K C R'

119 - lJIIL,(K) <- v) - 911 L,(K)

+ v) - H(., v)IIL,(K) + v) - bIIL,(K)

Since h - H on R", v) N v) on W" for almost all v E R"-'". Hence,there exists a sequence {vs}sEN, v, E R"-', such that v, -+ 0 as s -+ oo and

119 - tIIL,(K) <- IIh(-, v,) - 9IIL,(K) + IIH(., v8) - tIIIL,(K)

On letting s -4 oo, we establish that g - on R.Finally, if f is continuous, then

I If (u, v) - f (u, 0) I I L,.4(K) <_ mess K maKx If (u, v) - f (u, 0)1.

Hence, 11f v) - f (', 0) I IL,(K) -* 0 as v -+ 0 because f is uniformly continuouson K x B,, where Bi is the unit ball in R"-m. Thus, f 0) is a trace of f .

Theorem 1 Let Z(R") be a semi-normed space of functions defined on R"such that

1) Z(R") (::L1°c(R")and

2) C00(R) fZ(R") is dense in Z(R").Suppose that 1 < m < n and for each compact K C R'" there exists cl(K) > 0such that V f E COO(R") fl Z(R") and Vv E R"'" satisfying Ivi 5 1

ci(K) IIfiIz(R-). (5.2)

Then V f E Z(RI) there exists a trace off on W.

1 One may include the case m = 0, considering a number g satisfying h(v) - g as v - 0.

5.1. NOTION OF THE TRACE OF A FUNCTION 199

Idea of the proof. Consider a function f E Z(R) and a sequence of functionsfk E C°°(R") n Z(R"), k E N such that fk --' f in Z(R") as k -* oo. Applying(5.2) to fk - f prove that Vv E R"-" : IvI < 1 there exists a functiongv defined on Rm such that v) -+ g,, in L'°C(Rm) as k - oo. Defineh(u, v) = g. (u), (u, v) E R", and prove that the functions h and g =- go satisfyDefinition 1.Proof. Let B,., b,. be open balls in R"', R"-.m respectively, of radius r centeredat the origin. By (5.2) with fk - f, replacing f and BN, N E N, replacing K,it follows that v) - v) -+ 0 in Ll (BN) as k, s -1 0o for all v E B1 andall N E N. By completeness of Ll(BN) there exists a function q,,,N E Ll(BN)such that in Ll(BN). Consider any function 9v - 9,,,N onBN for all N E N. Such a function exists because g,,,N - g,,,N+1 on BN. Thisfollows by passing to the limit in the inequality

pII9v,N - gv,N+1II L1(BN) IIgv,N - fk(., v)II L1(BN) + Ilfk(', V) - 9v,N+l IIL1(B,N+1)'

Clearly, fk(.,v) -> g° in Ll°`(R") as k -+ oo and, hence, for the function h,defined by h(u, v) =gn(u), (u, v) E R", we have fk(.,v) -* v) in L'°C(Rm)for all v E B1.

On the other hand, f v) v E B1. This followssince by the Fatou and Fubini theorems and condition 1)

J (liminfJ Ifk(u,v)- f(u,v)Idu)dvB1 BN

< lim inf f (f I fk(u, v) - f (u, v) I du) dvk-+oo

B1 B,v

= lim J Ifk(u,v)-f(u,v)Idudv=0.k- oo

BN x B1

Thus f v) is equivalent to v) on Rm for almost all v E B1. Conse-quently, by Fubuni's theorem, 2 f is equivalent to h on Rm x B1.

Furthermore, by the continuity of a semi-norm, on letting s -+ oo in (5.2),where f is replaced by fk - f we get

Ilfk(', v) - v)II L,(K) 5 c1(K)Ilfk - f IIZ(R").

2Fbr, let e" = {(u,v) E Rm x 61 : f(u,v) # h(u,v)} and e, (v) = {u E Rm : f(u,v) #h(u,v)}. Then meas"e" = f (measme,"(v))dv = 0.

B,


Therefore

Ilh(', v) - 9IIL,(K) = Ilh(', v) - h(., O)IIL,(K) <- IIh(', v) - fk(', v)II L,(K)

+Ilfk(', v) - fk(', 0)II L,(K) + Ilfk(', 0) - h(', 0)II L,(K)

< 2 cl (K)Ilfk - f IIz(R") + meal K maKx lfk(u, v) - fk(u, 0)1.

Given e > 0, we choose kE E N such that for k = k,. the first summandis less than z. Since fks is uniformly continuous on K x Ell, there exists,y = -y(e) > 0 such that for lvI < y the second summand is also less than z and1l9(', v) - 9ll L,(K) < e. Hence v) - 9II L,(K) -4 0 as v -> 0 and h(., v) -* gin L'

I

-(R-).Thus, by Definition 1, g is a trace on R" of the function f. 0

Remark 1 On replacing f by fk in'(5.2) and letting k -> oo, we establish thatVf E Z(R")

IltrfIIL,(K) <_ cl(K)Ilfllz(R").

Moreover, it follows that dfk E C°O(R")nZ(R"),k E N, satisfying fk - f inZ(R") as k a oo we have f in L1°°(Rm).

Corollary 1 In addition to the assumptions of Theorem 1, let the followingcondition be satisfied

3) if f E Z(R" ), then Vv E Rn-- f v) E Z(R") and

v)IIZ(R") = Ilf IIZ(R")

Suppose that for each compact K C R"-'" there exists c2(K) > 0 such thatVf E C°°(R")nZ(R")

llf 0)IIL,(K) <- c2(K) Ilf llz(R"). (5.3)

Then V f E Z(R") there exists a trace on R"'.

Idea of the proof. Given f E Z(R"), apply (5.3) to the function f,,, definedby f v), which by condition 3) lies in Z(R"), and verify thatinequality (5.2) is satisfied for all v E R"-'". 0

5.2. EXISTENCE OF THE TRACES ON SUBSPACES 201

5.2 Existence of the traces on subspacesTheorem 2 Let 1, m, n E N, m < n and 1 n-mP

i.e., if, and only if,

for 1 n - m for p=1, (5.4)

Wy(R"-m) C3 C(R"-m).

Idea of the proof. If (5.4) is satisfied, write the inequality corresponding toembedding (5.5) for functions f (u, ) with fixed u, and take Lp norms withrespect to u. Next use Theorem 1. If (5.4) is not satisfied, starting fromExample 8 of Chapter 1, construct counter-examples, considering the functionsfe(u, v) = Ivlp 771(u) 7k(v) if I < n PM and g.1(u, v) = I in lvI I" nl (u) *i2(v) ifI = "pm, 1 < p < oo. Here 77, E Co (R"),, E Co (R"-m) are "cap-shaped"functions such that nl = 1 on B1, rh = 1 on B1, where B1, B1 are the unit ballsin Rm, R"-m respectively.Proof. Sufficiency. Let (5.4) be satisfied. First suppose that 1 < p < oo. ThenVf E C°°(R") nWW(R"), by Theorem 12, we have that for almost all u E Rm

if (u10)1 <- M, (11f (u,17) IIL,,,(R"-m) + E II(D(°1) f)(u, n) IIL,,,(R"-m))171=1

where ry = ('y.+1, ..., ryn) E N _ and M1 depends only on n - m, p and 1. ByFubuni's theorem both the left-hand and the right-hand sides are measurablewith respect to u on Rm. By Minkowski's inequality and Fubuni's theorem weget on taking LP-norms

11f (U, 0)IIL,,.(Rm) <- M1 (II 11f (U, n)IIL,.,,(Rn-m)IIL,,,(Rm)

+E II II(D(°,ry)f)(u,rl)IIL,.,,(Rn m)IIL,,.(Rm)) <-M1IIfIIw;(Rn).

I'YI=1

Consequently, by Corollary 1, it follows that each function f E W, (R") has atrace on Rm.

Necessity. Let I < n pm and I - n pm < [3 < 0. Then, by Example 8 ofChapter 1, fp E WP(R"). On the other hand for each g E L;0c(Rm) and V E .B ,by the triangle inequality,

Ilfs(, v) - 9IIL,(B,) ? IvI" 117h 111,1(%) - II91IL,(B,) _+ 00


as v -a 0. Hence the trace of fp does not exist. If l = "p'",1 < p < oo and0 < -y < 1 - p, then, by Example 8 of Chapter 1, g, E Wp(R"), but a similarargument shows that the trace of gg on R" does not exist. 0

Remark 2 Assume that (5.4) is satisfied. By Remark 1 it follows that foreach f E Wp(R") the trace tr f E Lp(Rm) and

IltrfllL,(R-) <_ c3IIfIIw;(R"), (5.6)

where c3 > 0 depends only on m, n, p and 1. Moreover, if f E Coo (R") n WW(R" )are such that fk -+ fin Wp(R"), then f(., 0) -* tr f in Lp(R"`).

Thus, if we consider the trace space

tr..Wp(R") = {tr f, f E Wp(R")}

_ {g E Li°c(R") : 3f E Wp(R") : tr f = g},

then

tr1mWp(R") C Lp(R"). (5.7)

The problem is to describe the trace space. In order to do this we need tointroduce appropriate spaces with, in general, noninteger orders of smoothness.

5.3 Nikol'skii-Besov spacesIt can be proved that for I E N,1 < p < oo the definition of Sobolev spacesWp(R") is equivalent to the following one: f E Wp(R") if, and only if, f ismeasurable on R" and 3

IIf IIL,(R") + SUPIIOhf II `,(R") < 00.

hER",h00 Ihl

This definition can easily be extended to the case of an arbitrary positive l: onemay define the space of functions f, measurable on R", which are such that

Ilf IIL,(R") + sup IIAhf II11400) < 00,hER",h#0 Iht

where aENand0<I<<a.This idea will be used in the forthcoming definition. However, for reasons,

which will be clear later, in the case of integer lit will be supposed that I < a

3 One of the implications has been established in Corollary 8 of Chapter 3.

5.3. NIKOL'SKII-BESOV SPACES 203

(as in the case of noninteger 1). 4 Moreover, an additional parameter will beintroduced, providing more delicate classification of the spaces with order ofsmoothness equal to 1.

Definition 2 Let l > 0, or E N, a > 1, 1 < p, 9 < oo. The function f belongs tothe Nikol'skiT-Besov space BB,e(R") if f is measurable on R" and

IIf IIB, (R") = IIf IILp(R") + Ilfllblp,,(R") < 00,

where

if1<0<ooand

Ilfllbp.,(R") - (l (Iloh jhl1 R^))BIhn)I

Rn

Ilfllap,,(R") = supIlohfll ,(Rn)

hER",hoO Ihl

This definition is independent of a > I as the following lemma shows.

Lemma 1 Let l > 0, 1 < p, 9 < oo. Then the norms 6 II IIB,,,(R") correspondingto different a E N satisfying a> I are equivalent.

Idea of the proof. Denote temporarily semi-norms (5.9) and (5.10) correspond-ing to a by II II(°). It is enough to prove that II II(°) and II II(°+') are equivalenton L (R") where a > 1. Since Iloh+'fIILp(R") 2IIAAfIIL,(R"), it follows that

II < 211 ll(°) To prove the inverse inequality start with the case0 < I < 1, a = 1 and apply the following identity for differences

ohf = 2 1&21.f - 2 Ohf, (5.10)

which is equivalent to the obvious identity 6 x - 1 = 2(x2 - 1) - 2(x - 1)2for polynomials. To complete the proof deduce a similar identity involvingA"f, Ashf and ph+l f.Proof. 1. Suppose that 0 < l < 1 and Ill 11(2) < oo. By (5.11) we have

IlohfllL,(R") <- 2Ilo2hfllLp(R") + 2IIohfIlLp(R") (5.11)

4 The main reason for this is Theorem 3 below, which otherwise would not be valid.5 See footnote 1 on page 12.6Here x replaces the translation operator Eh where h E R" ((Ehf)(y) = f (y+h), y E R').


First let 8 = oo. Denote W(h) = IhI-1llohf 1IL,(R^) (Clearly, W(h) < oo for all

h E R", h 0 0. Then it follows that

cp(h) < 2'-',p(2h) + 2-'1If lI(2)

Consequently, Vk E N

W(h) <2'-'(2'-'cp(4h)+2-'IIfIl(2))+2-'IIfII(2) <...

< 2('-')ktp(2k h) + 2-' II f II(2) (1 + 21-1+...+ 2(1-1)k)

< 2('-1)ktp(2kh) + (2 - 21)-1IIf 11(2).

Let k be such that 2klhl > 1, then cp(2kh) <- I1o2khf IlL w.) 5 211f IIL,(R^)Hence,

co(h) <- 2(-')"+1llf IIL,(R-) + (2 - 21)-' Ilf II(2).

On letting k - oo, we get 11f11') < (2 - 21)-1IIfII(2) Thus,

(2 - 21)Ilf11(1) <- Ilfll(2) 5 211f110). (5.12)

If 1<8<coo, we set Ve>0r

(F) - (J (ll.hf DDL,(R") l0 dh )id

Ihi' ' FhI"(hl>c

Since 11Ahf IIL,(R^) <- 211f IIL,(R^), +G(r) < oo for all e > 0. From (5.12) it follows,after substituting 2h = 77, that

',(e) < 2'-'V5(2e) + 2-' 11f 1j(2),

and a similar argument leads to the same inequality (5.13).2. If then

(x - 1)° = 2-0(x2 - 1)° + (x - 1)0-2-0 (X2 - 1)°

= 2°(x2 - 1)-0 + P°-1(x)(x - 1)°}',

where

P°-1(x) = -2-°(x - 1)-1((x + 1)° - 2°) = -2-0 ) (x - 1)8-1.s=1

3

Hence,An f = 2'A &f + P°-,(Eh)Or+l f


and, since IIEhIIL,(R^)-*LP(R^) = 1,

II' hf IIL,(R") <- 2-°Ilozhf IIL,(R") + 2-°-1(3° - 1)IIOn+1f IIL,(R^)

The rest is similar to step 1.

We shall prove next that the norm 11 ' IIB,',(R") is equivalent to a similarnorm containing the modulus of continuity

U4(6,f)P = SUP IIAhfIIL,(R")Ihj<d

To do this we need several auxilary statements.

Lemma 2 (Hardy's inequality) Let 1 < p < oo and a < 1 . Then for eachfunction f measurable on (0, oo)

Ilto t f If I dXDIL,(o,oo) (; - a)-1IIx°f (xW IIL,(o )

0

(5.13)

Idea of the proof. Substitute x = yt, apply Minkowski's inequality for integralsand substitute t = y.Proof. We have

t i

IIta 1 If(x)I dx t°If (yt)l dyt f L,(0,oo) = II f L,(0oo)

0 0

I1

<- f Ilt° f (yt) II L,.,(0,c) = f y-°+o dy IIx°f (x) IIL, o,.)0

a)-'IIx°f (x)II L,(o,oo).

Remark 3 The constant a)-' in (5.14) is sharp. One may verify thisconsidering the family of functions fj where 0 < 6 < p(, - a), defined by

fs(x)=Ofor0<x< 1 and fs=x-°-19 forx> 1.7 Since f is measurable on (0, oo), the function F(y, t) :=if (yt)I is measurable on (0, oo) x

(0, oo) and we can apply Minkowski's inequality for integrals.


Corollary 2 Let 1 < p < oo and a < n - 1. Then for each function f mea-surable on Rn

llta vn n f If l dxllLp(0,oo)

<C 4

1 1 IxI np' f(X) 11Lp(R-), (5.14)

Ixl <t


Idea of the proof. If n = 1, apply inequality (5.13) and its variant for the case,in which in the left-hand side the integral over (0, t) is replaced by the integralover (-t, 0) and in the right-hand side the norm 11 - IIL9(o,.) is replaced byII IILD(-,,,o). If n > 1, take spherical coordinates, apply Minkowski's inequality,inequality (5.13) and Holder's inequality.Proof. Let n > 1. Then by (5.13)

llta vn n f If I dxllLv(O,oo)Izl<t

t

= Ilt° 1 f (J?-'If de)vntn

Lpj(O,-)S.-t 0

o f llta-("-')t f

Sn-, 0

< (vn(n - p - a))-' f lleaf (ef)IILn.,(0,.) dSn-'

Sn-1

<- (vn(n - 1 - a))-'u ( f ( f Q I f (g)I p) dSn-'S.-, 0

-c411 lxla n=f(x)IIL,(R.).

Next we generalize the trivial identity

(Ohf)(x) = 04) (x)+(Ah-nf)(x+n),

where x, h, rl E Rn to the case of differences of order o > 1.


Lemma 3 Let a E Rl, h, r) E R" and f E Lia(R" ). Then for almost all x E R"

(Ahf)(x) = E(-1)°-k (k) ((Ok, f)(x + (a - k)h)k=1

kf)(x + krl)). (5.15)

Idea of the proof. Replacing the translation operators Eh and Es by y and zrespectively, it is enough to prove the following identity for polynomials

(y r(-1)17-k (k) (zk - 1)°y°-k - (-1)k (k) (y - zk)°. 0 (5.16)k=1 k=1 /

Proof. Identity (5.16) is equivalent to the identity

(-1)°-k (0k

II

I (zk - 1)°y°-k = (-1)°-k I k) (zk - y)°k=0 k=0 \

which is clear since both its sides are equal to

k,m-0

Corollary 3 Let a E IV, h, 77 E R1, I < p < oc and f E Lp(R" ). Then

IlohfIIL,(R^) k (k) (IloenfliLp(R^) + (5.17)

Idea of the proof. Apply (5.16), Minkowski's inequality for sums and the invari-ance of the norm II . IIL,(R^) with respect to translations. 0

Lemma 4 Let a E N, 1 < p < oo. Then for all functions measurable on R"and dh E iR"

II' f IIL,(R^) <_ v, hI" l II' f IILp(R^) A, (5.18)

Inl<Ihl


k+m(-1)( k) (m) z

kmy°-k. 0

0101


Idea of the proof. Integrate inequality (5.17) with respect to 77 E B(2, l1).

Proof. If n E B(2,1l), then 1, h - kv E B(0, IhI), k = I,-, o. Hence, bysubstituting a = , h - _ e respectively, we have

v,(I2I)"IIohfIIL,(R^) <- 2 (k) \k/n J IIo{fIIL,(R^)SIfISIhI

Thus

II hf - 2(2ovnlhn-

1) f o

I,iI5lhI

Corollary 4 Let o E N,1 < p < oo afnd f E Lp(Rn). Then

' ±17w°(t, f )p <- vn l II1 f IIL,(R^) 1771n'17

I+iI<t

Idea of the proof. Direct application of inequality (5.18).

(5.19)

We note also two simple inequalities for modulae of continuity, which followby Corollary 8 of Chapter 3:

w°(b,f)p <_ 2°IIfII1,(Rn) (5.20)

and

w°(5,f)p: c45ÌIf1Iw;(R^),

where 1, o E N, l < a,1 0. Ifs E N, it follows, with s° replacing (s + 1)1, from the identity 8

e-1 -1

d,=0 ,,=o

and Minkowski's inequality. If s > 0, then

w°(s5,f),<_w°(Qs]+1)5,f)p<_Qs)+1)°w°(5,f)p (s+1)°w°(5,1)9.

8 It follows, by induction, from the case 8 = 1, in which it is obvious.


Lemma 5 Let l > 0, a E N, a > l,1 < p, 9 < oo. The norm

),d00

IIfIIBl(R^) = IllIILp(R")+ (f t1

0

1 1

is an equivalent norm on the space B,,e(R").

(5.22)

Idea of the proof. Since, clearly, Ilohf1lL9(R^) < wo(jhl, f)p, the estimate

11f11 Bp B(R*) < M1II f II B B(R^), where M1 is independent of f , follows directly

by taking spherical coordinates. To obtain an inverse estimate apply inequali-ties (5.19) and (5.14).Proof. In fact, by (5.19)

0l )r f (wa(61f)p)8

) < M2IItn_1-,vn n f`0 ` Inl<e

InI-"Ilonf IILp(R^) d'jIIL. (50)'

where M2 is independent of f.Since I > 0, the assumptions of Corollary 2 are satisfied and by (5.14)

IIf il(l) R") <_ Ill IIL,(R") + M311 InI-1 a llonf IILp(x")IIL,(0,.) < M4II f IIB;(R"):B' .O

where M3, M4 are independent of f .

Since the modulus of continuity is a nondecreasing function, it is possibleto define equivalent norms on the space BB,e(Rn) in terms of series.

Lemma 6 Let l > 0, a E Pi, a > l,1 < p, 9 < oo. The the norms

II tii(2) _ 11 ell ._ . - ( 1 e 1 800

k=1

and

k, J)) kl (5.23)

Ilfll(BP'(..) = IllIIL,(R^)+flp1e

(5.24)Lk=1

are equivalent norms on the space Bp9(R").


Idea of the proof. Apply (5.21) and the following inequalities for nondecreasingnonnegative functions p and a E R:

100

csEka-1`p(k) < rx' (x) _ <cs2k 1W(k)

and

k=2 k=1

00

C7 2-kap(2-k) f -< cg 2-k°tp(2-k),fxP(x)00

k=2 °

xk=1

where c5,..., c8 > 0 are independent of W.

Remark 4 The norm II IIB;,,(R.) is the "weakest" of the considered equivalent

norms on the space BP,B(R") and the norm II IlBlpe R,) (or any of its variants

II . II (Bp (xn) or II II(B? B(R)) is the "strongest" one, since the estimate II IIB;,,(Rn) <

M1 II . (I(' (R") is trivial, while the inverse estimate II' II ',(R") < M211 IIBP,,(R") isnontrivial. For this reason, estimating II IIBB,,(R") from above, it is convenient to

use this norm itself, while estimating some quantities from above via II II Ba,,(R.),

it is convenient to use the norm IIIIB),(Rn)

This observation will be applied

in the proof of Theorem 3 below.

Lemma 7 Let l > 0,1 < p, 0 < oo. Then Bp a(R") is a Banach space. 9

Idea of the proof. Obviously B,,e(R") is a normed vector space. To prove thecompleteness, starting from the Cauchy sequence {fk}kEN in BB,e(R"), deduce,using the completeness of Lp(R") and 10 Lpg(RIn), that there exist functionsf E L,(R") and g E Lp,e(R2") such that fk - f in Lp(R") andlhl-I-*fk(x) -+g(x, h) in Lp,B(R2n ). Choosing an appropriate subsequence {fk. }aEN, prove thatg(x, h) = Ihl-1-I f (x) for almost all x, h E R" and thus fk - f in B, e(R").

9 See footnote 1 on page 12.10 Lp,,(R2") is the space of all functions g measurable on Re", which are such that

II9IIL,,.(Ra..) = II 00.


Lemma 8 Let I > 0. The norm

=11(1+IKI2`)2(Ff)(OIIL2(R-) (5.25)

is an equivalent norm on the space B2,2(Rn).

Idea of the proof. Apply Parseval's equality (1.26) and the equality

(F(Def))(f) = 1)°(Ff)(C) = (tie' 2)° (sin h2

for f E L2(R"). \ /Proof. Since a + <_ f + v < 2 a2 + b2, a, b > 0, the norm IIIIIez.x(R")is equivalent to

(IIf IIL2(Rn) + f IhI-21IIohf IIL2(R.)

R"

_ (fR^

where f IhI-21-"sin' 2t dh.

R"

If n = 1, then after substituting h = ', we have

00

AI(S) = MIIfI21, M1 = f ItI-2-'sin2o 2 dt < oo,-00

since l > 0 and a > 1. If n > 1, we first substitute h = Afq, where A( is arotation in R" such that h = If Ini, and afterwards q _'I. Hence

An(t) = JI77I-21 nsin2o dq = M"ICI21, Mn = J

ItI-'-"sin2o 21 dt.R^ R^

If t i , = It11Tk, k = 2,..., n, then ItI = It1I 1 + ITI2, where ITI = (E Tk) .

k=2

Hence, applying (4.116), we haveao

Mn=M1 f 1+In2-21 nd-r=M1an-1f 1+p2-2-2 de<oo.R^-1 0


To complete the proof it is enough to note that K1(1 + It 11) <_ 1 + 22°A"(1;)K2(1 + IeIV), where K1i K2 > 0 are independent of t;.

Corollary 5 If I E N, then

WZ(R").

The corresponding norms are equivalent. Moreover, =IIfIIw;(R")

Idea of the proof. Apply Lemmas 8 of this chapter and of Chapter 1.

Next we state, without proofs, several properties of the spaces BP 9(R" ),which will not be directly used in the sequel, but provide better understandingof the trace theorems.

R.emark 5 If 1 > 0,. 1 0, 0 < e < 1, 1 < p, 9, 91, 92 < oo, then

Bpei(R") C Brpe(R") C BPI-P'81

Hence the parameter 9, which is also a parameter describing smoothness, is aweaker parameter compared with the main smoothness parameter 1.

Remark 6 If I E N, 1 <p:5 oo and p 96 2, then for each 9,1 50< oo,

Bp,9(r) T Wp(R").

Moreover, if 1 E N, 1 min {p, 2}, 92 < max {p, 2}, thecorresponding embeddings do not hold.

Remark 7 The following norms are equivalent to II f I I BD,,(R" )

IIfIIBk)0(R") = IIfIIL,,(R")+IIfII,a (R")+ k=5,6,7,8,


where

II!II ,,(R.) _ 2 ( f (IIz DtifIILa(RI))8 dh

1a1=mI I

1I I

lhlGN

fo,(n) ti- J t

0

H

Ilfll() -v'(f 1_te)(POW IILa(R^) 0dtbpl.#(RI) - ti-m ! t

J=1 0

n H woj t+ 8x hp(30) B itIlflld' R - L(f ( w P

I It=1 0

Here m E 1%, m < l < a+m, 0 < H < oo , ee is the unit vector in the directionof the axis Oxi and wQa gyp) is the modulus of continuity of the function cp oforder a in the direction of the axis Ox,. If 0 = oo, then, as in Definition 2, theintegrals must be replaced by the appropriate suprema.

There also exist other eqivalent ways of defining the space B,,e(R ): withthe help of Fourier transforms (not only for p = 0 = 2 as in Lemma 8), withthe help of the best approximations by entire functions of exponential type, bymeans of the theory of interpolation, etc.

Remark 8 It can be proved that

Wp(Ye) 1 Boo,P (r), l > p, 1 < p < oo,

and

°n(1-1)<I<n, I<p<q<oo.P P 4 P

These embeddings are sharp in terms of the considered spaces: the second lowerindex p can not be replaced by 8 < p.

Remark 9 In the sequel we shall use only the spaces Bpi (W) = Bpy(R"). Onecan easily verify by changing variables that in this case IlfllBp(R.) is equivalentto

IIfllB9,(Rp)=IllIILp(R")+(f f I(A'f)(x,y) (A' f)(y)

9-11


5.4 Description of the traces on subspacesWe recall that by Corollary 8 of Chapter 3

Ilohf C IhIÌIf IIu4(Itw) (5.26)

or

II (oAf)(x)< c9 IIfIIwo4(a"),

Ihlt Ilcp,.(a^) -where l E N, 1 < p < oo and c9 = nt-1.

Lemma 9 Let I E N, 1 < p < oo and f E w,(R"). Then Vh E R" for almostallxER"

tlhl

(oef)(x) = f (5.27)

0

) is the weak derivative off in the direction of e,where { = Ihl and (ofW

Kz(p,r)=(xQ*...*xe (r), 0<p<oo, -oo<r<oo,t

and xQ is the characteristic function of the interval (0, p).

Idea of the proof. Starting from Lemma 5 of Chapter 1, prove, by induction,that for almost all x E R"

Ihl IhI

(Ahf)(x)= f ... f (C'f )w(x+t(rr+..r))dr1...drl,

0 0

and apply the following formula

00 00 00

00 -00 -00

where K,(r) = JV(r), (p, V) E L1°°(R) and tb has a compact support.

I

5.4. DESCRIPTION OF THE TRACES ON SUBSPACES 215

Corollary 6 Under the assumptions of Lemma 4 Vh E R" and for almost allxER"

I( !1 Ìhl &

I \°et)(x)< (l - 1)!fr1(.)(x+r)dr. (5.28)

0

Idea of the proof. It is enough to notice that K1(p, r) = 0 for r < 0 or r > 10and to prove, by induction, that K1(p,r) < for 0 < r < lp.

Lemma 10 Let l E N,1 p. Then t1 f E wp(R") for almostall xER"

I!

(°f1(x) " < cl0 IIf I1w;(lt^) (5.29)hIhI

where )) cla > 0 is independent of f.

Idea of the proof. Take spherical coordinates and apply Corollary 6 and Lemma2.0Proof. After setting h = pt;, where p =IhI and Il E S", substitutingp = i and applying (5.28) and (5.14) we get

I = II IhI-1(°hf)(x)IIL9,h(1t") - I IIpt+

(A' f)(x)IILp.a(0'-)IILp.E(s"-')

!Q

(< (l 11)! II IIp1+' rr'

).(X drll Lp.e(O,oo)II0

1 1 - A

/ r'-1I(01 )w(x+ST)I drIILp,.(0,.)IILp.E(S"-1)P)!11111.-1+1+P

r0

(l - a)(l - 1)! II IIr 81;1)

Since < nt-1, where a E 1% and Ia! = 1, we have, for almost all x E R",

\81;1/w(z)I - I'9xj+)(z)Ij1=1 j)=1

11 If n = I = 1,p > 1, then cls = p' and (5.29) is equivalent to (5.13), where a = 0 and fis replaced by f;,.


E a',I(D°f)w(x)I < n-' E I(D°f)w(Z)I.I°I=1 i°I=1

Consequently, by Minkowski's inequality,

n1_1l1_aD° x + TI (l - p)(l

v

1). 101=1

11

T nD`(f)w( )

Lp..(O.oo) Lp.1(S°-1)

1 13 1 1 1 ID

_ (1-p)(1 D 1)1I II(D°f)w(x+y)IILp.V(o:') _

(l np)(l 1)!IIfIIw,(').

Theorem 3 Let 1, m, n E N, m <n, 1 <p:5 oo and l > "p"'. Then

"ttA,,,WP(W") = Bp pm(R'n). (5.30)

R.emark 10 Assertion (5.30) consists of two parts: the direct trace theorem,stating that V f E WD(R") there exists a trace g E B1 -

apm (Rn' ), and the ex-

tension theorem, or the inverse trace theorem, stating that Vg E B, '-"P (R'n)there exists a function f E WW(W) such that f I

R-g. Actually stronger

assertions hold. In the first case it will be proved that the trace operatornm

tr : W, (Rn) -iBp1_p

(R"') (clearly linear) is bounded. In the second caseit will be proved that there exists a bounded linear extension 12 operatorT : Bp °m (R"") -3 Wp(R").

Idea of the proof of the direct trace theorem. Start with the case m = 1, n = 2.If I = 1, apply for f E COO (R2) nWp (R2) the identity

f(u+h,0)- f(u,0)= f(u+h,0)- f(u+h,h)

+f(u+h,h)- f (u, h) + f (u, h) - f(u,0) (5.31)

and inequality (5.29) with l = 1. If l > 1, deduce a similar identity for differ-ences of order 1. In general case take, in addition, spherical coordinates in R'"and R". .Proof of the direct trace theorem. 1. Let Vf E COO(Rn) n Wp(R"). It is enoughto prove that

IIf('10)11 z(R-) < Ml IIfIIW (R"), (5.32)

12I.e., Vg E Bp P (R'"),9 is a trace of Tg E W, (R") on am.


where Ml is independent of f. In fact, if f E Wp(R"), then we consider anyfunctions fk E C' (R") fl W,(R"), such that fk -+ f in WW(R") as k -> oo.By a standard limiting procedure (see, for example, the proof of Theorem 1

t_ n-mof Chapter 4), it follows, since the space Bp ' (RI) is complete, that there

exists a function g E By ' , (R"`) such that g in Bp n m'(RI). ByRemark 2 the function g is a trace of the function f on R'". Moreover,

IltrfIlBD (RT)< M IIfIIWW(R") (5.33)

Since inequality (5.6) is already proved, it is enough to show that the inequality

IIf(',0)IIbpn-m(Rm)

< M2IIfIIWo(R")

holds V f E CO0 (R") n W, (R" ), where M2 is independent of f .2. Let l = 1, m = 1, n = 2, 1 < p < oo and f E C' (R) n Wp(R2). By

(5.31) and (5.26) we get

II(Au,hf)(u,O)IILp..(R) < II(Av,hf)(u+h,0)IIL,,,,(R)

+II(Au,hf)(u, h)IIL,,.(R) + II(ov,hf)(u, 0)IIL,,U(R)

< 2 II(Av,hf)(u, 0)IIL,,r(R) + IhI . II BL (u, h)IILP.r(R)

Hence, applying Fubuni's theorem and inequality (5.29), we get

II Ihl-'II(ou,hf)(u,0) IIL,..(R) IIL,,h(R)

< 2 11II IhI-'(Av,hf)(u,0) IIL,.,,(R) IIL,.,,(R) +II2U-

2p' 11112(U, v)II L,.,(R)IIL,.%(R) + IIOUIIL,(RI) <

2p'IIfIIwD(R')

3. Let next l > 1, m = 1, n = 2. The following identity

(4,,hf)(u, 0) (O;,,hf)(u + Ah, 0)A=0

(A) (D;,,nf)(u, Ah) (5.34)A=1


is an appropriate generalization of (5.31) for differences of order 1 > 1. 13 By(5.26), as in step 2,

II(^'-.!.

,1f)(u,0)IIL,..(R) 5 M3(II(ov,hf)(u,0)IIL,..(R)

+Ihl'II

ah)IILp..(R)/,

where M3 = 2'. Hence, by inequality (5.29)

IIf(',0)IIiP= II lhI-'II(Dub(R) ,hf)(u,0)IIL,..(R)IIL,.h(R)

M4(II II IhI-'(Au,hf)(u, 0)IILy.,,(R)IILyr(R)

+ II II(u, ,\h)

X=I IILo..(R)IILs.a(1d

r l l<M6 (II 8v' II out (u, .\h)

<IILa(R2)) MsII f Il, (Ra),

where M4, M5 and M6 are independent of f.4. In the general case, in which 15 m < n, 1 > 1, we apply the identity

(Au,hf)(u, 0) _ )' (1) (Ov lhlnf)(u +) h, 0)a=o

(-1)1- (%X ,hf)(u, Alhlrl), (5.36)A=I

where r) E S"-", which also follows from (5.35) if we replace x by E4,h andy by EvJhl,,. Taking spherical coordinates in Rm and using equality (4.116), weget

l1f(',0)'1bv Rm) M7(IIIIIhI-t+ fO'IhInf)(u,0)IILp.h(Rm)IIL,,.(Rm)

13 This follows from the obvious identity for polynomials

(-1)'(Z - 1)'(v -1)' + (x - 1)'(1 - (-1)'(v - 1)')

a)Za(va=a \ / A=1

(5.35)

if x is replaced by Ed.h and v by Eh.


+E II II (D(7,olf)(u, Alhl77)IIL,,,,(Rm)IIL,,u(Rm))171=1 A=1

=a.M7(lllle `+°

+::ll lien a-'171=1 A=1

Here M7 is independent of f. Taking Lp norms with respect to q E Sn-m-1

and applying inequality (5.29), we get

_i 1

Ilf(,0)Ilbv"°m(Rm)

G On°mQn,M7(ll II Ihl-`(oV,h)(U, O)IIL,,h(w m)IIL,,u(R=)

+ IIII(D(7,o)f)(u,Av)IIL,,(R^-m)IIL,,.(R=))

I71=1 A=1

< M8(II 1 Il(D(0,0)f)(u,v)IIL,,,,(R--m)IIL,,,.(RV)

I0I=1

+ E IID(7io)(u, )tv)IIL,(R")) <_ M9 U 114(R-),IIt4(Rn),171=1 A=1

where M8 and M9 are independent of f.

In the proof of the second part of Theorem 3 we shall need the followingstatement.

Lemma 11 Let I E N, l > 1. Suppose that the functions A, v E havecompact supports and satisfy the equality 1a

A(z) - (-1)1_k(k) knv\/, z E Rn. (5.37)k=1

Then bf E Li°C(Rn) for almost all x E Rn

f (Ohf)(x)A(h) dh = f (Oh f)(x)v(h) dh. (5.38)

Rn R-

14 We note that from (5.37) it follows that f z'A(z) dz = 0, s = 1, ...,1- 1.R"


Idea of the proof. Notice that from (5.38) it follows that

f A(h) dh = (-1)1+1 J v(h) dh, (5.39)

Rn Rn

expand the difference Oh f in a sum and use appropriate change of variablesfor each term of that sum.Proof. By (5.37) and (5.38)

f(f)(x)(h) dhRn

_ E(-1)1-k (k) J f (x + kh)v(h) dh + (-1)h f (x) J v(h) dhk=1 Rn R.

1 f f(x+z)v(k)dz- f(x)J A(z)dzk=1

an Rn

= f (f (x + z) - f (x))A(z) dz = f ('&hf)(x)A(h) dh.Rn Rn

Let w E Co (R") and let w6 where 3 > 0 be defined by wa(x) = -nw(17). Wedenote by the operator defined by A6,W f = w6 * f for f E L a(R" ). (If,

in addition, suppw C B(0,1) and f wdx = 1, then A6,_ = A6 is a standardRn

mollifier, considered in Chapters 1 and 2).

Lemma 12 Let I E N,1 0 is independent off and 5.

Idea of the proof. Notice that

(A6,%f)(x)-f(x) = f (f(x-z8)-f(x))A(z)dz = f (Di:6f)(x)v(z)dz (5.41)R's R-

15 If I = 1, then A = v and (5.40) coincides with (1.8).


since by (5.39) f adz = 1, and apply Minkowski's inequality for integrals andR"

(5.21). 0Proof. Since the functions J1(-a) and v(-a) also satisfy (5.37), equality (5.38)still holds if we replace µ(h) and v(h) by a(-a) and v(-6). After substitutingh = -zb we obtain (5.41). Let r > 0 be such that suppv C B(0, r). ByMinkowski's inequality for integrals and (5.21)

II A6,Af - f IIL9(R') IIA` zbf IIL,(R")jv(z)I dz < SUP IIAhf II1IIL,(W')

R"Ihl<rd

= Wr(rb, f)PIIUIIL,(Rn) < (r + 1)ÌIvIIL,(Rn) W,(b, f)p.

Corollary 7 In addition to the assumptions of Lemma 12, let u E C0 (R" )If f /tdx = 1, then Vf E Lp(R")

Rn

IIA6,a.µf - f IIL,(Rn) < C12 W10, f )p,

and if f µ dx = 0, then V f E Lp(R° )R"

(5.42)

II A6,A.pf IIL,(Rn) < C13 W10, D p, (5.43)

where c12, c13 > 0 are independent off and b.

Idea of the proof. Inequality (5.42) is a direct corollary of (5.40) because inthis case f (A * µ) dx = f A dx f ,u dx = 1. If f µ dx = 0, starting from the

Rn R" R' R"equality

(A6,a.µf)(x) = f ( f (f (x - zb - Cb) - f (x))A(z) dz) /i(C) df,R. R.

argue as in the proof of Lemma 12. OIdea of the proof of the inverse trace theorem. Define the "strips" Gk by

Gk = {v E R"-m: 2-k-1 < Ivi < 2-k}, k E Z.

Consider an appropriate partition of unity (see Lemma 5 of Chapter 2), i.e.,functions tpk E Co (RI), k E Z, satisfying the following conditions: 0 < k < 1,

00E '+Gk(v) = 1, v # 0,

kc-0o

Gk C SUPP?bk C {v E 1r_- : 8 2-k-1 < Ivl 2-k}


C Gk-1 U Gk U Gk+1 (5.44)

and

I(D7Y'k)(v)I :5C14 2kI"I, k E Z, v E Ir-m, 7 E N-'", (5.45)

where c14 > 0 is independent of v and k.

Keeping in mind Definition 2 of Chapter 2, for g E Bpn

m° (R'") set

(T9) (u, v) = L 16k(v)(A2-k,W9)(u), (5.46)k=1

wherew = A * A (5.47)

and the function A is defined by equality (5.37), in which n is replaced by mand v E C0 (Wn) is a fixed function satisfying 16 f vdu = (-1)1+1

Prove that g is a trace of Tg on RI by applying Definition 1 and property(1.8). To estimate IIT9IIL,(Rn) apply inequality (1.7). Estimate IID°T9IIL,(R^),

where a = (fl, y), Q E IV, -y E Alo-m and IaI = 1, via 1191j(',)--. . To do thisB, a (R")

differentiate (5.46) term by term, apply inequalities (2.58), (5.42) and (5.43)and the estimate

IIDyV)kIIL,(R--m) < C152k(IYI-1 ), (5.48)

where c15 > 0 is independent of k, which follows directly from (5.45).Proof. 1. By the properties of the functions 1Iik it follows that the sum in (5.42)is in fact finite. Moreover,

s+1

(Tg)(u,v) _ *k(v)(A2-'%,Wg)(u) on Ii"" x G, (5.49)k=s-1

and (Tg)(u, v) = 0 if IvI > 18. Hence Tg E C°°(R" \ R"`) and Va = (Q, ry)where ,B E N , 7 E

Nò-m

00

(D°(T9))(u, v) = F (D7 k)(v)D,6((Az-4w9)(u))k=1

By (5.39) and the properties of convolutions it follows that f w du = 1. If 1 = 1, then

A = v. In this case one may consider an arbitrary w E Co (R'") satisfying f w du = 1.R-


00

_ E(D"*k)(v)2k101(A2-k,A.DsA9)(u) (5.50)k=1

since, by the properties of mollifiers and convolutions,

D0(A2-k,A.A9) = 2k101A2-k,D8(A.A)9 = 2k101A2-k,A.DDA9

2. Let IvI < 16. By (5.44) tik(v) = 0 if k < 0. Hence

Ek(v) = 1. (5.51)

k=1

Let s = s(v) be such that 2-'-' < IvI < 2-'. Then by (5.51), (5.44), (5.42) andMinkowski's inequality

8+1

(T9)(', v) - 90 IIL,(Rm) = II 0A:(v)(A2-k,A.A9 -9)IILp(Rm)

k=s-1

s+1 s+1

_ 0k(v)IIA2-k,A.A9 - 9IILp(Rm) < MI E wt(2-k, 9)pk=s-i k=s-1

8+1

< M2 L.r2k((-"pm)wl(2-k, 9)p

k=s--1

s+1

< M3 I vI t n pm E 2k(1- p )wl (2-k, 9)p,k=s-1

where M1, M2, M3 are independent of g and v.

Since the function g E BB ° (WTh ), by Lemma 6 it follows that the quantitye D! )w` (2-k, 9)p -+ 0 as k - oo if 1 < p < oo and is bounded if p = coo.Hence

II(T9)(', v) - 9(')IIL,(Rm) = o(Ivl'-'), 1 < p < oo (5.52)

and

II(T9)(', v) - O(Ivl') (5.53)

as v -+ 0 (hence s -+ oo). In particular, by Definition 1, if follows that g is atrace of Tg on R.


3. By (1.7)

00

Etkk(v)IIA2-k,W9IIL,(Rm)

<_ M4II911L,(R"),

k=1

where M4 = IIWIIL,(Rm) Since (Tg)(u,v) = 0 if IvI >_ ]s, we have

Ms II9IIL,(Rm), (5.54)

where M5 = M44. Let a = (,Q, -y), where Q E I pi"', ry E Rip"-m and Ial = 101 + 11'I = 1. First

suppose that 1 < p < oo and ,Q 56 0. Since the multiplicity of the covering{V)k}kEZ is equal to 2, by (2.58) we have

00

IID°TgllL,(R^) < 2' (E IID"'Pk"IL,(1^-m).2kl0ip IIA2-1-,a.D8A9ll gym))k=1

Since f DOA du = 0, by (5.43) and (5.48) we haveRm

00 1M62k(1-PPm)P,l(2-k,g)p)' = M6IIgII(g;_!-

k=1B, (Rm)

(5.55)

where M6 is independent of g.00

If Q = 0, then -y i4 0 and by (5.51) E(D"V)k)(v) = 0 for v satisfyingk=1

0 < IvI < 16. Hence

00

(D(p,") (T9))(u, v) _ ,(D"eik)(v)((A2-k,A.a9)(u) - 9(u)), 0 < IvI < 16.

k=1

Furthermore, *k(v) = 0 if IvI > 16 and k > 2. Therefore

(D(°'1)(T9))(u,v) = (D"t l)(v)(A2-lA,.%9)(u), IvI > &

Consequently, by (2.58), (5.42), (5.48)

00 à(p") 1-1 IITY.I. Up II A _IIP 1D

IID (T9)IIL,(Rmxb ) < 2 ° ( L


M7II9II(3;_B, v (R")

and by (1.7)

IID(si7)(T9)IIco(R'"X`B )

= IID"'1IIL,(R.-m)IIA2-I,A.AgIIL,(R-) <- M8II9IIL,(Rm),

where M7 and M8 are independent of g.If p = oo, then the argument is similar. For example, if 6 54 0, then

IIA2-"A.Daa9IIL (R"')kEN

< Mg sup2klwi(2-k,g),,. = M9II9IIBl (Rm),kEN

where M9 is independent of g.From (5.54) - (5.57) it follows that

(5.56)

(5.57)

IIT9lIw;(R"} <- M1o IIghI(',__m(5.58)

B, (R-)

where M10 is independent of g.

Corollary 8 Let 1, m, n E N, m < n, l < p:5 00, 1 > n pm and let the operatorT be defined by (5.46). Then

TgIRm = g; D°(T9)I R," = 0, 0 < IaI < l - npm. (5.59)

Idea of the proof. Establish, as in step 2 of the proof of the second part ofTheorem 3, that, in addition to (5.52) and (5.53),

o(IvI'-I°I- ), 1 < p < 00, (5.60)

and

II D"(T9)(.,v)II L (Rm) = O(IvI`-IaI) (5.61)

asv-+0.Proof. Let a = (13,'y), where p E N, y E 1V0-m, and 2-1-1 < IvI < 2-8. If,6 96 0, then by (5.49) and (5.43)

'+I

E I(D"0k)(v)I2k'0IIIA2-1,a.D0a9IIL,(R-)k=s-1


s+1

< Ml 2k1°I wt(2-k, 9)pk=s-1

s+1

<E 2k(1-%m)

1(2-k, 9)p,L,k=a-1

where Ml, M2 are independent of g and v.If Q = 0, then by (5.49) and (5.51)

(5.62)

s+1

II (D(°'')(T9))(., v)IIL,(R-) = II(Dry'l')(v) (A2-k,a.a9- 9)DL,,(R'")

k=a-1

a+1

< M3 2kI01 IIA2-11,A.19 - 91IL,(st-)k=a-1

and by (5.42) we again obtain (5.62).

Relations (5.60) and (5.61) follow from (5.62) as in step 2 of the proof ofthe second part of Theorem 3. 0

The following stronger statement follows from the proof of the second partof Theorem 3.

Theorem 4 Let 1, m, n E N, m < n, 1 'p"`. Then there existsa bounded linear extension operator

l- -mT : Bp ' (R"') -4 WD(W) n COO(r\ R"`) (5.63)

satisfying the inequalities

Il lvl'°'-lD°(Tg)IIL,(Rn) < c1611911Bp a:; s(R,,,), lal > 0, (5.64)

and

11IvI-`(T9-9)IIL,(Rn) 5 c1711911Bo npm(Rm), (5.65)

where c16i c17 > 0 are independent of g.In (5.64) the exponent 1al - I can not be replaced by la1- l -e for any r > 0

and for any extension operator (5.63).


Idea of the proof. Consider the extension operator (5.44) used in Theorem 3.To prove (5.64) apply, in addition, the inequality 2--2 < IvI < 2-k+I forV E supp zfik. To prove the second statement apply Remark 11.Proof of the first statement of Theorem 4. 1. Let a = ((3, y), where 6 E N',yEl -',IaI>0ands=lal-l=l,6l+I'yl-1.Thenasfor(5.55)weobtain

II Ivl'(D' (T9))llL,(R")

00

< 21-1p II Ivl'D"OkIIL,(R"-m) IIA2-k.A.Dsa9llLP(Rm))P

k=I

M1 1 2-ksP2k(I'YI-" pm )P2kj1IP W1(2-k, 9)p) (5.66)k=1

00

= M, (2k(' P m ) W1(2-k, 9)P) P) D = MI I I9I I (3)_

k=l \ B, P(R-)

where M, is independent of g. The proof of the appropriate analogues of (5.56)and (5.57) is similar and we arrive at (5.64).

2. Furthermore, as for (5.56) and (5.57)

00(T9)(u, v) - g(v) = VGk(v)((A2-k,A,A9)(u) - 9(u)), 0 < IvI < 6'

and

k=I

(T9)(u,v) - g(v) = 01(v)(A2-y,A.A9)(u) - 9(u), IvI ? 16Hence by (5.42)

II Ivl-'((T9)(u, v) - 9(v))II( )L, R'" x B

<_ 2' a II IvI-'V)kllLP(R"-m) IIA2-k,A.,\9 - 9IILP(Rm))k=I

< M2 11911 '-

and

II IvI-'((T9)(u, v) - 9(v))IILP(Rmx`B>

<_ II(-% ) (IIA2--,A.A9llL,(Rm) + II9IIL,(R")) < M3 II91IL,(Rm),

where M2 and M3 are independent of g, and (5.65) follows.


Remark 11 Let m,nEN, m< n, R+={x=(u,v)ER" :v>0}17,IEN,1 0. We shall say that the function f belongs to the weightedSobolev space if f E Lp(R+), if it has weak derivatives Da f on R+for all a E N satisfying I aI = I and

II IvI'DofIIL,(R+) <00. (5.67)

101=1

We note that the set C°O(R") n W, (R+) is dense in WPv1,

(R+). This isproved as in Lemma 25 of Chapter 6.

Suppose that 1- s - "P"' > 0. Then

tr,. Wp, lol'(R`+) = Bp D (R"'). (5.68)

The idea of the proof is essentially the same as in Theorem 3.The proof of the extension theorem is like that of Theorem 4. If in (5.65)

IaI = 1, then the same argument shows that

II IvI'D°Tg IIL,(R-+):5 MI 11911

etc.

To prove the direct trace theorem one needs to follow, step by step, the

proof of the first part of Theorem 3 and apply the inequality

II Ihl'(uhf)(0) II _ <_ Mz II f IIw

IhI LD.A(Rn m) D.

where 1 0 and M2 is independent of f, instead of(5.29) (with n - m replacing n and x = 0). The last inequality, as (5.29), isalso proved by applying inequality (5.15).

Proof of the second statement of Theorem 4. Suppose that (5.64) holds with

j al - l - e replacing j al -1, where e > 0. Let g E BB D= (R^') \ Bp ` ^ Dm (1r).Then Tg E Wp' 1v1,1 _,_, (1) where 11 E N, 11 > l + e. Since g is a trace of Tg,

t+,-n-Mby (5.67) g E Bp D (R'n) and we have arrived at a contradiction.

We note that from (5.65) it follows, in particular, that TgIR = g. Thismay be deduced as a corollary of the following more general statement.

"We recall that v = (xm+1, ..., z") >0 means that zm+i > 0, ..., z" >0.


Lemma 13 Let 1, m, n E N, m < n,1 <p< oo,l > ' n'" if p > 1 andI > n - m if p = 1. Suppose that A is a nonnegative function measurableon R"-"`, which is such that 1IAIILp(&) = oo for each e > 0. Moreover, let

f E I40c(R"), for -f E N_' satisfying I ryl = I the weak derivatives f existon R" and

IIAf IILp(Rn) + F, II D;p°'7)f IILp(R-) < 00-1-f1=1

Then f IRm = 0.

Idea of the proof. Using the embedding Theorem 12 and the proof of Corollary20 of Chapter 4, establish that there exists a function G, which is equivalent tof on R" and is such that the function IIG(., v)IILp(Rm) is uniformly continuouson RI-1.Proof. Let us consider the case p < oo, the case p = oo being similar. ByTheorem 6 of Chapter 4 f E LL(R"), hence, f E W,(R") and for almost allu E R' we have f (u, ) E W,(R"'m). By Theorem 12 of Chapter 4 there existsa function E C(R"-'") such that Vv E R"-'

Ig4(v)I <- M1( IIf(u,')IILp(R"-m) + F, II (Dw°,7)f)(u, )IILp(R°-m))

171=1

where M1 is independent of f and u. Let G(u, v) = u E R", v E Rn-'.Then G - f on R" and

II IIG(u, v)IILp,.(R")Ilc,(R^-m) <- M1 ( IIf IILp(R") + E II D.(°'7)fIILp(R^))171=1

As in the proof of Corollary 20 of Chapter 4, let fk E Co (52) and 16

Ilf - fkllLp(R^) + E II D.(O'7)f - D(°'7)fkII4(w.) -+ 0171=1

as k - oo . Then, by the triangle inequality,

II IIG(u,v)IIL,,.(R=) - Ilfk(u,v)IIL,,.(Rm)

<-11 IIG(u,v) - fk(u, V)IILp,.(Rm)IICC(Rn-m) -i 0

as k -+ oo. Since the functions Ilfk(u, )IILp,.(Rm) are uniformly contin-uous on R"', the function H(.) = IIG(u, )IIL,,,,(R=) is also uniformly

18 The existence of such fk is establised as in Lemma 2 of Chapter2.


continuous on R"-'". So there exists urn H(v) = A. If A > 0, then

2II' < IIAHIIL,(R"-m) = IIARL,(R-) for sufficiently small e > 0. This ist' 0

impossible because IIakIL,(a.) = oo and IIafIIL,(R^) < oo. Hence A = 0, i.e.,

lim v)IIL,(R-) = 0 and by Definition 1 f IRm = 0. 0

The next theorem deals with the case p = 1, 1 = n - m, which was notconsidered in Theorem 3.

Theorem 5 Let m, n E N, m < n. Then

tr, Wl -'"(R'") = L1(Rm). (5.69)

Idea of the proof. The direct trace trace theorem follows from Theorem 2 and, inparticular, from inequality (5.6). To prove the inverse trace (- extension) the-orem it is enough to construct an extension operator T : L1(Rm) -> Wi (R"`+' )and iterate it to obtain an extension operator T : L1(Rm) - Wl -'(R"). How-ever, it is more advantageous to give a direct construction for arbitrary n > m.Start from an arbitrary sequence {6k}kEZ of posivite numbers 6k satisfying

M6k+1 <5 z , E 6k < 1 and consider the sets Gk = {V E R : µk+1 < IvI < µk},

k=000

where pk = E 6,. (Note that < 26k.) Verify that from the proof of Lemma=k

4 of Chapter 2 it follows that there exist functions'Pk E C0 (R"-'") satisfyingthe following conditions: 0 < V)k < 1,

00

E ?Pk(v)=1,v#0,k=-oo

GkCsuppz'kC{VER: uk+1-64 SIvISpk+ 4}

C Gk-1 U Gk U Gk+1 (5.70)

and

IID7VGkIIL1(R"--) < M16k-m-I7I -y E fq m, I'I'I 5 n - m,

where M1 is independent of k.ForgEL1(Rm)set

(5.71)

"0

(Tg)(u,v) = 1:Ok(v)(A6,,,g)(u), (5.72)k=1


where w is the same as in (5.46). Prove as in the second part of the proof ofTheorem 3 that

IIT91Iw, -m(Rn) <- M2 (II9IIL,(Rm) + Ew,(6k, 9)1), (5.73)k=1

where M2 is independent of g and 6k . Since wi(6k,g)1 --> 0 as k -+ oo, choose6k depending on g in such a way that, in addition, wl(6k,g)1 < 2-k II9IIL,(Rm)Hence

IIT9IIw; -m(R^) <_ M3 II91IL,(Rm), (5.74)

where M3 = 2M2. 0Proof. 1. Since 45k < µk - a < µk + a <

4

6k and I7I n - 7n, inequality(5.71) follows from (2.10):

IID"'GkIIL,(R°-m) = IID"A6k , XkIILI (F14+1-ak4 l SIUISaktl+6k-

5IVI:5Ak+;)

< M4 36k}1) + 6k 36k))

= M5(6k+m-I"I k-m-I"I) < 2M5 6k

where M4 and M5 depend only on n - m.2. Let Iv1 < µl - 4 . By (5.70) vyk(v) = 0 if k < 0 and hence

E Vik(v) = 1-k=1

Let s = s(v) be such that v E Ga. Then by (5.75) and (1.9)

a+1

II(T9)(', v) - II E l)k(v) (A2-k,W9 - 9) IILt(Rm)k=a-1

a+1

E IIA2-k,W9 - 9IILI(Rm) -+ 0k=a-1

(5.75)

(5.76)

as v -+ 0 (hence s -> oo). Thus by Definition 1 g is a trace of Tg on Rm.


3. Since (Tg)(u, v) = 0 if IvI > pi + a and pi +a < P0, we have

00

IIT9IIL1(R^) = ll F,0 kilA66,,W911L,(Rm)llL,(R^-m)k=1

00

< M6 II E'Pkll I19liL,(Rm) < M7µoII9IIL,(Rm) <- M7II9IIL,(Rm), (5.77)

where M6 and M7 are independent of g and 6k.4. Let a = (p, y), where 0 E No, y E loo-m and IaI = IQI + IvI = 1. If

130 0, as for (5.55) we obtain

00

IID°T9IIL,(R^) <- IID'kIIL,(R^-m)bk "IIIA6k .oa,9IIL,(Rm)

k=1

00 00

< Ms ` 9)1 = Me W1(bk, 9)1,k=1 k=1

where M6 is independent of g and 5k.

(5.78)

If,6 = 0, then starting from (5.56), where now 0 < IvI < pi + AL, and (5.57),where IvI > /'l + a , we have as for (5.58) and (5.59)

00

IID(0-")(T9)IIL,(R-.fi "4) s E IID"0k11L1(a"-m) IIA6,...a9 - 911L1(Rm)k=1

00

< Mg F, W1(6k,9)1 (5.79)

and

k=1

IID(°'')(T9)IIL1(Rmx°a = IID'',0lIIL,(R^-m) IIA6,,a.a911L,(Rm)

<- M10 II9IIL1(Rm), (5.80)

where M9 and M10 are independent of g and 5k. So we have established (5.73). 0

Remark 12 If m = n - 1, then in fact

II0kIIL,(R) = 26k, II01IIL,(R) = 4, 2 (5.81)

Given e > 0, this allows one to construct, choosing appropriate 6k = dk(g) , anextension operator T : L1(R"-1) -+ W1(R") satisfying IITII 5 2 + e.


Remark 13 The extension operator T : L1(Rm) -+ W°-'(R") defined by(5.65) is a bounded nonlinear operator since dk depend on g. It can beproved that a bounded linear extension operator T : L1(Rm) Wl -m(Rn)does not exist. However, there exists a bounded linear extension operatorT : L1(R'n) -+ Biym(Rn) acting from L1(Rm) into slightly larger spaceB12m(Rn) than Wi -" `(Wn) (see Remark 6).

Theorem 6 Let m, n E N, m < n. Then there exists a bounded nonlinearextension operator

T : Li(Rm) -+ Wj -m(R°) nc-(Rn \ Wm) (5.82)


II IvI1°1-(n-m)D°(Tg)IIL,(R-) <- C18II9IIL,(Rm),

and 19

IaI > 0, (5.83)

II IvI-(n-m)(T9 - 9)IIL,(RmXB,) < C19 II9IIL,(Rm), (5.84)

where cls, c19 > 0 are independent of g.In (5.83) the exponent Ial - (n - m) can not be replaced by Ict - (n - m) - £

for any e > 0 and for any extension operator (5.82).

Idea of the proof. As in Theorem 5 consider the extension operator (5.72) . Toprove (5.83) and (5.84) note, in addition, that IvI < 26k on supp'bk and

IIIvI171-(n-m)(D7iGk)(v)IIL,(R"_m) < M1, 7 E po-m, (5.85)

where M1 is independent of k. The second statement of the theorem is provedas the second statement of Theorem 4.Proof. 1. Since 4dk<.uk - a < µk - a < 48k and ID7(AaaXk)(v)I M2 dk171,

where M2 is independent of v and k, by (2.10) we have

sup Ivi7I(D' 'k)(v)I = max sup IvI171I(D7(Ak .Xk)(v)I,vER^-"

{g

by+,_Ivi<16w+,

sup Ivl171I(D7(AXk)(v)I} < M3,:5105 14 JA;

19 By Lemma 13 from (5.84) it follows directly that TglR'" = g.


where M3 is independent of k. Hence

IIIvII7I-(n-m)(D7iPk)(v)IIL,(Rn-m) G M3 II IvI-(n-m)IIL1( 6k+1<-Ivl<_76k)l 96k

3ak- 1= M3 Qn-m f dp = M3 an-,n in

bk+16k+1

and (5.85) is established with M1 = M3 an-m In 6.2. If a = (#, -y), where /3 E IT, y E No" and / 0, then as in step 4 of

the proof of Theorem 5

II Iv IIQI-(n-m)DQT9I1

L1(R. )

00

G M5 IIIVIIAI+IYI-(n-m)V)kIIL1(R^-m) bk lal

IIA6k,A.D$A91IL1(Rm)k=1

00

G M6 E 11 W1(6k,9)1k=1

M6M1 Ewl(bk,g)1 G M6MI119IIL1(Rm)k=1

The case a = (0, 'y) where y 96 0 is similar.3. As in the second step of the proof of Theorem 4

(79)(u, v) - g(u) _ ''+Gk(v) 9(u)), 0 < IvI < {ul - 4k=1

and

(T9) (u, v) - g(u) = W1(v) (A61,A.A9) (u) - 9(u), µl - 4 G IvI G 1.

Hence by (5.42)

II IvI-(n-m)((T9)(u, v) - 9(u))IIL1(RmxB1)

511 IvI-(n-m)ok(V) ((A6k,A.A9)(u) - 9(u)) II00 L1(R-)k=1

+11 IvI-(n-m),pl(v)(A61,A.A9)(u)lJL1(R°) + IIIVI-(n-'n)9(u)IIL1(R^,x{

61gv1<1})


00

<- E IIIvI-(n-m)V)k(v)IIL,(Rm)

W1(bk,9)1 + II IvI-("-')'01(v)IIL,(Rm) II9IIL,(Rm)k=1

+IIIvI-(n-m)

IIL,(;6,<IvI<1)II9IIL,(Rm)

< M7(II9IIL,(Rm)+EWI(bk,9)I) S 2M7II9IIL,(Rm),00

k=I

where M7 is independent of g. 0

Remark 14 Here we give the proof of the second part of Theorem 8 of Chapter2. Let 11 = R+ x E R" : x" > 0 }. First suppose that I > v and g E

Bp ° (R"-1) \ Bo 5(R"-1). By Theorem 3 there exists a function f E W,(R" )

such that f IRn- = g. Suppose that there exist functions ,p, E C0D(R+) n

WW(R"+), which satisfy property 4) and are such that II D°'p, x1°I-1-EIIL,(R+) < 00for all a E FA satisfying Ial = m > 1 + E. By Lemma 13 from (2.86) it follows

that W,IRn,

= f IRn_I = g. Since W. E WP (R+), where m E N, m > I+E,1+E-1

by the trace theorem (5.68) g E Bp ' (R"- I) and we arrive at a contradiction.If I = p = 1, the argument is similar: one should consider g E LI(R"-I) \

B1 (R"-1) and apply Theorem 5 instead of Theorem 3.

Let 1 , m, n E I i i , a E I + ) o ' . Suppose that IaI < 1 - n Dm for 1 < p < oo andIal < I - (n - m) for p = 1. By Theorem 6 of Chapter 4 and Theorem 2 itfollows that Vf E Wy(R") there exist traces tram Dwf. We note that thesetraces are not independent. In fact, let a where Q E N", y E Pla-m.Then20 tr5m D,(vfl'7) f = DO (trim Dw°'') f ). For this reason we consider only weak

°derivatives Dw') f and introduce the total trace of a function f E W,(R") bysetting

f = (tr1m Dw°''') f) , 1 < P:5 00, (5.86)I7I<I-nn Dm

and f ( (°,7) f) p =I.

(5.87)TTRm = trAm DwI7I_!-("-m),

20 If f E w; (R-) fC°°(R"), then this formula is clear. If f E WW(R"), it can be ob-tained by choosing a sequence of functions fk E W, (R") f l C°O (R"), which converges to f in(Wi)1oa(R"), and passing to the limit in the definition of the weak derivative.


In particular,

Tr. , f = (f 1 <p < oo.

We also define the total trace space by setting

Trim Wp(R") = jlritm f, f E Wp(R")}

Theorem 7 Let 1, m, n E N, m < n, 1 < p < oo. Then

fi BP I7I-n m (5.88)p(Rm), 1<p<00,

and

Tr.- WW(R") = TT Bi-17I-(n-m)(Rm) X fj L+(R-). (5.89)

171<+(n-m) 171=+-(n-m)

Idea of the proof. The direct trace theorem follows from the first part of Theo-rem 3. To prove the extension theorem (i.e., the inverse trace theorem), giventhe functions g., where I7I < 1- "p' for 1 < p:5 oo and I7I < 1 - (n - m) forp = 1, lying in the appropriate spaces, set

(T{g7})(u, v) _7 (T1g7) (u, v), 1<P,500, (5.90)

171<+- ^ Pm

where v7 = xm+1 ' ' ' xn", 7! = 7m++! .. 7,,! and the operator Tl is defined by(5.46) and

(T {97})(u, v) _ F, U, (T19-,) (U, v) + v+ (T297)(u, v), (5.91)171<+-(n-m)

7171=+-(n-m)

7.

where Tl is defined by (5.46) while T2 is defined by (5.72). Apply (5.59), (5.64)and (5.83).P r o o f . 1. Suppose that 1 < p:5 oo and let 7, µ E and I7I, IuI < 1- n

PIf

7 < p, i.e., 7j < 1A I, j = m + 1, ..., n , then by Leibnitz' formula D.(°'") (v7Tlg7)

is equal to 7! D.(°'"-7)(Tlg) plus a sum of the terms containing the factor v°where o 0. Otherwise Dw°")(v7Tig7) is a sum, each term of which contains


the factor V where a 54 0. Hence tr D.( ') (v7T1gg) is equal to -y! tr D.(0 A--Y) (Tlg)

if ry 1, by (5.54) we have

IIT{97}II4(R") < E IIT197IIL,(R") <_ M1 E lI97IIL,(Rm),

where M1 is independent of f .

171 <t- "

3. Finally, let a = (#,p), where Q E T, p E t -m and lal = 10 1 + Ip = lBy Leibnitz' formula

D°(v°T197) _ cd,7,pv7-p+vD(p,7)(T1g7)

0<v<p,v>µ-7

for certain cp,7,µ E N0. Hence by (5.64)

IID°(v°Ti97)IIL,(R") <_ M2 E0<v<jj,v>µ-7

M3 119,11 --M

and, consequently,

IIT({97})Ilwo(R") <_ M4 E II97IIBPP

(5.92)

I71<1--P

where M2, M3, M4 are independent of 9.,-4. If p = 1, then

IIT{97}IIw;(w )

< MaH<t-(n-m) 171=1-(n-m)

where Mr, is independent of g7. 0

(5.93)

Remark 15 As in Theorems 4 and 6, in addition to (5.92) and (5.93), we havethe following estimates

II Ivl1°1-tD°(T{g7})IIL,(1t") <_ a20 II97ll s (Rm), (5.94)


where 1 < p< oo, Ial > (- npm, and

IIIvl1"1-1(D(0,1')(T{97}) - 9N)IILo(R") < C21 1197IIB (Rm (5.95)

171<l_nD

where 1 0 and for any extension operator T. We alsonote that by Lemma 13 from (5.95) it follows directly that D(0'µ)(T{g7})IRm =gp Similar statements hold for p = 1.

Remark 16 From the concluding statements of Theorems 4, 6 and Remark15 it follows that the extension operators defined by (5.44), (5.72), (5.90) and(5.91) are in a certain sense the best possible extension operators, namely, inthe sense that the derivatives of higher orders of Tg, T{g7} respectively, havethe minimal possible growth on approaching R1.

5.5 Traces on smooth surfacesLet fl C R" be an open set with a C1-boundary. We should like to extendDefinition 1 to the case, in which W", IIi"" are replaced by (1, O) respectively.

We start with the case of a bounded elementary domain Sl C R" with aC1-boundary with the parameters d, D, M. Thus Q has the form

S2={xER":an <x"<V(2),.EW},

where .t = (x1, ..., x"-1), W = {-* E ll'-1 : a; < xi < b i = 1,...,n- 1}, -oo <a, < b, < oo, i = 1, ..., n - 1, -oo < an < b" < oo, and satisfies the definitionof Section 4.3.

Suppose that f E L1(f ). In the spirit of Definition 1 we say that thefunction g E L1(r), where r = {x E IIt" : x" = cp(s), 2 E W}, is a trace of thefunction f on r if there exists a function h equivalent to f on 12 such that

h( +tea) -1 g(.) in L1(r) as t - 0-, (5.96)

where e" = (0,..., 0, 1). Since (&) (2) I < M, x E W, i = I,-, n - 1, we havedri

(1 + (n - 1) M2)-#IIF(z,W(i))IIL,(w) <- IIFIIL,(r)

= J IF(2,,p(2))I (1 +F ((a`p)(z))2)-' d a < IIF(-,w(2))IIL,(w)8xi

w '=1

5.5. TRACES ON SMOOTH SURFACES 239

Consequently f(5.96) is equivalent to

J If (x, W(2) + t) - g(t, V(x))j dx -r 0 as t -a 0 - . (5.97)

w

Let the transformation y = 4b(-t) be defined by

9=z, yn=xn-<p(9), xEit, (5.98)

then

and

t(it)={yE R" <0,9EW}

I(I')_{yER' :yn=0,9EW}- W.Relation (5.97) means that g(V-1)) is a trace of f (V`)) on P(r).

Next suppose that an open set it C R" is such that for a certain mapA, which is a composition of rotations, reflections and translations, the setA(it) is a bounded elementary domain with a C'-boundary and I' is such thatA(r) = {x E IV : x" = -t E W}. Then we say that g is a trace off on rif g(A(-1)) is a trace of f (A(-')) on A(r) in the above sense.

Finally, suppose that it C R" is an arbitrary open set with a Cl-boundarywith the parameters d, D, x and M, and let Vj be open parallelepipeds satisfy-ing conditions 1) - 4) in the definition of Section 4.3. From the proof of Lemma3 of Chapter 2 it follows that there exists an appropriate partition of unity, i.e.,there exist functions 10j E C' (R) such that 0 < Vj < 1,supp' j C (6j) , j =

a

T,-s, E Vj(x)=1onIlandj=1

I (D°'1Gj)(x)1 5 c22 d-1Q1, x E W, a E No, j= 1, s, (5.99)

where c, > 0 is independent of x, j and d.

Definition 3 Let SZ C R be an open set with a Cl-boundary and f EL1(il f B) for each ball B C W1. Suppose that f = fj, where supp fj C Vj

j=1and f j E L1(it n Vj ). If the functions gj are traces of the functions f j on

aV n m, j = 1, then the function E gj is said to be a trace of the function

j=1f on M.


Remark 17 One can show that Definition 3 does not depend on the covering

{V; } and on the representation f = P_ fj and satisfy the requirements to theP_j=1

notion of the trace analogous to the requirements 1) - 4) at the beginning ofSection 5.1.

Next we need to define the spaces Bp(8S2) where 1 > 0, 1 < p < oo. Wefollow the same scheme as in defining the notion of the trace. If f2 is a boundedelementary domain with a C'-boundary and the function f is defined on r, wesay that f E B,(r) if f (2, V(2)) E Bp(W) and set

IIfIIBp(r) = IIf(4)(-1))IIBp(a(r))

II Il s; (W) is defined as in Definition 2, where n is replaced byn - 1, R" by W and, in (5.8), (5.9), Ilohf IILp(Rn) by IIonf IILp(W.,hi)

If SZ is such that for a certain map A, which is a composition of rotations,reflections and translations, the set A(Q) is a bounded elementary domain witha C'-boundary, then f E Bp(1') if f (A(-')) E BB(A(r)) and

IllIIBp(r) = IIf(A(-'))lle;(a(r)) = Ilf(A(-'))IIaD(n(r)),

where A = 4i(A).

Definition 4 Let l > 0, 1 < p < oo and let SZ be an open set with a C'-boundary. We say that f E B,(81)) if f t; E BB(V1(l 8S2), j = T,-s, and

IIfIIBp(en) _t IIfo;IIB;(V; ))p

i8

p < oo. (5.100);=1

Here A; = 4sj(A;) and 4i; is defined by (5.98), where W, W are replaced by Bpi,W; respectively.

Remark 18 In the case l = k - p, where k E N and 1 < p < oo, which willbe of interest for us, from Theorem 8 below it will follow that, for open sets Chaving a C*-boundary, this definition is independent of {V;} and {iii}. As forthe general case, see Remark 19.


For the function f defined on an open set 0 C R we write fo for itsextension by 0 to R". If f E WP(SZ) and supp f C S2, then, by the additivityof the Lebesgue integral and the properties of weak derivatives, fo E Wp(R")and Ilfollw;(R.) = IIfIIwo(n) We shall need an analogue of this statement forthe spaces B,(S2), which has the following form.

Lemma 14 Let 1 > 0, 1 < p:5 oo and b > 0. Then for each open set SZ C R"and V f E B' (Q) satisfying supp f C 06

IIfoIIB;(R") < C23 IIf IIB,(n)

where c23 > 0 is independent off and Q.

Idea of the proof. Note that for IhI < 2

IlohfOllL,(R") < I IOhpJ IIL,(fl0. 1) + IIOhpJOIILy(`ltelhl) - IIL9(S2nlhl)

Proof. From the definition of the spaces B,(c) we have

(f p dh nIIfoIIb',(R") < (J (Ihl_ÌIAhf IIL,(n,jhl)) Ihln)

R.

(IhI_'IIAhfOIILP(°sZe,hj))vdh V

+( f hI )I

IhI? I&

iS IIf IIb,,(n) + 2°IIf0IIL,(R^) ( f Td h_ pi)'

< M1 IIf IIB;(n)

Ihl?6

where Ml depends only on n, i, a, p and 6, and (5.101) follows.

(5.101)

0

Theorem 8 Let I E N, 1 p, (5.102)

and

tre.W1(SZ) = Li(8Il). (5.103)


Idea of the proof. 1. To prove the direct trace theorem establish, by Theorems

3 and 5, that the trace gj of f O, exists on Vj n&2, and gj E By ° (vi n an) ifl>1and g,EL1(Vn8f)if I=p=1.

2. To prove the inverse trace (_- extension) theorem, given a function

g E B,, ° (8SI), consider the functions on W? , extend them by zeroto R"'' preserving the same notation, and set

d

T9 = E (Aj), (5.104)j=1

where To is a modification of the extension operator (5.46) for I > 1, respec-0

tively (5.72) for I = p = 1. Namely, the sum E in (5.46), (5.72) must bek=1

00replaced by E , where ko is such that

k=ko

supp Toh C (supp h)a x B(0, d). 0 (5.105)

Proof. 1. By Corollary 18 of Chapter 4 f tljj E WW(Vj n 11) and by Lemma16 of Chapter 4 (f1/ij)(A 1)) E WI (A,(Vj n ()). Since supp (ft/ij)(A(-1)) CAj(Vj n n), the extension 21 by 0 to R" of the function (fiPj)(A(71)) is suchthat E W,(R"). Hence by Theorem 2 there exists a trace hj ofthis function on R"-1 and therefore on Wj* = Aj(Vj n 80). This means that

gj = hj(Aj) is a trace of ft/ij on V n BSI. So by Definition 3 g = E gj is aj=1

trace of f on 852. Moreover, if I > 1, then by Theorem 3

1101 ,-1 = Ilhjll ,-1 llhjIl -1 < M1BP P(Vjn0n) BP P(Wj) BP (Re-1) - P

Finally, since gj = gt/ij, by Definition 4, Corollary 18 of Chapter 4, (5.99),Minkowski's inequality for sums and (2.59), we have

11911By ;(OA) - ( 11 11

P

B` Pv)Pj=1 a jnacl)

< Ml Mz IIfIIWi(Vj ))P1=1 j=1

21 We preserve the same notation for the extended function.


< M 2 'P ' + E (tIIDwIILp(V,nn)) P) < Mp x1II f II w;(a)i=1 1a1=1 t=1

where M1 and M2 are independent of f.

If l = p = 1, then, by Theorem 5, in the above argument Bp P should bereplaced by L1.

2. If To is defined by (5.72), then by (1.4) supp Toh C (supp h)2-k0 xB(0, 2-10+ ').Hence (5.105) follows if 2-k0+1 < d. If To is defined by (5.72), thensupp Toh C (supp h)36''o and (5.105) follows if, say, 45ko < d.

Let gi = Since by Lemma (5.105) supp(Togi)(A,) C Vi n fIand supp Tog, C Ai (Vi n fl), by Lemma 16 of Chapter 4 we have

II(To9i)(Ai)IIWp(R°) = II(To9i)(Ai)IIwL(Vjnn)

< M3 IITo9jIIwy(Aj(Vjnn)) = M3 IITo9,Ilwp(Rn),

where M3 is independent of g and j.By the proofs of the Theorems 3 and 5 gi is a trace of Tog, on R"-' , hence

(g0i)(Aj(-1)) is a trace of To((gtii)(Aj(-1))) onWj* = A,(Vifl8f ). Consequently,

gtpi is a trace of To((gt/ii)(AJ-')))(Ai) on vi n 8t and, by Definition 3, g =

P, gii is a trace of Tg on 8f2.j=1

Suppose that I > 1, the case I = p = 1 being similar. By Theorem 3 andLemma 14 we get

IITo9jIIww(R") < M4II9tIIay 1P(R^-1)

1

< Ms II (9Di)(A )II ,-1 = Ms -1J BP P(W;) 8P P(vjnen)

where M4 and M5 are independent of g and j. So

a s

IIT9IIww(R-) = II t(To9i)(Ai)II ww(R^) < M3 t IITo9JIIww(Rn)i=1 i=1

a

< M3 Ms t II90iII 1-1 = M3 MsII9II ,-i . 0i=1 BP P(Vjn8n) 8P P(8n)

Remark 19 We note that in Theorem 8 the coverings IV-1 and the partitionsof unity {iii} could be different in the first and the second parts of the proof.


1- 1From this fact it follows that Definition 4 of the spaces Bp ' (60), l E N, I >

p, does not depend on {Vj} and {Oj} for the open sets with a C'-boundary.Consider two coverings {V,k} and partitions of unity {1Pj,k}, k = 1,2, andlet II ' II(" i be the norms defined with the help of {Vj,k}, {',j,k}. Then by

Bp '(8n)(5.102) 'f E Wp(Q)

IIfII('i-1 _< M1 IIT2f llwo(n) :5 M2 IIf ii(2;-i , (5.106)Bp '(80) Bp '(8n)

where T2 is defined by (5.104) for {Vj,2} and {0j,2} and M1, M2 are independentof f. Similarly we estimate (I II ' II(1;_i Hence the norms

By '(8n) Bp '(8n)

II ' and II ' (I(2_i are equivalent.Bp '(8n) Bp '(8n)By this scheme it is also possible to prove, applying the trace theorem (5.68),

the independence of Definition 4 of {Vj} and {7ij} for the spaces B,(a1l) withan arbitrary l > 0. In this case one should verify that an analogue of (5.68)and Theorem 8 holds for the spaces Wp,Q, (1l), where e(x) = dist (x, 09S1), andreplace (5.106) by

Ill IB,(en) < M2 IIT2fIIw;,,(n) <_

whererEN,r>1+1,s=r-l-1 and8S2EC'.

For an open set Q C Rn with a C'-boundary let v(x) be the unit vectorof the outer normal at the point x E Oft Hence v(x) =where yj are the angles between v(x) and the unit coordinate vectors ej. Forf E W, (11) the traces of the weak derivatives D,f exist on asp if Ial < 1- 1.We define the weak normal derivatives by

a'f a'faLs w E COS COS -Yj, (ax

... axj...... j,=1 7i j.

The total trace and the total trace space are defined by

respectivelyTree W (1l) = JTren f, f E WW(Q)}.


Theorem 9 Let 1 E N,1 < p < oo and let 0 C R" be an open set with aCI-boundary. Then

1-1 1-a- ,

= Bp ° (852), 1 < p< coo,Tr80 WW (1l) 11a=0

and

(5.107)

1

l1-2

menW1(1l)=flap s-1(OC) x L,(852). (5.108)8=0

Idea of the proof. Combine the proofs of Theorems 7 and 8. 0

Remark 20 If p > 1, then as in Remark 15 one may state that there exists abounded linear extension operator

1-1

T:flBp' °(852)-,Wp())nC°°(52),a=0


II

18k(T{9a}) IIC 1-1` 8Uk Lo(ft)

c24 119a11B,, k > 1, (5.109)3=0

and

8't(T 1-'IIPfc-1(

{k'}) - 9k) II S 025E 119311 1-.-1 0 < k < 1, (5.110)8s, Lp(f1)

8=0By °(on)

where e(x) = dist (x, 8f2) and c24, c25 > 0 are independent of ga.In (5.109) the exponent k - l cannot be replaced by k - l - e for any e > 0.If p = 1, then a similar statement holds. (We recall that in this case the

extension operator T is nonlinear.)

Remark 21 The problem of the traces on smooth m-dimensional manifoldswhere m < n - 1 may be treated similarly, though technically this is more

n-1

complicated. Suppose that 52 C R" is an open set such that SZ = U Fm,m=0

where Fm are m-dimensional manifolds in the class C' and rm n r1, = 0 ifm & p. (Some of I'm may be absent.) Let, for example, 1 n - pl, then for each f E W, (1) the trace off on rm exists.


Moreover, the traces of the weak derivatives Daf also exist if dal < l - npm.For this reason the total trace and the total trace space are defined by

n-pl<m<n-1

Tra, Wp(0) = {Tre., f E Wp(f2)}

respectively. Here D ,w =( - are weak derivatives with respectw

to an orthonormal set of the normals u1 i ..., vn_m to rm. The appropriategeneralization of (5.107) has the form

Tre.Wp!1G(l) = -1rj rl Bp p (O).nm=0 I0'I<1- n-m

P

Similarly one may generalize (5.108).This statement plays an important role in the theory of boundary-value

problems for elliptic partial differential equations, because it explains whatboundary values must be given and to which spaces they can belong.

Chapter 6

Extension theorems

The main aim of this chapter is to prove that under sertain assumptions on anopen set S2 C R" there exists an extension i operator

T : Wp(SZ) -+ Wp(R"),

which is linear and bounded. The existence of such an operator ensures thata number of properties of the space WI (R") are inherited by the space Wp(fl).Examples have been given in Section 4.2 (Remark 11 and the proof of Theorem3) and Section 4.7 (Corollaries 20, 24 and the second proof of Theorem 13).

6.1 The one-dimensional caseWe start with the simplest case of Sobolev spaces W, (a, b), in which it is possibleto give sharp two-sided estimates of the minimal norm of an extension operatorT : WI(a, b) - W,'(-oo, oo).

Lemma 1 Let -oo < a < b < oo. If f is defined on [a, c) and is absolutelycontinuous on [a, b] and [b, c], then f is absolutely continuous on [a, c].

Idea of the proof. Derive the statement directly from the definition of absolutecontinuity on [a, b) and [c, b].Proof. Given e > 0, there exists b > 0 such that for any finite system of disjointintervals (a;l), f3;1)) C [a, b] and (a;2), (2)) C [b, c] satisfying the inequalitiesE(p;') - a;')) < d, j = 1, 2, the inequalities E If (a;')) - f(Q;f))I < 2, j = 1, 2,

i This means that (T f)(z) = f (z), if z e i2.

248 CHAPTER 6. EXTENSION THEOREMS

hold. Now let (ai, Qi) C [a, b] be a finite system of disjoint intervals satisfyingai) < J. If one of them contains b, denote it by (a*, Then

If(ai) - f(Qi)I 5 If(ai) - f(Qi)I + If(a") - f(b)I

+I f (b) - f (,6*) l + If (ai) - f (Qi) I < e.

(If there is no such interval (a', Q'), then the summands If (a') - f (0*) 1 andIf (b) - f (,Q')I must be omitted.)

Lemma 2 Let I E N,1 < p < oo, -oo < a < b < oo, f E 14, (a, b) andg E WP(b, c). Then the pasted function

h=l9

on (a, b),on (b, c).

belongs to WP (a, c) if, and only if,

f (3)(b-) = g.(') (b+), s = 0, 1, ..., l - 1, (6.2)

where f,(,,')(b-) and gw')(b+) are boundary values of and gw") (see Remark 6of Chapter 1).

If (6.2) is satisfied, then

IIhIIWW(a,c) 5 IIf II W,(a,b) + 11911Wp(b,c) (6.3)

Idea of the proof. Starting from Definition 4 and Remark 6 of Chapter 1, applyLemma 1.Proof. Let f, and gi be the functions, equivalent to f and g, whose derivatives

exist and are absolutely continuous on [a, b], [b, c] respectively.Then f;') (b) = and g( )(b) = g.(') (b+), s = 0,1, ...,1 - 1. If (6.2) issatisfied, then the function

on [a, b],on [b, c]

is such that h('-1) exists and is absolutely continuous on [a, b]. Consequently,the weak derivative h;;) exists on (a, b) and

1(1) on (a, b),

hwIf92) on (b, c).


Hence, inequality (6.3) follows.If (6.2) is not satisfied, then for any function h2 defined on [a, b], coinciding

with fl on [a, b) and with gl on (b, c], the ordinary derivative h(21-1)(b) does notexist. Hence, the weak derivative

4-1)does not exist on (a, c) and h is not in

Wy1)(a,c).

Lemma 3 Let I E N, 1 < p < oc. Then there exists a linear extension operatorT : W, (oo, 0) -+ WW(-oo, coo), such that

IITIIw;(-.,u)-,wa(-...) < 81. (6.4)

Idea of the proof. If l = 1, it is enough to consider the reflection operator, i.e.,to set

(T1f)(x) = f (-x), X>0. (6.5)

If l > 2, define (T2)(x) for x > 0 as a linear combination of reflection anddilations:

(T2f)(x) _ ak(TIf)(Qkx) = E akf(-Qkx), (6.6)

k=1 k=1

where Qk > 0 and ak are chosen in such a way that

(T2f )(W8) (0+) = As) (04 3 = 0,1, ---,l - 1. (6.7)

Verify that IIT2jIwl(oo,o)-.wW(_c ,ao) < oo and choose Qk k = 1, ...,1, inorder to prove (6.1).Proof. Equalities (6.7) are equivalent to

Eak(-Qk)' = 1, s = 0, 1, ..., l - 1. (6.8)k=1

Consequently, by Cramer's rule and the formula for Van-der-Monde's determi-nant,

11 (Qi - Qj) Iii=..1

akI<i<j<!

= -II C8 - Qj)

1<i<j<I

11 (Qi + 1) 11 (-1 - Qj)1

k = 1, ..., 1. (6.9)la (Qi - Qj) fl (Qk - Qj) 1<jS1,j*k Q, - Qk'

1<i<k k<j<I


If Nk = k, k = 1, ...,1, then

ak = (-1)k-lk

21)

(1)

l+k \i k

and

I«kl411(k).Therefore, setting y = -f3kx, we have

IIT2fIIWp(0,oo) = IIT2fIILp(0,aa) + II(T2f)(ti)IILD(O,ao)

k=1 k=1

k=1 k=1

< (8' -1)IIfIIWp(-oo,0)-

Hence, inequality (6.4) follows if we take into account Lemma 2 and, in partic-ular, inequality (6.3).

Remark 1 It follows from the above proof that the inequalities

IIT2IIw; (-oo,o)_,wo (-oo,oo) 81, m E N0, m < 1,

also hold.

Corollary 1 Let I E N,1 < p < oo, -oo < a < oo. Then there exists a linearextension operator T : W,(a, b) -+ Wp(2a - b, 2b - a), such that

IITIIwp(a,b).ww(2a-b,2b-a) 5 2.8'. (6.10)

Idea of the proof. Define

Eakf(a+Qk(a-x)) forxE(2a-b,a),k=1

(T3f)(x) = f(x) for x E (a,b), (6.11)

1

E ak f (b + 13k(b - x)) for x E (b, 2b - a),k=1

where ak and Qk are the same as in (6.6), observe that T3f is defined on(2a - b, 2b - a) since 0 < Qk < 1, and apply the proof of Lemma 3.


Corollary 2 Let I E N,1 < p < oo, -oo < a < b < oo. Then there exists alinear extension operator T : W'(a, b) -r W'(a - 1, b + 1) such that

IITIIw,(a,b)_w,(a-1,b+1) <_ 2.8'(1 + (b -a)-'+7).

Idea of the proof. Let 6 = min{1, b - a} and define

(6.12)

!

ak,6f(a+6,3k(a-x)) forxE (a- 1, a),k=1

(T4f)(x) = 1(x) for x E (a.b), (6.13)

it

E ak,6 f (b + b,3k (b - x)) for x E (b, b + 1),k=1

l

where (3k are the same as in (6.11) and ak,6 are such that ak,6(-al3k)' = 1.k=1

1. Observe that by (6.9) Iak,6I < (b - a)-'+' Iak I and apply theproof of Lemma 3.Proof. As in the proof of Lemma 3

IIT4fIIWw(b,b+1) 5 i Iak,61. IIf(b+6,3k(b-x))IILp(b,b+1)k=1

+ Iak.6I00k)1IIf,(`)(b+S0k(b - .C))IIL,(b.b+l)k=1

< ( Iak,61(6#k) °) IIf IIWI(b-6pk,b)k=1

< 1)11fllW,(a,b) <-(8'b-l' -1)IIf!Iw

(.,b)

and

IIT4f IIw;(a-1,b+1) 5 IIT4f IIww(a-1,a) + IIT4f IIw;(a,b) + IIT4f IIw;(b,b+1)

< 2.8'6-'+ 111 II W, (a,b).

In order to estimate the norm of an extension operator T : WP(-oo, 0) -+WP(-oo, oo) from below we prove the following statement, which reduces thisproblem to a certain type of extremal boundary-value problems.


For given ao, ..., al_1 E R let

)

GP1(ao,...,al_)) = inf IIfIIwy(o,oo)w

(6.14)

Gp,l (ao, ..., al-1)

p(O.-I:I e

is defined in a similar way with (-oo, 0) replacing (0, oo). Let

Qp1 = supGpl(ao,at,...,aj_j)

laol+...+la,_,I>o Gp,l(ao, at, ..., al-1)

G,+,, (ao, a,) ... at-,)sup + (6.15)

loot+...+la,_,I>o Gp 1(ao, -a), ..., (-1)1-1at-1)

The latter equality follows if the argument x is replaced by -x in the definitionof GN,1. Moreover, it follows from (6.15) that for 1 1, 1 E N, Qp,) = 1. (6.16)

Lemma 4 Let I E Nl,1 < p < oo. Then

/i

1 1+ Qv t11 p< if IITIIwy(-oo,0).,wW(-00,00) < 1+ Qp,a.

(If p = 00, then (1 + QP 1) o must be replaced by Qo,1.)

(6.17)


1

(Ilf Ilwo(_oo,o) + IITf Illpvo(o,o.)) p <- IITf Ilw;(-00,00) <- Ilfllwn(-oo,o) + IITf Ilwo(o,oo)

(6.18)In order to prove the first inequality (6.17) apply also the inequality

IITf IIw;(o,oo) > Ga 1(ao, ...) at_,), (6.19)

which, by the definition of G+,, holds for all ao, ..., a1_1 and for each extensionoperator T. In order to prove the second inequality (6.17) define, Ve > 0, theextension operator Te setting TE f = g£ for x E (0, oo), where gE E Wp(0, 00) is

any function, which is such that ffk)(0-), k = 0,..., l - 1, and

II9EIIw;(0,oo) 5 Gp 1(f (0-), ..., f( .1-1)(0-)) + e 11f Ilwo(-oo,o) (6.20)


Proof. 1. The second inequality (6.18) is trivial since

IIh11L°(-00,00) <_ IIh11L°(-ao,o) + IIhIIL°(o,00)

The first inequality (6.18) follows from Minkowski's inequality for finite sums,because

IIhIIL°(-o0,0) + IIhIIL°(0,oo))

+{Ilhu')IIL°(-00,0) + Ilh(w)IIL°(0.c)) ° >_ { (IIh11L°(-o,o) + Ilh(')IIL°(-oO,0))p

p

+(Ilhllr,°(o,.) + (Ilhllwp(-,,,,,o) +

2. It follows from (6.18) and (6.19) that for each ao, ..., a(_) E R such thatIaol + ... + Ia(-) I > 0

IITIIwp(-.,o)-.w;(-.,.) = supIIT! IlWp(-00,00)

tIIJ IIWp(-.,O)

> (1+ sup(IITfllwp(o,oo)\\p

JIIf IIw op(-oo,JEWp(-06.0): )

> 1+ G+ p 1 °p,(ao,...,a(_,)) sup1lfllwl/EWp(-W-0)' °(-00,0)

(G-( ao, ... , ai_t Y)

and we arrive at the first inequality (6.17).3. Given e > 0 by (6.18) and (6.20) we have

IITehI <_ 1 + supI19eIIWp(0,oo)

fEW;(-oo,o),/'w Ilf II Wp(-oo,o)

v,t (ao, ... , a->' )<1+e+ sup sup =1+Qpi+ea0,...,o1-1ER: /EWW(-m,0): IIf II Wp(-00,0)

and the second inequality of (6.17) follows. 0


Corollary 3 Let 1 - 20 for eachextension operator T. On the other hand it is clear that for the extension

ioperator T, defined by (6.5) IITi IIw;(-oo,o)-+wo(-oo,oo) = 2' .

Remark 2 Note also that if the norm in the space W,(a, b) is defined by

b

vIIfDwp(a,b) = (f (If(x)IP+Ifw')(x)I")dx)

a

(see Remark 8 of Section 1.3), then

)P)o,inf (1 + (QPI)

where QI; is defined by (6.14) - (6.15) with II . II') replacing II - II. This followsP

from the proof of Lemma 4 and the equality

IITillw;(-oa,ao) = ((IITf Ilyvo(,,,o))P + (IITillwo(o.00))P)'.

Lemma 5 Let I E N, 1 < p:5 oo and f E Wp(0, oo). Then

IlfIlw (o,00)Il f k.(0+)-

xkilL,(o,a,).k=0

(6.21)

Idea of the proof. Apply Taylor's formula and Holder's inequality.Proof. Let f E Wp(0, co). Then for almost every x E (0, oo)

!-1 (k)

f (x) = EfW

(k±)xk

+ (1 11)! f(x - u)'-1 du,ka0 0

where the k = 0, 1, ... ,1 - 1, are the boundary values of the weakderivatives f). (See formula (3.10) and comments on it in Section 3.1). Hence,by the triangle inequality for each a > 0

II f (k)(°+) xk ll < IIf IIL +1 1

ill (x-u)'-'f.() (u) dullk! Lp(o,a) - ( 1)! L,(O,a)'

k=0 0


By Holder's inequality

x(lx-1)P .f (x - u)1-' f (()(u) du

IILP(O,a) < II (I - 1)p' + 1 / IIJW''ll/.P(0,2)IILP(O.a)

0

<((1-1)p'+1) '11X' P- PIILP(0,a)IIfW`'IILP(0.a)

= a`(lp) P((1 -1)p' + 1) allfw''IILP(O.a) <_ Ilfw`'IILP(o,a)

Consequently,

1-1

II f k,(k! )xk IIL ro.a)<- 11f 11L,(0,.) +

k=O

Setting a = ' 1!, we get (6.21).

Corollary 4 For all I E N,1 < p:5 oo, ao,... , at_1 E R

1-1 LGp j (ao, ... , at_1) > E k! xkLP(O,. !)

(6.22)k=O

Lemma 6 Let I E N,1 < p < oo. Then for every extension operator T

Wp,(-00, 0) -+ WP' (-00, 00)

IITIIW;(-aa.0).wa(-oo,oo) ? 2'-21-p. (6.23)

Idea of the proof. For 1 = 1, 2 inequality (6.23) is trivial since IITII ? 1 for eachextension operator T. Assume that I > 3 and set

0 for-oo<x<-a,f1(x) _(x + a)1 for -a < x < 0,

where a = Y11Proof. By (6.14), (6.15), (6.22) and the triangle inequality we have

/1

11-1 U,1nik

`CP,1

______1(0), ... , f1(1-1'(0)) > II k=0 k! IILP(0,a)

IIfIIIW;(-oo,0) IIhIIWW(-oo,0)


= II(x+a)'- x1jjL,(o,.) > II(x+a)'IIL,(o,a) - IIX'IIL,(o,a)

II (x + a)' II w;(-a,o) Ilx' I I wa(o,a)

(21p+1-1)a-1 2'-1 2'-1 _z1 2 s_ , >

1> 1 0>2 1 P.

1 + (!p+ 1)p 1a(1-° ,+(P+1),

°) 3

Hence by (6.17) inequality (6.23) follows.

Remark 3 Note also that there exists a constant cl > 1 such that 2

IITIIw;(-ao,o)_*w,,(-o,.) >- c1, l > 2, 1 0 Then there exists a "cap-shaped" function 71 E Co (1R) such that 0 < n < 1, 77 = 1 on (a, b), supp r) C(a-e,b+E) and

117(k) (x) I < (41)ke-k, x E R, k = 0, ... ,1. (6.25)

Idea of the proof. Set

77 _ * w * ... * w *X(a- ,b+ ),

1 times

(6.26)

where X(a-j,b+j) is the characteristic function of the interval (a - 2, b + 2),

w(x) = 1 - Ixl if Ixl < 1, w(x) = 0 if Ix) > 1, w is any nonnegative infinitelydifferentiable kernel of mollification (see Section 1.1) and y is a sufficientlysmall positive number. Apply Young's inequality (4.138) and the equality

II (w * Ilw +, IIL,o(R). (6.27)

Proof. Leta=L(l+4)(1+y)-1. By Section 1.1nECo (R),0<n<1,77 = 1on (a-2+a,b+2-Q) (a,b) andsuppr) C [a-z-o,b+2+v] C (a-L,b+z).Moreover,

1177(')11L. (R) <- II(we,7 * ... * w * X(a-I,b+j))wk)IIL (R)

2 Inequailty (6.24) does not hold Vl E N because of Corollary 4 for p = oo.


= II (W) * ... * (w ) #W c * ... *W*(w2+

` , * X(a--,b+-)YIIL (R)2+7 +1 +7 +, 2

k - 1 -times I - k times< (2(1+7))k-1

11W11k-1

IIWIIL,(R) IIW2 +, IIL_(R) < 2.4k-'(1 + Y)kE-k

E J

Choose -y > 0 satisfying e7 < 2, then (1 + y)k < ek(l + e < 2 lk and soobtain (6.25).

Finally we note that (6.27) follows from

b+ 12

W e # e x W x - 'dya-i

z-b-1

r E E)l-( J 2+,(z)dz =W, (x-b-2)-low,+

2--a-3

since the terms of the right-hand side have disjoint supports.

Corollary 5 In the one-dimensional case `dl E N there exists a nonnegativeinfinitely differentiable kernel of mollification p satisfying (1.1) such that

I1(k)(x)I < (41)k, x E R, k = 0, ...,1. (6.28)

Idea of the proof. Define q by (6.26), where a = b = 0 and E = 1, and applythe equality If *9IIL,(R) =IIfIIL,(R) - II9IIL,(R) for non-negative f,g E L1(R).

Lemma 8 There exists c2 > 0 such that for all 1, m E N, m < 1,1 <_ p, q <oo, -oo < a < b < oo and V f E W, (a, b)

Ilfwm)IILq(a,b) < c12(b-a)4 P((b a)mIIfIIL,(a,b) + (ba)1-m

(6.29)

Idea of the proof. Apply the integral representation (3.17) with (a,,0) = (a, b)

and w(x) = b µ (2(b o b)) , where the function µ is a function constructed inCorollary 5.Proof. The numbers oaf,,, defined by (3.20) satisfy the following inequality

IQa mI1

1-1 l - 1( )

21-1, (s-m)! ` k

k=O(s-m)!


Consequently, taking into account (3.5) and Remark 5 in Chapter 3, we havethat for almost all x E [a, b]

b 1-1

IA-) (x) I <- f (E (32 ), (b - a)'-m (41)'\b ?

a)'+1(41)')If I dy

b b(b a i '"-1+( ) ' f If( .1)Idy < (b- a)-m-116t1m C` f IfI dy

a 8--m

_\

b

+(b - a)1-m-1 IM-1-1 (11 m-1

l f Ifwj)I dy(1-m-1)!0

< (16e)' (b - a)-p

((b a)m Ilfllt,(ab) + (bl

a)1-mIlfw`)IIL,cab) (6.30)

and inequality (6.29) follows with c2 = 16 e. 0

Remark 4 Inequality (6.29) is an improved version of inequality (4.55). 0

Corollary 6 If, in addition to the assumptions of Lemma 8, b - a < 1, then

Ilf,v IIL,(a,b) <- C11' (b - a) m+o p IlfIIwW(a,b)' (6.31)

If, in addition to the assuptions of Lemma 8, b - a > 1 and q > p, then

II W'-(0)- (6.32)

Idea of the proof. Inequality (6.31) is a direct corollary of (6.29). In order toprove (6.32) apply (6.29) and Lemma 7 of Chapter 4. 0P r o o f . Let b - a > 1 and q > p. Choose intervals (ak, bk), k = 1, ... , s, in such

a way that bk - at = 1, (a, b) = U (ak, bk) and the multiplicity of the coveringk=1

{(ak, bk)}k=1 is equal to 2. By (6.29)

IIfwm)IIL,(a.,b.) <- 01 (1m IIfIIL,(ar,b.) + lm-`

Hence, by Lemma 7 of Chapter 4

2a 021 (1m IIfIIL,(a,b) + 1--1 IIf,(°)IIL,(a.b))

and (6.32) follows. 0


Lemma 9 Let I E N, 1 < p < oo, - oo < a < b < oo, b - a < 1. There existsa linear operator T : W, (a, b) - W, (-oo, oo), such that

c'l'IITIIw;(a.b),w;(-oo,aa) <_ 3 !_ , , (6.33)

(b - a) V

where c3 is a constant greater than 1.

Idea of the proof. Consider the operator

(Ts f) (X) = (T4 f)(x)rl(x), x E R, (6.34)

where r) is the function constructed in Lemma 7 for e = 1 and T4 is definedby (6.13), assuming that (T5 f)(x) = 0 for x (a - 1, b + 1) and applyCorollary 6.Proof. It follows from the Leibnitz formula, (6.25), (6.32) and (6.12) that

IITsfIIWw(-aa,aa) = IIrlT4fIILp(a-1,b+1) + II(17T4f)(w)IILp(a-l,b+l)

<- IITafIILp(a-l,b+l) + (m) II(T4f)('')IILp(a-1,b+1)m-0

< IIT4fIILp(a-1,b+1) + ( (MI ) (41)' m(2C2)'lm) IIT4f II W;(a-1,b+1)m=0

< (1 + (16c21)') IIT4f IIww(a-l,b+l) <- 4 (1 + (16c2l)') 81 (b - a)-'+a IIf IIw;(a,b)

< c'3 l' (b - a)-'+7 Ill I I WW(a,b) ,

where c3 = 32 (1 + 16 c2). Hence we obtain (6.33). 0

Lemma 10 Let I E N,1 1. There existsa linear extension operator T : WI (a, b) -+ WI(-oo, oo) such that

ItIITII WW(a,b)-,w;(-oo,oo) < L14 (1 + ! )

(b - a)- (6.35)



Idea of the proof. Consider the operator

(T6f)(x) = (T3f)(x) i(x), (6.36)

where i is the function constructed in Lemma 7 for e = b - a and T3 is definedby (6.11), and apply Lemma 8.Proof. It follows from the Leibnitz formula, (6.25), (6.29) and (6.10) that

IIT6fIIW,(-oo,oo) = IInT3fIILp(2a-b,2b-a) + II(77T3f)(w)IILp(2a-b,2b-a)

< IIT3fIILp(2a-b,2b-a) + (m)M=0

:5 IIT3fIILp(2a-6,2b-a)

+mF, (m)

(41)1--(b - a)m-1 C2-((b t a) m IIT3f IILp(2a-b,2b-a)

+(bl

a)f mII(T3f)(w)IILp(2a-b,2b-a))

IIT3f IILp(2a-b,2b-a) + (4l)1 (u(1)rn)

(b - a)-' IIT3f IILp(2a-b,2b-a)

+4' ( lm) Cam` II(T3f)W(I)IILp(2a-b,2b-a)

M=0

< (1 + (4 (1 + c2))' (1 + 1'(b - a)-')IIT3f IIWo(2a-b,2b-a)

5 2(1+(4(1+c2))181(1+1'(b-a)-')IIfIIw;(a,b)

5 c'4(1+1'(b-a)-')IIfIIww(a,b) <- c'4 (1+1'(b-a)-'+t)IIlIIW (a,b)

where c4 = 16 (1 + 4 (1 + c2)). Hence we obtain (6.35).

Remark 5 It follows from the proofs of Lemmas 9 and 10 that for all-oo < a < b < oo there exists an extension operator T such that

IITIIWa (a,b)-Wo (-oo,oo) < C5 (1 +m

m_ )+ m E Np, m<-1, (6.37)(b - a)



Now we consider estimates from below for the minimal norm of an extensionoperator.

Lemma 11 Let 1 E N,1 _ 8I (e)'l'(b-(6.38)Remark 6 We shall give two proofs of Lemma 11. The first of them is adirect one: as in the proof of Lemma 6 it is based on the choice of a functionf E W,(a, b), which is the "worst" for extension. The second one is based onLemma 12 below, in which a lower bound for the norm of an arbitrary extensionoperator via the best constants in the inequalities for the norms of intermediatederivatives is given. In both proofs the polynomials Qt-);p of degree 1-1 closestto zero in Lp(0,1) are involved, i.e., Q1-1;P = xt-1 + a1_2x1-2 + ... + ao and

IIQ1-1;pIIL°(o,1) = inf Ilx'-' + b1_2x1-2 + ... + boIIL°(o,)).b0.....bl_2ER

We recall that 2-1+1R(_1(2x - 1), where R,,,, is the Chebyshevpolynomial of the 1-st type: Rm(x) = 2-m+1 cos(m arccos x). Moreover,

IIQ1-1;pIILp(o,1) 5 IIQi-1roolILp(0,1) IIQ1-1;ooIILoo(0.1) = 8.4-'.

Idea of the first proof of Lemma 11. In the inequality

°_oo,ao) >11TfIIw°(-oo,oo)

IITII = IITIIw°,(a,b)-,wi(IIfIIW;(a,b)

set _ ! 1 (ll1)1

Q1-1,P b-a),

apply inequality (4.50) and the relation

inf IIhllw;(-oo,a) >- 1. 0he Wj (-oo.a):

h(a-)=I

First proof. It follows from (6.40), (6.41) and (6.39) that

IITII - (I IIQt 1;PIILo(,1)II9IIw;(-oo,oo)

(6.39)

(6.40)

(6.41)

(6.42)


2 41-1(1 - 1)! (b - a)-1+-,

II9IIw;(-oo,oo)

where g = T f . By inequality (4.50)

II9w`-1)IILP(-oo,oo) 2 II9IIL,II9w`'IILp(Too,oo) < 2 II9IIWW(-oo,oo)

Consequently

(1-1) (1)ir

II9w II W, (-oo,oo) = II9w IIL,(-oo,oo) + II9w IILp(-oo,oo) 5 2 + 1 II9II W;(-oo,oo)

and

IITII - 4 ' -7

r

' ( 1+

21)1 II9w)-1)IIW, (-oo,oo)

Since fw1-1) = 1 and g E Wp(-oo, oo), by Lemma 2, gw-1)(a-) = 1. Hence by(6.42)

tII9w(-))Ilwp(-oo,oo) AEwvnf IIhlIww(-oo,a) 1

h(.-)=l

Thus by Stirling's formula

41_I (1- 1)! (b - a)-1+ 21r(l - 1) 4 1-1 1-1 '+IITII? +2 it+2 (e) (l-1) (b-a)-

e tl1(

)-1+J-1(1- I (e ba

4(7r+2) l

>2,

(4)111(b-a)-'+p >0.12(4)111(b-a)-1+

4(7r+2)f e 1 e

and we obtain (6.38) with 0.12 replacingFinally we note that (6.42), by Holder's inequality, follows from (3.8):

0+1 a+1

1 = Ih(a-)I 5 f IhI dy + f IhwI dy 5 IIhhIwp(-oo,oo) 13o a

Now for 1, n E N and 1 0 andof E Wp(R")

II DmfIILg(R") < A If II W (R') (6.43)


It follows from Chapter 4 that p < q < oo and 1,61 < l - n(I - 1) orq=ooandl6I <1for p=oo, I/3I <1-p fort <p<oo,I/3I <1-n forp = 1. Furthermore, for an open set S2 C R" and (q, 0) E M,,,,,,, we denote byC* (S2, p, q, 1,/3) the best (minimal possible) value of C, for which V f E WP (S2)

IIDwfIIL,(n) <- CIIfIIw;(n). (6.44)

Lemma 12 Let 1, n E N, 1 sun"C' (Q, p, g,1.0)

W n -+W R0( ) p( ) -(9.Q)Ebf+.n.v C'(R", p, q, 1, l3)

(6.45)

Idea of the proof. Prove (6.44) by applying an arbitrary extension operator Tand inequality (6.43) where A = C' (R", p, q,1, (t).Proof. For all (q, /3) E Mt,n,a

IIDWfIIL,(n) 5 IID1(Tf)IIL,(R-) s C'(R, p, q, 1,13) IITfIIw/(Rn)

< C'(R", p, q,1,13) IITIIw;(n)-,wo(Rn) Ilf II WW(n)-

Hence,

C'(S1,p,q,1,/3) <_ C*(R", p, q,1,Q) IITIIwp(n)-.w1(R")

and (6.45) follows.Idea of the second proof of Lemma 11. Apply Lemma 12 with /3 = 1 -1, q = 00and inequality (4.53). Use the function f, defined by (6.41) to obtain a lowerbound for C* ((a, b), p, oo,1- 1,1).Second proof. By (6.45) for every extension operator T : W,(a, b) -*W, (- 00, oo)

p-.,.)C* ((a, b), p, oo,1,1-1)

IITII = IITIIwa+(n,b).w+( C ((-oo,oo),p,oo,1,1-1).

It follows from (6.44), with f defined by (6.41), and (6.39) that

C' ((a, b), p, oo,1,1- 1) > b-a_+ II b)

= (l - 1)! (b -a)-1+iT

> 141(1- 1)! (b - a)-i+

QQ1-1;ehIL,(o,1) 8


From (4.53) C'((-oo, oo), p, 00, 1, l - 1) < 2ar. Hence, applying Stirling'sformula as in the first proof of Lemma 12 , we get

()111(b - OIITII ?41

2(127r 1)1(b - a)-1+7 ? 8 f e

Finally, we give a formulation of the main result of Section 6.1.

Theorem 1 There exist constants c6, c7 > 0 such that for all 1 E N, 1 < p < 00and -oo < a < b < oo

cs 1 +it

1 + 11 '_ ) . (6.46)iTf IITII wo(o.a),wp(-oo.oo) ((b a)(b - a)

Idea of the proof. Apply Lemmas 3, 6, 9, 10 and 11.Proof. If b - a = oo, then (6.47) follows from (6.4) and (6.24). If b - a < oo,then (6.47) follows from (6.33), (6.35) and (6.38).

Remark 7 If p = oo, then the statement of the Theorem is also valid for thespaces C 1(a, b), i.e., there exist c8, c9 > 0 such that

( l1 l ll lc8 (1 + (b - a)1-1 ) < if c1s (1 + (b - a)1-1) . (6.47)

The estimate from below is proved in the same manner as for the spaceW.(a, b). When proving estimates from above, the operator T2 defined by(6.6) must be replaced by T2 defined by (T2 f) (0) = f (O-) and (T2 f) (x) _1+1 1+1

E ak f (-,9kx), x > 0 , where ,Bk > 0 and E ak(-Qk)' = 1, s = 0,1, ... ,1.k=1 k=1

In that case (T2 f)(')(0+) = f (')(0-), s = 0,1, ... ,1, which ensures that T2 f EC1(-oo, oo) for each f E G 1(-0o, 0). Moreover, (IT2IIUi(_00 o)iZ!l(_. .l < 161.

The rest of the proof is the same as for the space WW (a, b).

6.2 Pasting local extensionsWe pass to the multidimensional case and start by reducing the problem ofextensions to the problem of local extensions.

6.2. PASTING LOCAL EXTENSIONS 265

Lemma 13 Let I E N, 1 0. If s = oo, suppose, in addition, that the multiplicity of thecovering x - x({Uj};=1) is finite.

Suppose that for all j = !-,s there exist bounded extension operators

T; : Wp(Sa n U,) -> WP(L(6.48)

where Wp(S2 n U,) _ (f E Wp(S2 n Uj) : supp f c S2 n U,}. Ifs = oo, supposealso that sup IIT, 11 < oo. Then there exists a bounded extension operator

jEN

T : Wp(S2) -* W,(R" ). ' (6.49)

Moreover,

1ITh 5 cIo sup IITifl, (6.50)

where c10 > 0 depends only on n, 1,

j=6 and x.

If all the T, are linear, then T is also linear.

Idea of the proof. Assuming, without loss of generality, that (U,)5 n 11 0 0construct functions Oj E C°°(R"), j = !-,s such that the collection { ?}'=1 isa partition of unity corresponding to the covering {Uj)J=1, i.e., the following

sproperties hold: 0 < '0j < 1, supp zlij C Uj, t O2 = 1 on S2 and Va E No

j=1

satisfying jal < 1, M1, where M1 depends only on n, l and 6.For f E Wp(SZ) set

Tf = > ej Tj(f+,j) on W. (6.51)j=1

(Assume that OjTj(fOj) = 0 on c(U,)). 0Proof. 1. Let rlj E C°°(R") be "cap-shaped" functions satisfying 0 < ri, <1, nj = 1 on (U,)s, rj = 0 on °((Uj)g) and ID°r)j(x)h 5 M26-1°1, a E No,

a

where M2 depends only on n and a. (See Section 1.1.) Then 1 < E rl? < xj=1


a a

on U (U;) Further, let E C6 (R"), 1 on 12, 77 = 0 on `(U (Uj) s ). One1=1 7=1

can construct functions tkj by setting iOi _ ?7t t) & 77j2)- on U (U,)1 assuming

that O, = 0 on c(U(U;)1).:=1

2. The operator T defined by (6.51) is an extension operator. For, let x E Q.If X E supp V)j for some j, then 1,b (x)(Tj(f 1G1))(x) = OJ2 (x) f (x). If x 0 supp le,,

e

then -0,(x)(T,(f-G;))(x) = 0 = 1,,(x)f(x). So (Tf)(x) = E'+GJ2 (x)f(x) _j=1

f(x).3. Let a E No and Ia1=1. If s E N, then

Do(Tf) _ ED.(?PjTi(f ik1)) on It". (6.52)i=1

If s = oo, then (6.52) still holds, because on `(U (U;)¢) both sides of (6.52)

are equal to 0 and dx E U (U;)1 the number of sets (Uj) j intersecting thei=1

ball B(x, 1) is finite. Otherwise there exists a countable set of Uj,, s E N,satisfying (UJ,)t n B(x, z) 36 0. Hence x E U,i,, and we arrive to a contra-diction since x({U1},q0=1) < oo. Consequently, there exists S. E N such that

supp (1G,T,(ftli,)) n B(x, z) 54 0 for j > sy. So

Tf OiTT(fOj) on B(x,1).7=1

Hence,00

Dm(Tf) _ E Dw(1G1Ti(f ,Oi)) _ E D"('O;T;(f O,)) on B(x, 2).

i=1 i=1

Therefore by the appropriate properties of weak derivatives (see Section 1.2)(6.52) with s = oo follows.

4. Let aENi, and a=0orjal=1. In (6.51),for all xEW1,and in (6.52),for almost all x E R", the number of nonzero summands does not exceed x.Hence, by Holder's inequality for finite sums,

ID."(Tf)IP < W-' t IDw(0tT, (f1, ))IPj=1

6.2. PASTING LOCAL EXTENSIONS 267

almost everywhere on R" and

f IDD(Tf)I°dx < W- > f ID°('jTj(fbj)) IPdx.R. i=1 Rn

Therefore, taking into account Remark 8 of Chapter 1, we have

IITfIIW;(Rn) < M3 t II P Ti(fuj) flwp(R')j=1

where M3 depends only on n, l and x. Since supp u'j C U, applying Corollary18 of Chapter 4, we have

IIVGjTj(ftj) IIwp(si) <_ M 4 I I T?(fps) IIwp(u,) <_ M 4 I I Tj II IIf Vj IIWP(snU )

< Ms IITiII III IIwp(nnu,),

where M4 and M5 depend only on n, l and 6. Now it follows, by (2.59), that

IITf Ilw;(Rn) < M6 sup IIT,II (E IllIIW (nnu,))i j=1

v

<M7sup IIT;II( f (IfI + E JDWfIP)dx?=1 nnu, l01=1

< M8 SUP IlTill If IIww(n),iwhere M6, M? and M8 depend only on n, 1, 6 and x.

Remark 8 Suppose that in Lemma 13 the operators Ti satisfy the additionalcondition

fEfVlpnUj)==> suppT,fcUj. (6.53)

In this case the operator T may be constructed in a simpler way with the help

of a standard partition of unity J=1, i.e., E of = 1 on ). We assume that3=1

T,(f p1)(x)=0ifxEUj and set

,

Tf T,(f,,i) on R. (6.54)j=1


The operator T is an extension operator. For, let x E 1. If x E Uj, then(T.i(f Vi))(x) = b1(x) f(x), and if x V Uj, then (T,(f'j))(x) = 0 =O9(x) f(x).

Thus (T f)(x) = E Oj(x) f (x) = f (x). Note also that for f E Wy(0), becausei=1

of (6.53), we have T,(fiP) E Wp(Rn) and II Ti(f'Pi) 11 2' (f0,) Ilwp(u,).

Further we consider a bounded elementary domain H C R" with a C1- orLipschitz boundary with the parameters 0 < d < D < oo, 0 < M < oo, whichby Section 4.3 means that

H={xEitt":an<xn<cp(2), 2EW}, (6.55)

where 2 = (xi,...,xn_1), W = {2 E Rn-1, ai < xi < bi, i = 1,...,n - 11,-oc < a, < b, < oc, diam H < D,

an + d < ;p(2), x E W, (6.56)

and

1m°aaIID°Wllc(w) < M (6.57)

or

I(v(2) - W(g)1 <- M 12 - 91, 2,9 E W, (6.58)

respectively. Moreover, let V = {x E Rn : ai < xi < bi, i = 1,... , n - 1, an <xn < oo}.

Lemma 14 Let I E N, 1 < p < oo. Suppose that for each bounded elementarydomain H C Rn with a Cl- or Lipschitz boundary with the parameters d, D andM there exists a bounded linear extension operator

T : Wp(H) -* WP(V), (6.59)

where W,(H) = If E W, (0): supp f c Fn V} and 1IT11 < c11, where c11 > 0depends only on n, 1, p, d, D and M.

Then for each open set Q C Rn with a C1-, Lipschitz respectively, boundarythere exists a bounded linear extension operator (6.49).

Idea of the proof. Apply Lemma 13 with Uj = Vj, where Vj, j = 1, s are openparallelepipeds as in the definition of an open set with a Cl- or a Lipschitzboundary.

6.3. EXTENSIONS FOR SUFFICIENTLY SMOOTH BOUNDARIES 269

Proof. By the assumptions of the lemma for all j = 1, s there exist boundedextension operators

T; : WP(.\i(I n V;)) -> WP(aj(ti's))

Let (A, f)(x) = f ()19(x)) and define

T(1)

=Aj1TA9.

It follows from the proof of Lemma 16 of Chapter 4 that A9 : W, n 17) ->Wp(A(Sl n V9)), A;1 : Wp(a9(V9)) -a Wo(LJ) and II A, II, II A9 II-I do not exceedsome quantity depending only on n and 1. Hence.

Tj1i i: (sl n l;) -+ WI(V )

and

II 2 ') II <_ II A-' T'. II ' IIA3 II S M1 II T, II,

where M1 depends only on n and 1.If 11 is bounded, then s E N and by Lemma 13 there exists a bounded

extension operator (6.49). If 1 is unbounded, then s = oo and by the definitionof an open set with a Cl- or Lipschitz boundary each bounded elementarydomain a9(SZnV9) has the same parameters d, D, M. Hence, by the assumptionsof the lemma II T.i II 5 c11. Moreover, in this case the multiplicity of the covering{ V? 1,9_1 is finite. Thus Lemma 13 is applicable, which ensures the existence ofa bounded linear operator (6.49).

6.3 Extensions for sufficiently smooth bound-aries

Lemma 15LetlEN,1<p<ooand S2={xER": ai<x,<bi,i=1,. .. , n}, where -oo < ai < bi < oo. Then there exists a bounded linearextension operator (6.49).

Idea of the proof. Apply Lemmas 9-10 n times.

Lemma 16 Let I E N, 1 0 depends only on n, I and M.


Idea of the proof. Let H- _ {x E R"; -00 < x, < tp(x), x E W}, (Tof)(x) =f (x) for x E H, (Tof)(x) = 0 for x E H-\H. Moreover, let (Af)(x) = f(a(x)),where (a(x))k = xk, k = 1, ... , n - 1, (a(x))" = xn + W(r) and (T2 f) (x) =

1

i akf (x, -Qkxn) for I E W, xn > 0, where flk > 0 and ak are defined byk=1(6.8). Set

T = A-' T2 A To (6.60)

and apply Lemma 16 and Remark 25 of Chapter 4.Proof. If f E W,(H), then Tof E Wp(H-) and IITofIIww(H-) = IIfUIww(H)Hence, IIToIIIVP(H)4wp(H-) = 1. Since A(H-) = Q- = {x E Rn : x E W, xn <

0) and o= (x) - 1, by (4.126) and (4.148) we have

IIAIIw;(H-)_wp(Q-) <- M1 Max II D°,II c(w) 5 M1 M,

where M1 depends only on n and 1.Since (a(-')(x))k = xk, k = 1, ... , n - 1, (a(-')(x))n = x" - cp(x) , the same

estimate holds for IIA-'IIw.(q)-,w;(q) where Q = W x R. Finally by Lemma 3,

IIT2IIwa(Q-)-.wg4) <- 81. Thus,

IITIIWO(H)-.Wy(V) < IIA-'II IIT2II' IIAII II2'oll <_ C12,

where C12 depends only on n, I and M.

Remark 9 Note that1

(T2Af)(x) = E akf (x, xn - (1 + fk)lxn - V(x)))k=1

(6.61)

onH+= {xERn: IEW,xn> w(z)}, where 1k>0andaksatisfy (6.8).

Theorem 2 Let I E N,1 < p < oo and let 11 C Rn be an open set with aC(-boundary. Then there exists a bounded linear extension operator (6.49).

Idea of the proof Apply Lemmas 14 and 17.

Remark 10 If p = oo then Lemmas 13 -16 and Theorem 2 are also valid forthe space C'(0). Thus, for each open set with a C'-boundary there exists abounded linear extension operator T : ?71(fl) -+ ?7'(R'). (See also Remark 7.)

6.4. EXTENSIONS FOR LIPSCHITZ BOUNDARIES 271

6.4 Extensions for Lipschitz boundariesLet

Q = {x E Rn : X. < V(:i), x E K'-'), (6.62)

where V satisfies a Lipschitz condition on IIt1-1:

IV(z) - V(9)I < MIA - yI, x, y E lW"-'. (6.63)

Lemma 17 Let ! E N,1 < p:5 oo. Suppose that for each domain S2 defined by(6.62) - (6.63) there exists a bounded linear extension operator

T : t4p(S2) --> 14'p (R"), (6.64)

where y(1) = (f E W1(11) : suppf is compact in Iltn } and IITII <_ c13, wherec13 > 0 depends only on n, p, l and M.

Then for each open set 1 with a Lipshitz boundary there exists a boundedlinear extension operator (6.49).

Idea of the proof. Prove that for each bounded elementary domain H C Rnwith a Lipschitz boundary there exists a bounded linear extension operator(6.59) such that IITII < C13 and apply Lemma 14.13Proof. 1. Let H be defined by (6.55), (6.56) and (6.58). Denote by thefollowing extension of the function W in (6.55):

'P(al, x2, ..., xn_1) for x1 < at,'0(x1) x29...,xn-1) _ W(x1,x2,...,xn_1) for al < xl < b1, (6.65)

P(bl, X2,..., xn_ 1) for b1 < x1.

Then 10 satisfies a Lipschitz condition on Wl = {x E fly"-1 : oo < xl < oo,a, < xi < b;, i = 2,..., n -1 } with the same constant M as the function W. For,if, sayxEW,yEW1 andyl >b1, we have

I'(xl,x2,...,xn-1)- '(y1,y2,...,yn-1)I = IW(xl,x2,...,xn-1)-V(bi,y2,...,yn-1)I

:5 I W(xl, x2, ..., xn_1)-p(b1, x2, ..., xn-1)I+IW(b1, x2, ..., xn-1)-co(bt, y2, ..., tin-1) Ia

< M(y1 - x1) + M((x2 - y2)2 + ... + (xn-1 - yn_1)2)' < MIx - 91.

Repeating this procedure with respect to the variables x2i ..., xn we obtain afunction, which coincides with cp on W and satisfies a Lipschitz condition on


with the same constant M as the function cp. We denote it also by W andconsider the domain Il defined by (6.62) and the operatorT satisfying (6.64).

2. For f E WW(H) let To f be the extension of f by zero to Q. Since

suppf \ H (1 V, we have To f E W, (Q) and IIToIIw;(n) = II!Ilwp(H) Hence

IIToIIww;(H)-.wp(n) = 1. Next we observe that TTo : 4' (H) -* Wp(R") and

IITT011rVP(H)-,WP(V) :5 IITIIWo(n)- w (R^) < C13.

Thus Lemma 14 is applicable and the statement of Lemma 17 follows. 0

Our next aim is to construct a bounded linear extension operator (6.64) for9 defined by (6.62), (6.63).

LetG=R"\S2={xEl(t":xn>cp(x)}. We set

Gk = {x E G: 2-k-' < en(x) < 2-k}, k E Z,

where

N(x) = xn - V(x)

is the distance from x E G to 8G = 8SZ in the direction of the axis Oxn.First we need an appropriate partition of unity. 3

Lemma 18 There exists a sequence of nonnegative functions 10k satisfying thefollowing conditions:

1)

001

k=-oo

forxEG,for x V G,

(6.66)

2) G = U suPP''kk=-oo

and the multiplicity of the covering {supptpk}kEZ is equal to 2,

(6.67)

3) Gk csupp'Ok c Gk_, U Gk u Gk+,, k E Z, (6.68)

4) ID°lPk(x)I 5C14(a)2k1a1, x E R' , k E Z, a E K, (6.69)

where C14 (a) > 0 depends only on a.

3 In Lemmas 18- 25 below fl is always a domain defined by (6.62) - (6.63) and G = Ir \1I.


Idea of the proof. Apply the proof of Lemma 5 of Chapter 2.

With the help of the partition of unity constructed in Lemma 18 we definean extension operator in the following way:

f (x) for x E S2,00(T f)(x) _ E 'Pk(x) fk(x) for x E G, (6.70)

k=-oo

where

Here a

fk(x) =J

f(x - 2-k2,x, - w(z)dz

Q1^

= A-12kn f w(2k(t - y), A-12k(xn - yn))f (y) dy. (6.71)

it.

A=16(M+1) (6.72)

and w E Co (R") is a kernel of mollification satisfying

suppwC{xEB(0,1):xn>2} (6.73)

and

f w(z) dz = 1; f w(z)z° dz = 0, a E Non, 0 < jal < 1. (6.74)

B(0,1) B(0,1)

Now let us show that the operator T is well defined. First, we assumethat Ok(x)fk(x) = 0 for x V supplik even if fk(x) is not defined. On theother hand, if x E suppok, fk(x) is defined. This is a consequence of thefollowing inequality, which holds for x E supp ik and z E supp w since by(6.68) Bn(x) < 2-k+1 and by (6.73) IzI < 1,zn >- 1

:

xn - A2-kzn -1V(x - 2-kz) = xn -'P(x) + 1P(s - 2-kz) - A2-kzn

< 2-k+1 +M2 -k121- A2-kzn2-k+1 + M2-k I zj -A2 -kZ" < 2-k 2+M-A) 2<0.

(This means that the point (a - 2-kz, xn - A2-kzn) E ft)

4 One can choose any larger fixed quantity depending only on M.


Furthermore, by Lemma 18, Vx E G the sum in (6.70) is in fact finite: foreach x E G it contains at most two nonzero terms. Moreover,

m+1

Tf = E V)kfk on Gm. (6.75)k=m-1

Thus T is a linear extension operator defined for functions f E Lt0c(Sl).If x E 811 the values (Tf)(x) are not defined by (6.70). When consider-

ing the spaces W.' (RI) this is of no importance, because measn 8Sl = 0. Inthose cases, in which the functions f are defined and continuous in Sl, we shallnaturally assume that (T f)(x) = f (x) for x E U.

Remark 11 Because of the factor A in (6.71), fk is an inhomogeneous mol-lification of f with the steps 2-k, ...,2-1,A2-*.., 2-k, A2-k with respect to the variablesX't, , xn-1, xn. For x E R, r > O, h > 0 consider an open cylinder centered atthe point x of radius r and height h

C (x, r, h) _ (y E lr : y E B(2, r), I xn - y, ,l <2Al.

Because of (6.73) the value fk(x) is determined by the values f (y) for y be-longing to the cylinder

Cyk = /r((a,x" - 4A2-k),2-k, 'A2-k),

which is centered at the point (2, xn-a

A2-k) translated with respect to x in thedirection of the set Q. This follows since in (6.71) w(2k(2 - y), A-12k(xn - yn))can be nonzero only if 2k1 - yI < 1 and .1 < A-'2k(xn - yn) < 1. For thisreason (T f)(x), x E G, can be looked at as an inhomogeneous mollification ofthe function f, for which both the step and the translation are variable. Thusthe extension operator T is closely related to the mollifiers with variable stepsconsidered in Chapter 2.

Note also that on G the operator T is an integral operator:

(Tf)(x) =J

K(x, y)f(y) dy, x E G,n

with kernel

00K(x, y) =A-' E 1Pk(x)2knw(2k(2A-'2k(xn (6.76)

k=-ao


Lemma 19 Let f E L, °°(Q), x E G and x' xn - 4A[!n(x)). Then thevalue (T f)(x) is determined by the values f (y) for y belonging to the cylinder

Cx = C(x*, 4Bn(x), 4APn(x))) C CZ C Q. (6.77)

Idea of the proof. Apply (6.76) and Remark 11.Proof. Let x E G. Choose the unique m E N such that x E G,n. Then zkk(x) = 0if k V {m - 1, m, m + 1} and the value (T f)(x) is determined by the valuesfk(x) where k = m - 1, m, m + 1. By Remark 11 those values are determined

m+Iby the values f (y) for y E U C=,k. Hence J .,t - JI < 2-' < 4e,, (r). and

k=m-I

gAPn(x) < A2-m-2 < xn - yn <.42-m+I < 4A 71(.,r)

Consequently Ixn - aen (x) - N1 < 2APn (x) and y E C?. Moreover. Vy E C,

Ay) - yn = 4P(y) -'A-t) +V(z) - xn + xn - .4 (x) - y,, +

(-4M - 1 + q)en(x) > 39n(x) (6.78)

because of (6.72). Therefore CZ C Q. Note also that similarly

jo(y) - yn < 10A[Jn(x). (6.79)

Lemma 20 Let f E L;°c(11). Then T f E COO (G) and Vc E 1%j

00

D°(Tf)(x) _ Ia!

I (D°-a,i,k)(x)(D3fk)(x). (6.80)0<8<0

. (a - a) k=_oo

Idea of the proof. Apply Remark 11 and Lemma 18.Proof. By Remark 11 Vk E Z and Vx E supp ?k we have Cz,k C 0. Con-sequently, by the properties of mollifiers (see Section 1.1) fk E C°°(H). ByLemma 18 Vx E G there exists a ball, centered at x, which is contained in nomore than 3 sets supp Ik. Hence the series (6.70) can be differentiated termby term any number of times, and by the Leibnitz formula equality (6.80)follows.

Lemma21 LetlEN,1 <p<oo and for kEZ

Gk = Gk-1 U Gk U Gk+1 = {x E G : 2-k-2 < 'on(X) < 2-k+l }

and

S2k = {x E f : 2-k-2 < Ig (x)I < b2-k+I}


where b = 10A.Then Va E Wo satisfying Ial 0 depends only on n, l and M.Moreover, Va E l there exists a function 6 g0, independent of k, such that

IID*fk-gullLp(Gk)<C162k(IaI-')IIfjj.',(?i,)' (6.82)

where e16 > 0 depend only on n, 1, M and a.

Remark 12 It is important for the sequel that g, should be independent ofk and the multiplicities x(; and xn of both coverings {Gk}kEz and {Qk}kE2he finite and bounded from above by quantities, which depend only on M.This follows since these mulitlicities coincide with the multiplicities of the one-dimensional coverings {(2-k-22-k+l)}kE2, ((2-k-2 b2-k+l)}kEz respectively,

and because the multiplicity of the covering {(p2-k-1 v2-k+1)}kez, where 0 <

p < v, does not exceed loge For, the inclusion x E (p2-k-2 v2-k+l) isequivalent to - loge x - loge p < k < - loge x - loge v. Hence the length logeof this interval is greater than or equal to the number of those k, for whirl1 7E (p2-k-2v2-k+l). Thus xG < 3 and xn < log2(8b).

Idea of the proof. Observe that Yx E Gk

C=,kCC.CS2k. (6.83)

To prove (6.81) for a = 0 apply Minkowski's inequality, the substitution x -2-k2 = y, x - Az = y,, and (6.83). To prove (6.82), in addition, expand thefunction f (2 - 2-12, x,, - A2-kz,,) under the integral sign, applying the integralrepresentation (3.38). Taking into account Remark 12 of Section 3.4, replacein (3.38) the ball B by the ball B., = B(x', C C= and w by

w.(y) = (4Pn(x)) °p((4Qn(x))-'(x - y)), (6.84)

where p is any fixed kernel of mollification satisfying (1.1). Apply also ananalogue of inequality (3.56). OProof. 1.The first inclusion (6.83) follows since for y E Cx,k we have Ix - yl <2-1

< 4 e,,(x) and

aA2_k+' < x - y,, < A2-k < 4Agn(x).

5 If jal > 1, (6.82) holds for g, = 0.


Consequently, as in the proof of Lemma 20, Ix. - a A e(x) - y"I < 2 A e" (x).The second inclusion (6.83) follows since inequalities (6.78) and (6.79) holdVyEC=.

2. First let a = 0. By Minkowski's inequality

IIfkIIL,(d,)

< f 11f (.t - 2-k2, x,, - A2-kz")IIL,.,(d,) Iw(z)I dzB(0,1)

IIf Cr.k) Iw(z)Idzsuppw

since by Remark 11 (x - 2-k2, x,, - A2-kz") E CZ,k for z E suppw. Hence, by(6.83)

IIfkIILP(ck) < C151'f

where c15 = IIwIIL,(R-) and we have established (6.81) for a = 0.3. Let E R" and let us consider the polynomial in of order less

than or equal to l - 1

P(e, x) = f(E,(_1)171Dy,

f( - y)7 wx(y)]) f (y) dy,B. 171<1

which is closely related to the first summand in the integral representation

(3.51), where B, u) are replaced by B=, w= respectively. Writing u(z) for (x -2-kz,x - A2-kz"), by (3.51) we have

f(2 - 2-k2, x" - A2-kz") = f(u(z)) = P(u(z),x)

+ (Dwf)(y) w7,.(u(z), y) dy = P(u(z), x) + r.r(u(z), x).171= V, (s)

Iu(z) -y)n_1

171=i

Note that by (6.83) u(z) E Cz and hence Vu(s) C C. Furthermore,

fk(x) = f P(u(z), x) w(z) dz + E f r.1(u(z), x) w(z) dzB(0,1) 171=18(0,1)

R0,k(x) + E R,,k(x). (6.85)

171=1


4. The function P(u(z), x) is a polynomial in the variables ofdegree less than or equal to l - 1:

P(u(z), X) = P(x, x) + > cO(x) z'o,o<IHI<c

where co(x) are independent of z. Note that by (6.74) Ro,k(x) = ro(x, x) andset

go (x) = P(x, x). (6.86)

5. Since w= is defined by (6.84), from inequality (3.57) we get that Vy E Vu(Z)

(,)n-1,7I =1,I w. (u(z), y) I < M' d

where M, depends only on n and 1, d = diam B., = 8 e (x) and by (6.77)

D < diam C= < 10 A Lyn(x). Hence, Vy E V.(,)

Iw7..(u(z),y)I < M2,

where M2 depends only on n, l and M. Consequently,

I r7(u(z), x) I G M 2 f I (Dwf)(y) I Iu(z) -vIt n dy

cc

Let Xnk be the characteristic function of 1 k and c7(y) = I Xnk(y), y ER n.(We assume that '7(y) = 0 for y V ft) Then

IRr.k(x)I < M2 f (f t '(Y) I u(z) - yI'-n dy) Iw(z) dz.

B(0,1) C.

We set 77 = u(z) - y. Since both u(z),y E C., we have Ii7I < diam C.10 A o (x) < 20A2-*. Hence,

I Rr.k (x) I

< M2 f ( f It7(x - 2-kz - f , X. - A2-kzn - r!n) IiI` nd77) Iw(z)I dz.

B(0,1) B(0,20A2-k)(6.87)

Since,

II,t7(' - h)II Lo(ck) <_ II07(' - h)IIL,(R") = II'D7IIL,(R") = 11D-7f IIL,(nk),


applying Minkowski's inequality, we get

II ,kIILv(ck) <- M3 II D of IILp(nk) f 11711 -n d77M4 2-k! II Dwf IILP(nk),

B(0,20 A 2-k )

(6.88)where M3 = M2 IIwllL,(R") and M4 > 0 depends only on n, I and M.

Inequality (6.82) with a = 0 follows from (6.85), (6.86) and (6.88).6. Now let IaI > 0. Taking into account Lemma 3 of Chapter 1, we

differentiate (6.71) and get

(D°fk)(x) = f (Dwf)(x - 2-kz, x - A2-kzn) w(z) dzB(0,1)

= A-°"2klal f f ( - 2-kx, xn - A2-'Z,,) (D°w)(z) dz.B(0,1)

Inequality (6.81) with IaI > 0 follows from the first equality and inequality(6.81) with a = 0.

Next let

c (_1)Ia1+171

P. (C x)= fBy (3.52), as in steps 3-4, it follows from the second equality for D°fk that

(D°fk)(x) = ga(x) + L, A-a"2kIOIRry (x),171=1-la1

where

ga(x) = A-a"2klal f P.(u(z),x)(D°w)(z)dzB(0,1)

and R(°k (x) is obtained from R7,k(x) by replacing Dw f, I by D°+7 f, I - IaIrespectively. By (6.74) f zO(D°w)(z) dz = 0 if IaI > I, or IaI < l and ,0 O a.

B(0,1)

If IaI < l and Q = a, then f z°(D°w)(z) dz Hence g° = 0 forB(0,1)

IaI>Iand

ga(x) = A-Q"2k1*1Ds (Pa(u(z),x))I:.O= Pa(x,x) (6.89)

for IaI < 1. With this choice of ga inequality (6.82) with IaI > 0 follows as instep 5. O


Remark 13 In the above proof the functions g° defined for a = 0 by (6.86)and for 0 < Jai < l - 1 by (6.89) are the first summands in the integral repre-sentations (3.51), (3.52) respectively, where B, w are replaced by Bz, w= respec-tively. Since Vx E Gk we have B(x'(k), M12'k) C B= = B(x'(k), 4 CB(x' (k), M2 2-k), where x' (k) _ (x, x" - 2 A 2-k) and Ml, M2 > 0 dependonly on n. These inclusions explain why one may expect estimate (6.82) tohold with appropriate g°,k. The choice of the ball B(x',4pf,(x)), independentof k and "compatible" with B(x'(k), M2 2-1*), allows us to construct a functiong°, for which inequality (6.82) holds and which is independent of k.

Remark 14 In the proof of Lemma 22 (Section 4) we have applied property(6.73) for Jal < l - 1. The fact that it holds also for Jai = l allows us toprove the following local variant of (6.82) for p = oo : dx E Gk and Va E Nsatisfying lal 0 and a > 0 depend only on n, l and M. Here gO is independentof k and is defined by (6.89) with I + 1 replacing 1.

Estimate (6.90) follows from (6.87), where I is replaced by I + 1 and 17I _I + 1, if to observe that Vz E supp w and do E Ck the point (2 - 2-k2 - q, x -A

2-kx" - i,,) E B(x, a2-k) where a = 22A .

Lemma 22 Let I E N, a E P)al, jai < l and 6 f E C°°(Si). Then the derivativesD°(T f) exist and are continuous on R".

Idea of the proof. By Lemma 7 T f E C°°(R" \ lin). Let x E 80. First show,by applying (6.90), that

lim D°(Tf)(y) = (D°fl)(x), Jal < 1. (6.91)y-+:,yEG

Applying (6.91) and the definition of a derivative prove that (D°(T f))(x) _(D0fl)(x) first for Jai = 1 and then, by induction, for all a E Ni,' satisfyingJai<IEN.Proof. 1. Let 1°,o _ D°-Ot,ik DO fk. Then by (6.80)

k=-oo

D°(Tf)(y) = Eal

1°a(y), y E G, (6.92)#!(a - 1

61.e., there exists a domain Al 17 and a function f, E C00(01) such that f, = f on f2.


where 4j = E D°-p?,kD,9fk. Let x E O 1, i.e., x First we00k-oo

study the difference1°°(y) - D°.f1(x)

m+1

E 'Ok(y) f ((D°f1)(9-2-kz, (D°f1)(±, co(x))]w(z)dz,k=m-1 B(I,1)

where m is such that y E G. (m is defined uniquely). Let it = (y - 2-k4. y -.42-kzn). then lu - 2-1 :5 I:r - yl + 2-k + .42-A < Ix - yl + (A + 1) 2" <Ix - yl + 4 (A + 1) Since on(y) = y" - (s) = Y. - X. + i0ft) - (J) <(11 + 1) Ix - yl we have lu - xl < M, ly - TI, where M1 depends only on M.Consequently,

11.°(y) - (D°f1)(x)1 <_ 312 sup I(D°f1)(u) - (D°.f1)(.r)I -4 0lu-zl<Al, Ix-yI

as y - .r. y E G. (Here 1112 depends only on tl and .11.)W

Furthermore, when 0 0 ct we have E (D°-,1tjk)(y) = 0 andk=-oo

00

I°a(y) = E g9(y))

m+1

(Do-8? k)(y)((Dafk)(y) - gv(y)). ((;.931k=m-1

where gp is the function constructed in Lemma 21 (see (6.89)). Applying (6.901we get

II°p(Y)I <- 11132-m(J+1--I°uIlfllCI+i(fZnB(x,o.2-m+'))

< M4 Ix - yll+1-I°I IIfIIC'+'(nnB(=,Arslx-yI))

where M3, M4, M5 depend only on n,l and M.Therefore 1°p(y) -+ 0 as y -+ x, y E G, and this proves (6.91).2. It follows from what has been proved in step 1 that the function Tf

is continuous in R" and (T f) (x) = f 1(x), x E O. Now we shall prove that(T ) (x) = 8 (x) for x E 852.azi

Consider the one-dimensional set ex = 0 fl lr1), where l=1) is a straight linepassing through the point x and parallel to the axis Ox1. Let x2, ..., x,, be fixed


and W(x,) = (Tf)(xl,...,xn), xi E R, W,(x,) = fl(x1,x2,...,x"),xl E F.--Consider

Note that

a(Tf)(x) = lim '+G(y1) - 0(x,) = lim '(yi) - 01(x1)ax, yl_xi Yi - x, y1-x1 y1 - x,

limy, -4x, ,yl EG

CY0 - 01 (XI) = lim wl(yl) - 01 (XI) = Of, fx).Yi - XI y,-+x, y, - xi ax, l

Let y, V ex. Denote by yl the point in F. lying between x, and y,, which isclosest to y, . We obtain '

V'(yl) --01(x1) _ afl P(y1) - V41) + t'1(yi) - i1(xl) Of,

Yi - XI ax, (x) Y1 - xi - 8x, (x)

W,(SI)yi - Y, + V'1(Yi) - 01(x1) yi - xi afl (x)Yi - xl yl - xl Yi - xi ax,

) Yi -Yi_ (a(Tf)(.F1,

x2, x,,) -afl

(x,,x2, . . ., x" )J...,ax, ax, Yi - xi

+(101(yl) - 101 (XI) - of 1 (x)) yi - xi .yi - x, ax, y1 - x1If y, - x1 the first summand tends to zero because of (6.91) since(c,, x2, ... , x") E G and l;, lies between x, and y,, and the second summandtends to zero because (yl, x2i ..., x") E S2. This proves that 2M (x) = a (x).The continuity of 2 F. follows again from (6.91).

Similarly one can prove the existence and continuity of the derivativesa" , i = 2,..., n (when i = n, the situation is simpler since n fl ii") is a half-line), and, by induction, of the derivatives of higher orders. 0

Lemma 23 Let 1 E N, 1 0 depend only on n, l and M, and

II(x" 2OIIfII-P(1)), IaI > 1, (6.96)

where c2o > 0 depend only on n, 1, M and a.

"If any neighbourhood of x contains infinitely many interval components of es, thenyl # x. Otherwise, for a point yi, which is sufficiently dose to x, we have yj = x,, and theargument becomes much simpler.


Idea of the proof. 1. To prove (6.94) first observe that, as in the proof of Lemma13 of Chapter 2,

IIT!IIL,(n) 5 2( IIfkIIL,(ak))k=-oo

(6.97)

then apply inequality (6.81) and the fact that the multiplicity of the covering{Stk}kEZ is finite.

2. To prove (6.95) apply (6.92) and (6.93). Estimate I. as in step 1. Toestimate I,# where Q 34 a apply inequalities (6.69) and (6.82). In the case ofinequality (6.96) use also the inequality x - cp(a) < M, 2-k on Gk where M1is independent of.k. 0Proof. 1. Since the sum (6.70) for each x E G contains at most two nonzeroterms by Holder's inequality

xIITf 11%(G) 5 2P-' f (E I kfklP) dx.

C k=-x

Furthermore,

x m+l x m+1 x k+1 xf=f= f=jG k.-x k=-xGm k=m-1 m=-xk=m-lGm k=-xm=k-1GT k=-xCk

and inequality (6.97) follows since 0 < 1'k < 1. Consequently, by (6.81)

IITf IIL,(G) 5 2 c1s (E If IILn(nk)) 2 clam Ilf IIL,(n),k=-x

where xn is the multiplicity of the covering {Stk}kEz. which, by Remark 12,does not exceed log2(8b).

2. Suppose that a E No satisfies Ial = 1. Then we consider equality (6.92).As in step 1

IIIaaIIL,(G) <- c21IID.*fIIL,(n)

To estimate IIIoBIIL,(G) where fl # a we can apply (6.93). First of all

IIIaaIIL,(G) <_2IID°-00k(D8fk

- g9)IIPLP(6k))00k=-oo


Furthermore, it follows, by (6.69), (6.82) and Remark 13, that

IIIaf hIL,(G) S M2 ( (2kla-ol 2-k(!-IpuIIf Ilwp(nk))p)

00k=-oo

00 S

= M2( IIfIIwYk=-oo

< M3 E ( IIDwf IILP(?ik)) ° < M4 xx E IID*f IIL,(n) <- M5 11fIal= k=-oo I0I=1

where t112, ..., M5 > 0 depend only on n, l and M, and inequality (6.95) follows.The proof of inequality (6.96) is similar. Let Ial > 1. Since ga = 0, for allsatisfying 0 < /3 < a we have

II(xn -P(t))Ial-'IasIIL,(c)

00 1

< 21 > 11(x. - p(z))Ial-'Da-d'0k (D' fk - 9s)IIp ) °Lp(ak)k=-oo

00 1

< M6 ( E (2-k(lal-')2kla-sl 2-k(1-IaI) Illllwp(nk))') " 5 M7 11fk=-oo

where M6, M7 depend only on n, 1, M and a.

Lemma 24 For each polynomial pt of degree less than or equal to 1, Tp1 = pi.

Idea of the proof. Expand the polynomial pi(z - 2-k2, x - in (6.71)and apply 8 (6.74) and (6.66). O

Lemma 25 Let I E N, 1 < p < oo, f E Wn(Q). Then there exists a sequenceof functions fk E C00(S2) such that

fk -+ f in WP' (9), 1 < p < 00 (6.98)

and

as k -+ oo.

fk -+ f in WW'(SZ), IIfkIIw-,(n) -4 IIf Ilw-,(n) (6.99)

8 If in (6.74) jai < m, then Lemma 25 is valid for polynomials of degree less than or equalto m. This lemma is similar to Lemma 15 of Chapter 2.

is. 1. EXTENSIONS FOR LIPSCHITZ BOUNDARIES 285

Idea of the proof. By Lemma 2 and Remark 2 of Chapter 2 it is enough toassume that supp f is compact in R". Set

fk = A6k (f (. + k en)),

where en = (0, ..., 0, 1) and A6k is a mollifier with a non-negative kernel definedin Section 1.1 with step bk, which is such that dk < dist (supp f. 0Q +and apply the properties of mollifiers (see Sections 1.1 and 1.2).

Theorem 3 Let I E N,1 < p < cc and let S2 C R" be an open set with aLipschitz boundary. Then there exists a bounded linear extension operator

T : lVP(1l) - ii P(Rn) n c- cm (6.100)

such thatIIe'°I-'D°(Tf)IILp(`n) < X21 IIfIIW,(n), Ial > 1, (6.101)

where e(x) = dist (x, 8SZ) and c21 > 0 is independent of f .There exists an open set S2 having a Lipschitz boundary such that in (6.101)

the exponent at - I cannot be replaced by lal - I - e for any e > 0 and for anyextension operator (6.100).

Idea of the proof. Apply Lemmas 17, 23, 25 and note that for a domain S1defined by (6.62), (6.63) e

X. - c(x) < P(x) < X. - (6.102)1+M -To prove the last statement consider 11 = It = {x E R : x" < 0) and argueas in Remark 12 of Chapter 5. 0Proof. First let fl be a domain defined by (6.62), (6.63) and f E V (St). ByLemma 25 there exists a sequence of factions fk E C°°(SZ) satisfying (6.97),(6.98). Consequently, by Lemma 23

IITfkllwp(Rn) < MI llfkllwy(fl),

where Ml depends only on n, l and M. Passing to the limit as k -; oo we es-tablish this inequality with f replacing fk. Applying Lemma 17 we get (6.100).

6 The second inequality is obvious. To prove the first one we note that p(x) > UK (x), whereUK = dist (x, 8K) and K C G is the infinite cone defined by yn > op(z) + MI± - gl, y E R".The desired inequality follows since B(x, (1 + M)-I (x,, - V(:t))) C K, which is clear becauseby E B(z, (1+M)-'(xn-w(2))) we have yn-W(s)-Ml2-gl = yn-xn+xn-iv(y)-Ml2-gl >-(1 + M)Iz - yl + (xn -,'(2))) > 0.


In the case of inequality (6.101) the argument is similar. One shouldonly take into consideration (6.102) and (6.96) and note that the appropri-ate weighted analogue of Lemma 17 is also valid.

Finally, let S2 = R°- and suppose that for some e > 0 and for some extensionoperator (6.100) we have IIx;,°1-1-`D°(T f)IIL,(R .) < oo for all f E Wp(R"-) andfor all a E l satisfying Ial = m > I + e. First suppose that l > 1. Let

q E Bp ° (R"-') \ BB` ' (R`). By Theorem 3 of Chapter 5 there exists afunction f E 4i'p(R")such that f g. By Lemma 2 T f I *_ = f g.

R R R

Since T f E W' (R+), by the trace theorem (5.68), g EBB1+E

° (R"-1) andn.=n

we have arrived at a contradiction. If I = p = 1, the argument is similar: oneshould consider g E L1 (R"`') \ Bi (R"-') and apply Theorem 5 of Chapter 5instead of Theorem 3 of that chapter.

Remark 15 The extension operator constructed in the proof of Theorem 3satisfies (6.100) and (6.101). So, by the last statement of that theorem, it isthe best possible extension operator in the sense that the derivatives of higherorders of T f on `S2 have the minimal possible growth on approaching 011.

Remark 16 The extension operator constructed in the proof of Theorem 3 issuch that for all m E 1 satisfying m < 1 we have T : WD (11) -* WD (R").

Remark 17 Now we describe an alternative way of proving of the first state-ment of Theorem 3. Let 0 be defined by (6.62) and (6.63). It is possible to getan extension operator (6.100) by "improving" the extension operator (6.61) .To do this we replace x" - p(2), which in general is only a Lipschitz function,by the infinitely differentiable function A(x) = 2(1 + M)ej (x), where p# is theregularized distance constructed in Theorem 10 of Section 2.6. By (6.102) wehave

x" - W(2) < A(x) < 2 (1 + M)(x" - cp(z)) (6.103)

and

ID°0(x)I S c, (x" - a E K. (6.104)

So we set1+1

(Tf)(x) = Eakf(x,x" - (1 +Qk)o(x)), (6.105)k=1

1+1 1+1where ,8k > 0 and E ak(-008 = 1, s = 0, ...,1. (Hence E ak(1 + ,Qk)' _

k=1 k=10, s = 1, ...,1.) By using formula (4.127), expanding for f E Cc0(i) the deriva-tive Dsf(x,x" - (1 + 6k)A(x)) by Taylor's formula with respect to the point


(I, cy(I)) E OcI and applying (6.103), one can prove that Lemmas 22 and in-equalities (6.94) and (6.95) are valid for this extension operator as well. Therest is the same as in the proof of Theorem 3.

The extension operator (6.105) cannot be "the best possible" because, ingeneral, T f V C°°(`1Z). On the other hand in (6.105) it is possible to replace

1+1 00

the sum E by the sum E and 10 choose 13k > 0 and ak in such a way thatk=1 k=1

00 00

E IakI II3kl' < oo and E ak(-13k)' = 1 for all s E N0. This gives an operatork=1 k=1(independent of 1) such that (6.100) is satisfied for all 1 E N.

Remark 18 The extension operator (6.29) -- (6.30) in contrast to the extensionoperator described in Remark 17, is also applicable to the spacesdefined in Remark 26 of Chapter 4, i.e., for I E N, 1 < p < oo and an open setS2 C R" with a Lipschitz boundary

T : Wy.....(l) -4 W,.....f(IItn). (6.106)

To prove this for n defined by (6.62) - (6.63), following the same scheme, oneneeds to prove an analogue of (6.95) for wP 1(Sl). This can be established withthe help of an integral representation, which involve only unmixed derivatives

e , j = 1, ..., n.W

Remark 19 The supposition "0 has a Lipschitz boundary" in Theorem 3 issharp in the following sense: for each 0 < ry < 1 there exists an open set SZwith a boundary of the class 11 Lip-1, which is such that the extension operator(6.100) does not exist, as the following example shows.

Example1 Letn>1,IEN, 1<p<ooandQ,={xER":III<1,IaI"<X, < 1} where 0 < ry < 1. Then i E Lip y, but 8l 0 Lipl. Suppose thatthere exists an extension operator (6.100), even nonlinear or unbounded. ThenVf E W,(11,) we have T f E WP(R"). It follows, by the embedding theoremsfor WW(R ), that Tf E Lq(R"), hence f E where q = "P if I < p,

00to Or by an appropriate integral. In that case (T f)(x) = f f (2, x" - A,

1

00 00where 0 E C°°([1,oo)) satisfies f Itb(a)IA'dA < oo for all a E No, f t,b(A)da = 1 and

1 100

f 9P(A)A'da=0forall$E N.1

11 To obtain the definition of such sets one should replace in (4.89) It - 9I by 12 - 9I''.


qE(1,oo)isarbitrary ifl=p,p>1andq=ooifl>D,p>1orl>n,p=1.Consider the function f6(x) = xn where 5 E R \ N0. Then f6 E W,(C1) if, andonly if, 6 > l - v + np 1(1 - y because

1 1

r / (6-l)p+n-1

II.f6IIWo(l.,) < 00 bJ

I xn6-1)P dt) dxn = vn_1 xn dxn < 00.

0

jtj<sn0

(This is also true for l = 0, i.e., for Lp(54).) Let I < p, the cases l = a and l > v

being similar. If-v+"p1(1-ry) =l-a+np1(1-,-y) <8 < -v+"q1then f6 E Wp(O,) but f6 f LQ(SZ7), and we have arrived at a contradiction.

Remark 20 If Sl has a boundary of the class Lip y, where 0 < y < 1, then itis possible to construct an extension operator

T : Wp(SZ) -+ Wpf(R"), (6.107)

where for noninteger yl Wnf(R") - Bp1(R"). The exponent -fl is sharp. Sothe extension (6.107) is an extension with the minimal possible deterioration ofsmoothness. Moreover, if a bounded open set 0 C R" has a continuous bound-ary, then there exists an extension operator, which preserve some smoothness,i.e., for some

T : Wp(Sl) -+ (6.108)

Here BPU)(R") is the space with the generalized smoothness, defined with thehelp of a function which is positive, continuous, nondecreasing on (0, oo)and can tend to 0 arbitrarily slowly. To obtain the definition of the spacesBp(,o(R") one should replace Ihi' by \(Ihl) in (5.8)-(5.9) with 0 = oo andsuppose that lim \(t)t-' = 00.

t 40+

Chapter 7

Comments

The first exposition of the theory of Sobolev spaces was given by S.L. Sobolevhimself in his book [134) and later in his other book [135].

There are several books dedicated directly to different aspects of the the-ory of Sobolev spaces: R.A. Adams [2], V.G. Maz'ya [97], A. Kufner [85],S.V. Uspenskii, G.V. Demidenko & V.G. Perepelkin [150]. V.G. Maz'ya &.S.V. Poborchii (100). In some other books the theory of Sobolev spaces is in-cluded into a more general framework of the theory of function spaces: S.M.Nikol'skii,[114], O.V. Besov, V.P. Il'in & S.M. Nikol'skil [16], A. Kufner, O.John & S. Fucik [86], E.M. Stein [138], H. Trielel [144], (145]. Moreover, inmany other books, especially on the theory of partial differential equations,there are chapters containing exposition of different topics of the theory ofSobolev spaces, adjusted to the aims of those books. We name some of them:L.V. Kantorovich & G.P. Akilov [76], V.I. Smirnov [128], M. Nagumo [107),O.A. Ladyzhenskaya & N.N. Ural'tseva [88], C.B. Morrey [105], J. NeL'as [108],J: L. Lions & E. Magenes [92], V.M. Goldshtein & Yu.G. Reshetnyak [64], D.E.Edmunds & W.D. Evans [56], V.N. Maslennikova [96], E.H. Lieb & M. Loss [91].Throughout the years several survey papers were published, containing expo-sition of the results on the theory of Sobolev spaces: S.L. Sobolev & S.M.Nikol'skiT [136), S.M. Nikol'skii [113], V.I. Burenkov [20], O.V. Besov, V.P. Il'in,L.D. Kudryavtsev, P.I. Lizorkin & S.M. Nikol'skii [15], S.K. Vodop'yanov, V.M.Gol'dshtein & Yu.G. Reshetnyak [152], L.D. Kudryavtsev & S.M. Nikol'skii [84],V.G. Maz'ya [98]. We especially recommend the last two surveys containingupdated information on Sobolev spaces.

We do not aim here to give a detailed survey of results on the theory ofSobolev spaces and their numerous generalizations, and we shall give only briefcomments tightly connected with the material of Chapters 1-6.

290 CHAPTER 7. COMMENTS

Chapter 1

Section 1.1 The proofs of the properties of mollifiers Aa can be found inthe books S.L. Sobolev [134], S.M. Nikol'skii [112] and E.M. Stein (138].

Section 1.2 The notion of the weak derivative plays a very important role inanalysis. It ensures that function spaces of Sobolev type constructed on its baseare complete. Many mathematicians arrived at this concept, friequently inde-pendently from their predessors. One can find it in investigations of B. Levi [89]at the beginning of the century. See also L. Tonelli [142], G.C. Evans [55], O.M.Nikodym [109].

S.L. Sobolev [131], [132] came to the definition of the weak derivative fromthe point of view of the concept of generalised function (distribution) intro-duced by him in (129], (130] and of the generalized solution of a differentialequation. An approach to this notion, based on absolute continuity, was devel-oped by J.W. Calkin [52], C.B. Morrey [104] and S.M. Nikol'skii [112]. See thebook S.M. Nikol'skii [114] (Section 4.1) for details.

Lemma 3 is taken from [24]. Lemma 4 is due to S.L. Sobolev [134].Section 1.3 S.L. Sobolev has introduced the spaces W,(S1) in (131], (132]

and studied their different properties in those and later papers. (Some factsconcerning these spaces, for particular values of parameters, were known ear-lier. See, for example, the papers B.Levi [89] and O.M. Nikodym [109].) Inhis book [134] S.L. Sobolev has pointed out that these spaces are essentiallyimportant for applications to various problems in mathematical physics. Thisbook has given start to an intensive study of these and similar spaces, and toa wide usage of them in the theory of partial differential equations. NowadaysSobolev spaces have become a standard tool in many topics of partial differen-tial equations and analysis. S.L. Sobolev himself worked out deep applicationsof the spaces Wp(11) and their discrete analogues to numerical analysis. (Seehis book [135] on the theory of cubatures.)

Chapter 2

Section 2.2 Nonlinear mollifiers with variable step were first consideredby H. Whitney [153] (their form is different from the mollifiers considered inChapter 2), and later by J. Deny & J.-P. L. Lions [53] (the mollifiers B8) andN. Meyers & J. Serrin [102] (the mollifiers Ca).

For a general lemma on partitions of unity, including Lemmas 3-5 see V.I.Burenkov [28]. That lemma is proved in the way which differs from the proofs

CHAPTER 7. COMMENTS 291

of Lemmas 3-5 in Chapter 2. The idea of constructing the functions ?Pk byequality (2.10) has its own advantages: it is essentially used in the constructionof the partition of unity in the proof of Theorem 5 of Chapter 5 satisfyinginequality (5.71).

Section 2.3 For the spaces C'(S2) Theorem 1 was proved by H. Whit-ney [153], for the spaces Wp(1l) where 1 < p < oo - by J. Deny & J: P. L.Lions [53] and N. Meyers & J. Serrin [102]. The case of the spaces WW(Q) isnew. Theorem 2 was proved by the author [24]. The statement mentioned inRemark 12 is proved in the same paper.

Section 2.4 For the spaces Cf(Sl) Theorem 3 was proved in [153]. Theorem3 (for 1 < p < oo) and Theorem 4 were proved by the author 124], [30].

Section 2.5 The linear mollifiers E6 were introduced by the author [22]. Inthe case SZ = R" \R n the linear mollifiers H6 with variable step (see Remark 26)for some special kernels w were considered and applied to the problem of exten-sion of functions from R"' by A.A. Dezin [54] and L.D. Kudryavtsev [82], [83].V.V. Shan'kov [126], [127] considered the linear mollifiers H6 with variable stepand applied them to investigation of the trace theorems for weighted Sobolevspaces.

Theorems 5 - 9 are proved by the author [221, [30].E.M. Popova [118] has proved that inequality (2.87) in Theorem 8 is sharp

in a stronger sense, namely, the factor POI-I cannot be replaced by BI°I-Iv(p),where v is an arbitrary positive continuous nonincreasing function, satisfyingsome regularity conditions, such that lim v(u) = oo.

4-40+

Theorem 8 was generalized in different directions by the author [24], [30],V.V. Shan'kov [126], [1271, E.M. Popova [118]. See survey [35] for details.

For a fixed e Theorem 10 was proved by A.P. Calderdn & A. Zygmund [51](see detailed exposition in the book E.M. Stein [138]). For an arbitrarye E (0, 1) a direct proof of Theorem 10, without application of Theorem 9,was given by the author [21]. Later L.E. Fraenkel [59] gave another proofand considered the question of the sharpness of inequality (2.96). For thedomain 11 defined by (6.62) and (6.63) Yu.V. Kuznetsov (87] (see also O.V.Besov [11]) constructed a regularized distance pa, satisfying (2.93), (2.96) and,in addition, the inequality (e-) (x) < -b, x E S2, where b is a positive constant.

Chapter 3

Section 3.1 The idea of choosing the function w in the integral represen-tation (3.17) in an optimal way, which has been discussed in the simplest case


in Remark 4, was used by the author in [29], [33], [34]. It gave possibilityto establish a number of inequalities with sharp constants: for the norms ofintermediate derivatives on a finite interval in [29], [33] and for the norms ofpolynomials in [34].

Section 3.2 In the case of bounded fl Lemma 4 was proved by V.P.Glushko [63].

Section 3.4 Theorem 4 is due to S.L. Sobolev [1311-[1331. However, inthose papers the first summand in (3.38) has the form of some polynomial inx, , ..., x of order less than or equal to 1-1. The explicit form of that polynomialwas found, and the tight connection of Sobolev's intergal representation to themultidimensional Taylor's formula was pointed out in O.V. Besov [9], [10],Yu.G. Reshetnyak [121] and V.I. Burenkov [23]. The proof in the text followsthat of [23].

With the help of the integral representation (3.38) where I = 1 M.E. Bogov-skii [17], [18] constructed an explicit formula for the solution v E Wp (Sl),1 <p < oc, of the Cauchy problem: div v = f , where f E Lp(Q), f f dx = 0, for

nbounded domains star-shaped with respect to a ball.

The proof of the integral representation (3.67) on the base of (3.69) is given,for example, in the books M. Nagumo [107] and E.M. Stein [138].

For an arbitrary open set 1 an integral representation for functions f Ewi(fl) nWi (Sl), where 2k > I has been established by V.G. Maz'ya [98].

Finally, we note that in many cases it is important to have an integral repre-sentation, which involve only unmixed derivatives (see, for example Remark 17of Chapter 6). A representation of such type was first obtained by V.P. Il'in [73].In other cases it is desirable to get an integral representation via differences.Integral representations of both types may be deduced, in the simplest case,

a

starting from the elementary identity (Aj ) (x) = (Aa f) (x) - f (ei (At f) (x)) dt,e

where As is a mollifier considered in Section 1.1. Detailed exposition of thistopic can be found in the book O.V. Besov, V.P. Il'in & S.M. Nikol'skil [16](Sections 7-8).

Chapter 4

Section 4.1 Lemma 1 is a variant of Theorem 2 of Section 7.6 in the bookS.M. Nikol'skii [114]. We discuss in more detail the case of semi-Banach spaces(see Lemmas 2-3).

Section 4.2 Inequality (4.49) for p = oo is due to A.N. Kolmogorov [77].E.M. Stein (138] proved that ci,,,,,1 = cl,,,,,,o and ct,m,p < q,m,00 for p E (1, oo).


Theorem 4 and Corollaries 10, 11 contain all the cases, known to the author,in which the constants are sharp. If (b - a) > (p' + 1)', in (4.57) the sharpvalue of the constant multiplying IIffIILD(a,b) is not known.

Section 4.4 For open sets with quasi-continuous boundaries inequalities(4.105) and (4.107) in Theorem 6 are proved in the book J. Netas [108]. Thefirst proof and application of a theorem similar to Theorem 8 was given by R.Rellich [120].

In V.I. Burenkov & A.L. Gorbunov [43] it is proved that in inequality (4.112)c31 < M'1181, where M depends only on n.

Formula (4.127) for weak derivatives is proved, for example, in the bookS.M. Nikol'skiT [114] (Section 4.4.9).

One can find the detailed proof of the Marcinkiewicz multiplicator theorem,formulated in footnote 21, in [114] (Sections 1.5.3-1.5.5).

Section 4.5 Theorem 10 was proved by G.H. Hardy & J.E. Littlewood [66]for n = 1 and S.L. Sobolev [131], [132] for n > 1. The proof discussed in Section4.5 is taken from L.I. Hedberg [68]. One can find proofs of the properties of themaximal functions, formulated in footnote 22, in the books E.M. Stein [138]and E.M. Stein & G. Weiss [140]. The proof of the Theorem 11 in the case

< - is a modification of the proof given by L.I. Hedberg [68]. In the caseQ = f Theorem 11 was proved by D.R. Adams [1]. Counter-example in thecase,6 > was constructed in J.A. Hempel, G.I. Morris & N.S. Trudinger [69].

Section 4.6 Theorem 12 is due to S.L. Sobolev [131], [132], [133]. Thestatement of Remark 33 was established by V.I. Burenkov & V.A. Gusakov [44].

Section 4.7 Theorem 13 for p > 1 was proved by S.L. Sobolev [131], [132],for p = 1 - by E. Gagliardo [61]. The case in which p = 1 and in (4.149) q. isreplaced by q < q. was also considered in [131], [132], [133] (see Remark 36).The second proof of Theorem 13 is a modification of the proof given in [61].For further modifications of this proof see V.I. Burenkov & N.B. Victorova [49].

The statement of Remark 38 was proved by V.G. Maz'ya [97] and H. Federer& W.H. Fleming [58] for p = 1, and by E. Rodemich [123], T. Aubin [4] andG. Talenti [141] for p > 1. (For detailed exposition see [141].) The statementof Remark 39 was proved by V.I. Burenkov & V.A. Gusakov [45], [46].

The compactness of embedding (4.16), under assumptions (4.169), wasproved by V.I. Kondrashov [78].

Theorem 15 was independently proved by V.I. Yudovit [154], S.I.Pokhozhaev [117] and N.S. Trudinger [146]. The sharp value of ca~ in (4.170)for the case of the space W,, (Sl), was computed by J. Moser [106].

In Theorems 12 -13 sufficient conditions on (I weaker that the cone condi-tion, and in some cases necessary and sufficient conditions on Q, in terms of


capacity were obtained by V.G. Maz'ya [97], [98], [99). The case of degener-ated open sets St is investigated in detail in V.G. Maz'ya & S.V. PoborchiT [100).

Chapter 5

Section 5.1 Definition 1 is close to the definition of a trace given in the bookS.M. Nikol'skii [114]. Theorem 1 is similar to Lemma 6.10.1 of that book andto Theorem 10.10 of the book O.V. Besov, V.P. II'in and S.M. Nikol'skii [16].

Section 5.2 Theorem 2 is an updated version of the theorem proved byS.L. Sobolev [133], [134].

Section 5.3 The spaces Bp.(R") - Hp (R") were introduced and studiedby S.M. Nikol'skiT [110), the spaces B,9(lf"), where 1 < 0 < 00, - by O.V.Besov [7], [8]. Of possible equivalent norms we have chosen, as the main norm,the norm (5.8), which contains only differences. This definition appeared to beconvenient in the approach which is used in the proofs of the direct and inversetrace theorems in this book. In this section we prove only those properties ofthe spaces B,,e(R"), which are necessary in order to prove the trace theoremsfor Sobolev spaces. Detailed exposition of the theory of the spaces B,,e(R")can be found in the books S.M. Nikol'skii [1141 (including the case I < 0),O.V. Besov, V.P. Il'in & S.M. Nikol'skiT [16] and H. Triebel [143], (144) (for-00<1<oo,0<p,9<oo).

The usefulness of the simple identity (5.12) was pointed out by A. Mar-choud [95]. The proof of Lemmal is a modification of known proofs. We notethat it works for all 1 < p, 6 < oo and does not use the density of Co (R") inB,,B(R") for 1 < p,0 < oo.

In the one-dimensional case the proof of the inequality (5.19), based onan integral representation via differences, is given in [16] (Section 16.1). Theidentity (5.16) and the proof of (5.19) are taken from [36].

One can find the proofs of the facts stated in Remarks 5-8 in [114] and[16].

Section 5.4 Lemma 10 may be considered as one of possible generalizationsof Hardy's inequalities (5.13), (5.14). The proof of the direct trace theoremfor Sobolev spaces (the first part of Theorem 3) is based on the identities fordifferences (5.31), (5.43) and (5.36) and Lemma 10. In the case l = 1, m = n-1it is due to E. Gagliardo [60]. In the rest of the cases it seems to be new.

Theorem 3 was proved by the efforts of many mathematicians: N. Aron-szajn [3], V.M. BabiS & L.N. SlobodetskiT [6], E. Gagliardo [60], O.V.Besov[7], [8], P.I. Lizorkin [93], S.V. UspenskiT [147], [148], V.A. Solonnikov [137].The final step was done by O.V. Besov (71,18). Theorem 3 was preceeded by a


similar theorem for the spaces Bp,,,(R") established by S.M. Nikol'skil [110].The trace theorem (5.68) was proved by S.V. Uspenskii [1491.Theorem 5 is due to E. Gagliardo [60]. Nonexistence of a bounded linear

extension operator was proved by J. Pe4tre [116]. Existence of a bounded linearextension operator T : L1(R") -+ B"(W), where 0 > 1, was established inV.I. Burenkov & M.L. Gol'dman [41].

The extension operators constructed in the proofs of Theorems 4, 6 and Re-mark 15 in the case of Sobolev spaces WP(R") are the best possible (see Remark16). In the case of Nikol'skii Besov spaces BP,Q(R") the best possible extensionoperators were constructed by L.D. Kudryavtsev [83], Ya.S. Bugrov [19] andS.V. Uspenskii [149].

Section 5.5 Detailed exposition of the trace theorem in the case of smoothm-dimentional manifolds, where m < n - 1, is given in the book O.V. Besov,V.P. Il'in & S.M. Nikol'skii [16] (Chapter 5). The trace theorem in the case ofLipschitz (n-1)-dimentional manifolds was proved by O.V. Besov [11], [121 (seealso [16], Section 20). In more general case of the so-called d-sets, 0 < d < nthe trace is studied in the book A. Jonsson & H. Wallin [75].

Chapter 6

Section 6.1 The idea of defining an extension operator by (6.6) is due toM.P. Hestenes [70]. Estimate (6.4) can be found in V.I. Burenkov & A.L. Gor-bunov [43]. Lemmas 5-6 are proved by V.I. Burenkov & G.A. Kalyabin [471.Inequality (6.25) is taken from V.I. Burenkov A.L. Gorbunov [42], [43]. Forb - a = 1 Theorem 1 is formulated in V.I. Burenkov [31], in the general case itis proved in V.I. Burenkov & A.L. Gorbunov [43].

Section 6.3 Theorem 2 is proved independently by V.M. BabR [5] andS.M. Nikol'skil [111].

Section 6.4 If 1 < p < oo, then the existence of an extension operator(6.100) for Lipschitz boundaries was proved by A.P. Calder6n [52]. His ex-tension operator makes use of an integral representation of functions. (In thesimplest case this possibility was discussed in Remark 2 of Chapter 3.) Toprove (6.100) LP estimates of singular integrals are used, which is possible onlyif1<p<oo.

For 1 < p < oo the existence of an extension operator (6.100) is provedby E.M. Stein [138]. The idea of his method is discussed in Remark 17. Theconstruction used in [138], which is independent of the soomthness exponent 1,is given in footnote 10. Another construction of an extension operator of suchtype is given by V.S. Rychkov [124]. In the case of the halfspace the existence


of an extension operator T, independent of I and satisfying (4.100) for every1 E No, follows from earlier papers by B.S. Mityagin [103] and R.T. Seeley [125].

The best possible extension operator, satisfying inequality (6.101), is con-structed by the author [25], [26]. It satisfies also (6.106). Further generaliza-tions of the methods and results of Section 6.4 for anisotropic Sobolev spacesare given in V.I. Burenkov & B.L. Fain [39], [40].

There is an alternative way of constructing the best possible extension op-erator. One may start from an arbitrary extension operator T (6.100) andimprove it by applying the linear mollifier ES with variable step of Chapter 2,constructed for `Q, i.e., by considering the extension operator defined by ESTon `S2. See V.I. Burenkov & E.M. Popova [48] and E.M. Popova [119].

For open sets f with a Lipschitz boundary the multidimensional analogueof Theorem 1 is proved in V.I. Burenkov & A.L. Gorbunov [42], [43].

The problem of extension with preservation of Sobolev semi-norm II ' ]Iw,(n)is considered in [27], [28].

The condition 8S2 E Lip 1 in Theorem 3 is essential, as Example 1 shows,but it is not necessary. For a wider class of open sets satisfying the so-calledE - 6 condition the existence of an extension operator (6.100) was proved forI = 1, n = 2 by V.M. Gol'dshtein [65] and in the general case by P.W. Jones [74].

We emphasize that the important problem of finding necessary and sufficientconditions on 0 for the existence of an extension operator (6.100) is still open.Answers are known only in some particular cases. If fl is a siply connecteddomain, then for I = 1, n = 2, p = 2 in S.K. Vodop'yanov, V.M. Gol'dshtein &T.G. Latfullin [151] and V.M. Gol'dshtein & S.K. Vodop'yanov [65] it is provedthat the e - 6 condition is necessary and sufficient. In the case 1 E N, n = 2and p = oo necessary and sufficient conditions for simply connected domainsare found by V.N. Konovalov [80], [81].

The existence of an extension operator (6.107) is proved by the au-thor [25], [26]. The fact that for bounded open sets fl with continuous bound-aries the extension by zero from 11 to R" satisfies (6.108) for some M() is provedin V.I. Burenkov [32].

Another types of extensions with deterioration of the class in the case 8) ELip'y into Wq(R") where q < p and into a weighted space Wpm(R°)) wereobtained by B.L. Fawn [57], V.G. Maz'ya & S.V. Poborchil [100].

Finally, we note that the idea of constructing extension operators with thehelp of appropriate partitions of unity, which is used in [25], [26], [39], [40], [74]and in Section 6.4, goes back to H. Whitney [153).

Bibliography

[1] D.R. ADAMS, A sharp inequality of J. Moser for higher order deriva-tives, Ann. Math. 79 (1978), 385-398.

[2] R.A. ADAMS, Sobolev spaces, Academic Press, 1975.

[3] N. ARONSZAJN, Boundary values of functions with finite Dirich-let integral, Conference on Partial Differential Equations, Studies inEigenvalue Problems, University of Kansas (1956).

[4] T. AUBIN, Problemes isoperimetrique et espaces de Sobolev, C. R.Acad. Sci. Ser. A. 280 (1975), 279-281.

[5] V.M. BA]3I1 On the extension of functions, Uspekhi Mat. Nauk 8(1953), 111-113 (Russian).

[6] V.M. BABIe & L.N. SLOBODETSKIL, On boundness of theDirichlet integral, Dokl. Akad. Nauk SSSR 106 (1956), 604-607 (Rus-sian).

[7] O.V. BESOV, On a certain family of function spaces. Embedding andextension theorems, Dokl. Akad. Nauk SSSR 126 (1959), 1163-1165(Russian).

[8] O.V. BESOV, Investigation of a certain family of function spaces inconnection with embedding and extension theorems, Trudy Mat. Inst.Steklov 60 (1961), 42-81 (Russian); English transl. in Amer. Math.Soc. Transl. (2) 40 (1964).

[9] O.V. BESOV, Extension of functions in Lp and Wn, Trudy Mat. Inst.Steklov 89 (1967), 5-17 (Russian); English transl. in Proc. SteklovInst. Math. 89 (1967).

298 BIBLIOGRAPHY

[10] O.V. BESOV, The behaviour of differentiable functions at infinityand the density of functions with compact support, Trudy Mat. Inst.Steklov 105 (1969), 3-14 (Russian); English transl. in Proc. SteklovInst. Math. 105 (1969).

[11] O.V. BESOV, Behaviour of differentiable functions on a nonsmoothsurface, Trudy Mat. Inst. Steklov 117 (1972), 3-10 (Russian); Englishtransl. in Proc. Steklov Inst. Math. 117 (1972).

[12] O.V. BESOV, On traces on a nonsmooth serface of classes of differen-tiable functions, Trudy Mat. Inst. Steklov 117 (1972), 11-21 (Russian);English transl. in Proc. Steklov Inst. Math. 117 (1972).

[13] O.V. BESOV, On the density of functions with compact support in theweighted space of S.L. Sobolev, Trudy Mat. Inst. Steklov 161 (1983),29-47 (Russian); English transl. in Proc. Steklov Inst. Math. 161 (1983).

[14] O.V. BESOV, Integral representations of functions and embeddingtheorems for a domain with the flexible horn condition, Trudy Mat. Inst.Steklov 170 (1984), 12-30 (Russian); English transl. in Proc. SteklovInst. Math. 170 (1984).

[15] O.V. BESOV, V.P. IL'IN, L.D. KUDRYAVTSEV, P.I. LI-ZORKIN & S.M. NIKOL'SKII, Embedding theory for classes ofdifferentiable functions of several variables, Proc. Sympos. in honourof the 60th birthday of academician S.L.Sobolev, Inst. Mat. Sibirsk.Otdel. Akad. Nauk. SSSR, "Nauka", Moscow, 1970, 38-63 (Russian).

[16] O.V. BESOV, V.P. IL'IN & S.M. NIKOL'SKII, Integral repre-sentation of functions and embedding theorems, 1-st ed. - "Nauka",Moscow, 1975 (Russian); 2-nd ed. -"Nauka", Moscow, 1996 (Russian);English transl. of 1-st ed., Vols. 1, 2, Wiley, 1979.

[17] M.E. BOGOVSKII, Solution of the first boundary value problem forthe equation of continuity of an incompressible medium, Dokl. Akad.Nauk SSSR 248 (1979), 1037-1040 (Russian); English transl. in SovietMath. Dokl. 20 (1979).

[18] M.E. BOGOVSKII, Solution of some vector analysis problems con-nected with operators div and grad, Trudy Sem. S.L.Sobolev, Novosi-birsk 1 (80) (1980), 5-40 (Russian).

BIBLIOGRAPHY 299

[19] YA.S. BUGROV, Differential properties of solutions of a certain classof differenital equations of higher order, Mat. Sb. 63 (1963), 69-121.

[20] V.I. BURENKOV, Embedding and extension theorems for classes ofdifferentiable functions of several variables defined on the whole space,Itogi Nauki i Tekhniki: Mat. Anal. 1965, VINITI, Moscow, 1966, 71-155 (Russian); English transl. in Progress in Math. 2, Plenum Press.1968.

[21] V.I. BURENKOV, On regularized distance, Trudy N1IREA. Issue 67.Mathematics (1973), 113-117 (Russian).

[22] V.I. BURENKOV, On the density of infinitely differentiable func-tions in Sobolev spaces for an arbitrary open set, Trudy Mat. Inst.Steklov 131 (1974), 39-50 (Russian); English transl. in Proc. SteklovInst. Math. 131 (1974).

[23] V.I. BURENKOV, The Sobolev integral representation and the Tay-lor formula, Trudy Mat. Inst. Steklov 131 (1974), 33-38 (Russian):English transl. in Proc. Steklov Inst. Math. 131 (1975).

[24] V.I. BURENKOV, On the density of infinitely differentiable functionsin spaces of functions defined on an arbitrary open set, In: "Theory ofcubatures and applications of functional analysis to problems in mathe-matical physics", Novosibirsk, 1975, 9-22 (Russian).

[25] V.I. BURENKOV, On the extension of functions with preservationand with deterioration of the differential properties, Dokl. Akad. NaukSSSR 224 (1975), 269-272 (Russian); English transi. in Soviet Math.Dokl. 16 (1975).

[26] V.I. BURENKOV, On a certain method for extending differentiablefunctions, Trudy Mat. Inst. Steklov 140 (1976), 27-67 (Russian); En-glish transl. in Proc. Steklov Inst. Math. 140 (1976).

[27] V.I. BURENKOV, On the extension of functions with preservationof semi-norm, Dokl. Akad. Nauk SSSR 228 (1976), 779-782 (Russian);English transl. in Soviet Math. Dokl. 17 (1976).

[28] V.I. BURENKOV, On partitions of unity, Trudy Mat. Inst.Steklov 150 (1979), 24-30 (Russian); English transl. in Proc. SteklovInst. Math. 150 (1979).

300 BIBLIOGRAPHY

[29] V.I. BURENKOV, On sharp constants in the inequalities for thenorms of intermediate deriavtives on a finite inteval, Trudy Mat. Inst.Steklov 156 (1980), 23-29 (Russian); English transl. in Proc. SteklovInst. Math. 156 (1980).

[30] V.I. BURENKOV, Mollifying operators with variable step and theirapplications to approximation by infintely differentiable functions, In:"Nonlinear analysis, function spaces and applications", Teubner-Textezur Mathematik, Leipzig, 49 (1982), 5-37.

[31] V.I. BURENKOV, On estimates of the norms of extension operators,9-th All-Union school on the theory of operators in function spaces.Ternopol. Abstracts (1984), 19-20 (Russian).

[32] V.I. BURENKOV, Extension of functions with preservation ofSobolev semi-norm, Trudy Mat. Inst. Steklov 172 (1985), 81-95 (Rus-sian); English transl. in Proc. Steklov Inst. Math. 172 (1985).

[33] V.I. BURENKOV, On sharp constants in the inequalities for thenorms of intermediate derivatives on a finite inteval. II, Trudy Mat.Inst. Steklov 173 (1986), 38-49 (Russian); English transl. in Proc.Steklov Inst. Math. 173 (1986).

[34] V.I. BURENKOV, On sharp constants in the inequalities of differ-ent metrics for polynomials, In: "Theory of approximation of func-tions." Proc. of international conference on approximation of functions."Nauka", Moscow (1987), 79 (Russian).

[35] V.I. BURENKOV, Approximation of functions, defined on an arbi-trary open set in R", by infintely differentiable functions with preserva-tion of boundary values, In: "Methods of functional analysis in math-ematical physics". Peoples' Friendship University of Russia, Moscow(1987), 20-43 (Russian).

[36] V.I. BURENKOV, On an equality for differences and on an estimateof the norm of the difference via its average value, 13-th All-Unionschool on the theory of operators in function spaces. Kuibyshev. Ab-stracts (1988), 37 (Russian).

[37] V.I. BURENKOV, Compactness of embeddings for Sobolev and moregeneral spaces and extensions with preservation of some smoothness (toappear).

BIBLIOGRAPHY 301

[38] V.I. BURENKOV & P.M. BADJRACHARIA, Approximation byinfinitely differentiable vector-functions with preservation of image fordifferent function spaces, Trudy Mat. Inst. Steklov 201 (1992), 152-164(Russian); English transl. in Proc. Steklov Inst. Math. 201 (1992).

[39] V.I. BURENKOV & B.L. FAIN, On the extension of functionsin anisotropic spaces with preservation of class, Dokl. Akad. NaukSSSR 228 (1976), 525-528 (Russian); English transl. in Soviet Math.Dokl. 17 (1976).

[40] V.I. BURENKOV & B.L. FAIN, On the extension of functionsin anisotropic classes with preservation of the class, Trudy Mat. Inst.Steklov 150 (1979), 52-66 (Russian); English transl. in Proc. SteklovInst. Math. 150 (1979).

[41] V.I. BURENKOV & M.L. GOL'DMAN, On the extension of func-tions in Lp, Trudy Mat. Inst. Steklov 150 (1979), 31-51 (Russian);English transl. in Proc. Steklov Inst. Math. 150 (1979).

[42] V.I. BURENKOV & A.L. GORBUNOV, Sharp estimates for theminimal norm of an extension operator for Sobolev spaces, Dokl. Akad.Nauk SSSR 330 (1993), 680-682 (Russian); English transl. in SovietMath. Dokl. 47 (1993).

[431 V.I. BURENKOV & A.L. GORBUNOV, Sharp estimates for theminimal norm of an extension operator for Sobolev spaces, IzvestiyaRoss. Akad. Nauk 61 (1997), 1-44 (Russian).

[44] V.I. BURENKOV & V.A. GUSAKOV, On sharp constants inSobolev embedding theorems, Dokl. Akad. Nauk SSSR 294 (1987),1293-1297 (Russian); English transl. in Soviet Math. Dokl. 35 (1987).

[451 V.I. BURENKOV & V.A. GUSAKOV, On sharp constants inSobolev embedding theorems. II, Dokl. Akad. Nauk SSSR 324 (1992),505-510 (Russian); English transl. in Soviet Math. Dokl. 45 (1992).

[461 V.I. BURENKOV & V.A. GUSAKOV, On sharp constants inSobolev embedding theorems. III, Trudy Mat. Inst. Steklov 204 (1993),68-80 (Russian); English transl. in Proc. Steklov Inst. Math. 204 (1993).

[471 V.I. BURENKOV & G.A. KALYABIN, Lower estimates of thenorms of extension operators for Sobolev spaces on the halfiine (toappear).

302 BIBLIOGRAPHY

[48] V.I. BURENKOV & E.M. POPOVA, On improving extension oper-ators with the help of the operators of approximation with preservationof the boundary values, Trudy Mat. Inst. Steklov 173 (1986), 50-54(Russian); English transl. in Proc. Steklov Inst. Math. 173 (1986).

[49] V.I. BURENKOV & N.B. VICTOROVA, On an embedding theo-rem for Sobolev spaces with mixed norm for limiting exponents, Mat.Zametki 59 (1996), 62-72 (Russian).

[50] A.P. CALDERON, Lebesgue spaces of differentiable functions anddistributions, Proc. Symp. Pure Math. IV (1961), 33-49.

[51] A.P. CALDERON, A. ZYGMUND, Local properties of solutions ofelliptic partial differential equations, Studia Math. 20 (1961), 171-225.

[52) J.W. CALKIN, Functions of several variables and absolute continuity.I, Duke Math. J. 6 (1940), 176-186.

[53] J. DENY & J.L. LIONS, Les espaces du type de Beppo Levi, Ann.Inst. Fourier, Grenoble 5 (1953-54), 305-370.

[54] A.A. DEZIN, On embedding theorems and the extension problem,Doki. Akad. Nauk SSSR 88 (1953), 741-743 (Russian).

[55) G.C. EVANS, Complements of potential theory. II, Amer. J. Math55 (1933), 29-49.

[56] D.E. EDMUNDS & W.D. EVANS, Spectral theory and differentialoperators, Clarendon Press, Oxford, 1987.

[57] B.L. FAIN, On extension of functions in Sobolev spaces for irregulardomains with preservation of the smoothness exponent, Doki. Akad.Nauk SSSR 285 (1985), 296-301 (Russian); English transi. in SovietMath. DokI. 30 (1985).

[58] H. FEDERER & W.H. FLEMING, Normal and integral currents,Ann. Math. 72 (1960), 458-520.

[59] L.E. FRAENKEL, On regularized distance and related functions,Proc. Royal Soc. Edinburg 83a (1979), 115-122.

[60] E. GAGLIARDO, Caratterizzazione delle tracce sulla frontiera rela-tive ad alcune classi di funzioni in n variabili, Rend. Sem. Mat. Univ.Padova 27 (1957), 284-305.

BIBLIOGRAPHY 303

[61] E. GAGLIARDO, Propriety di alcune classi di funzioni in piu vari-abili, Ricerche Mat. 7 (1958), 102-137.

[62] E. GAGLIARDO, Ulteriori propriety di alcune classi di funzioni inpiu variabili, Ricerche. Mat. 8 (1959), 24-51.

[63] V.P. GLUSHKO, On domains starlike with respect to a ball, Dokl.Akad. Nauk SSSR 144 (1962), 1215-1216 (Russian); English transl. inSoviet Math. Dokl. 3 (1962).

[64] V.M. GOL'DSHTEIN & YU.G. RESHETNYAK, Foundations ofthe theory of functions with generalized derivatives and quasiconformalmappings, "Nauka", Moscow, 1983 (Russian); English transi. , Reidel,Dordrecht, 1989.

[65] V.M. GOL'DSHTEIN & S.K. VODOP'YANOV, Prolongementdes fonctions de classe Lp et applications quasiconformes, C. R. Acadsci. Ser. A 290 (1983), 453-456.

[66] G.H. HARDY & J.L. LITTLEWOOD, Some properties of frac-tional integrals. I, Math. Zeit. 27 (1928), 565-606.

[67] G.H. HARDY & J.L. LITTLEWOOD & G. POLYA, Inequalities,Cambridge, 1934.

[68] L. HEDBERG, On certain convolution inequalities, Proc. Amer.Math. Soc. 36 (1972), 505-510.

[69] J.A. HEMPEL, G.P. MORRIS & N.S. TRUDINGER, On thesharpness of a limiting case of the Sobolev imbedding theorem, Bull.Austral. Math. Soc. 3 (1970), 369-373.

[70] M.R. HESTENES, Extension of the range of a differentiable function,Duke Math. J. 8 (1941), 183-192.

[71] V.P. IL'IN, On an embedding theorem for a limiting exponent, Dokl.Akad. Nauk SSSR 96 (1954), 905-908 (Russian).

[72] V.P. IL'IN, On a theorem due to G.H. Hardy and J.L. Littlewood,Trudy Mat. Inst. Steklov 53 (1959), 128-144 (Russian).

[73] V.P. IL'IN, Properties of some classes of diffrentiable functions ofseveral variables defined in n-dimensional regions, Trudy Mat. Inst.Steklov 66 (1962), 227-363 (Russian).

304 BIBLIOGRAPHY

[74] P.W. JONES, Quasiconformal mappings and extendability of func-tions in Sobolev spaces, Acta Math. 147 (1981), 71-88.

[75] A. JONSSON & H. WALLIN, Function spaces on subsets of R",Mathematical Reports. Harwood Acad. Publ. New York. V. 2, 1984.

[76] L.V. KANTOROVICH & G.P. AKILOV, Functional analysis innormed spaces, 2-nd ed., "Nauka", Moscow, 1977 (Russian); 3-rd ed.,"Nauka", Moscow, 1985 (Russian); English transi. of 2-nd ed., Perga-mon Press, Oxford, 1982.

[77] A.N. KOLMOGOROV, On inequalities for upper bounds of succes-sive derivatives of an arbitrary function on an infinite interval, UchenyeZapiski MGU. Mathematics 30 (1939), Vol. 3, 3-16 (Russian).

[78] V.I. KONDRASHOV, On certain estimates of families of functionssatisfying integral inequalities, Dokl. Akad. Nauk SSSR 48 (1938), 235-239 (Russian).

[79] V.I. KONDRASHOV, Behaviour of functions in LP on the manifoldsof different dimensions, Dokl. Akad. Nauk SSSR 72 (1950), 1009-1012(Russian).

[80] V.N. KONOVALOV, Description of traces for some classes of func-tions of several variables, Preprint 84.21. Math. Inst. Ac. sci. Ukraine.Kiev, 1984 (Russian).

[81] V.N. KONOVALOV, A criterion for extension of Sobolev spaces W.(")on bounded planar domains, Dokl. Akad. Nauk SSSR 289 (1986), 36-39(Russian); English transl. in Soviet Math. Dokl. 33 (1986).

[82] L.D. KUDRYAVTSEV, On extention and embedding of certainclasses of functions, Dokl. Akad. Nauk SSSR 107 (1956), 501-505 (Rus-sian).

[83] L.D. KUDRYAVTSEV, Direct and inverse embedding theorems. Ap-plications to the solution of elliptic equations by the variational method,Trudy Mat. Inst. Steklov 48 (1959), 1-181 (Russian); English transl. ,

Amer. Math. Soc., Providence, R.I., 1974.

BIBLIOGRAPHY 305

[84] L.D. KUDRYAVTSEV & S.M. NIKOL'SKII, Spaces of differen-tiable functions of several variables and the embedding theorems, Con-temprorary problems in mathematics. Fundamental directions. V. 26(1988). Analysis - 3. VINITI, Moscow, 5-157 (Russian).

[85] A. KUFNER, Weighted Sobolev spaces, John Wiley and Sons, 1983.

[86] A. KUFNER, O. JOHN & S. FUeIK, Function spaces, Academia,Prague, 1977.

[87] YU.V. KUZNETSOV, On regularized distance for one class of do-mains, VINITI, Moscow, No 2582-74 Dep (1974), 1-12 (Russian).

[88] O.A. LADYZHENSKAYA & N.N. URAL'TSEVA, Linear andquasilinear elliptic equations, "Nauka", Moscow, 1964 (Russian); En-glish transl. , Academic Press, 1968.

[89] B. LEVI, Sul principio di Dirichlet, Rend. Ciec. Mat. Palermo 22(1906), 293-359.

[90] E.H. LIEB, Sharp constants in the Hardy-Littlewood-Sobolev and re-lated inequalities, Ann. of Math., 118 (1963), 349-374.

[91] E.H. LIEB & M. LOSS, Analysis, Amer. Math. Soc., 1997.

[92] J.-L. LIONS & E. MAGENES, Probli mes aux limites non ho-mog6nes et applications, vol. I, Dunod, Paris (1968).

[93] P.I. LIZORKIN, Boundary properties of functions in weighted classes,Dokl. Akad. Nauk SSSR 132 (1960), 514-517 (Russian); Englishtransl. in Soviet Math. Dokl. 1 (1960).

[94] P.I. LIZORKIN, Generalized Liouville differentiation and the func-tion spaces Embedding theorems, Mat. Sb. 60 (102) (1963),325-353 (Russian).

[95] A. MARCHOUD, Sur les deriv6 s et sur les differences des fonctionsde variables rdelles, J. Math. pures et appl. 6 (1927), 337-425.

[96] V.N. MASLENNIKOVA, Partial differential equations, Peoples' Friend-ship University of Russia, Moscow, 1996 (Russian).

306 BIBLIOGRAPHY

[97] V.G. MAZ'YA, Classes of domains and embedding theorems for func-tion spaces, Dokl. Akad. Nauk SSSR 133 (1960), 527-530 (Russian);English transl. in Soviet Math. Dokl. 1 (1960).

[98] V.G. MAZ'YA, Sobolevspaces, LGU, Leningrad, 1984 (Russian); En-glish transl. , Springer-Verlag, Springer Series in Soviet Mathematics,1985.

[99] V.G. MAZ'YA, Classes of domains, measures and capacities in thetheory of spaces of differentiable functions, Contemprorary problemsin mathematics. Fundamental directions. V. 26. (1988) Analysis - 3.VINITI, Moscow, 158-228 (Russian).

[100] V.G. MAZ'YA & S.V. POBORCHII, On extension of functions inSobolev spaces to the exterior of a domain with a peak vertex on theboundary, Dokl. Akad. Nauk SSSR 275 (1984), 1066-1069 (Russian);English transl. in Soviet Math. Dokl. 29 (1984).

[101] V.G. MAZ'YA & S.V. POBORCHII, Differentiable functions onbad domains, World Scientific Publishing, 1997.

[102] N. MEYERS & J. SERRIN, H = W, Proc. Nat. Acad. Sci. USA,51 (1964), 1055-1056.

[103] B.S. MITYAGIN, Approximative dimension and bases in nuclearspaces, Uspekhi Mat. Nauk, 164 (1961), 63-132 (Russian).

[104] S.B. MORREY, JR., Functions of several variables and absolute con-tinuity. II, Duke Math. J. 6 (1940), 187-215.

[105] S.B. MORREY, JR., Multiple integrals in the calculus of variations,Springer-Verlag, 1966.

[106] J. MOSER, A sharp form of an inequality by N. Trudinger, Ind. Univ.Math. J., 20 (1971), 1077-1092.

[107] M. NAGUMO, Lectures on contemprorary theory of partial differen-tial equations, Kyoritsu Suppan, Tokyo (1957) (Japanese).

[108] J. NEInAS, Les mcthodes directes en thdorie des equations a lliptiques,Academia, Prague, 1967.

[109] O.M. NIKODYM, Sur une classe de fonctions considdree dans leprobltme de Dirichlet, Fund. Math. 21 (1933), 129-150.

BIBLIOGRAPHY 307

[110] S.M. NIKOL'SKII, Inequalities for entire functions of exponentialtype and their applications in the theory of differentiable functions ofseveral variables, Trudy Mat. Inst. Steklov 38 (1951), 244-278 (Rus-sian); English transl. in Amer. Math. Soc. Transl. (2) 80 (1969).

[111] S.M. NIKOL'SKII, On the solutions of the polyharmonic equationby a variational method, Dokl. Akad. Nauk SSSR 88 (1953), 409-411(Russian).

[112] S.M. NIKOL'SKII, Properties of some classes of functions of severalvariables on differentiable manifolds, Mat. Sb. 33 (75) (1953), 261-326(Russian).

[113] S.M. NIKOL'SKII, On embedding, extension and approximation the-orems for differentiable functions of several variables, Uspekhi Mat.Nauk 16 (1961), 63-114 (Russian); English transl. in Russian Math.Surveys 16 (1961).

[114] S.M. NIKOL'SKII, Approximation of functions of several variablesand embedding theorems, 1-st ed., "Nauka", Moscow, 1969 (Russian);2-nd ed., "Nauka", Moscow, 1977 (Russian); English transl. of 1-st ed.,Springer-Verlag, 1975.

[115] J. PEETRE, The trace of Besov spaces - a limiting case. TechnicalReport. Lund (1975), 1-32.

[116] J. PEETRE, A counter-example connected with Gagliardo's trace the-orem, Comment Math. Prace Mat. 2 (1979), 277-282.

[117] S.I. POKHOZHAEV, On the Sobolev embedding theorem in the casept = n, In: "Reports of scientific-technical conference of ME)", Moscow,1965, 158-170 (Russian).

[118] E.M. POPOVA, Supplements to theorem of V.I. Burenkov on approx-imation of functions in Sobolev spaces with preservation of boundaryvalues, In: "Differential equations and functional analysis". Peoples'Friendship University of Russia, Moscow (1985), 86-98 (Russian).

[119] E.M. POPOVA, On improving extension operators with the help ofthe operators of approximation with preservation of the boundary val-ues, In: "Function spaces and applications to differential equations",Peoples' Friendship University of Russia, Moscow (1992), 154-165 (Rus-sian).

308 BIBLIOGRAPHY

[120] R. RELLICH, Ein Satz uber mittlere Konvergenz, Nachr. Akad. Wiss.Gottingen, Math.-Phys. Ki., (1930), 30-35.

[121] YU.G. RESHETNYAK, Estimates for certain differential operatorswith finite-dimensional kernel, Sibirsk. Mat. Zh. 11 (1970), 414-428(Russian); English transl. in Siberian Math. J. 11 (1970).

[122] YU.G. RESHETNYAK, Certain integral representations of dif-ferentiable functions, Sibirsk. Mat. Zh. 12 (1971), 420-432 (Russian);English transl. in Siberian Math. J. 12 (1971).

[123] E. RODEMICH, The Sobolev inequalities with best possible con-stants, In: "Analysis Seminar at California Institute of Technology'(1966).

[124] V.S. RYCHKOV, On restrictions and extensions ofBesov and Triebel-Lizorkin spaces with respect to Lipschitz domains (to appear).

[125] R.T. SEELEY, Extention of COO-functions defined in halfspace, Proc.Amer. Math. Soc. 15 (1964), 625-626.

[126] V.V. SHAN'KOV, An averaging operator with variable radius andthe inverse trace theorem, Dokl. Akad. Nauk SSSR 273 (1983), 307-310 (Russian); English transl. in Soviet Math. Dokl. 27 (1983).

[127] V.V. SHAN'KOV, An averaging operator with variable radius andthe inverse trace theorem, Sibirsk. Mat. Zh. 6 (1985) (Russian); Englishtransl. in Siberian Math. J. 6 (1985).

[128] V.I. SMIRNOV, Course in higher mathematics, Vol. 5 rev. ed., Fiz-matgiz, Moscow, 1959 (Russian); English transl. , Pergamon Press,Oxford and Eddison-Wesley, Reading, Mass., 1964.

[129] S.L. SOBOLEV, Le probleme de Cauchy dans l'espace des fonc-tionelles, Dokl. Akad. Nauk SSSR, 3 (1935), 291-294.

[130] S.L. SOBOLEV, Metode nouvelle A resourdre le probli me de Cauchypour les equations linedaires hyperboliques normales, Mat. Sb., 1 (43)(1936), 39-72.

[131] S.L. SOBOLEV, On certain estimates relating to families of functionshaving derivatives that are square integrable, Dokl. Akad. Nauk SSSR,1 (1936), 267-270. (Russian).

BIBLIOGRAPHY 309

[132] S.L. SOBOLEV, On a theorem of functional analysis, Dokl. Akad.Nauk SSSR 20 (1938), 5-10, (Russian).

[133] S.L. SOBOLEV, On a theorem of functional analysis,. Mat. Sb. 46(1938), 471-497, (Russian). English translation: Amer. Math. Soc.Translations, 2(34) (1963), 39-68.

[134] S.L. SOBOLEV, Applications of functional analysis in mathematicalphysics, 1-st ed.: LGU, Leningrad, 1950 (Russian). 2-nd ed.: NGU,Novosibirsk, 1963 (Russian). 3-rd ed.: "Nauka", Moscow, 1988 (Rus-sian). English translation of 3-rd ed.: Translations of MathematicalMonographs, Amer. Math. Soc., 90, 1991.

[135] S.L. SOBOLEV, Introduction to the theory of cubature formulae,"Nauka", Moscow, 1974 (Russian).

[136] S.L. SOBOLEV & S.M. NIKOL'SKII, Embedding theorems, Vol.1, Proc. Fourth All-Union Math. Congress, 1961, "Nauka", Leningrad,1963, 227-242 (Russian); English transl. in Amer. Math. Soc. Transl.(2) 87 (1970).

[137] V.A. SOLONNIKOV, On some properties of the spaces My of frac-tional order, Dokl. Akad. Nauk SSSR 134 (1960), 282-285 (Russian);English transl. in Soviet Math. Dokl. 1 (1960).

[138] E.M. STEIN, Functions of exponential type, Ann. Math 65 (1957),582-592.

[139] E.M. STEIN, Singular integrals and differentiability properties offunctions, Princeton Univ. Press, 1970.

[140] E.M. STEIN & G. WEISS, Introduction to Fourier analysis on Eu-clidean spaces, Princeton Univ. Press, 1971.

[141] G. TALENTI, Best constant in Sobolev inequality, Ann. Mat. PuraAppl., 110 (1976), 353-372.

[142] L. TONELLI, Sulla quadrature delta superficie, Rend. R. Accad. Lin-cei 6 (1926), 633-638.

[143] H. TRIEBEL, Interpolation theory, function spaces, differential oper-ators, Berlin, VEB Deutscher Verlag der Wissenschaften 1978, Amster-dam, North-Holland, 1978.

310 BIBLIOGRAPHY

[144] H. TRIEBEL, Theory of function spaces, Birkhauser-Verlag, Basel,Boston, Stuttgart, 1983; and Akad. Verlagsges. Geest & Portig, Leipzig,1983.

[145] H. TRIEBEL, Theory of function spaces. II, Birkhauser-Verlag, Basel,Boston, Stuttgart, 1992.

[146] N.S. TRUDINGER, On imbeddings into Orlicz spaces and some ap-plications, J. Math. Mech. 17 (1967), 473-483.

[147] S.V. USPENSKII, An embedding theorem for the fractional orderclasses of S.L. Sobolev, Dokl. Akad. Nauk SSSR 130 (1960), 992-993(Russian); English transl. in Soviet Math. Dokl. 1 (1960).

[148] S.V. USPENSKII, Properties of the classes Wpr) with fractionalderivatives on differentiable manifods, Dokl. Akad. Nauk SSSR 132(1960), 60-62 (Russian); English transl. in Soviet Math. Dokl. 1 (1960).

[149] S.V. USPENSKII, On embedding theorems for weighted classes,Trudy Mat. Inst. Steklov 50 (1961), 282-303 (Russian).

[150] S.V. USPENSKII, G.V. DEMIDENKO & V.G. PEREPELKIN,Embedding theorems and their applications to differential equations,"Nauka", Novosibirsk, 1984 (Russian).

[151] S.K. VODOP'YANOV, V.M. GOL'DSHTEIN & T.G. LAT-FULLIN, A criterion for extension of functions in class LZ from un-bounded planar domains, Sibirsk. Mat. Zh. 34 (1979), 416-419 (Rus-sian); English transl. in Siberian Math. J. 34 (1979).

[152] S.K. VODOP'YANOV, V.M. GOL'DSHTEIN & YU.G. RE-SHETNYAK, On geometric properties of functions with first gener-alized derivatives, Uspekhi Mat. Nauk 34 (1979), no. 1 (205), 3-74(Russian); English transl. in Russian Math. Surveys 20 (1985).

[153] H. WHITNEY, Analyticic extension of differentiable functions definedin closed sets, Trans. Amer. Math. Soc. 36 (1934), 63-89.

[154] V.I. YUDOVIG, On certain estimates related to integral operatorsand solutions of elliptic equations, Dokl. Akad. Nauk SSSR 138 (1961),805-808 (Russian); English transl. in Soviet Math. Doki. 2 (1961).

Index

Absolute continuity 20local 20

Arcela's theorem 136

Boundarycontinuous 1490- 149 -150Lipschitz (Lip 1)149-150Lip -y 288quasi-continuous 150quasi-resolved 150resolved 149

Cone condition 93Conic body 93Continuity with respect to transla-tion

for LP spaces 37

for Sobolev spaces 38

Differential dimension 32

Embedding 119compact 135continuous 119operator 119theorems 137-138, 146, 181,186, 194, 196

sharp constants 137-138,184-185, 191-192

Equivalence ofnorms 30

semi-norms 123Extension

operator 216, 247theorems 216, 226, 230, 233,241, 245, 264, 270, 285

Fourier transform 33Function of "cap-shaped type" 17.256

Hardy's inequality 205Hardy-Littlewood-Sobolev inequal-ity 178Holders inequality 14, 141, 188

for mixed norms 188

Kernel of mollification 15

Lipschitz condition 17

Marcinkiewisz multiplicator theo-rem 177Maximal function 178Minkowski's inequality 14

for integrals 14, 49for Sobolev spaces 35

Modulus of continuity 16, 51Mollifier (A6) 15

linear, with variable step (E6,H,, R,) 64, 79 - 80nonlinear, with variable step(Bj, Ca) 45, 47

312

Multidimensional Taylor's formula100

Nikol'skii-Besov spaces 203, 240

Parseval's equality 33Partition of unity 42-44, 222, 230,272Potential

logarithmic 116Newton 116

Rearrangement of a function 190Regularized distance 78

Sobolev space 28 - 29semi-normed 29

Sobolev's integral representation 83,104, 109, 111-112, 114 -116Space

BP,B(P") 203By(O l) 240C(Sl) 11C6(1) 11C(c) 12C'(1) 12C6(c) 12& (0)13C°°(S2) 13C°°(S2) 280CO -(n) 13L9(S2) 12LP (S2) 12

INDEX

L,,. (0) 12Ll(S2) 30w,( S2) 13, 29WW(Q) 13, 28-29(W')i°c(12) 49

Wp(f2) 13, 51

(W 1)o(Q) 13LVp(S2) 13

(Re) 228WP", (S2) 244

W1,--'(S2) 176

Star-shapedness with respect to apoint 92ball 92

Traceof a function 197-198, 239

total 235, 244space 202

total 235, 244theorems 216, 230, 241, 245

direct 216, 230, 241, 245inverse 216, 226, 230,233, 241, 245

Weak differentiation 19, 22closedness 23,commutation with mollifiers 24under the integral sign 23

Young's inequality for convolutions166, 192

http://www. teubnerdeISR\ 3-8S.-,C68-'

B. G. Teubner Stuttgart Leipzig v 8

Date post:	11-Mar-2015
Category:	Documents
Upload:	grigorij-volovskij
View:	273 times
Download:	9 times

Sobolev Spaces on Domains

Documents