Lecture Notes in Mathematics A collection of informal reports and seminars

Edited by A. Dold, Heidelberg and B. Eckmann, ZUrich

Series: Institut de Mathematique, Universit~ de Strasbourg Adviser: M. Karoubi and P. A. M%er

256

Carlos A. Berenstein Harvard University, Cambridge, MA/USA

Milos A. Dostal Stevens Institute of Technology, Hoboken N J/USA

Analytically Uniform Spaces and their Applications to Convolution Equations

Springer-Verlag Berlin .Heidelberg • New York 1972

A M S Subjec t Classif icat ions (1970): 4 2 A 6 8 , 4 2 A 9 6 , 35 E99, 46F05

I S B N 3-540-05746"3 Springer-Verlag Berl in • He ide lbe rg • N e w Y o r k

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Preface

About twelve years ago Leon Ehrenpreis created a theory which

culminated in what he called the fundamental principle for linear

partial differential equations with constant coefficients. This

theory yields practically all results about PDE's and systems of PDE's

as corollaries of a new Fourier type representation for their solu-

tions. The possibility of such a representation is essentially the

content of the fundamental principle. However the whole subject

remained unpublished until recently, when two special monographs

appeared, both giving complete proofs as well as a great number of far

reaching applications. Nevertheless, in view of the amazing

complexity of the whole material, neither of these monographs enables

the reader to penetrate rapidly into the heart of the subject. The

main purpose of the present volume is to provide such an introduction

to this beautiful field which represents a promising area for further

research. In order to achieve this goal, the authors decided to treat

only the case of one PDE. Indeed, all the basic ideas appear in this

case, while one can still avoid building the huge machinery which is

necessary for the proof of the general statement (cf. the first four

chapters in either of the aforementioned monographs). In proving the

main theorem (see Chapter IV below), the authors adopted the original

approach of Ehrenpreis. HoweveLin the concrete presentation and

choice of the material they mainly followed their previous publications.

w--7~reviated in the sequel as PDE's.

~V. P. Palamodov, "Linear differential operators with constant coefficients," Moscow 1967; L. Ehrenpreis, "Fourier analysis in several complex variables," Wiley-Interscience 1970. The latter monograph contains many applications going far beyond PDE's.

Some of the results appear here for the first time (for a more

detailed account, see the section of bibliographical remarks at the

end of this volume).

Let us now characterize very briefly the contents of these

lecture notes. The main idea of the subject consists in a systematic

use of Fourier transforms in the study of convolution operators acting

on different function spaces. However, instead of dealing directly

with concrete function spaces and their duals, one considers a large

class of spaces satisfying certain natural conditions (the class of

analytically uniform spaces). The definition and basic properties of

these spaces can be found in Chapter I. Chapter II is devoted to one

important family of analytically uniform spaces, namely the class of

Beurling spaces. This chapter also serves as an illustration of the

difficulties one has to overcome in proving that a given space enters

the scheme defined in Chapter I. Another class of analytically

uniform spaces is treated in Chapter III. Moreover this chapter

contains an application of these spaces to certain convolution equa-

tions. The basic ideas of Chapter III can also serve as motivation

for Chapter IV, where the fundamental principle is established. The

concluding section contains the bibliographical remarks as well as

some further comments concerning the results discussed in the text.

The present lecture notes were originally based on a course

given by the second author at the University of Strasbourg (Strasbourg,

France) in the Spring of 1970; and, on a similar course given later by

the first author at Harvard University. However in its final form the

text differs rather substantially from both of these courses.

~Formulas, definitions, theorems, etc., are numbered throughout each chapter separately. Thus, for instance, "Lemma 3.II" refers to Lemma 3 in Chapter II, while "Theorem 2" means Theorem 2 of the same chapter in which the reference appears; raised numbers such as 2 refer to the section of the bibliographical remarks.

It is a great pleasure for the authors to express their

sincere thanks to Professor P.-A. Meyer of the University of

Strasbourg for his kind invitation to publish this volume in the

Springer series, "Lecture Notes in Mathematics"; and, to Professor

Leon Ehrenpreis of Yeshiva University in New York for his constant

support and interest. The authors also extend warm appreciation to

Mr. William Curley and Dr. Marvin Tretkoff for carefully checking

the manuscript; and, to Miss Mary Jo Vogelsang and Miss Connie Engle

for their excellent typing job.

C.A.B. , M.A.D.

~The first author was supported by the U. S. Army Office of Research (Durham); the second author wishes to thank "Institut de recherches math~matiques avanc~es" in Strasbourg (France) for various forms of support.

CHAPTER

I.

II.

III.

Contents

Definition and basic properties of analyticall X

uniform spaces

§i. Introduction ......................................

§2. General properties of analytically uniform spaces.. 7

Examples of AU-spaces

§I. The Beurling spaces ~ , ~ .................... 26

§2. The Beurling spaces ~ , ~e .................... 58

Spaces of approximate solutions to certain

convolution equations

§I. The spaces ~B(L;~) .............................. 69

§2. A uniqueness theorem for convolution equations .... 77

IV. The fundamental principle

§i. Formulation of the theorem and auxiliary lemmas.. 90

§2. Proof of the theorem ............................. 107

Bibliographical remarks and other comments ...................... 122

Bibliography .................................................... 128

CHAPTER I

Definition and Basic Properties of Analytically Uniform Spaces

§i. INTRODUCTION

About fifty years ago a new trend appeared in mathematical

analysis, and since then it has been flourishing until the present

day. To characterize its earlier period, it suffices to recall the

names of S. Bochner, K. O. Friedrichs, J. Hadamard, F. John,

I. Petrovskij, M. Riesz, S. Sobolev, and N. Wiener. Their work signi-

ficantly changed such classical areas of mathematics as the theory of

partial differential equations and Fourier analysis.

The next stage of this development was marked by the appear-

ance of the celebrated treatise of L. Schwartz, "Th4orie des distribu-

tions" (1950-51; cf. [46]). The importance of this work for analysis

was twofold. First of all, the classical concept of a function was

broadened by introducing more general objects called distributions (or

generalized functions) on which the standard operations of analysis

can easily be defined. The second and equally important achievement

of this theory was the extension of Fourier analysis to certain classes

of these generalized functions. As a consequence, the classical

theory of Fourier series and integrals became applicable to many func-

tions which are large at infinity (more exactly, to generalized func-

tions of polynomial growth, cf. [4611). Since then the theory of dis-

tributions combined with complex variable techniques developed into a

powerful tool in mathematical analysis.

Once the proper framework had been found it became possible

to formulate properly, and later to solve, many of the basic problems

of PDE's. 2 The pioneering work in this field is connected above all

with the names of L. Ehrenpreis, L. HSrmander, and B. Malgrange. To

give a better idea about the type of problems we have in mind, we shall

briefly discuss one of them.

First, let us recall some basic definitions and the corres-

ponding notation. Let ~ be a non-empty open subset of ~n. A sequence

{Ks}s> 1 is said to exhaust ~(notation: K s ~ ~), if all K s are

non-empty compact subsets of ~, K s ( int K s (s > i) and U K s = ~. - s > 1

For each s K i, ~(Ks) is defined as the space of all C~-functions

with support in K s . ~(Ks) is endowed with the Fr6chet topology of uni-

form convergence of functions and their derivatives on the set K s .

Then we define

(1) ~ ( ~ ) = l i m i n d ~ ( K s ) * S -~ o0

Hence ~(fl) is an (dF)-space *; and, it is not difficult to see that the

definition of ~(~) does not depend on the particular choice of the

exhausting sequence {Ks}. 3D'(~) denotes the dual space to ~(~), i.e.

the space of all Schwartz distributions on ~. If a = (al,...,an) is

any multiindex of nonnegative integers, [~] will denote its length,

i . e . I~1 = C ~ l + ' - ' ÷ a n ; a n d , f o r D = - 1 ~ - - ~ - 1 , . . . , - and any v e c t o r

~ = ( g l . . . . ,~n ) c 1~ n , we s e t

(2) D a ( - i ) ]a] 3 ] a l ~a a a = a ; = ~i "'' ~n

~x I ... 3Xn n

If P(gl,...,~n) is a polynomial, P(D) = P(D I ..... Dn) denotes the cor-

responding partial differential operator.

~--this we shall always mean a strong inductive limit of Fr~chet spaces.

Now the problem can be formulated as follows:

Given an arbitrary P as above~ when is the e~uation

(3) P(D)u = f

solvable in ~'(~) for any f ~ ~'(~)?

At a first glance, this would seem to be a typical problem

in the functional analysis. In abstract terms one could formulate it

in the following way. Let T be a continuous injective mapping of an

~5)-space E into another ~r)-space F. When is the adjoint mapping

T': F' ÷ E' surjective? The Hahn-Banach theorem shows that it suffices

to find conditions under which T is an open mapping. (Indeed, to show

I

that T'u = f has a solution u in F' for each f ~ E, we first

observe that this equation already defines u on the range of T,

because for each ~ = T¢, <~,u> = <~,f>. If the functional

~ <~,u> is continuous on TE, it can be extended to the whole space

F; but the continuity of ~ ~ <~,u> follows from the continuity of

T -I, i.e. the openness of T.) However, since the mapping T generally

is not surjective, the openness of T cannot be proved by standard

methods. We are thus led to the following problem: Given an injective

continuous mapping T: E ÷ F, where E and F are (~Jr)-spaces, find

necessary and sufficient conditions for T to be open. However this

turns out to be a difficult problem, a fully satisfactory solution of

which has not yet been found. 3

All of this indicates that in order to solve equations such

as equation (3), one has to combine functional analysis with yet diffe-

rent methods. The above-mentioned problem concerning equation (3) was

completely solved by H~rmander [26,28]. Combining functional analysis

with Fourier transforms, H~rmander found necessary and sufficient con-

ditions for the solvability of equation (3) in ~'(~). On the other

hand, the most systematic use of Fourier analysis in this field was

made by L. Ehrenpreis who found a unified way of studying different

problems of the above type by Fourier transform. ~ Before concluding

this section, let us sketch very briefly the motivation which underlies

Ehrenpreis's approach.

Given any ¢ ~ ~(~) we define the Fourier transform of @ by

(4) ,,¢'(qb) (¢) = ~ ( ~ ) e_i<x,~>#(x)dx

~n

(~ = ~ + in c cn).

The space of all functions 6, ~ ~ ~(~), will be denoted by ~(~).

The topology ~(~) on ~(~) is defined by requiring that ~:~(~) ÷ ~(~)

be an isomorphism of locally convex spaces. One important consequence

of formula (4) is that the elements of ~(~) are entire functions, i.e.

~(~) is a subspace of the space ~ of all entire functions in cn The

space ~ will always be considered in the topology of uniform conver-

gence on compact subsets in cn; therefore, ~(~) is continuously

embedded in ~. Then, however, problems of type (3) can be viewed as

problems of mappings between various subspaces of~ carrying a finer

topology than the relative topology of ~. Since in all concrete situa-

tions these subspaces are characterized by different types of growth

conditions, a good way of describing their topology seems to be the

following:

~n. Let us call a ma~orant any positive continuous function on

If 3%f= {k} is some non-empty family of majorants, we set

~method also has its limitations; see Example 3 below and the Bibliographical remarks.

qf(¢) [ o~(J{) = {f ea¢: {[fNk def= sup _ _ < ~o (Vk eX)}

¢ e cn k(¢)

J/(k; e) = { f e ~ ( J 4 ) : I l f l ]k < e} ; JF(k) = J r ( k ; 1 ) .

Naturally, for some ~ we may obtain ~(X) = {0}. The space M(X) is

equipped with the topology ~(X) generated by the norms l].]Ik, k e~;

~(~) then becomes a Hausdorff locally convex space. Since each k is a

positive continuous function on cn, the space ~(~) is continuously

embedded in og. In particular, each M(~) is a complete space.

Now it is natural to pose the following

Problem. Assume that (E,gE) is a given space of entire functions,

which is continuously embedded in o¢. Is it possible to represent the

space (E,~E) in the form (M(~),~(~))? In other words, given a con-

crete space (E,~E) with the above properties, the question is whether

there exists a family of majorants X such that

(A) E = ~ ¢ ( x )

(B) $~ = ~(~ )

If such a family ~ exists, we shall call the space X(~) a

complex representation of the space E.

Some explanatory remarks should clarify the foregoing pro-

blem. First, the problem should not be understood as the question of

whether each space, which can be continuously embedded into ~, allows

a representation with properties (A) and (B) (cf. Example 3 below).

Instead, the problem consists of finding complex representations of

concrete spaces which are important in applications. Moreover, even

23] ) .

(B).

d e n t .

if we already know that some space has a complex representation we may

ask for another one; namely, we can look for a new family~ which sa-

tisfies some other conditions in addition to (A) and (B) (cf. [4, 17,

Obviously, ~ is not uniquely determined by conditions (A) and

It is also easy to see that conditions (A) and (B) are indepen-

Ehrenpreis was first to recognize the importance of such

complex representations for solving linear equations of convolution

type in various spaces of distributions. He also found complex repre-

sentations for most of the known function spaces [20, 21, 23]. The

present lectures are intended as an introduction to these topics.

7

§2. GENERAL PROPERTIES OF ANALYTICALLY UNIFORM SPACES

In the sequel all topological vector spaces are always

assumed to be Hausdorff and locally convex. We shall call them l.c.

spaces. Given an l.c. space E, let E~ be the strong dual of E and

<'''>E the bilinear form defining the duality between E and E{. For

all other terminology and facts related to l.c. spaces, cf. [24, 32,

43].

Definition i. An l.c. space W is called an analytically uniform space

(AU-space) of dimension n provided the following conditions are satis-

fied:

(i) W is the strong dual of some l.c. space (U,~u).

(ii) There exists a continuous analytic embedding w of the

n-dimensional complex space Cn into W such that the range

~(¢n) is a total subset of W. In particular, for each

S ~ U,

A

S(z) = <S,~(z)> U

A

is an entire function in Cn. The mapping ~: S ÷ S is

obviously linear and injective. Let U d~f {S: S s U}, ^

~ = ~(~U). Hence ~: U ÷ U is an isomorphism of l.c.

spaces.

(iii) There exists a family J( = {k} of majorants (cf.§l), which ^

defines a complex representation of U, i.e. J6 such that

(B) ~ (JO = ~ 0 •

Each ~ with properties (A) and (B) is called an analytically

(iv)

(6)

(v)

(vi)

Remarks:

8

uniform structure (AU-structure) for the space W.

space U is called the base of the AU-space W.

The

There is an AU-structure ~ = {k} such that if we form

for each N > 0 the family ~N = {kN}' where

kN(Z ) d~f max

Iz z'l<N

{k(z')(l + ]z'[)N},

then ~N is again an AU-structure for W.

There exists a family J~ = {m} of majorants with the follow-

ing properties. For each m s ~ and k sTY,

m(z) = £Y(k(z)). Hence all sets

A

ls(z) l A(m,cO def= (S s U: sup < c~ ~ (c~ > O; m s~)

z c cn m(z)

are bounded in U; moreover, we require that the family

(A(m;~)} defines a fundamental system of bounded sets in the

space U. Each family~ with these properties will be called

a bounded analytically uniform structure (BAU-structure)for

the space W.

Similarly as in (iv), all modifications ~N of some Jfare

again BAU-structures for W.

I. The families ~ and Jr are not uniquely determined.

Obviously, the uniqueness could be achieved by requiring

that these families be the maximal families with properties

described in Definition I.

*Families ~N will be called modifications of ~.

9

A

2. Given S ~ U, the entire function S will be called the

Fourier transform of S. This terminology is fully justified,

because in all the examples we shall consider, W will be a

space of functions or distributions in the variable

x, x E ~n, and ~ will be the exponential mapping,

i<x,z> ~: Z ~ e

It follows from condition (iv) above that multiplication by ^

a polynomial defines a continuous endomorphism of U. In the case of a

distribution space W, multiplication by a polynomial P(z) in U corres-

ponds to the partial differential operator P(D) acting on W. This

suggests the following definition.

Definition 2. If F is an entire function such that the multiplication

by F is a continuous endomorphism in U, i.e. F ~ L(U,U), F is called

a multiplier of the space U; and, the continuous operator ~F defined as

the adjoint of the mapping S ÷ ~-I(F(~(S))) is called a convolutor

df the space W. The set ~(W) of all convolutors of W is a subspace

of L(W,W); and,~(W) is given the corresponding relative (compact open)

topology of L(W,W). Each convolutor ~F such that the corresponding

multiplication by F defines an open endomorphism of U, will be called

invertible in W, and the multiplier F slowly decreasing in U.

Remark 3. One can verify as in §i that invertibility of C F implies

~F(W) = W. The terminology "slowly decreasing" comes from

the following condition which is sufficient for a convolu-

tor ~F to be invertible in every AU-space W:

There exist positive constants A and N such that for

all z E ~n there is a Pz' 0 < Pz--< N, such that

(7)

(8)

10

min iF(z,) I > A Iz'z'i=Pz -- (l÷I~I) N

Indeed, given k E %, condition (iv) shows that, for each

N > O, there is a k s X and C > 0, such that

Y(kN) c~(k;C) (cf. (5)). If FS s ~(k), inequality

(7) implies

I~(z)[ < A -I max (]F(z+z')~(z+z')[(l+ ]z[) N) ~ A-ikN(Z) , - iz'L!N

i.e. ~ ~ Y(k;C/A); hence the multiplication by F is an open ^

mapping of U. ~

Since condition (iv) of Definition 1 implies that all poly-

nomials are convolutors for any AU-space W, one can ask which polyno-

mials are invertible for a given space W. The answer is simple:

Proposition i. Every polynomial satisfies condition (7). Therefore

all equations (3) are solvable in any AU-space W.

The proof of a more precise version of this proposition

appears in Chapter IV (cf. Lemma 2, IV).

Remarks: 4. The above definition of a convolutor is correct only if

the mapping F + C F is one-to-one. This, however, is an

immediate consequence of analyticity of F.

5. Solvability of any equation (3) in some W is sometimes

referred to as solvability of the division problem in W.

Therefore the impossibility of solving the division problem

in some W implies that W is not an AU-space.

11

We have just seen that each partial differential operator*

P(D) defines a homomorphism of any AU-space W. Therefore it is natural

to expect that AU-spaces are nuclear. Actually much more is true s.

Theorem 1.

n u c l e a r .

If W is an AU-space and U its base, then both U and W are

In the proof we shall need a simple lemma on entire

functions:

Lemma i. Let Z(z) be a majorant in Cn and H e~. Set

A 1 = {z = (Zl,Z2,...,Zn) : maxlzjl !l}; ~(z) = sup {%(z')(l+Iz'I~n+l}; j z'-zca I

and dp(z) = ~r-n(l+Izl)-2n-iIdz where Idzl is the Lebesgue measure

in cn = ~2n. Then

I.(~) I P Ill(z)_____! (9) sup n < J dp(z)

z ~ ¢ ~ ( z ) - ~ ( z ) c n

Proof. The mean value property of harmonic functions implies

1 (IO) H(z) -

n IT

P

J H(z +c)ldCl •

A 1

Multiplying the integrand in (i0) by the function

~(z)/(~(z')(l + Iz'l)2n+l), which is ~I in the polydisk AI, we obtain

i I H ( z + ~ ) l l d ~ l < .... ,

A1 (~n

and this proves the lemma.

*i.e.P linear and with constant coefficients as we shall always assume.

12

The proof of Theorem I is based on the following criterion

for nuclearity of a strong dual E' of an l.c. space E (cf. [43], Propo-

sition 4.1.6). If A is a bounded, closed and absolutely convex subset

of E, let E(A) be the normed space E(A) def.= U XA with the norm PA X>0

defined by the unit ball A. The unit ball in the dual space E'(A) will

be denoted by A °. The set A ° is a compact space in the weak topology

g(E'(A),E(A)); and, we have a natural embedding ~:E(A) ÷ C(A °) where

C(A °) denotes the space of continuous functions on A °. Let ]i{ denote

the mapping of E(A) into C(A °) defined by ]1](x) = the absolute value

of the function t(x).

Nuclearity of a strong dual:

The space E' is nuclear if and only if E has a fundamental

system ~(E) of bounded closed absolutely convex sets such that for

each A ~ ~(E) there exists a B e ~(E) and a positive Radon measure

on A ° for which XA c B, for some X > 0; and, for each x ~ E(A),

(11) P B ( X ) <_ <l l l (x) ,~>c(Ao )

(Obviously, we can also write <]1](x),~>C(AO ) = y {<x,a>E{d~(a)')

A o

In our case E = U and ~(E) will be defined by means of any

family v*¢(satisfying conditions (v), (vi) of Def. i) as follows: Let

us set t(~) = ~2n+l : {m2n+l}m¢~' t2(~) = t(t(~))''''~= U ~(~); n>l

then ~(E) is the family of all sets A(m,a) (cf. Def. i), m ~.

Let A = A(m,~) be a fixed set in W(E). For each

let y(z) be the functional defined on S ¢ E(A) by <S,y(z)>

= S(z)/m(z). Then y maps cn continuously into A °. For each

let 6(s) be the element of C'(A °) defined by <f,6(S)>c(Ao ) = f(s)

all f E C(A°). Let us now consider the continuous mapping

z ~ (~n

S ¢ A 0 ,

f o r

13

60 y: ~n ÷ C,(AO). Integrating this mapping with respect to the mea-

sure dp(z) of Lemma i, we obtain a measure ~ c C'(A °) such that for

the elements of C(A °) of the form Ifl, f E C(A°), we have

(12) <Ifl ,~>C(AO) = cn

On the other hand, if S e E(A),

t<f,6(Y(z))>C(AO ) ldP(z)

< l (S) ,6 (y (z)) >C(A °) = <S,y(z) >E(A) g(z)

= ~ ;

hence by (12),

(13) <III(s)'~>C(A°) = ~ m(z------~ dp(z) cn

Let B = A(m2n+l;a ). Then B e ~(E) and inequality (9) holds by

Lemma i. However comparing (9) and (13) we obtain (ii), which proves

the nuclearity of W. The proof of the nuclearity of U is similar [5],

and uses a criterion which is "dual" to the one above [43].

The next corollary expresses some well-known properties of

nuclear spaces (cf. [43]).

Corollary I. Let W,U be as above. Let us consider on the space 0

(in addition to the norms [l.llk, k e X) the following systems of norms

(14) llglt~ k) = ~ IS(z) l dp(z) (k e 74) k(z) cn

(15) HSll~ k) [~n [S(z)]dP(z) ] ~ = (k ~ X)

k2(z)

14

Then each of the systems II'II k (k ~ X), (14) and (15) define~ the same

topology on 0. ~ In particular, the topology of 0 can be defined by the

the scalar products

{l(Z)~z(z)dP(z) (k E x ) = (16) [~l,~2]k kZ(z )

cn

Let us denote by W the dual of the space U. The space

corresponds to the space W by the formula <S,T> U = <S,T>~ for any

T ~ W and S ~ U. Let T E W be fixed. By Corollary I~ there exists

k, k ~ ~, such that T defines a bounded linear functional on the

pre-nilbert space (U,[.,.]k).

space. The mapping

Let U(k) be the completion of this

H ~ f i = n(z)

k(z) (i + Izl)n+~

is an isomorphism of the space U(k) onto a closed subspace U(k) of

L2(¢n). If T is the image of ~ in this isomorphism, then T can be

extended to the functional ~ defined on the whole space L2(¢ n) by

setting, for instance, ~ = 0 in the orthogonal complemeat of (k) L2(¢n). Thus we have <H,T>~ = <H,~>L2 ~ Let F(z) be the (¢n)

function in L2(¢ n) generating the functional T, i.e.

in

~ S G(z)F--C~Ytdzl (17) <G,T>L2C¢n) ¢n

for all G ~ L2(¢n). Applying this representation to the elements

G ~ L2(¢ n) of the form G(z) = S(z), S ~ U, we obtain

~thus also on U.

Corollarz__~. For any T E W, there exists a majorant k ~ 5~ and a

function F(z) E L2(¢ n) such that T can be written as the Fourier

integral

F(z) Idzl (18) T = S w(z)

cn k(z) (I + Iz[) n+~

The integral in (18) is to be understood in the functional sense.

Remarks:

(19)

6. In many examples of AU-spaces the integral in (18) con-

verges as a Lebesgue integral. Moreover, each Fourier

representation (18) can be written in the form

I

S ~(z)(1 + Izl) n+~- d~(z)

cn

where ~ is the Radon measure in cn given by

(2o) d~(z) =

(i+ Izl) 2n+l

For S s U, formulas (17) and (!9) imply

(21) O ~(z~ d~(z) <S,T> U = j

cn

Actually it can be shown (cf. [19]) that for each T, there

is a Radon measure ~ * and a k ~ ~ such that (21) holds

without assuming (iv), (v), (vi). Then, however, one has to

assume that for all S ~ U and k ~ ~, S = ~(k). Such

spaces will be called weak AU-spaces ,.

~not necessarily of the form (20).

16

7. Very often it is not necessary to integrate in the inte-

gral of (21) over the whole space cn, but only over a

smaller set 6. If ~ can be taken the same for all elements

T c W, 6 is called a sufficient set for W. By the maximum

principle, each ~ of the form 6 = Cn~K, where K is an

arbitrary fixed compact set, is sufficient for every

AU-space W. However it frequently happens (cf. [23,51])

that there are sufficient sets of smaller dimension than

dim W:

Examp!e i. It is shown in [23] that the space ~ is itself an

AU-space with the base U~ such that U~ is the space of all entire func-

tions of exponential type. For the sake of simplicity, let n = i.

Using the Phragmdn-Lindel~f principle one can prove [19,23] that the

union of any two non-parallel lines in the complex plane is a suffi-

cient set for ~. Moreover, it has recently been proved by B.A. Taylor

[51] that the set of all lattice points (m +in~ (m,n integers) in the

plane is a sufficient set for~; in particular, every entire function

f can be written as

f(z) = ~ am, n e (m+in) z m,n

8. Another interesting problem of this kind is to find

subsets 6 in Cn which would be sufficient for representation

of all elements of a given subspace W o of W. Given an

AU-space of distributions and P(D) a linear partial diffe-

rential operator with constant coefficients, set

Wo def= ~f ~ W:P(D)f = 0~. Then the main result on AU-spaces,

the so-called fundamental principle (cf. [18,23,41] and

Chap. IV below), asserts that the set Vp = (~:P(~) = 0~ is

17

(essentially) a sufficient set for W . o

9. It is clear that if W is an AU-space and A an equicon-

tinuous subset of W, then there exists a majorant k e

and a constant C > 0 such that the Fourier representation

(21) holds with the same k for all T e A; and, for the total

variation of d~ we have JJd~ lJ ! C. This suggests the

question whether the converse of this statement is also true.

More exactly, let W be a reflexive l.c. space satisfying

conditions (i) and (ii). Furthermore, assume that there is

a family of majorants ~ = {k} such that: (iii ~) every

equicontinuous set can be represented uniformly (with respect

to k's) by Fourier integrals of the form (21); (iv*) every

Fourier integral of the form (21) is an element of W; and,

(v*) for each k ~ X, the set

¢n

is equicontinuous. Is the space ~'(X) then a complex

representation of W'? In particular, is W an AU-space?

The answer is in general negative, since it suffices to

take as U any reflexive proper subspace of a reflexive space

U 1 which is an AU-base of some W I. Then W = U~ is the

counterexample. This indicates that even for concrete

spaces W the converse of Corollary 2 may be difficult to

prove. In many cases this problem is equivalent to an

approximation problem in ~(X) (cf. [23], p. 461-462).

There are some other properties of U and W which follow

from Theorem i.

18

Corollary 3 Let W be an AU-space with base U. Then, in addition to

being nuclear, the spaces U and W always possess the following proper-

ties:

(a) U is complete, semireflexive and the bounded sets in U

are metrizable and relatively compact (therefore also

separable) .

(b) W is barreled, separable and the bounded sets in W are

precompact. Moreover, if U is barreled,* then (U,W) is a

(reflexive) pair of Montel spaces, and the bounded subsets

of W are also separable.

(Proof: That U is complete and W separable follows from Def. i.

As a nuclear complete space, U is semireflexive [43]; hence W

is barreled [32]. Metrizability of bounded sets in U is proved

in [5]. The rest follows easily.)

Remark i0. Actually, the reflexivity of the pair (U,W) was included

in the original definition [18,19]. In this case condition

(i) can be dropped from Def. I. However, as the next

example shows, there are non-reflexive AU-spaces.

Example 2. Let I'I by any norm in sn. For each integer Z > 0, set

s c s ( ~ n ) : B£ = {x:Ix I ~ £}. Denote by ~£, s=0,1,..., the space {f g o

supp f C_B£ with the natural topology of a Banach space,

Let us set

= s u I D C ~ f ( x ) I •

s ~F = lim proj ~s ~s = lim ind ~ Z '

W~act, a less restrictive condition is still sufficient [5 ].

On the other hand we have (cf. §i),

~(Bz) -- lim proD ~s ~ = lim ind ~(Bz) , s Z

It is obvious that as sets, ~F = ~ -- Co(~n)" The identity mapping

÷ ~F is continuous [46]. If A is a bounded subset of ~F' then A

is bounded in every ~s. Since the spaces ~s are strict inductive

s o i.e limits, A must be bounded in some.~z In particular, A c~Z ' " S O

all f ~ A have the support contained in BZ However, since the O

S S relative topology of ~s on ~Z coincides with the topology of ~Z and O O

A is bounded in ~s, A is also bounded in ~ for all s _> 0; hence A O

is bounded in ~B~) and thus also in ~. Therefore the bounded sets in

and ~F coincide, and the space ~ def ~ , = (F) b , called the space of

distributions of finite order, has the relative topology of the space

!

~'. By [46], ~ is dense in ~'. Thus (~F)b = (~')b =~" As a

projective limit of nuclear spaces, ~F is also nuclear [43]. (Since

it can be shown [23] that ~ is an AU-space with the base ~F' the

nuclearity of both ~F and ~ also follows from Theorem i.)

We can summarize the properties of ~F and ~ in the following

table (cf. Corollary 3):

I ~ Space

!Property ~

nuclear complete semi re f l ex ive r e f l e x i v e ba r re l ed bornological

~F

yes

yes

yes

no

no

no

yes

no

no

no

yes

yes

The space ~F is not barreled since it is not reflexive. However,

20

being complete but not barreled, ~F cannot be bornological [32].

Finally to see that ~ is bornological we proceed as follows: ~'s,

the strong dual of ~s, is metrizable (~'s = lim proj ~'S(Bz)), and

thus bornological. However, since ~F is dense in each @s, we have by

, , = ~! , [24] p. 148, Th. 1.6 ~ lim ind s and bornologicity of ~ S

follows.

Let WI,W 2 be AU-spaces of dimensions nl,n 2 and with bases

UI,U 2 and AU-structures ~i' ~2 respectively. Set W = W 1 @ W2,

U = U 1 @ U2, 7( = {kl(Zl)k2(z2):k I e ~I' k2 s 7#2} and

~(Zl,Z2) = ~l(Zl) @ ~(z2). Let U be the completion of U in the

topology ~(~). If one of the spaces UI,U 2 is barreled, then all

topologies compatible with the tensor product U 1 @ U 2 coincide with

~(~) [5], and the completion W of W in the finest topology on

W 1 @ W 2 (i.e. in the l-topology of Grothendieck on W 1 @ W2, cf. [43])

is an AU-space with the base U and the AU-structure 7~(cf. [5]). The

space W will be called the AU-product of W 1 and W 2. This remark is

useful in various problems involving AU-spaces in several variables,

because the possibility of decomposing an AU-space into a tensor pro-

duct of 1-dimensional AU-spaces very often turns out to he of primary

importance. Actually, if,in addition to this tensor property of W,

its base U satisfies some further conditions, then the main result of

the theory, the so-called fundamental principle, can be established

(cf. Chap. IV). Let us summarize these conditions in the following

definition.

Definition 3. An AU-space W of dimension ~ is called a product

localizable (or PLAU-) space provided the following holds:

WTh-~ ~ F is not bornological also follows directly from the previous

discussion of bounded sets in ~ and ~F"

21

(vii) There are 1-dimensional AU-spaces Wj, with Uj, K(J~ de (j)

(j=l,2,...,n) as in Def. i, such that W is the AU-product of

the spaces W.. Moreover, there is a BAU-structure ~of W J

such that each m ¢ df is of the form

m(z I ..... z n) = ml(zl)m2(z2)...mn(Zn) ,

where m. ~ ~(J). J

The next condition must hold for each Wj, and for this reason we write

there, for any fixed j=l, ,n, W, U,X,~ in place of Wj, Uj, ~(J) . . .

d¢ (j) respectively.

(viii) The family X can be chosen so that, for each ~ > 0 and

each m ¢~, there exists m* ~ such that for any

= + " E ¢ , z o x o IY o

for which

m(Zo) [¢(z) [ min [~(~)[

I¢-Zol!~

and

I~(z) I sup { r e a l

there are entire functions ~(z) and ~(z)

< m*(z)

m(~ +iYo) )< m*(z)

min l~(~+it) i -- It-Yol <_

for all

It follows from the above discussion of AU-spaces that in a

certain sense these spaces represent the largest class of l.c. spaces

which can be studied by means of the Fourier transform. Nevertheless

it is interesting to observe that there exist spaces whose duals can

be described by Fourier transforms, but which do not enter the scheme

22

of Def. I.

Example 3. (The space of real analytic functions on the line. 6)

Given ~ > 0 and K = [-n,n], let A be the space of n E,n

a l l f u n c t i o n s c o n t i n u o u s on Kn, ~ = {z ~ ¢ : d i s t ( Z , K n ) ~ E} and ho lomor -

p h i c i n s i d e Kn, E. ~ , n i s a Banach space and ~(Kn) = l im i n d E ÷ 0 c,n

is a strict inductive limit. For each m > n, the natural injection

~(Km) + ~(Kn) is a compact mapping and thus J~ = lim proj J~(Kn) is n -+ oo

an (fj4)-space. (cf. [24], p. 109). In particular, Y4 is reflexive.

Obviously, g is the space of all real analytic functions on the line.

We claim that ~ is not an AU-space. In the proof we shall need a

simple lemma on interpolation.

Lemma 2. Given positive numbers Sn ~ 0 and complex numbers z n such

that en[Zn[/- ~ and [Zn+l[ ~ max(4[Zn[,n2), there exists an entire

%1%1 function ~(z) such that ¢(Zn) = e and, for each c > 0,

e ~Izl for some C > 0 and all z. I¢(z)[ ! c

Proof. Let us set

Z f(z) = (i - ~- )

k=l n

This product is convergent and represents an entire function of order

< i. We claim that

c (22) [f'(Zn) I >

Iznl

Indeed,

II ( t - (Zn /Zk) ) f ' (z n) = - z n

k~n

For k < n,

Therefore,

co

k#n

Ii - (Zn/Zk) I ~ i;

23

and, f o r k > n , I t - (Zn/Zk) [ > 1-4 n ' k .

co

Z n

- k > n Zk

j=l

> 2 3/~ = C

> 0

and (22) follows. Now we define ~ as

Snl ZnJ f ( z ) (z~) ~n (23) ~(z) = e f,,,, (Zn) Z-Zn

n

where ~n = [enlZn I] + i.* First we observe that ~(z)/f(z) is

analytic in {z:IZ-Zn[ > 1 (Vn)}. In fact, for such points z,

where we denoted

t.l n h(z) = Z d z

~n

d 1J n

< ! I d~ n t z [~n - - C

n

EnlZnt 1"~ n = e l nl However the series

is obviously convergent everywhere. This shows that

is entire. For the order p~ of the function ~ we find p~ ~ i. If

pc < I, we are done. If p~ = I, we find that ~ is of minimal type,

and the lemma follows.

Using this lemma we shall prove that there is no family 3~ ^

for which ~' = M(K). Assume the contrary. Then ~' = M(K) for

some ~. First we claim that for each k ~ 7( and Z > 0, there is an

~T-denotes the integral part of the number a.

sZ > 0 such that

(24)

24

exp(e~lxl + ~[Yi) = O'(k ( z ) )

If it were not so, one could find sequences En~ 0 and IZnl +

= > 0) for (the latter one growing arbitrarily fast and Yn Im z n _

which

(25) exp(SnlX l + %yn ) ~ nk(z n)

-i(Z+el)Z Let ¢ be the entire function of Lemma 2 and F(z) = e ¢(z).

Then, by the Pdlya-Martineau theorem [39], F c ~'(K~,), for some

~' > Z+~l' whence F s ~'.(Let us recall that ~' is an inductive

limit of ~'(Kg), because A is dense in each ~(Kg); cf. [24], p. 143.)

Therefore, IF(z)] ~ Ck(z) with some C > 0. However, by (23), (25)

(g+E1)y n+ enlZnl I F ( Z n ) I = e ~ nk(z n) ,

which is a contradiction.

Next we claim that (24) holds with some E independent of Z.

> 1 be arbitrary but fixed. Then

Let

(26)

T Ixl+ tyI <__~tyl <__%lxl+~lyl . . . if ~llXl!2~ly];

e I % e I e 1 -g- Ixl +~lyl ! ylxl +Tlxl ! qlxl+lyl ... if ~llXlZ 2~lyl.

Thus, for each Z and z, we have

c 1 exp(x-lxl + }lyl) : #(k(z))

By Corollary 2 (of. Remark 6), every real analytic function can be

25

written as a Fourier integral,

(27) h(s) = ~ eiSZ ~

¢

E 1 Now, if s = ~+ iT is such that I TI !-3- and la I is bounded, the

integral (27) still converges. This shows that every real analytic

function can be analytically extended to the whole strip ITI !-~-,

which is obviously false.

Comparing Example 3 with Proposition i (cf. Remark 5) leads to the

following unsolved

Problem. Is the division problem solvable in the space ~ of real

analytic functions? ~

*Added in the proofs: For n = 2 this has just been answered in the

affirmative ny Ennio De Giorgi and Lamberto Cattabriga (cf. their

forthcoming paper, "Una dimostrazione diretta dell' esistenza di

soluzioni analitiche nel piano reale di equazioni a derivate

parziali a coefficianti costanti").

CHAPTER II

Examples of AU-spaces

§i. THE BEURLING SPACES ~ , $' ...... CO --CO--

In this section we shall study an important class of function

spaces ~ and their duals ~ depending on a parameter co taken from a

certain family~defined below. This class was first considered by

A. Beurling [8]. The Schwartz spaces ~ and ~' represent a special,

and in a well-defined sense, extreme case of Beurling spaces (cf.

Remark 1 below). I

Definition I. /~ denotes the class of all real valued functions co,

defined on the space ~n, such that

(~) 0 : co(O) : lim co(x) < Lo(~+n) < ~(~) + co(~) (V~,~ ~ ~n) ; x ÷ 0

.. coG,) de (B) Jn(co) = (i + I[[) n+l < oo ;

Cn

(Y) for some real number a and a positive number b,

co(~) > a + b l o g ( 1 + I~19 e v e E l~n) .

Definition 2. Given m ¢ N~ and K any compact set in ~n, let ~co(K)

be the vector space of functions ¢ c LI(~ n) with support in K and

such that, for all X > 0,

27

(co) I~1 = Y IS(E)]eXm(E)dg < o~ (x) I¢1~ : x ,n

The space ~m(K) is equipped with the topology generated by the system

of norms {l-l~m)}~> 0" ~co(K) is obviously a Fr~chet space, and by

°

(y), the elements of ~(K) are Co-functzons (cf. [9]). Let {Ks} s > 1

be any sequence of compact sets exhausting ~n. The space

= ) is then defined as the inductive limit ~m ~m (~n

(13 ~ = lim ind ~m(Ks) S + ~

The definition of ~co is actually independent of the sequence Ks p sn.

Therefore we shall always take for K s the balls K s = {x:Ix I jR s}

where {R s} is some fixed sequence such that O < R s / +~. The space

~co is called the Beurling space of M-test-functions. Similarly, the

dual $' of % is called the space of all Beurlin$ M-distributions , °

~(ind) will denote the topology of C (cf. (i)).

Remarks: i. Actually, conditions (~), (8) and (y) imposed on func-

tions co are very natural. Thus, condition (~) guarantees

that ~co is an algebra under the pointwise multiplication and

for all X > 0 and ¢,~ in ~ . Restriction (S) is obviously m

a Denjoy-Carleman type of condition, i.e. (B) is equivalent

to the non-triviality of the space ~ (cf. [9] and Chap. III,

o~proofs, cf. [9].

28

§2 below). Condition (y) is equivalent to the inclusion

~w c C~(~ n) Moreover, if we set ~o(~) = log(l + I~I) O "

then it is easy to see that ~ is just the Schwartz space O

3; and, for any ~ ~ ~, the space ~ is densely embedded

into ~ = ~ . Therefore the Schwartz space ~is the O

largest possible space of Beurling test functions. This

also shows that the Fourier transform $ of any Beurling test

function is an entire function. Moreover, for each ~ ~ ~Z,

~, D ~,. There are some other function spaces which can

also be obtained as ~ for some special choice of ~. Thus,

for instance, by taking w(~) = I~I I/Y, y > i, we obtain

the Gevrey classes ~B n ~ where B is the sequence (kY}k> 1

and SB is the space studied in Chapter III below.

2. If K is a compact set in ~n, the supporting function H K

o f K i s d e f i n e d a s

(3) HK(r~ ) = max < x , n > ( r l ¢ ]~n) . x E K

It is shown in [9] that on each ~(Ks) the system of norms

(I'l(~)}l> 0 defined in (1) is equivalent to either of the

two systems { If'If ~) }~ > 0 and { III" III l,s (m) }A,s >0' where the

corresponding norms are defined as follows:

( ~ )

( ~ )

(~) il*ll x = sup. n ( I ~ ( ¢ ) [ e ~ ( ~ ) )

I[1¢111 (~) = sup [ I ~ ( ~ ) I e ~ P ( ~ ( ¢ ) - H K (~)-~1~1]

where C = ~+i~ ~ ~n. Since the spaces ~e are defined in

29

terms of the Fourier transform, we shall often transfer

different notions from ~ to 2~ W without mentioning it

explicitly. Thus, for instance, it is clear how to define

the norms (~)-(~) for f entire, f ~ ~ .

tion the following result will be useful:

and such that Ill flll (~) < ~ for some a and

and for all ~ > 0. Then

supp ¢ C {x: Ix[ <_ R s

This is, of course, the Paley-Wiener theorem for the spaces

In this connec-

Let f be entire

s = s fixed, O

f = $, where ¢ ~ ~ and

(The converse is trivial.)

~ [9].

3. As can be easily seen, the Beurling spaces have the

same geometric properties as the Schwartz spaces ~, ~': For

! .... each ~ e~K, the spaces ~, ~a are bornological Montel (and

thus also barreled and reflexive) spaces, etc.

For later purposes it will be convenient to replace each

by another function 5 defining the same space ~w:

Lemma i. For each a e ~, set ~ = I + p,a where p is a fixed

C=-function with supp p = {x:Ix ] ~ g} and S O(~)d~ = i; the posi- o ,n

tive number e is taken so small that a(~) < 1 for I~] < e. Then

is a C~-function such that O

(~i)

and, for any multiindex

such that

i = (ii,...,in) , there is a constant T > 0 I

Moreover, since for all ~,

coincide.

3O

l~(g) - ~(g) I ! z, the spaces < and ~

Proof. By the subadditivity of ~,

~(~ + ~) = 1 + ~ ~(~ + q - t) p(t)dt o

< 1 + f w(n t ) p ( t ) d t + f" ~ ( ~ ) p ( t ) d t

<-- ~(q) + S [w(E~ - t ) + ~ ( t ) ] p ( t ) d t

Property (a2) follows from (B). Indeed,

tD I~ (~ ) t < 1 + S ~(g,-,t) (1 + t g - t t ) n + l l D l p ( t ) t d t - ( l + i g _ t l ) n + l

<__ 1 + T[Sn(~) (i + I g l) n+l, etc.

Remark 4. From now on, ~ will always stand for its modification

defined above.

The main objective of this section is to prove the following theorem.

t Theorem i. The space ~ of Beurling u-distributions is an AU-space.

The theorem will follow from Propositions i, 2, and 3 below, which are

interesting in their own right. We start by introducing some

31

additional topologies on the space C"

somewhat geometric fashion:

The first one is defined in a

Topology ~ ( ~ ) .

s equences r . x J

For any positive constants C,X, and arbitrary

and aj ~ ~, j=0,1,...,ao=0, let

(4) Aj = {~ ~ ~n : a jm(~) <__ ]nl <__ a j + t ~ ( ~ ) }

and

(5) i~(C,X,{rj},{aj}) : {¢ s ~ : sup ~ ~ kj

[l~(~)exp(X~(~)-rjI~l)]

< C for j=0,1,...}

Each set ~ of this form is absolutely convex and absorbing. Indeed,

let ¢ be any function in ~ . Then by Remark 2,1here exists a posi-

tive constant A such that, for any 6 > 0,

t $ ( t ) l <_ C 8 exp(-60~(t) + Ajql)

• > A; hence for ~ > X, the for some C~ > 0. For Jo large, r)o -

function (ccil) ~ satisfies the inequalities defining the set ~(C,X,

{rj}, {aj}) for J K Jo" In the remaining strips, we have

a3o InJ i . ~(~), since ~ ~ i. Therefore, by choosing ~ ~ X + Aajo

I~(~)I ! C~exp(-6m(~) + Alql) ! C6exp(-~(~))

for r s AI U...U Ajo_l. Hence (CC;I)~ s ~(C,X,{rj},{aj}).

This shows that there exists an l.c. topology ~(~) on ~

32

having for the basis of neighborhoods of the origin the system of all

sets ~ of the form (5).

Topology %(%).

numbers, H s ~ ~,

sequence (~s}s>l

Let {Hs}s>l be any concave sequence of positive

Hs/S ÷ 0. Fix a positive number ~ and a bounded

of positive numbers. Then the series

{6) oo

k ( ~ ) = k ( ( H s } ; ( C s } ; p ; ~ ) = [ ~ s e X p [ - ( s + p ) o J ( ~ ) + Hs l r l l ] s = l

is locally uniformly convergent in ~n and defines a majorant in the

sense of Chap. I. ~ = ~(~) will denote the system of all such

series k. For each k c ~, the set ~(k), defined by

is clearly absolutely convex and absorbing. Hence, all sets ?/(k),

k c ~, define on ~ an l.c. topology which will be denoted by ~(~).

Topology ~(Z). This topology is determined by the basis of neigh-

borhoods ~(k), k g ~(w), defined as follows:

(8) ;v(k) i~ c ~@ : there exists a positive integer N(~) N

such that 0 can be written as

N = ~ ~j' ~j ~ C' and for all ~,j,

j=l

l~j (~) I < ~jexp[-(j+~)w(~) + Hjlnl]).

Proposition i. For each ~ c Tg,

~w(ind) = ~ (g) = ~(~) = ~(Z)

33

The proof will be divided into three steps:

I. ~(ind) = ~(Z). Each set %C(k) of the form (8) absorbs all

bounded sets in the space ~ = (~w,~w(ind)). Indeed, each bounded

set M in the latter space is bounded in some ~(Ks) , i.e. for some

positive constant C~ and all k > 0,

s u p ( l ~ ( ~ ) l : ¢ ¢ M,K ¢ c n } ! CXexp[-k~(C) + HK (q) + ~ ] s

If we take N so large that

Inl H K (n) + 4 ! HNt~I ,

s

and set k = N + p, then cM c ~(k) for some positive c. Since

the space ~ is bornological (cf. Remark 3), this proves that the

topology ~w(Z) is coarser than ~(ind). To prove the opposite

relation, let Z be a convex neighborhood of the origin in the topology

~w(ind). Then, for some ~s ~ 0 and positive integers k s ,

Z n - ~ ( K s) __ (¢ : II¢H k, ! 6s) s

> ~' as Let us define a new sequence of integers ks - s

f follows. First, we set kl = kl' k2 = k~ and denote by Pl the seg-

ment in the plane (~,R) with endpoints (0,0) and (X2,R I) (R s are the

numbers of Def. I). Let p~ be a halfray originating at the point

(k2,Rl) and with slope being half the slope of PI" Let A be the

point on p~ for which A = (v,R2) ; ~3 the integral part of

1 + max{k~,~}; and, p 2 the segment with endpoints (~2,RI) and (k3,R2).

Continuing in the same way we obtain the broken line Pl U P2 UP3 U...,

>s ~hose equation in the (t,R)-plane is R = ~(t). Obviously ~s - '

34

and ~ is a concave function such that ~(t) ~ ~, ~(t)/t ÷ 0.

Furthermore, let ~ = ~, H s = ~(s) and ~s = 6s 2.s We claim that

for k = k({Hs};{Ss};g ) (cf. (6)), 3~f(k) c~.

N : N(qS),

Let ~ be any element in 7g(k). Then, for some integer

can be decomposed into the sum

N N = ~ ~j = ~ 1 (2j~)

j=l j=l 2 ] '

where for each ~ e Cn and j=I,2,...,N,

t ~ j ( c ) l < ej e x p [ H j l n [ - ( j + ~ ) c o ( g ) l

Hj < j < X . By the above construction, < R s when ks - s+l

by the Paley-Wiener theorem (cf. Remark 2), supp ~j C K s .

II~j[IX~ _< 2-J6..j For the remaining indices j, 1 _< j < kl,

l~j(~)l <_ ~j 2-je-~(~) <__ 2 J6le

Therefore

Moreover,

This shows that, for all j=l,2 ..... N, 2J~j s ~. Convexity of

yields ~ ~ ~ , and the equality ~(ind) = ~(Z) follows.

2. ~u(~ ) = ~(3~). In order to show that ~(~) is coarser

than ~(g), one has to find, for each k ~ •, a set

?~(C,~,{rj},{aj}) contained in ~(k). We claim that it suffices to

1 HI = H for s > 1 choose C = min(l,~l) , X = ~+I, r ° = ~ ' rs s

and a sequence a ~ ~ such that S

(9) a > S --

s - log ~s+l

Hs+ 1 H s

Indeed, given any ~ s ~(...), we shall show that

35

]¢(E) I <_ k ( [ ) f o r a l l [ s c n . If ~ e i o ,

H 1 I¢(~) l <_ ~ t e x p [ - x ~ ( ~ ) ÷ Inl ~-]

then

< k(~)

If ~ a Aj, j ! i, then by (9), the function I$(~)] is bounded by

the term in the series k, for which s=j+l.

To prove the converse we begin by constructing an auxiliary

function p(t), t a ~, which will be a differentiable, convex and

even function on 2- Set r ° = s o = s ° = p(0) = 0, s I = I, and

construct the function p first on the inter~al [-rl,rl] so that, in

addition to the above properties on this interval, p will also satisfy

the following conditions:

( i ) p ( r 1) = s 1 ;

(ii) if (~I,-I) is the normal vector to the graph of the function

p at the point (rl,Sl) , then al > al and for some integer ql'

a I = aql In particular ql > i. If the integers s I < s 2 < ... < s m

and the function p(t) on [-rm,rm] have already been defined so that

(10) p ( r j ) = sj , ( j = l , . . . , m )

and if (~j,-l) is the normal vector to the graph of p at the point

(rj,sj), then for some qj,

(ii) ~. = a > a. J qj J

It is clear that the construction can be continued to finally yield

function p such that

( 1 2 ) B. d e f { x : I x j l < r j } = {x : p ( t x ] ) < s . } j = _ _ j ,

36

a n d

(13) a. ~ ~. J

If the sequence {H s} is defined by the conditions

(14) p (H j ) = j ( j c z) ,

then by (13), the sequence {Hi}j> 1 is concave,

by (I0), Hsz rz Set

. -" ~ H /j ÷ ~, Hj , j and

(15) ~ = I + a I

and choose ~s positive so that

o@

< C (16) ~ Ss T

s = l

For any ~ in Cn we write

for some integers ~ and q,

= ~ + in = ~ + i@~(~). Then,

(17) ~ ~ IOl < < t e t < i e ~ ~ A aZ+l and aq _ aq+l, . . q

Suppose first @ ~ 0. We claim that

(18) k ( ~ ) < C e x p ( - i w ( ~ ) + r i + l l ~ l )

Indeed, if k is written as

k = s~

s= l s>s~

then

(19)

s~

s=l . . , < exp(-Xm(~) + r z ln [)

For the estimate of the second sum, the geometric properties

of the function p(Ixl) have to be used. First, as this function is

symmetric, relations (12) and (17) imply that there exists a point x

such that, rz 2 Ixl < rz+ 1 and (@,-i) is the normal vector to the

graph of the function p(Ixl) at the point (x,p(Ix[)). Moreover the

vectors x and @ are collinear, i.e. x = Ixl@/1@I. The convexity of

p then implies the inequality

(20) [ (Y,P([Yl) ) - ( x , p ( l x [ ) ) ] (@,-1) <_ 0

for any Y c ~n.

p ( t y l ) = s and

In particular, for y = Hs@/l@ I (s > sz), we get

(21) (H s - I x l ) l e l ~ s - p ( I x l ) 2 s s~ ,

whence

(22)

co

¢seXp{(X-s-P)~(~) + (Hs-r~+l) [q [} S = SL4-1

2 ~ %exp{-so~(¢) + ( H s - l x l ) l e l ~ ( C ) }

[ 1 - s z ~ ( ~ ) c 2 ~s e 2

(by (21) :)

Inequality (18) now follows from (19) and (22). The assumption @ ¢ 0

is automatically satisfied if q ~ i. In this case we claim that

(23) k(~) <_ Cexp(-X~($) + rqlq ]) (~ ~ Aq; q >_ 1) .

38

Here we have to distinguish two cases: ~ = 0 and ~ K i. For

= 0, r£+ 1 = r I ! rq and (23) follows from (18). If Z ! i, then

by (ii), ~Z = aq . However, by the construction, ql > I; hence

a . Thus qg > g. Comparison of bounds in (17) gives aqg q

~+i ! q£ ~ q, and (23) follows again from (18). The only region

which still has to be checked is A o. There, however, Inl ~ al~(~),

and (16) and (17) give

(24) k(~) < Ce -%~°(~) (~ s Ao) .

Inequalities (23) and (24) prove the inclusion

~(k) c U(C,~,{rs},{as}) ' which completes the proof of part 2.

3. ~(ind) = ~(g). Since ~m(ind) is barreled (cf. Remark 3),

the topology ~(g) is clearly coarser than ~(ind).

~ (ind) .

Now let ~ be a closed convex neighborhood in the topology

0 and ~ ~ Then, for some Cs s '

ra S ~ ( B ( s ) ) ~ {¢ e .~w(B(s)) : sup I$(£)le < ¢s ' ~ c ~n

where we denoted B(s) = {x : iX[l = IXll+...+[Xnl ! s}. We claim

that for a convenient choice of the parameters, ~(C,~,{rj},{aj}) c~.

Let~ be the set of all lattice points M = (m I .... ,mn) in ~n such

that [M[ ~ n/~. Furthermore, let us set

for M e ~, and

S 0 : {x : IX I < (n+l) fn}.

{~M } , a M c ~, By [9] there exists a partition of unity subordinate

39

to the covering {S M : M c .A~U{0}} of ~n. Then, for any positive

g, we have by Remark 2,

(25) ]~M(~)] < C , M exp ( -6w(~) + HSM(n ) + ~lnl)

As above , we have s e t

1 (21MI1 aM ~ )

2

for each ~ a ~(...). The parameters defining the neighborhood

~(...) will be chosen so that, for every M,

(26) 2tMII+ n ~M ~ e ~ .

-IMI I (Let us recall that 22 = 2n). Then convexity of ~ will imply

Obviously, for M c~, supp(~M¢ ) c B(2IMII); and,

supp(~0~ ) c B(n2+n). Since ¢ E ZZ(...), we obtain

1~0(~31 ! f l~(~- t ) l l a o ( t ) ] d t ,n

(27) <_ CC6, 0 ~ e x p [ - X ~ ( ¢ - t ) - ~ ( t ) ] d t

<-- CC°,0 e - d ~ ( ¢ ) S e x p [ ( ~ - X ) m ( ¢ - t ) ] d t ,

where we used the subadditivity of w.

(28)

Now let us set d = n2+n

n+l X = v 2 + ~

n + n

and

40

where b is the constant from condition (y), Def. i.

C > 0 so that

We also fix

S dt (29) 2nccv ,0 (l+jtl)n+l ~ e 2

n2+n ~n n +n

Then inclusion (26) is verified for M = 0.

In order to estimate those terms in (26) for which

JMJl = s > 0, we have to shift the integration in the convolution

A M ~M ¢ from ~n to the variety F~ = {T = t+in ~ cn : ~ = _ ~-[ as~(~_t) '

t ~ ~n }. By the Cauchy-Poincar4 formula, this can be done provided

as~(~-t)

(30) lim Jtln'l ~ J~(C-t+i~v) ~M(t-i~v) Jdv = 0 Itl ÷ o

However, relation (30) follows from obvious estimates:

Itl n-1

as~(~-t)

S ... <_ C¢,pJtJn-lasm(~_t)exp[-p~(~-t) + Rasm(~-t)] x

O

x Cd,Mexp[-d~(t) + (s+l)as~(~-t)] ÷ 0 .

1 Here we used the inclusion supp ~ c {x : Ixl <_ R - T},

+ 1 d n = + (s+l)a s , = ~- . Therefore, notation p Ra s

and the

A (31) ~M¢ (~) = $ (~-T)~M(T) dT

r~

, _ ~-t l)n2+n(l r In the last integral Jd~ I < (1 + J + as )n, where T is

the corresponding constant from Lemma 1. Therefore, using (25) and

the fact that C-T ~ A s for T e l-G, we get from (31) (cf. (i))

(3z)

A lC~M~(~)l < CCv,M(1 + T a s ) n e -g~(C) x

× ~ (l+,~-t,)n2+nexp{~(~-t)~g-~+as(~r s+HSM(- ~))] } dt

sn

Let g = V2s"

be chosen. Clearly~

The s e q u e n c e s { r j } and {a j} s t i l l r e m a i n to

HS M 5 ~ and n / n < IMI ~ IMII _ ( - M / I M I ) = - I M I + ~ _ = s < n l M I .

Thus, if

def s 15 /~ r s = E iV '

then 0 < r s ~ ~, and

(33) 1s ¢~ < IMI iV m S

From here we obtain (cf. (32))

(34) 1 1 4-+ r s + HSMC-M/IMI) < - 1-~

Let Q be the constant

dt Q = Ce (n+ l ) 2a max c6, M ) n"+l ' {M: IMI1 = s} ~n (l+Itl

where a is the constant from (y), Def. i.

that

If a s is chosen so large

(35) a s

42

> 32(g - X + (n+l)2b-l) ,

then, by (34) and (35), inequality (32) yields

(36) laM¢(~)l <__ Q(I + Tas)nexp " 37 - V2s ~(~)

By taking a s even larger we can achieve that

a

n _ 2 -s-n (37) Q(I + Tas) exp(- 3~) < e2s

and (36) then implies (26).

So far all parameters defining the neighborhood

U(C,k,{rj},{aj}) have already been chosen except for the first few

values of a. and r.. However these can be defined arbitrarily as J J

long as the sequences {a.},{r.} will remain positive and strictly J J

increasing. The proof of part 3 is complete and Proposition 1 follows.

Corollary. Given any function k in the family ~C=~f(~),

co

k(~) = ~ EseXp[-(s+~)~o(¢) + Hsln I] , s = l

there exists another majorant k e Y~(~),

~(~) = ~ ~ s e X p [ - ( s ÷ ~ ) ~ ( ~ ) ÷ Hsln ]] s = l

such that, if ¢ ~ ~ and I~(~)I <__ k(~) for all ~ E cn, then

there are functions ~j (j=I,...,N; N=N(¢)) in ~ such that

N ¢ = ~ Cj a n d

j=l

[$j(C) I _< s.exp[-(j+~)~(~)j + Hjlnl] (j=I,...,N; ~ scn)

This statement is similar to a lemma due to A. Macintyre

(cf. [Ii], p. 80). An interesting problem would be to find any esti-

mate for Hs,Ss,~ in terms of Hs,gs,D. This would probably follow

from a constructive way of proving that the topology ~w(Z) is coarser

than ~(~). A similar problem was studied by B. A. Taylor in [49] who

who used the technique of L2-estimates of the ~-operator (cf. [29]).

In our case, this does not seem to work.

Next we want to show that the family ~(~) determines

completely which C~-functions are elements of ~ (cf. property (A) in O

Chapter I).

A

Proposition 2. For each ~ and ~= J~(~) as above, ~(X(~)) = ~

The proof of this statement depends on a lemma (see Lemma 2

below) which will be useful on several occasions in this section. We

shall employ the following notation: ~(or ~+) denotes the class of

all functions h which are concave, increasing to +~, continuously

differentiable on [0,~) and such that h(0) > 0 (or h(0) > 0 resp.)

1 and 0 < h'(s) ~ 2s+l for all s ~ 0.

Lemma 2. Let h be a function in ~ and p its inverse.

a > 0 and b > I,

Then, for all

(s8)

eah(s)_b s

s=0

eas_bp (s)

s=0

<_ (5 + 2a)e ah(a)

44

In the second inequality we assume h(O) = O.

Proof. First, let us show that

(39) ~ e ah ( s ) -b s <_ (S + 23-a)e ah(a) s=O

In the proof of (39) a simple version of the Euler-Maclaurin formula

will be needed:

Let a,B be integers, ~ < 8, f(x) a continuously differentiable

function in [a,~] and ~(x) the function of period 1 such that

O(x) = x - ½ for 0 < x < i. Then

B

(40> ~ > + ~ c o + ~ + . . + ~ ~1 + ~ ( ~ ~ f ~x~dx + .~ ~ , C ~ 0 ( x ~ x .

From here follows

(41) s--0

S e t t i n g bs = t

N

= 1 + ~ eah( s ) -bSds e a h ( s ) - b s ~eah(0) + ~ a h ( N ) - N O

N

+ y (ah' ( s ) - b ) e a h ( s ) - b S o ( s ) d s

O

in the l a s t i n t e g r a l we ob ta in

(42)

N Nb

1~('")1 <- z@~ [ah'(~)+blexp[ah@)-t]dt O O

<- z-~- ~2( +I + b) e ah(t)-tdt

oo

a+l S eah (t) 5_ --2-- - t d t " o

(Here we used the obvious inequality h(~) £ h(t).)

tion (41) implies

By (42), equa-

c~

(43) ~ e ah(s)-bs <_ } eah(0)+ @ ; eah(t)-tdt s=O

0

Now it remains to estimate the last integral which can be written as a

f ( . . . ) +f ( . . . ) . However, 0 a

a a

y(...) ~ e ah(a) ; e

O O

-s < eah (a)

Moreover,

co

y ( . . . )

a

co

a ~ ) eah (s) - [eah(s) -s ]~ + ah ' ( s Sds a

co

a y 5_ eah(a)-a + 2a--a%-i- (''') a

Hence,

(44) ;e ah(s)-sds £ 3e ah(a)

0

and (39) follows from (43) and (44).

The proof of the second inequality in (38) is similar:

N eas_bp (s) ( 4 5 )

s=O

}(i+ eaN-bp(N)) + ~eas-bp(S)ds

O

N + y [a- bp' ( s ) ]eas -bp(s )~(s )ds

Furthermore, substituting ~ = p(s) in both integrals, we

obtain

( 4 6 ) s=0

eas-bp(s)

co

<-r*l eahCt)-tdt 0

46

Now it is clear that the second estimate in (38) follows along the

same lines as the first one.

In order to prove Proposition 2, let f be a fixed entire

function such that f = G(k) for each k E~C. In particular, if h

is any element in ~+, the sequence H s = h(s) (s=l,2,...) combined

with an arbitrary ~ > 0 and any bounded sequence {e s) defines a

majorant k = k({Hs};{es};~ ) ~ ~(~). Thus by (38),

1 nlHs-S~(~) I f ( ; ) l e ~c°(¢) < Ck(¢)e ~c°(¢) <_ C 1 ~ e

s= l

< C1(5 + } lq 0 e lq lh(Ir l t )

21nlh(Inl) ! Cze

for some constants C,CI,C 2 depending only on k and f. If we set

g(In[) = sup{Inl- lZog[C21(~(¢) + I f (~ ) l ) ] :V(n / l~ l ) ; V~; V~}, then

the last inequality can be written as

g(I,~l) (47) ]n~ h(ln]) - su < 2

Since h was an arbitrary element in ~+, we conclude from (47) (of.

• ~n [17]) that g < B for some B > 0, i.e for all ~ ~ and any

> O,

(48) t f ( ~ ) l < c e Blnl-~c°({)

with C = C 2. The Paley-Wiener theorem (cf. Remark 2) then implies

f = $ for some ~ ~ ~; and~this completes the proof of Proposition

2.

To complete the proof of Theorem l,it remains to exhibit a

suitable BAU-structure for ~'. Let ~ be the class of all sequences

(Cj - . I ~. }j~l of positive numbers such that 1 = m(O) < Cj+ 1 Cj

Given any {Cj} ~ ~ and arbitrary A > O, C > O, set

¢(g) = inf(C n- nm({)) and n

( 4 9 ) m ( ~ ) = m ( { C j } ; A ; C ; ¢ ) = Ce ~ ( ] ¢ ] ) + A [ n ]

Let~(®) be the family of all such functions m. Furthermore, let

(~c resp.) be the class of all positive functions A(t) (t ~ i) for

which X(t) def A(t)/t ÷ ~ when t ÷ ~ (and k concave resp.). Given

any A s ~ (A ~ ~c resp.) and arbitrary A > 0, C > 0, let

-A(co(~)) +A[ rq [ ( 5 0 ) m ( ¢ ) = m ( k ; A ; C ; g ) = C e

Denote ~(£) (~(Zc) resp.) the family of all such functions m.

Proposition 3. Each of the three families -/~(@), ~(£), j~(~c) is a

BAU-structure for the space ~' and satisfies condition (vi) of Def.

l,I. a

Proof.

Indeed, define Vj = {~ : Cj+ 1 >__ w(~) + Cj},

{Vj}j>_I exhausts ~n; and, for ~ ~ Vj\Vj_I,

continuity of the function ¢ then follows.

(6), and m an arbitrary element of J~(@).

Let us first observe that each ~ ~(~) is continuous.

V o = 9. The sequence

~(~) = C. - j~(~). The J

Let k be any series as in

> A Then, for some Sl, H s _

for all s >__ Sl; and,

48

e s e X p [ - ( s + ~ ) c o ( ¢ ) - ~ ( ¢ ) ] = [ ~ s e X p [ - ( s + ~ ) ~ o ( ¢ ) + max(noJ(~) - Cn) ] s>_s I s>__s I n

>_ ~ ~seXp ( - C [ ~ ] + l+s ) s >s I

Hence, denoting the last term by C/C, we obtain m(~) < Ck(~) for all

~. C o n v e r s e l y , l e t B be a b o u n d e d s e t i n N . Then by ( 1 ) , B i s a W

b o u n d e d s u b s e t o f some .g (K s ) , w h e r e K s = {x : [xl 2 R s }, i . e . 0 0 0

sup IIfll ~) < Cn (n=l,2, .) Let us choose A = R s - - t . • + 1 ,

f ¢ B o

C n > log Cn' and m = m({Cn};A;... ). Then B C A(m;~) for some

> 0; condition (vi) is easy to check.

Now let m E Af(a~) and k as above. We claim that

k(~)/m(~) >_ const. > 0. In fact, by (6) and (50),

(51) k ( ~ ) 1 = C- ~ ~ s e X p [ ( H s - A ) t~[ - ( s + ~ ) ~ ( ~ ) + ~ ( ~ ) k ( w ( ~ ) ) ] s = l

Thus, choosing s o so that HSo> A, and E > 0 so large that

(~(~)) >_ s o + ~, for I~I _> E, we obtain from (51),

> C - I ~ s m i n e : I¢1 < E > o

To prove that the family ~A~(~) is a BAU-structure, it suffices to find

for each m ~ ~f(@) a function m* s ~(£) such that

(52) m(~) < c o n s t , m*(~)

If m is given by (49) and {Vj} as above, let us fix an arbitrary

sequence of points gj,{j s aVj for J K i, and go = 0. Then

Igjl : ~ and ~(gj) = Cj+ I- Cj. Hence C n = C 1 +m({l)+...+m(gn_l),

and by the definition of ~,

Cn Cl+°J (~1 ) +- • • +m (gn_2) ( 5 3 ) = ~ - n < - ( n - l ) (g s V n - . V n _ l ) .

co ( ~ n _ 1 )

Now, let G' be any subclass of ® such that, for each {Cj} c 6, there

{Cj . > C Since we already know is a sequence } s @', for which CO _ j.

that~(~) is a BAU-structure, this will imply that the family J4(~') is

also a BAU-structure. Therefore, it suffices to prove (52) for all m

from any family,(6') with the foregoing properties. In particular,

let us choose as ~' the class of all sequences {Cj} in G, which grow

so rapidly that the last term in (53) is < 2-n for all n > 2. Let

m = m({Cj};...) be a fixed function in~(~'). Then for all ~ such

that w(g) = t, #(g) assumes the same value which we shall denote by

-A~(t). Let A(t) be defined for t > ~(0) as A(t) = A~(t) = c, I

where the constant c was chosen so large that A(t) > 0; and

A(t) def A(m(0)) for t s [0,m(0)). Then, by (53) #(g)/m(g) < 2-n

which implies A s ~. Now the inequality (53) follows for

m * ( g ) = C * e x p [ - A ( c o ( g ) ) + Alnl].

Finally, to show that Af(~c) is also a BAU-structure for ~'

it suffices to show (cf. the discussion following (53)) that, for each

X(t) = A(t)/t, A s ~, there exists a positive concave function X ~,

such that X* < ~ and lim X~(t) = ~. However, this is easy to see

(cf. [17], Lemma 6). Furthermore, the verification of the rest of

condition (v) of Def. l,I is straightforward. Thus Proposition 3 is

proved; and, this also completes (cf. Propositions 1,2) the proof of

Theorem i.

50

Our next objective is to prove that, for a large class of

functions ~,~ is a PLAU-space (cf. Def. 3,1). Let ~c = {~ ¢ ~:a

concave for [i ~ 0,..., ~n ~ 0, and ~ an even function in each

variable separately}. For each ~ E'~c, let ~l(t) = ~(nt,0 ..... 0),

...,Wn(t ) = a(0,...,0,nt). Then

1 n n

j =i j =i

Since for each ~ ~ ~c and a def n = ~ ~j, the spaces ~w and ~g j=l

coincide, we shall usually replace each ~ in ~c by its modification

which will be called w.

If P(t) is a decreasing convex function of t > 0,

each x,y ¢ i n ,

then for

(ss) P ( I x l ) + P ( l y l ) E P(Ixl + l y l ) + P(O) < P ( I x + y l ) + p (o ) .

Similarly, for Q concave and increasing on [0,~),

(s6) Q(lx+yl) + q(o) ~_ Q(lxt) + Q ( I y l ) ( x , y ~ gn)

Simple examples of functions ~ of ~ can be obtained by taking

QI,...,Qn arbitrary non-negative concave decreasing functions of

t > 0, and setting

n

w ( [ t , . - . , [ n) = ~ P j ( I [ j l ) j-1

Moreover, for Q as above, the function ~(~) = Q(I~I) i s in K .

Theorem 2. For each ~ c T~c, ~ is a PLAU-space. 3

Proof. First we have to verify condition (vii) of Def. 3,I. Since

w = Zmj~ it will follow that ~'~ is the AU-product of 1-dimensional

AU-spaces ~' provided we can show that: (a) each < is nontrivial; ~.~ ] ]

T and,(b) the AU- and BAU-structures of the spaces 9 . generate the ]

corresponding structures for ~ in the way described in Definition 3,I.

To check (a), let us take, e.g., j=l, and set S = {(tl,...,tn) :

t I h i, 2-1t I ! t k ! 2tl}. Then, (cf. condition (B) at the beginning

of this chapter)

2t 1

(log 4)n-i S~l(tl)dtl - ~ ~ dtk If 2 t k "'"

i tl k=2 tl/Z I

. . . . o) :S S

dt

S w (nt) dt < CIJ n(w) < ~ • i Co tntl~+l -

S

Thus Jl(~j) < ~ for all j, hence by [9], ~wj ~ 0 (Conversely,

let ~. be functions of one variable satisfying conditions (~),(B) for ]

n=l. Then the function ~(~l,...,~n) = ~l(~l)+...+w(g n) also satis-

fies (~),(B). Indeed, if 2(t) = ~l(t)+...+~n(t), then obviously

Jl(2) < ~. However, for some constants C2,C3,

S oj (~i) +... +Wn(~n ) ~ 2([~1) C2Jn(W) < d~ < in--- ~ d~ < C3JI(2) .]

- Igln+l - ]< -

I~I>_ l t~l >_1

(b) For any A ~ ~), A(t) = tk(t), we have

52

n n ~jk ) < ~k(~) < n ~ ~j j=l (~oj _ -- j =I X (~j) .

Therefore, if ~. denotes the family ~f(Zc) for the space ~., the fami- 3 j

ly ~ def= (m(~) = ml(~l)...mn(¢n) : mj ~ J(j, 1 _< j _< n} must be a

n BAU-structure for ~' Let D be the vector space @ ~ equipped

~" ~ j=l j

with the topology ~(D), defined as the unique bornological topology

compatible with a fundamental system of bounded sets of the form

A(m,c) = {~ ~ D : sup (~(~)I/m(~)) < c}. Obviously, the l.c. space ~ c ¢ n

D is isomorphic to a subspace of Co Thus the fact that the comple-

tion D of D is the whole space ~ follows from the density of D in

C .~ From the discussion preceding Def. 3,1 we then obtain that there

is an iU-structure ~on ~ of the form required by condition (vi) of

Def. 3,I; namely, we can define ~(as {k(¢) : k(¢) = kl(~l)...kn(~n),

kj ~ 7~j, V j}.

To verify (vii) we have to limit ourselves to the spaces ~ . J

Hence we fix j and call ~ = ~'3' -~(gc) = ~, etc. Let t = v(s)

be the inverse function of s = m(t) and

oo

Z , d e f {A(r ) = r X ( r ) ¢ Z : X ( r ) / v ' ( r ) ÷ 0; f t ' 2 A ( m / t ) ) d t < ~} . = C 1

It is clear that -/~(~*) is again a BAU-structure for ~' (cf. the end

of the proof of Prop. 3). Let us fix a A* in £*. We claim that

there exists a A in £ such that

1. A(~(t)) is a concave function of t > 0 (hence

A ( ~ ( t ) ) ~" +o~);

2. h ( ~ ( t ) ) < A * ( m ( t ) ) + c o n s t .

53

First, let us construct a continuous function

which H(5)-~ 0, and

H(6) (~ > O) for

S

(57) X,(s ) _ A*(S)s _> ls S H(6)~)' (6)d~ +

~(o)

I t s u f f i c e s to t ake It(s) = m i n ( X * ( 6 ) / u ' ( 6 ) : w(0) < 6 < s}.

S S

:--s ,,~ H(6)v ' (6)d6<_ I__S ~ X * ( 6 ) d ~ ~(o) ~(o)

< X*(s)

Then

On the other hand, since the set

bounded,

we have

{s : H(s) = X*(s)~'(s)} cannot be

H(s)v'(s) + ~, and (57) follows. Then, for g(z) = H(~(T)),

s t

H ( 6 ) v ' ( 6 ) d 6 = ~ g(x)dT

~(0) o

Thus, by (57), we can set

A(~(t))

t

= ~ g(z)dz + const.

O

Let ~** be the class of all such A's (i.e. A's constructed for all

h * E ~*). The family Y~ =~q(~**) is again a BAU-structure for ~i;

and, we shall show that the class /4satisfies condition (vii).

Given m e ~ and c > 0, we must first exhibit an

m' e ~ such that for any z ° = Xo+iY ° ~ ¢, there will be an entire

function ~(z) for which

(58) mCz O) l¢ (z ) I _< m ' ( z ) (z e ¢) min I¢(c) I

H o w e v e r , m ( ~ ) -- C e x p E x ( ~ ) + A I n l ] w h e r e X ( ~ ) -- - A ( ~ ( ~ ) ) , A ~ Z * *

54

It follows from the proof of Proposition 3 (cf. (53)) that e X(~) c L2;

by the definition of Z*,

(59) S t ~ d t < ~ 1

By [42], Th. XII, there exists a function g with compact support,

g ~ 0, and such that [~(~)I ~ m(~) for all ~ and ~(~) ~ 0. Since

~(0) = f g(t)dt > 0, there exist positive numbers ~, 6 < e, and c

such that, for [~I ~ 6, ~(~) > c. Set

(60) m' (¢) = CleX p[6Al~l + X ( ~ ) ] ; C 1 -- c- lc2exp[3Ac +X(0)] .

Given any z o = x o+ iYo, let

claim that the entire function

(58). For z = x+iy,

-- -3Asign Yo and B -- -3AlYol. We

¢(~) = ~(~(~-zo))exp[i~+B ] satisfies

(61) m(z o) I , (z) l <_ c2exp[x(x o) + AlYol

+ x(~(X-Xo)) + 6-Ac lYYo j+ ReCi~z+ e)]

By (55) we have

c6~) ×C~o) + ×c~XXo~ ~ ×¢~ Xo) + ~cfCXXo~ ~_ ×¢~x) + ×¢0~

since Re(i~z+~) ~ 3AIy[- 3AlYoJ, estimates (61) and (62) give

(63) 6x) + 6Alyl] m(z o) I , ( z ) l <_ c2exp[x( ~

where C 2 = C2e X(0)

estimate

Inequality (63) combined with the obvious

55

(64) min I¢(c) l > ce i~_zo l ! ~

-3Ae

implies (60).

Now, for m,~ fixed as above, we have to find an m'e~such

that for each z ° = Xo+iYo, one can find an entire function ~, for

which

(65) [~(z) ] sup -oo<~ <oo

m (~+iY o )

min l,(~+ir) I Ir- Yo I<--E

< m'(z)

for all z ~ ¢. Set ~(~) = X(0) -X(~) + log(l+~2). Then the space

~(~) is well defined and ~(~) ~ {0}. Take an arbitrary function

g s ~ such that supp g c (-A,A) and ~(~) >_ 0; and, let f be the

function in L2(~) for which ~(~) = exp[x(~)]. Let h = f.g. Then,

for each p, 0 < ~ < I,

lh(~+in) I <__ S If(t)~(g-t+i~)Idt <_ Cp S exp[x(t)-~5(~'t)+Alqj]dt -co

(66) <_ C exp[-~fi(~)+Alnl] ~ exp[x(t)+p~(t)]dt

<__ Cpexp [-p~(~) +AI q ] ]

In particular, this shows that h ~ ~ . On the other hand, by (55),

*If p were ~i, the last integral in (66) would not converge.

(67)

~(~)

>

56

f (~-t)~(t)dt = y exp [ x ( - t+~ ) ] ~ ( t ) d t -oo

eX(~) f exp[x(- t ) - x (O) ]~( t )d t = czeX(~)

We need an est imate of t ~ ( t + i ~ ) - ~ ( t ) l = I n l l ~ ' ( t + i f i ) l .

Cauchy formula, we obtain for each ~ > O,

Applying the

(68) I~(t+in)-~(t) [ < lnl max ]~(t+in+u) l lu I<_1

Alnl M~(t) ! T~lnle

Therefore, by (67) and (68),

(69) > I S ~(~-t)~(t)dt y ^ +" f(~-t) Ig ( t in) - ~ ( t ) [ a t

> C2eX(g) _ T l l q l e A ql ; e x p [ x ( ~ - t ) - 9 ( t ) ] d t

Using again the superadditivity of X (cf. (55)), we get

X(g-t) - ~(t) < X(~) - log(l+t2) • Then, for lq I sufficiently small, say

lql ! 6 for some 6 < e, (69) yields

cz eX(g) (70) l >- T

Now, let us define m' as

(71) m'(E) = C3exp[6AInl - ~tl(~)] ; C 3 = 2CCIC2 le3As ;

and, for z ° = Xo+iY o

where ~,~ are as above

57

fixed, we set ~(~) = fi(~(~-iYo))exp[i~+6],

(see (60)). Then, by (70),

(72) ]~(~+is)] = [ f i (~(¢+i (S-Yo)) ) ]exp[3A(s-Yo)Sign yo ]

C 2 > -2-- exp[×(~ ¢) 3A¢] ;

hence,

(73) sup {2C21m(~+iYo)eXp[3A~ - X(~ ) ] } _co<~<cc

AlYol C4e ; C 4 = 2ccile3AC

Finally, by (73),

] ~ ( z ) [ sup { . , . )

Alyol C4e Clexp [- ~ ~(~x) 6 + A~-ly-Yol+ 5 A ( l y l - l Y o [ ) ]

< m' (z) ,

and this completes the proof of Theorem 2.

58

§2. THE B EURLING SPACES ~ , ~.__

From now on each ~ w i l l be assumed s y m m e t r i c , i . e .

~ (¢ ) -- ~ ( - ~ ) . (We l i m i t o u r s e l v e s t o such ~ ' s o n l y f o r the sake o f

s i m p l i c i t y , c f . [ 9 ] . ) ~ s w i l l d e n o t e the s u b c l a s s o f ~ c o n t a i n i n g

a l l s y m m e t r i c f u n c t i o n s ~.

Definition 3. Let ~ be a function in ~s" Then $~ is defined as the

set of all functions ~ on ~n such that, for each compact set K, the

restrictions to K of ¢ and of some ~ in ~ agree. 5 The topology ~(~)

is given by the system of seminorms [~]X,K defined as

(74) [ ~ ] X , K = [~] (~) = i n f l ~ I x X,K ~=¢ in K

for all X > 0 and all compact sets K.

dual of ~.

~' will denote the strong

Remarks: 5. The spaces ~, ~'~ bear the same relationship to spaces

~ ~' that the spaces ~, ~' bear to the spaces ~, ~' In

particular, standard arguments show that ~ is a Fr~chet-

Montel space. Therefore ~, ~ are reflexive, barreled,

bornological, etc. Moreover, as a set, ~ can be identi-

fied with the subspace of ~' consisting of all elements

with compact support. Hence, the elements of ~ will be

called the Beurling u-distributions with compact support.

' there • '' For each ¢ ~ ~, 6 Paley-Wiene r theorem for ~ .

are constants C > 0, A > 0 and N real such that, for all

c ~ n

59

(75) I*(¢)1 < ceNco(¢)+Alnl ;

and, conversely, if g is an entire function satisfying ~ith

^, some C,N,A as above) the last inequality, then g c &co [9].

7. Let K be a non-empty compact subset of ~n.

denote the subspace of ~ defined by

I g co (K) will

!

~co(K) = {¢ ~ &':co s u p p ¢ C K)

If {Ks}s> 1 is a sequence of compact sets exhausting ~n,

then the bornologicity of ~ implies (cf. [46]) that

! = ~co lim ind ~'(Ks) S + oo

8. Another system of norms on ~co can be defined as

follows: for each X > 0 and ~ ~ ~co, let

(76) (co) i l(co)

Both systems (75) and (76) define the same topology.

(Indeed, given ~ > 0 and K compact, then for ~ ~ 1 on K

and ~ ~ ~co' [g]X,K = [~g]X,K ~ ~g]x,~" Conversely, given

X > 0, ~ e ~co, let K = supp ~. For each ~ > 0 there

is a ~ ~ ~co, ~ = g on K and ]~]X ~ [g]X,K + ~' Then

~¢ z ~g and = ~ I ¢ [ x ( [ g ] ~ , K + e ) ,

etc.). From here, it is easy to conclude that the space

~co can be characterized as the space of all multipliers of

the space 4' i.e. as the space of all complex valued func-

tions ~ such that each mapping M~ : ~ ~ ~ is an

endomorphism of the space ~ . Thus Sw is a subspace of

L(~w,D~) and the topology of ~w is the ~nduced) topology of

pointwise convergence in L(~ ,~a) (cf. [9]). For our

purposes a similar description of the space $' will be

needed:

Proposition 4. The space ~' is the space of all convolutors of the

~ . ' such that the space , i.e $~ consists of all distributions ~ E ~

mapping ~: ~ ~ ~ * ~ is in L(~,$~) Moreover the topology of ~'

coincides with the compact open topology induced on 3w from L(~,~). s

Proof.

compact. Let Pn be a regularizing sequence in ~ , i.e.

~n = ~ ~ Pn + ~ in ~'.~ We can assume that ~n ~ 0 for all n.

~n = ~ ~ Pn where {~n } is a new sequence in ~ defined as follows.

First, let Pl = PI" Since ~ is a convolutor of ~, there is an

r I > 0 such that supp ~I c Krl (cf. notation in Def. 2,II). Let

C K • and let x I c supp ~i" Define n 2 be so large that supp ~n2 ~ rl ,

P2 = Y2Pn 2 where Y2 is the constant defined by

Let ~ a 8' be a convolutor of ~ such that supp ~ is not

Let

I~1 (Xl) I Y2 = 32max [1, l ~,1 (Xl) i , i Vn 2 (x 1) l]max[1,1IPn2111]

Now let x 2 ~ supp ~2~Krl and supp ~2 c Kr 2"

C K and P3 = Y3 where supp Yn3 ~ r2 Pn3

Let n 3 be such that

min[l~l(Xl) I,I~ 2(x 2) I]

Y3 = 33max[l,[~l(Xl) l ,I~2(x2) I ,l~n3(xl) l,l~n3(x2)l]max[l,Ilpn3112 ]

etc. Since supp Pn c K r O

for all n. Let ~ = Z~j .

all integers m > i, p > i,

61

for all n, we also have supp Pn C K r O

This series converges in ~, because for

N > m ,

NiP NiP t j=N ~j < m -- j=N 3 J

Hence ~ ~ N and ~ = ~ * P ~ Nw" On the other hand, since

~j (Xk) = 0 for j < k, we get from the above construction of

[*(Xk) l >__ ]~k(Xk) l - }~ I~j (Xk) l j > k

>_ I ;k (Xk) t Z j>k ~" J~k(Xk) l > 0

yj's,

Therefore supp~ cannot be compact and this is a contradiction.

Let ~c.o. be the topology induced on ~ by the compact open

topology of L(~,~). First, let us show that ~c.o. is coarser

than ~(~)' . By Remark 7 it is enough to show that, for any compact

set K, the injection ~'(K)~ ÷ (~, ~.o.) is continuous. Let ~Ybe

! a ~c.o.-neighborhood of the origin in ~, i.e. for some bounded set B

in ~ and a neighborhood ~ of the origin in ~, 7J'= ~d'(B,~) = {~ ¢ ~' :

~ B c ~}. For a fixed K, there is a compact set K 1 such that, for

¢ ~(K) and f ¢ B, supp(~ ~ f) C K I. Moreover we can find

X > 0 and ¢ > 0 for which {¢ ¢ ~ :supp ¢ c K I, [¢[X ~ ¢} C ~.

We need a bounded set A in ~(K) such that, if [<A,~> 1 ~ i, then

-1^ )eX~(¢) [¢ * BIX ~ ¢. Set A = {g : ~(~) = s f(¢ 0(¢) where f ¢ B,

and @ is an arbitrary measurable complex valued function such that

]O[ z i}. (In fact, it would be sufficient to take for O's only cer-

tain functions from C~(~n~ M) where M is a "thin" set.) If g ¢ A

and ~ ¢ ~w' then

lg*l~

<-- YS I~({-t)}(t)]eP°a({)dt d{

<

- 5Y ¢ I f ( g - t ) ~ ( t ) le !aw(g)+Xw(g t ) d t dg

-1 I~I

By Remark 8 this shows that A is bounded in ~ and hence also in ~ (~. W

Now let us show that for ~ ~ A o, ]~ , BIX ! e. Actually, if

¢ c A o and f e B, there is a @(g) as above such that

= c <g,¢> •

Since g e A, the result follows.

v It remains to show that the topology ~c.o. (gm) is finer

I than ~(gw). Let ~ be a neighborhood in g(~), i.e. ~ = B ° for

some B bounded in ~w' If ~ is an arbitrary lattice point in ~n, let

K s be the cube of edge 2 and with center at ~. Let {~} be a parti-

tion of unity subordinate to the covering {Ks}. Let

= {2 [al+n ~ f : V(f E B;a)}

63

As can be easily seen by Remark 8, the set B is bounded in ~w" For

each f s B, we can write f = Z 2-1~l-nf s B There- ~, where f .

fore ~o c B o. We want to find a bounded subset A in ~w and a neigh-

borhood ~ of the origin in $ such that ~f(A,~) C ~o. Let ¢ be

1 on K (Therefore, ~g~¢ = Iglk functions in ~ such that ~ ~ ~,k

if supp g c K .) Since B is bounded, there are constants C~,~ such

< C for all g ~ B, Let C k = max C Let that ~g~ ,~ - ~,k i~i~ X ~,k"

us choose positive numbers 6 such that Ca,X~ - < Ck for all k,~.

Then A is defined as the set of all ¢ a ~ such that supp ¢ c K o

(= K for ~ = 0) and I¢Ik ~ c k for all ~ > 0. Denote by z_~

the translation ¢(x) ~+ ¢(x+a) and ~J~ = {¢ c ~w : max l¢(x) l < 6 }. x ~ K~ -- -~

Then ~is obviously a convex, closed and absorbing subset of ~; hence,

~is a neighborhood in ~ . Let ~P def~d~(A,~).= Then, for each

-I _~(~) for some a and ~ ~ A. Therefore, if S ¢7~ °, f s B, f = 6 a

~I -is = 1 * This shows that then l<f,S>I = ](S , ~)(-a)l i S -e .

~Pc B ° and Proposition 4 is proved.

Let ~ =~(~) be the family of all functions k constructed

as follows. Let ~o = C~-$+ (For the definition of classes ~ and

~+, cf. the text preceding Lemma 2.) Furthermore, let h be an increas-

ing function on the real line such that inf h(s) > -~, and the res-

triction of h to [0,~) is in ~o" Next we pick a function ~(s)

defined for s real, 0 < g(s) < i, and so rapidly decreasing to 0

when s + -~ that the function @e(s), defined as the inverse

function of -for ~(-s) (s > 0), is in ~. Finally let ~ be an arbi-

trary positive number. Then the series

(77) k ( E ) = k ( h ; ~ ; ~ ; ~ ) = ~ ~ ( s ) e x p [ I ~ l h ( s ) - ( s + ~ ) a ( ~ ) ]

*If ~ s A, then also ~(~) = ~(-~) s A.

64

is locally uniformly convergent in cn. 3~(~m) will denote the family

of all such majorants k.

Let J~ =~(~m) be the class of all functions

m(~) = Ce Nc°[~j+Alql ( c f , ( 7 5 ) ) .

! Theorem 3. The space ~ is an AU-space with basis ~, an AU-structure

7£(~w) and a BAU-structure J<(~) .

Proof. To begin with, let us first remark that conditions (i), (ii),

(iv), (v) and (vi) are obvious. Thus it suffices to prove (A) and (B)

Let F be an entire function such that for each of condition (iii).

k

-i ~ } (78) I v ( ¢ ) l <__ Ck(C) = C ~ + I = C(X_ + X+)

S = -co s=O

for some C > 0 and all ~ e cn. By (38),

< ( s + Z l q l ) e , n , h ( , n , ) , , I i I i Y ' + _

_oo = ~ e x p [ l o g ~( -6 ) + I n l h [ - ~ ) + (6-!a)m(~)]

6=1

oo

<_ ~ exp[ -o¢ (6 )+6co(¢ ) ] <_ (5 + 2co({))eC°(~)Oc (c°(~)) 6=0

Therefore,

(79) IF(~) I < C (l+co(~)) ( l+ In I )exp [ (co(~) + In I) (pa (co(~)) +h(]n I)) ] •

Let

65

G(~) d e f l o g { t F ( ~ ) l ( l + ~ 0 ( g ) ) - l ( l + ] n l ) - l } / ( ~ ( g ) + I'~1) =

The function G is bounded. Indeed, if it were not so, then for some

~n' ]~n ] ÷ ~ and ]G(~n) I ~ n 2. We can assume that ]gn] Z ~ and

]nnl ~ ~. (The remaining cases are even simpler.) Let ~,h be the

functions obtained by the linear interpolation of the values

~(0) = i, ~(~(~n) ) = n (n ~ i) and h(0) = i, h([nnl ) = n, respec-

tively. Then we find functions h* ¢ ~o' h e ~ h and pe~ ~,

p* < ~, and apply (79) to the majorant k = k(he;...)

IG(Cn )1 log C 2 1 < < + -- ÷ 0

-- 2 - 2 n n n

which is a contradiction.

of (47) .

I t r ema ins to v e r i f y p r o p e r t y (B) . Le t

be a f i x e d m a j o r a n t in ~. We c l a i m t h a t the s e t

! T . Y(k) = {, ~ ~w : ]$(C) 1 £ k ( c ) (V C)} i s a b a r r e l in ~m

' , t h i s w i l l imply t h a t ~ ( ~ ( ; ~ ) ) i s c o a r s e r t han ( ~ )

satisfies inequality (75) with some constants C,A,N.

each Z > 0,

The rest follows similarly as in the proof

k(~) = k ( h ; ~ ; ; ; ~ )

By Remark 5

Each $, % ¢ ~'

However, for

eN~+A]q] < e2N~+K~-MInl + e2Atn]-Z~+Mt~] = g I + E 2 ,

where M def inf h.

for s = -(2N+£),

E 1 2

and

E 2 2

Let ~ be so large that h(Z-~) ~ 2A +M.

-s~+I~lh(s ) k e < - ~ '

eI,lh(z-~)-z~ ~ k

Then,

This shows that ~(k) absorbs the distribution ~.

Now we have to show that ~(~(@~)) is finer than ~(£~).

By Proposition 4, this is the same as showing that ~(~(~)) is finer

than ~c.o. Let ~P = ~(B,~) be a neighborhood of the origin in the

latter topology. We can assume that B is defined by a function

m c Y~(([c) (cf. Proposition 3) and ~ is defined by means of some

c ~(~) (cf. (7)). If we can find a k c ~(~w) such that km ~ k,

the proof will follow. Let us write k = k(h;~;~;~), m = m(A;A;I;~)

and A(t) = tX(t); we want a k = k(h;c;~;~) such that

( 8 o )

k ( ~ ) -- (Z_ + Z + ) c ( s ) e x p [ [ n i h ( s ) - ( s + n ) c o ( C ) ]

co

<__ [ ~ ( s ) e x p [ I n [ ( h ( s ) - A ) + c o ( C ) ( ~ , ( c o ( g ) ) - s - ~ ) ] S--1

For all s large, say s K s A, h(s) ~ A. For s K 0 we choose

1 h(s) ~ h(S+SA)-A , c(s) = r ~ (S+SA) and ~ K ~+s A" Setting

s+s A = 6 we then obtain

1 (81) E+ . . . <-- 2- ~ ~ ( 6 ) e x P [ I n ] ( ~ ( 6 ) - A ) - ( 6 + ~ ) w ( g ) ]

6>_s A

i ½ k ( c )

It suffices to complete the definition of h and c for s < 0 so that

(82) E . . . <__ ~ ( s A ) e x p [ c o ( g ) ( l ( a ~ ( g ) ) - s A- ~ ) ]

Let h(s) = 0 for s < 0 and c(s) (s < 0) such that Pc <- ~

Then, by Lemma 2,

-i X < e -vc° (~)

S = - o o

67

el°gc(s)-s~(~) <_e-~(~)(s+2~(~))exp[~(~)~c(~(~)) ]

For ~ sufficiently large, the last inequality implies (82); this

together with (81) proves inequality (80) and thus also the theorem.

Remarks: 9. Comparing Theorem 1 to Theorem 3 we see that there is an

t interesting relationship between the spaces ~ and ~ . The

t family ~(~), which is the AU-structure for ~ discussed in

the previous section, can be obtained by taking the "Taylor"

parts of all k c ~(~); i.e., K(~) is comprised of all

functions k where k = Z+ for some k = Z + Z+ c ~(~).

Moreover, from here it would not be difficult to derive

another relationship between ~ and ~; roughly, it can be

described by saying that "outside a certain neighborhood of

the real subspace of cn, the topologies of ~ and ~ are the

same" (cf. [6]). Namely, let ~ c ~, ~(~)/a(Igl) ÷ 0

def {~ : I~I > ~(I~l)}. Then for for ]~I ÷ ~ and R = _ ,

each k c X(~), there is a majorant k I c ~(~) such that

0 < C 1 ! kl(~)/k(~ ) ! C 2 in R . This statement

generalizes to Beurling spaces a result of L. Ehrenpreis

[19,23]. It also has analogous consequences for the study

of hypoellipticity.

i0. There are other variations of describing the topology

o f ~ ' by means o f s e r i e s ( 7 7 ) . T h u s , f o r i n s t a n c e , one way W

! of defining the topology of ~ is the following. For each

! k ~ ~(~w) , let ~r(k) be the set of all ~ in ~ such that,

f o r some c o n s t a n t s N and C ( d e p e n d i n g b o t h on ¢ ) ,

for all ~. The system {~(k)} defines a basis of neigh-

borhoods in ~' [6~7]. (The constant N in (83) is not

necessarily a positive number, and is related to the order

of the distribution ~.)

II. It is very likely that under the same restrictions

on ~'s as in the previous section (cf. Theorem 2), one

could prove that ~ is a PLAU-space.

CHAPTER III

Spaces of Approximate Solutions to Certain Convolution Equations

§i. SPACES ~B(L;¢)

In this chapter we shall study another class of function

spaces which are closely related to certain convolution equations.

Roughly speaking, ~B(L;¢) will be the set of all "approximate

solutions" of a convolution equation, which satisfy certain "growth

conditions".

Let ~: sn 4 [0,+~] be a convex function such that ~(0) = 0,

¢(x I .... ,-xj,...,Xn) = ¢(x I .... ,xj,...,Xn) for any j, and

~(x)/Ix [ -* ~ for Ix I + ~. Let B = {bj} be a convex p-sequence, i.e.

bj -- exp (g(j)), where g: SP ~ $+ is a convex function such that 1 / I b j l

g ( x ) / I x I + ~ when Ixl ÷ ~. ( In p a r t i c u l a r , b j + ~ when j + ~ . )

Finally, let L -- (L 1 ...,Lp) be a vector with components L i ~ (I n)

1 _< i < p. If j = (Jl'''" 'Jp) is a multiindex, we set

LJ (z) -- T ~ ( L i ( z ) ) j i i= l

Definition I. For B, L, ~ as above, ~B(L;~) is defined as the space

of all C~-functions on ~n such that, for any E > 0 and any multi-

index ~, there is a constant C = C(f,E,~) so that, for all x ¢ ~n

and j = (Jl ..... Jp)'

I D a ( L J * f ) ( x ) l < C c l J l b . exp ( ¢ ( ¢ x ) ) - j

(1)

Let

def 1 ID ~ ............. q a , a ( f ) = ~ ~ 5up (LJ*f ) (x) l j . x exp [¢(¢x) ]

J

In the t o p o l o g y g e n e r a t e d by the seminorms q~,~, N B ( L ; ~ ) becomes 1 a F r g c h e t - M o n t e l s p a c e .

), the classes just defined are (roughly Remark I. For L = (~xi,..., ~x n

speaking) the Denjoy-Carleman classes and the Gevrey classes

[38,42]. The a's and ¢'s are introduced so that we obtain

(FM)-spaces.

Let ¢* be the Young conjugate of ¢ (cf. [53]), i.e.

(2) ~ (y) = max (<x,y> - ~(x)) , X

and I the series

lJl (s ) ~ ( z ) = Z

3 3

which is convergent for all z. Finally, let us recall that W

is called a weak AU-space, provided W satisfies conditions

(i), (ii), (iii) of Def. I, I with some AU-structure Y~

such that S = ~(k) for each S ~ U and k ~ 3~ (cf. Remark 6

in Chap. I, §2).

Theorem i. ~B(L;#) is a weak AU-space with an AU-structure Y/

containing all functions k(z) on Cn such that for arbitrary constants

N, c, d > O, k satisfies the estimate

(4) ( l + l z [ ) N X ( c L ( z ) ) exp [¢ (d. I m z ) ] = ~7(k(z ) )

Denote by @ ( N ; c ; d ; z ) t he f u n c t i o n in t h e l e f t - h a n d s i d e o f ( 4 ) .

M o r e o v e r , e v e r y e n t i r e f u n c t i o n F(z ) which i s bounded by some f u n c t i o n A T

@ ( N ; c ; d ; z ) i s an e l e m e n t o f ~ B ( L ; ¢ ) . 2

Proof: For the sake of simplicity we shall give the proof only for

p = I, i.e. for a single convolution operator. The general case can v

be proved along the same lines. We set U = ~B(L;~) and W = &B(L;~).

i<x,z> lie in W. First we have to check that all exponentials e

Indeed, for any fixed z, ~ and e,we have by (4),

(e i <x, z> (s ) qc~,~ ) _~ e([c~[ ; c ; d ; z ) < ~ ,

71

where c and d depend only on e. If T is an arbitrary element of !

~B(L;~), T must be bounded on some neighborhood ~ = ~(q~,e); in

particular, applying T to e i<x z> ' , we obtain from (5) that for some C

(independent of z),

(6) IT(z) l ! C@(t~[ ; c ; d ; z )

Now we are going to prove that the Fourier transform defines an

isomorphism of the vector spaces U and V, where by V we denoted the

vector space of all entire functions satisfying estimates of the form

* i <x ,z> (6). This will also prove that the set of exponentials e

is total in gB(L;¢); and, furthermore that ~ has property (A).

First we must establish an intrinsic description of the space U.

Lemma I. For each T ~ U there exist positive constants e,A,N and

entire functions Qj (z) satisfying

(7) T(z) = ~ Qj (z) £J (z) j=0

an d

(8) IQj(z)] ! A( l+ l z l ) N exp [~*(E1 Im z ) ] / c J b j

Conversely, if F(z) is an entire function which can be expanded as

in (7) with coefficients satisfying (8), then F = T for some T ¢ U.

Proof: Let us recall that the space ~(~) (cf. [4,23]) is the space

of C -functions satisfying conditions (I) with L = identity and bj

for all j ~ 1 and b o = I. The space ~(~) is equipped with the

natural topology defined by the seminorms

s u p I D~g (~)I Ig Ia , s x exp [~(¢x)]

this has been proved we shall, of course, write V = U.

Every f ¢ W can be mapped onto a sequence {fj} = f of functions in

g(~) by means of the mapping fj = LJ*f, j ~ 0; and, any such

sequence f satisfies, for all a,e,

(9) p a , s ( f ) = Z ( I f j l a , E / s J b j ) < J

I f W i s the space o f a l l s e q u e n c e s f , t h e n the seminorms (9) d e f i n e

a F r g c h e t t o p o l o g y on W; and, o b v i o u s l y f ~ f i s an i somorph i sm o f

1 . c spaces W and W. Th i s i somorph i sm shows t h a t w i t h e v e r y Ts U we

can a s s o c i a t e a (no t n e c e s s a r i l y un ique ) s equence {Tj}, Tj s & ' ( * ) ,

such t h a t

< T,f> = ~< Tj,fj> ; J

and, for some r,¢, A > 0 and for all j and g ~ $(~),

I< Tj,g> 1 j A max { I g l a , E : Ial ~ r} / sJbj

Conversely, any such sequence {Tj} defines an element T in U by the

formula

(i0) < T,f> = .~ < Tj,LJ*f> , J

and the lemma follows if we set Qj = Tj. (For the characterization

~ , • of (~) see [4,23] )

(Proof of Theorem 4 continued) I. T ~ T is injective: Assume

T(z) - 0. By Lemma 1 we can write

T(z) = ~ Qj (z) LJ (z) J

,w)d~f Let H(z ~ Qj(z)w j for z ~ ~n and w ~ ¢. Then H is an entire

J ~n+l function of (z,w) ~ , for the coefficients are entire and satisfy

the estimates (8). Moreover,

1 iH(z ,w)] < A ( I + I z l ) N )~(w) exp [$*(~- Im z)]

73

^ A

f o r some A , N , ¢ > 0 a n d a l l z . S i n c e H ( z , L ( z ) ) = T ( z ) ~ 0 , t h e

f u n c t i o n

(II) G(z ,w) - H(z ,w) w - ~ ( z )

i s e n t i r e , a n d I g ( z , w ) l < 4 max IH(~,w')l ( c f . Lemma 2 , I V ) . - I w - w ' I_<i

T h u s t h e f u n c t i o n G ( z , w ) c a n b e w r i t t e n a s a p o w e r s e r i e s i n w ,

(iz) co

G ( z , w ) = ~ Gj ( z ) w J • j = 0

We shall show that the entire functions Gj(z) satisfy uniform estimates

of the form (8). Since we can always find x > l such that

ix l max = ~ , k>0 J

we obtain

w' , , _ _ 2 2j+l inf ~x(F-)/lwr j : w ~ ¢, lw-w'l < i} <

_ _ cjb. J

Then the Cauchy estimates yield

IGj(z)I _< 2A(l+Izl) N exp [~*(~-~)]/~Jb.j (6= ~_)c

Comparing coefficients of equal powers of w in (ii) (cf. (12) and the

definition of H(z,w)) we obtain

A

T = G L, O O

Tj = Gj_ 1 GjL (j > i)

If Sj e ~'(~) is defined by Sj = Gj, then GjL = (L*Sj) ̂ and

< Sj*L,g> = < Sj,L*g> for every g E ~(~). Substituting the above

identities in (I0) we get, for every f ~ W,

eo

~F,f> = Z0 < Tj ,L]*f> J o o

= - <S o , L * f > + j = l

,LJ*f> -< S ,L j+l~ ( <s j- 1 j f> ) = 0.

A

Hence T = 0 and the injectivity of the mapping T ~ T is proved.

2. T. ~ T i s s u r j e c t i v e . The p r o o f w i l l be b a s e d on t h e same i d e a

u s e d i n t h e p r o o f o f t h e i n j e c t i v i t y o f T * T. The m a i n p o i n t

c o n s i s t s i n f i n d i n g an e n t i r e f u n c t i o n H ( z , w ) s a t i s f y i n g g o o d

e s t i m a t e s a n d s u c h t h a t H ( z , L ( z ) ) = F ( z ) w h e r e F i s an a r b i t r a r y

f i x e d e l e m e n t o f V; o r , i n o t h e r w o r d s , g i v e n a f u n c t i o n F a n a l y t i c

on t h e v a r i e t y { w - L ( z ) = 0} i n ~ n + l , we a r e s u p p o s e d t o e x t e n d F

t o an e n t i r e f u n c t i o n i n ¢ n + l w h i c h w o u l d s t i l l h a v e g o o d b o u n d s .

T h i s , h o w e v e r , i s a t y p i c a l p r o b l e m t o w h i c h t h e L 2 - e s t i m a t e s o f t h e

~ - o p e r a t o r ( c f . [ 2 3 ] , T h e o r e m 4 . 4 . 3 ) c a n be a p p l i e d .

L e t p be a C ~ - f u n c t i o n i n ~2 = ¢ s u c h t h a t 0 < o ( s ) < 1 f o r

a l l s ~ ¢ ; p = 1 f o r I s l j 1 / 2 ; p = 0 f o r I s [ ~ 1; a n d , f o r some

c o n s t a n t C > 0 , I a--gP[ < C. I f F i s an e n t i r e f u n c t i o n i n cn s u c h t h a t - ~ s -

(13) I F ( z ) t < A @ ( N ; u ; S ; z ) ,

H will be defined by

X(z,w) = F ( z ) ~ ( ~ - ~ ( z ) ) + ( ~ - { ( z ) ) u ( z , w )

Clearly, II(z,L(z)) = F(z) and we have to find the function u so that

H is entire, i.e., 7H = 0, or

-F (z)~[ ~ (w-L (z) ] gu (14) w - L(Z)

(This expression is well defined, for the numerator vanishes when

lw-L(z) I < I/2.) By virtue of the Paley-Wiener theorem, there are

constants D > i, M > 0 and B >0 such that, for all z,

A

Let us set C = (z,w), C = 2(n+I)DC and (cf. [29])

75

2 l~'ul 2= I ~ I + ~ la~ I _ aw az.

0

Then, using the fact that expression (14) vanishes for lw-L(z) I > i,

we derive from (13), (14) and (15) the inequality

(16) lauj 2 < [ C ( l + l z l ) M+N ~(alwl+c~) exp (BlIm zt+~'~(SIm z ) ) ] 2

= exp ( ¢ ( ¢ ) ) .

The function ~ defined by (16) is plurisubharmonic. Hence, by

Theorem 4.4.3 in [29], there exists a solution u of (14) satisfying

f def 2 f l u ( ¢ ) l 2 e - q ~ ( ¢ ) ( l + l ¢ l z ) - ( n ÷ 3 ) l d ¢ l _< ( x + [ ¢ 1 2 ) - ( n + l ) l d ¢ [ = K

Therefore H is entire and

(17) I Ill(C)1 2 e -¢(¢)(l+l~f2)-(n+3)Cl+lwlZ+IfCz)IZ)-lld~I _< 6~

Using Lemma I, I, we obtain from (17) the estimate

(18) IH(~)I _< ~o(l+Izl)P ~(2~lwl) exp (~*(~ Im z))

where p = 2M+N+2n+6, ~ = 2(B+B) and ~o is some positive constant.

Expanding H(~) into the power series, H(~) = ~ Hj (z)w j, we see that

by (18) ,

(19) ]HD(~) I _< ~o(l+Izl)P exp (~(~ Im z))/~Jbj ,

where E = I/4~. The surjectivity of the mapping T ~ 'F then follows

by Lemma l, because we can write

F(z) = H(z,L(z)) = [ Hj (z) LJ (z)

Finally, it is clear that V can be viewed as the l.c. space

o@(~C) with 3~C described in the statement of Theorem 4. We claim

76

3. T ~ T is an isomorphism of l.c. spaces U and V. Actually, since

U is reflexive, U is also barreled; Rence, T~-~T is continuous. On

the other hand, it can be shown [49] that V is the inductive limit of

the Banach spaces ~(N),

~(N) = {F s .d- : IF(z)l = d T ( @ ( N ; N ; N ; z ) ) }

In p a r t i c u l a r , V i s a b o r n o l o g i c a l s p a c e . H o w e v e r a l l c o n s t a n t s i n

(19) d e p e n d o n l y t h e c o n s t a n t s o c c u r r i n g i n ( 1 3 ) , b u t n o t on t h e

f u n c t i o n F i t s e l f . T h e r e f o r e , t h e m a p p i n g T ~ T maps b o u n d e d s e t s

i n t o b o u n d e d s e t s , and t h u s i t i s c o n t i n u o u s . T h i s c o m p l e t e s t h e

p r o o f o f T h e o r e m 1. a

77

~2. A UNIQUENESS THEOREM FOR CONVOLUTION EQUATIONS

In this section we shall give the first application of AU-

spaces by proving a uniqueness theorem for convolution operators that

generalizes the uniqueness theorem for the ~eat equation. ~ The

problem consists in the following. Suppose we know that

C ~ fCx t) e (¢n+l) satisfies the equation

(201 L*f(x,t) = ~f(x,t) + ~l*D~-lf(x,t) + ... + ~q~f(x,t) = 0

where ~l,...,~q are given distributions with compact support acting

on the x-variables, i.e. nj ~ ~,([n). Furthermore, assume that

f has zero Cauchy data, i.e. D~f(x,0) z 0 for j = 0,I,...

When can we conclude that f ~ 0?

It is well known that even for differential operators this

does not hold unless the hyperplane {(x,t): t=0} is non-characteris-

tic [27]. For the characteristic case one has to impose additional

restrictions upon f, e.g. certain growth conditions on f (cf. [4811.

Here we shall impose growth conditions on the x-variables only.

Let us set

~ ( x , t ) : * ( x ) + % ( t ) ,

where %o(t) = 0 for Itl < 1 and %o(t) = +~ for Itl h i. We shall

assume that the function f is in G(~). Actually, we can go even

further and study the case when f is not a solution to (20), but

satisfies this equation only approximately, i.e. when f s GB(L;~)

and D~f(x,0) ~ 0 for all multiindices ~ = (~],...,~n+l). Theorem 2

gives conditions on B and % which imply f ~ 0. Theorem 4 represents

an analogous result for an "overdetermined" system.

Remark 2. In the case when f actually solves equation (20), the

condition,

(21) D a f ( x , 0 ) - 0 f o r a l l a ,

4 is a c o n s e q u e n c e o f D ~ f ( x , 0 ) ~ 0 f o r 1 J j j q - 1 . I f f

s a t i s f i e s c o n d i t i o n ( 2 0 ) , f i s s a i d t o have z e r o Cauchy d a t a .

The v a r i a b l e s d u a l t o ( x , t ) ~ ~n+ l w i l l be d e n o t e d by

w = ( z , s ) e c n + l . F i n a l t y , i n t h i s s e c t i o n a c o n v e x

p - s e q u e n c e B i s a l w a y s a s sumed t o be o f t h e fo rm B = { b j } ,

b . = b ! 1) . . . b ! p) whe re a l l t h e s e q u e n c e s J J~ Jp

B (k) {b~ k )}~ a r e c o n v e x ( c f . § I ) .

We s h a l l a l so need some r e s u l t s from the t h e o r y o f

Den joy-Car leman c lasses [ 3 8 , 4 2 ] .

Definition 2. Let M be a fixed convex sequence. Then the

Denjoy-Carleman class gM is defined as the space of all functions

f E C ([0,i]) such that

[ f ( J ) ( x ) ] <_ c o c { b j f o r a l l j > 0 and x ~ [ 0 , 1 ]

with some constants Co,C 1 depending on f.

A class ~ of C~-functions (on [0,I]) is called

quas.iTanalytic, if no function f g 0 in ~ can vanish together with

all its derivatives at any point. If NM is a quasi-analytic class,

the sequence M will be called quasi-analytic.

Denjgy-Carleman Theorem. The following three conditions are equiva-

lent:

(a) ~M is quasi-analytic ;

(b) log -z am =

b - 1 / j ( c ) Z - = ~

j e l J

( c f . ( 3 ) ) ;

79

One way of generating quasi-analytic sequences is the

following. Let R(u) be a positive strictly increasing function of

u > 0 such that log (R(u)) is a convex function of log u and

satisfying for all j > 0,

uJ lim ~ = 0 ;

then, we define a sequence M = {m.} by ]

u j ( 2 2 a ) m. = max

J u>0 R-(E7

Then, as it is shown in [38], one can find positive constants a,B

such that

(22b) l o g ( R ( u ) ) - l o g ( l + u ) -B < l o g (),M(U)) < l o g ( R ( 2 u ) ) + or.

Therefore it follows that the class £M is quasi-analytic if and

only if

I log (R(u))u -2 du =

1

Given a convex sequence B and a positive integer q we can define a

new sequence {mj], denoted by B/q, by setting first R(u) = lB(U q)

for u > 0 ; and, defining m. as in (22). Then, we w~ll have - 2

b < m. < b

w h e r e [ j / q ] d e n o t e s a s u s u a l t h e i n t e g r a l p a r t o f j / q .

The f o l l o w i n g m o d i f i c a t i o n C = { c . } o f a g i v e n q u a s i - a n a t y t J

s e q u e n c e B w i l l be a l s o u s e d i n t h e s e q u e l ( c f . (33) b e l o w ) . I f

B = { b j } , l e t C = { c j } be d e f i n e d by

cj = max (bj ,j!) ( j = o , ~ , . . . )

It is clear that C is a convex sequence and

Xc(U) _< elUl ,. Xc(U] _< ~s(U)

However we claim that C is also quasi-analytic. Since B is quasi-

analytic we may assume j! > b. for infinitely many j; otherwise the - j

result would be clear. Then, there is a sequence of integers Jk such

= . ~ jk !. Since C is a convex sequence, that Jo I, 2Jk j Jk+l and bjk

I/j must be increasing" hence cj

J k + l - 1 / j - 1 / ( J k + l ) c . > (j Jk )C .

j = j k + l j - k + l - J k + l

- 1 / J k + l 1 . -1 1 >- ( J k + l - J k ) ( J k ! ) > g J k + l ( J k + l ) = I~ •

-i/j =~o and by the Denjoy-Carleman theorem, the sequence C is Thus ~ cj

q u a s i - a n a l y t i c .

I f gM' ~N a r e two q u a s i - a n a l y t i c c l a s s e s , t h e i r "sum"

&M + ~ N = {f+g: f c ~ M ' g s @N } i s no t n e c e s s a r i l y q u a s i - a n a l y t i c

[ 1 , 3 8 ] . N e v e r t h e l e s s , the convex r e g u l a r i z a t i o n y i e l d s a p a r t i a l

result in this direction:

Lemma i. If M = {j!} and N is quasi-analytic, then ~M + ~N is also

quasi-analytic.

Let us first sketch the intuitive idea which underlies both

the statement and the proof of Theorem I. Our objective will be to

find functions H(y,w), analytic in w and belonging to a fixed quasi-

analytic class on the interval 0 j y j l;and,moreover, such that

(i) The functions of the form H(l,w) form a total set in

( i i )

A !

~B(L;$).

For all j ~ 0,

supp ~JH(O,w). c {w s cn+l: s = O} .

^

If f c ~B(L;~), then by Theorem 1 and Remark 6, I there

exists a majorant k (in the AU-structure described in TI~.I)

and a Radon measure dv(w) such that

81

f(x,t) = I ei<(x,t) ,w > d~(w) kF~

~n+l

(iii) Furthermore, it will be shown below that the functions

de f f h(y) = <f(.),H(y -)> : H(y,w) dr(w) ,

cn+l

are in a fixed quasi-analytic class. Now if f has zero

Cauchy data, then by (ii), h(J)(0) = 0 for j = 0,i,... .

Then (iii) implies h(1) = 0 for all H, and by (i) we

obtain f = 0.

First we need the following lemma.

Lemma 2. Let ¢ = ~ + iT E ¢ denote the complex variable and ~(~) an

even positive convex function for which

(23) I~l a = ~7(~(T)) with some a > 1 .

Then there are non-zero entire functions F such that

(24) I F ( ¢ ) I = ~Y(exp ( - c l ~ l a + ~ ( c ' ~ ) )

where c , c ' a r e some p o s i t i v e c o n s t a n t s d e p e n d i n g on F. M o r e o v e r , f o r

any s u c h f u n c t i o n F, t h e s e t o f l i n e a r c o m b i n a t i o n s o f f u n c t i o n s o f

t h e f o r m e i ~ ¢ F ( ¢ + B ) (~,~ r e a l ) i s d e n s e i n ~ ' ( ~ ) ; h e r e f d e n o t e s

the Young conjugate of the function f. Since any of the functions

eieCF(¢+B) satisfies (24), the set of all functions satisfying (24)

is also dense in ~' (~*).

Proof: The existence of entire functions F ~ 0 satisfying (24) is

shown, for instance, in [33]. Assume that F is such a function and

set

f(¢) = F ( - ~ )

Then f ¢ ( f ) r ~ d ~ where S i s t h e S c h w a r t z s p a c e o f r a p i d l y

d e c r e a s i n g f u n c t i o n s [ 4 6 ] . We h a v e t o p r o v e t h a t t h e s e t ~ o f a l l

linear combinations of functions of the form e iax f(B-x) (~,B real)

~' is dense in (~) Let us first reduce this problem to showing that

oo

every function of the form g(x) f(B-x) (with g ~ C o and B real) is in

the closure of ~ . One can easily construct a sequence of functions

gm gm ¢ o' such that for every h ~ ~(~*), gm h ÷ h in ~(~*)

Then, if h is orthogonal to ./4~, i.e. <T,h> = 0 for all T ¢ j¢~ , we

shall have, for all m > 1 and B ~ i,

(2s) < g m ( X ) f ( f l - x ) , h ( x ) > = < f ( f l - x ) , g m ( x ) h ( x ) > = 0 .

oo

Since gm h E C o and f ¢ o ¢, equality (25) says that

(26) f~(gm h) (B) = 0 for all ~ ¢ ~ and m _> I.

Applying the Fourier transform to (26) we obtain

A

F(-~)gmh(~) = 0 for all ~ ¢ ~.

Since F is a non-zero analytic function, we obtain from here that

gm ~ = 0, for all m, i.e. h = 0. This shows that it suffices to prove

C ~ that all functions g(x)f(fl-x) B ¢ ~, g ¢ o' are in the closure of ~L.

Let ~M be the Schwartz space of all Ca-functions of poly-

nomial growth in ~ [46]. The topology of £v M has the following

property. Let p be a positive continuous function satisfying for

all m = 0,i,...,

lim p(x] - 0 ; Ixl+ Ixl m

and, let {hy)y be a net in ~M such that hy ÷ 0. Then

(273 s u p ( p ( x ) t h ~ k ) ( x ) t ) c ÷ 0 - ~ < X < ~ Y

for all k = 0,i,... .

mapping

We claim that for each fixed T E (~*) the

(28) h ~ hT

83

of ~M into &'($*) is continuous. Indeed, given hy + 0 in Gr M and

a bounded set {g} in ~($*), we have for every c > 0 and k > 0,

sup {t ( h y g ) (k) (x ) I e x p ( - ~ * ( E x ) ) } < C k sup ( I g ( J ) ( x ) l e x p ( - ~ * ( q ) ) } ; l < ' < k x x _3_

x sup {Ih~J) (x) I p (x)} x ; 1 <_j <_k

for some positive constant C k and

p ( x ) e x p [ $ , ex = ( - 7 ) - v * ( e x ) ]

Therefore hyg ÷ 0 in ~($*) uniformly with respect to {g}, and the

c o n t i n u i t y o f t h e m a p p i n g (28) f o l l o w s .

E v e r y f u n c t i o n g ~ C ° c a n be a p p r o x i m a t e d i n G M by l i n e a r

c o m b i n a t i o n s o f e i a x , a ~ ~ , f o r i t s u f f i c e s t o c o n s i d e r t h e R ie ma nn

sums o f t h e i n t e g r a l

1 I -ixv g (x) - 2~ e (v) dv

Hence, for any B real, the function g(x)f(B-x) is in the closure

of ~g. in ~'(~*). This completes the proof of Lemma i.

Since each ~j in (20) is a distribution with compact support,

there exist positive constants A, B and C such that, for all

j = 1,2,...,g, we have

l~j(z) I _< C(I+IR e zl)15 eAIIm z I j C (l+[Re zl)2B + 2 C_ e2AIIm z I

More generally, we can assume that we are given a function p(u), u > 0,

which is positive, continuous, strictly increasing and such that for

any 6 > 0 there are positive numbers 5' and 8" satisfying

(29) 5 p ( u ) < p(6 ' u ) + 6 "

/

(examples of p: p(u) e u m = ; p(u) = u , etc.). Moreover, we shall

84

assume that the Fourier transforms of the distribution~ pj satisfy the

estimates

(30) I U j ( z ) I 1 / j _< C 1 + C2lRe z] D + p ( l I m z I )

where CI, C 2 and D are some non-negative constants. Let us observe

that p(u) ~ ~ (except when all p. are zero, and in this case the J

answer is well known [23]). Therefore the inverse function p l(U)

of p is well defined. The function ~(u) will be assumed convex,

positive and such that the function

(31) R(u) = exp ( f ( p _ l ( U ) ) )

s a t i s f i e s t h e a b o v e c o n d i t i o n s on R ( c f . t h e i e x t f o l l o w i n g t h e

D e n j o y - C a r l e m a n t h e o r e m ) . T h e n by ( 2 2 a , b ) t h e r e e x i s t s a c o n v e x

s e q u e n c e M = {mj} s u c h t h a t f o r some p o s i t i v e c o n s t a n t C 3 ,

(32) XM(~) ~ C 3 R(u) ;

and~we may f u r t h e r a s s u m e ( c f . t h e m o d i f i c a t i o n C o f B p r e c e d i n g

Lemma 1) t h a t

(33) XM(U) ~ e Iul

Let us recall that M is quasi-analytic if and only if

f~ (p_ l (U) )U -2 du =

1

We can finally state

Theorem 2. Assume that ~ is as above and satisfies (23) with an a >D.

Furthermore, suppose that

(34) SM + SB/q is a quasi-analytic class.

Let ~(x) = ~(Ixl) and ~(x,t) be the function defined in the beginning

of this section. Then

85

[f E ~ B ( L ; ~ ) and DC~f(x,0) o (Va)}=> f -: o

Proof: Lemma 2 and the definition of ~ imply that the set of

functions of the form

(35) ei°sFl (Zl) . . . F n (Zn)

where o is real and F. satisfy condition (24), is total in &'(¢). i

T

In p a r t i c u l a r , t h i s s e t i s t o t a l in ~ B ( L ; ~ ) .

Le t us d e f i n e t h e f a m i l y Yf o f f u n c t i o n s H ( y , w ) , w ~ ~ n + l ,

0 < y < 1, by

Then

(36)

= (H(y ,w) = e i y a s F l ( Z l ) . . . F n ( Z n ) , F i as in (35 )} .

the set (H(l,w): H e 3~ } is total in ~B(L;~)

Moreover, for H E 3~ and j = 0,i,...,

J 3

$yJ H ( y , w ) = (ices) j H(y ,w) ;

therefore,

J

~. H(O,w) = F o u r i e r t r a n s f o r m o f 6 J) @ Tx, T E ~ x ( ~ ) 3yJ x '

i.e. ,

(37) aJ

~y3 H(0,w) acts only on the Cauchy data of

the functions in ~B(L;~).

Using the Fourier representation of f(x,t), it follows from the

estimates below that for every H ~ 5~ the function h(y) defined in

(iii) above is a C -function and

I ~ d'v (w) (38) h ( J ) ( y ) = ~JH (y ,w) ~ (j > 0) ay3

Let us set

A = {w = ( z , s ) : ] s ] < m a x ]2q~j(z)] l / j } l<_j <q

Then

1 q } . B = ¢n+l--A_C {w: l~(w) l >_~ Isl

Given ~ > 0, by (29) we can choose e',e" > 0 such that, for all u > 0,

(39) 8qep(u) < p(e'u) + 4E"

If w e A and I Im z I are so large that

4qep(JIm zJ) - 2e" > 0 ,

then, using (30), (32), (33), (39) and the convexity of XM, we obtain

_5 < )~M(2qeCI+ 2qeC2[Re zlD+ 2qelz(llm z]+ ¢"-¢")

1 _< ~- XM(4qeCl+ 4qeC2JRe zJD+2e '') + + ~M(4qsp(Jlm zJ)-2e")

<__ C 4 exp (4qeC2JRe zJ D) + 1 C3R(8qep(jim z])-4e")

_< C 4 exp (4qeC21Re zJ D) + + C 3 exp (~(e' Jim z[))

For the remaining points w in A we have

XM(eS) = C 5 exp (2qcC2JRe zJ D)

On the other hand, for y e [0,I], j = 0,i,..., and H E Y£, we get

~ H(y,w) l < c " l ~ [ J l s l J exp [[~ Ira s [ -c lRe zla+ ~ ( c ' l I m z l ) ] ~yJ

for some positive c, c', c" and ~ real. Set C 6 -- max {C4,C 5} and

> 0 so small that 4qeC 2 _< c. Then

i ~ j ~ " { z'a+~(c'IIm } • H(y,w) I < c"( )JmjXM(CS) exp ]o Im sl-clRe z[) ~yJ

87

_< C6c"( ) mj exp ]g Im s[+4qEC2[Re z[D-c]Re z Ia+~(c ' [ Im z[)

C H T ! + ~-- ( ) mj exp I~ Im sl+#(c IIm zl)+ #(e lira zl)

_< C7( ~-L)3mj k(w)

In the proof of the last inequality we have used the inequality D < a

and the fact that the function k dominates all the other factors.

The last estimates show that

(40 ) H ( F , w ) ~ ~ .

A

In the set B we have IL(w) l L ½1sl q ; and, since k contains a factor

larger than ~B(21L(w) I) ~ ~B(Islq),

S dv (w) (41) H(y,w) ~ ~ ~B/q B

Rela t ions (40) and (41) show tha t h ~ ~M + ~B/q" Since Daf(x,0) - 0

for all ~, we get from (37) and (38) that h(J)(0) = 0 for all j.

The quasi-analyticity of the class ~M + ~B/q then implies h(1) = 0,

i.e.

< H(l,w),f> = 0 for all H ~ ~ ,

which, as we showed in (36), implies f - 0. The theorem is completely

proved.

Instead of inequality (30) we can use the inequality

n D i n (42) l~J(z) ll/J ~ Co i=l i=l + ~ CilRe zil + ~ Pi(l Im zil)

with C o and C i positive, D i ~ 0 and Pi sastisfying the same conditions

as D; and,with ~i being even, convex and positive functions satisfy-

ing (23) for a i ~ D i. The function R(u) is chosen so that R is

dominated by all exponentials exp (Ti((Pi)_l(U))), and satisfies

88

the same conditions as in Theorem 2. Then, for the corresponding M

(cf. (22a)), the same proof yields the following

Theorem 3. Assume that (34) holds and set ¢(x) =

the conclusion of Theorem 2 holds.

n

Z ~i (xi)" i=l

Then

Similarly one can study 'overdetermined' systems of the form

qr ~r -I Ll*f(x,t) = D t f(x,t)+Mr,l*D f(x,t) +...+ ~r,qr*f(x,t)

( r = l , . . . , p )

where t h e d i s t r i b u t i o n s > r , j a c t o n l y on t h e x - v a r i a b l e . One has t o

i n t r o d u c e a new c o n v e x s e q u e n c e C d e r i v e d f rom the p - s e q u e n c e B by

means of

S(u) = aB ( u q l ' ' ' " ' u q p )

as in (22a). Let us observe that C is a quasi-analytic sequence

one of the convex sequences B(r)/qr is quasi-analytic; indeed, whenever

S(u) > ~ (u qr) for r = I, ..,p . - B(r)

Moreover, if (42) holds for all Mr,j, we can define M and ~ as in

Theorem 3 and obtain

Theorem 4. Let ~, M and C be as above. Assume that (34) holds for

the class ~M + ~C" Then any function f, f s ~B(LI,...,Lp;$), such

that D~f(x,0) -= 0 for all ~, is identically zero.

Example i. Let

and

L*f(x,t) = Dtf(x,t) + ~*f(x,t)

~(x) = [xl log cz+lxl)

Then for any ~ with compact support and any quasi-analytic sequence B,

89

the hypotheses of Theorem 2 are satisfied. In fact, let

P(u) = ~(u) = e A[u] for some A > 0. Then AM(U) = e lul and N = {j!}.

By Lemma I, the class ~M + ~B is quasi-analytic.

Example 2. Let us consider the heat equation

2 ~ ~2 L * f(x,t) = ~-~ f(x,t) f(x,t) (x,t) E

~x 2

then for ~(x) = Ixl 2 and any quasi-analytic sequence B we can repeat

2 the method of Example I, namely, set p(u) = $(u) = u , etc.

Example 3. A slight modification of the preceding operator is the

difference-differential operator

2 L ~ f(xl,x2,t ) = Dtf(xl,x2,t ) - D

x I f(x I ,x2+l ,t)

which can also be studied with the aid of Theorem 3. (Namely, we set

~l(Xl) = IX112, ~2(x2) = Ix21 log (l+Ix21) and take for B an arbitrary

quasi-analytic sequence, etc.).

CHAPTER IV

The Fundamental Principle

§i. FORMULATION OF THE THEOREM AND AUXILIARY LEMMAS

In this chapter we shall prove the main result on AU-spaces.

The motivation for the theorem can most easily be seen in the case

n = i. In fact, this case dates back to Leonhard Euler.

Let T be a distribution solution of a homogeneous linear

differential equation with constant coefficients,

dmT dm-iT

+ am_ 1 dtm- 1 + ... + a T = 0 . (i) am dt m - - o

Then T is a C~-function on the real line [46]. Moreover, T is an

exponential polynomial, i.e.

r JZ -I ie£x

(2) T(x) = [ [ c x j e £:i j=0 3

where e£ are the roots of the polynomial

(3) P(z) = am(iZ) m + am_l(iz)m-i + ... + a o ;

JZ is the multiplicity of ~£ ' Jl + "'" + Jr = m; and, the cj's are

constants depending on T.

On the other hand, since we know that the space ~'(~) is

an AU-space (cf. Th. i, II) , T must have a Fourier representation of

the form

(4) T(x) : i eixz d~(z) k(z) ;

(Corollary 2, I and Remark 6, I). If all the roots of P are simple,

relation (2) can be viewed as a Fourier representation of this kind,

but this particular representation has an additional property, namely,

91

the measure ~ occurring in (2) is such that

(5) supp ~ <_ Vp = {z: P(z) = 0} = {a I ..... a m }

Conversely, if (5) holds, then d~(z)/k(z) is a linear combination of

the Dirac measures ~I '~em; hence, representation (4) reduces to

r ia .x T(x) = [ cj e J ,

j=l

and T obviously solves the equation (i). However, if the roots of P

are not simple, the Fourier integrals (4) satisfying condition (5) no

longer furnish all the solutions of the equation. In this case we may

proceed as follows. Since

x j e ixz = i-J d j (eiXZ) ,

dz j

it is natural to introduce at every point eZ (i.e., at every

irreducible component of the algebraic variety Vp) the differential

operators

= i-J d j ------r , o . , (6) ~j,Z dz 3 ; j = 0,i, Ji"

Then we can write equality (2) in the form

r JZ-I (7) T(x) = [ [ :] 2 9 e ixz d~J,'z(z)=

i=l j=0 ,i k(z) '

with supp ~j,Z ~ {z = ~i}. Moreover, it is clear that now formula (7)

yields all the solutions of equation (i).

Next we can ask what can be said in the case n > i; or,

more exactly, whether one still has a similar description of the

"general" solution of a homogeneous linear partial differential

equation with constant coefficients. However natural this question

may appear, the reply has always been that it makes little sense to

look for general solutions when n > i. Rather one always looked for

particular solutions satisfying additional conditions (e.g. boundary

or initial conditions, etc)~ and,in order to determine the appropriate

conditions, it was necessary to classify the PDE's in the well known

fashion. Thus, Euler's approach (cf. (2)) has always been considered

as limited to ordinary differential equations. Nevertheless, it

recently turned out that Euler's method may indeed be generalized,

though in a very sophisticated manner, to partial differential

equations with constant coefficients, and also to systems of such

equations . Moreover, this approach became a powerful tool for

investigating different properties of such PDE's Very roughly,

this is the essence of the theorem discovered in 1960 by

L. Ehrenpreis [18] and called by him the fundamental principle.

The next section is devoted to the proof of this result. For the

sake of simplicity we shall limit ourselves to the case of one partial

differential equation. The main corollary of the fundamental princi-

ple reads as follows:

Let P(z) be a polynomial in z = (z I .... ,z n) ~ ~n; D as in

Chap. I; and W a suitable AU-space of distributions. Then each

T s W satisfying equation

(8) P(D)T = 0 ,

can be represented in the form

(9) T(x) = ~ ! ~ eiXZ d~j(z) j=l 3 k(z) '

]

where the V's are ~-subvarieties (cf. below) of the algebraic ]

variety Vp = {z: P(z) = 0} (Vj's are not necessarily all different),

and the ~.'s are certain differential operators associated with 3

equation (8). (The class of spaces W for which this statement is

proved below is the class of PLAU-spaces, cf. Def. 3, I.) Representa-

tion (9) is obviously the desired generalization of the Euler formula

(2) to the case n > i. However, it is not difficult to briefly

describe the main theorem itself.

93

Let W be an AU-space and T an element of W satisfying (8).

Then

<P(D)T,f> : <T,P(D) % f> = <T,P(z)f(z)> = 0 ,

for all f ~ U. Hence T is a continuous linear functional on the space

U/PU ; and, conversely, every continuous linear functional on U/PU

defines an element T of W satisfying equation (8). If the functions

FI,F 2 s U belong to the same coset modulo PU, their restrictions to

the set Vp must coincide. If F = FIIV P : F2[V p , then the function

is continuous on Vp and satisfies the same growth conditions as the

elements of U. Moreover, we also know that the function F 1 - F 2

vanishes at every point of Vp with (at least) the same order as the

polynomial P. This hints to the possibility of dividing the variety

Vp into a finite number of parts Vj for which there exist differential

operators ~j on Vj such that not only (FI-F2) IV P = 0, but also

(i0) (3j(FI-F 2)) IVj : 0 .

The theorem we are going to prove states that the converse is also

true, i.e. if FI,F 2 s U satisfy (i0) , then F 1 ~ F 2 (mod PU) ; more-

over, given any analytic function F on Vp (i.e. a restriction to Vp

of an analytic function) satisfyingon Vp the same growth conditions

as restrictions to Vp of functions in U, then there exists a function

F in U such that for all j,

3 ]

Let us call the set ~ = (Vj,~j)j a multiplicity variety. A system of

functions {Fj = ~jF}j , where F is an analytic function on ~ , is

called an analytic function on ~ The vector space U(~) is defined

as the space of all analytic functions on ~ bounded on ~ V by j J

functions k, k s ~ , where ~ is the AU-structure of W. These bounds

obviously define a natural l.c. topology on U(~). The above

*FIV P = restriction of F to Vp.

94

mentioned theorem can then be formulated as follows:

Fundamental principle.

Let W be a PLAU-space and P a polynomial. Then there is a

multiplicity variety ~ defined by means of the algebraic variety

Vp = {z: P(z) = 0} such that the mapping

xs a topological isomorphism.

As can be easily seen, this theorem implies the Fourier

representation for all solutions of (8) (cf. Corollary B below).

Remarks: I. The fundamental principle holds for systems of partial

differential equations with constant coefficients (cf. [23],

Chap. IV). However then the definition of the multiplicity

variety as well as the whole proof becomes more complicated,

although everything proceeds along similar lines as in the

case of one equation [23,41].

2. The idea of the proof consists in extending each

function F e 0(% 0 ) locally from V to the surrounding space,

and then, correcting these extensions so that they define

a function in 6 (in particular, special care must be taken

of the bounds). In other words, one must show the vanishing

of a certain cohomology group. Actually, the proof closely

follows the Cartan-Oka-Serre proof of the vanishing of the

cohomology groups Hi(~ n, ~), i > 0, where F is a coherent

analytic sheaf [25]. However, knowledge of the latter proof

will not be presupposed in the sequel.

Now we are ready to start with definitions and some auxiliary

facts which will be needed in the proof.

Definition i. If w is a point of Cn+l, we shall write w = (s,z) where

95

S ~ ~ and z s ~n. An analytic function P(w), defined in an open set

x B, B ~ ~, will be called ~ distinguished polynomial in s of

degree m, if we can write

P(s,z) = s m + Pl(Z)sm+l + ... + Pm(Z) ,

with the coefficients P.(z) analytic in B. 3

Remark. 3. Let us observe that by Definition l, a distinguished

polynomial P(w) i_nn s !9 not necessarily a polynomial in w.

On the other hand, by an appropriate nonsingular change of

variables any polynomial Q({) in ~n+l becomes a nonzero

multiple of a distinguished polynomial. Namely, let

m

Q({) : [ Qj ({) , j=0

where Qj (~) is a homogeneous polynomial of degree j. Let

Qm(a) # 0 for some a s ~n+l; in particular, a ~ 0. Next

we choose n arbitrary points b (j) s ~n+l such that the (n+l)

vectors a I, b (I) ,b (2) ,... ,b (n) are linearly independent.

Then there are points s e ~, z e ~n satisfying the system

of equations n

~i = ais + [ b(J) z j=0 1 ]

The change of coordinates ~ ~ w = (s,z) is clearly non-

singular and the polynomial P(w) defined by P(w) = Q(~) has

the same degree as Q. Moreover, if Pm(W) denotes the

homogeneous part of degree m in P, then Pm(W) = Qm({), and

Pm(S,0) = Qm(aS) = smQm(a)

This shows that

m P(s,z) = s Qm(a) + terms of degree ~ m-1 in s (Qm(a) @ 0),

which proves the above assertion.

The following lemma is known as the Weierstrass division

theorem [25]; however, the vanishing of the coefficients P.(z) at the 3

origin will not be assumed here. Later we shall discuss a local

version of this theorem.

Lemma i. Let F(s,z) be analytic in the open set ~ x B, B ~ ~n, and

let P(s,z) be a distinguished polynomial in s of degree m in ~ × B.

Then

F(s,z) = G(S,z)P(s,z) + R(S,Z)

where G(s,z) is analytic in ¢ × B, and

R(s ,z) = m-i

Z j=0

s j R (z) ]

with the coefficients R analytic in B. 3

def C = C(z) = max {IPj(z) J : j = 1 ..... m}.

Moreover, let

Then

JRj(z) J _< 2(I+C) max {IF(u,z)l: ]cJ _< 2(I+C) };

and, if r = max {]sJ ,i}, then

IG(s,z) j < (r+c) -m max {IF(<~,z) I : J<~J < 2(r+C)}.

Proof: First, let us observe that by the definition of

P(S,Z),

m + ... + Pm(Z)) + P(s,z) ; s =- (Pl(Z)S m-I

hence, we obtain

m+l s = [p2(z) - p2(z) ]sm-i + [Pl(Z)P2(z) - p3(z)]sm-2 + ...

+ Pl(Z)Pm(Z) + P(s,z)[s - PI(Z)]

More generally,

. . . . + P(s,z)Mj(s,z) , s j Qj,l(z)sm-i + + Qj,m(Z)

where

97

Qj+l,k = _Qj,iP m

- Qj,IPm ---

. . ,

] for 1 < k < m-l~

_ _ ~ ;

for k = m J and

Mj+I(S,Z) = sMj(s,z) + Qj,I(Z)

Thus, for j > m and 1 < k < m, we obtain - - i _ _

IQj,k(Z) I _< C(I+C) j-m

where C = C(z) is defined in the lemma. (Let us note that

Qm,k(Z) = - Pk(Z).) Furthermore, by using the recurrence formulae

given above, we see that

IMj(s,z) I _< (r+C) j-m {j >_ m)

where r : max {Is[ ,i} and M m = i.

Now we write

m-1 F(s,z) : Z Fj(z) sj = [ ([ Fj(z)Qq,kj

j=0 k=0 j

co

(z))s k + (j~0Mj(s,z)Fj(z))P(s,z).

The rearrangements of series we just made are admissible because of

absolute convergence which follows from the estimates

IFj(z) I _< 2-Jp -j max {IF(d,z)l: I~I < 2p}.

k Applying these estimates with p = (I+C) to the coefficients of s ,

we get

[Rj(z) I = I[ Fj <z)Qj, k ]

(z) I _< 2{I+C) max {[F{o,z) I : I~! < 2(I+C)}.

Similarly, if p = (r+C) , we obtain

IG(s,z) I = I[ Fj (z)Mj(s,z) [ _< (r+C) -m max {Im(o,z)I I~1 ~ 2(r+C)},

and the lemma is proved.

Remark 4. This proof does not extend immediately to the case when

the function F is defined only on an open set A × B, A ~ 6,

unless better estimates are available for Qj,k and Mj.

If P (0) = 0, this can be achieved by sufficiently shrinking ]

the set A × B.

Lemma 2. If P(s,z) is a distinguished polynomial in s of degree m in

an open set ~ × B, then for every z ~t B and 6 > 0 there exists 6 1 '

6 < 61 < 6 , and a neighborhood N of z such that

1 6 m 4-m -m IP(s,{) I > ~ (re+l) for Isl = 6 1 and 6 e N.

Proof. Given a point z, we divide the annulus 6/2 < Isl < 6

into m + 1 equal annuli. Then there must be at least one of them,

in which P(s,z) has no roots. Let D = {Isl = 6 1 } be the circle pass-

ing through the middle of this annulus. If ~l(Z),...,~m(Z) are the

roots of P(s,z) = 0, we have, for s s D,

6 IS-ei(z) I > 4(m+l)

m Since P(s,z) = I I (s-~i(z)) , we get

i=l

rain IP(s,z) I > [6/4(m+i) ] m D

By the continuity of P, there is a neighborhood N of z such that

min IP(s,~) I h ½ [6/4(m+I) ]m ~cN;ssD

Corollary. (Ehrenpreis [23]-Malgrange [35]). If P(z) is a polynomial

of degree m and F(z) an analytic function in the polydisk

A = {Izj I < 6 : j = 1 ..... m} such that

max IP<z~Ftz~l <M ,

then, for some constant C depending only on P,

with

99

max IF(z) [ < C6 -m

zea I

A 1 = {Izj [ < 6/2: j = 1,2 ..... m}.

Proof. There is a varaible, say z, for which we can write P as

m 1 P(z) = aml z 2 ..... Zn)Z 1 + ... + ao(Z 2 ..... z n) ,

where aml(z2, .... Zn) is a polynomial of degree m-m 1. Let z ° be a

fixed point in A I. Then for any z 2 ..... z n , Izjl < 6, the previous

lemma yields

m m 1

IF(z IO'z 2 ..... z n)aml(z 2 ..... Zn) I ! M 4 ~m+l) 6 -m I

The corollary then follows by induction.

Let us return to the local version of the Weierstrass theorem.

Lemma 3. If P(s,z) is a distinguished polynomial in s of degree m

such that for some 6 1 > 0 and an open set N C ~n,

min IP(s,z) I > £

ssD;zeN (D = {S: Isl = ~i }) ,

then for any analytic function F(s,z) on V = {s: Isl < 61} x N we can

write

F(s,z) = G(s,z)P(s,z) + R(s,z) ,

where G(s,z) is analytic on V, and

m-i R(s,z) = Z s j R (z) ,

j=0 3

with R.(z) analytic in N. Moreover, if 3

M = sup {IF(s,z) I : (s,z) ~ V} < ~

and

then

and

K = sup {IPj(z) ! : z ~ N, j : 1 ..... m} < r

IRj(z) [ _< is MK@I(1+61+...+@I -11

1 1 IG(s,z) [ _< ~- M + ~ MK~I(I+~I +. "'+6m-ll ] 2

Proof: Let 6' < 6 and 1

G(S,Z) - 1 i~ ] F(a,z) do 2~i P(d z) (d-s) la =~'

(Isl < ¢,, z s N)

Then, if 6' is sufficiently close to 61 , G(s,z) does not depend on

6' Hence we may assume that F is analytic for Isl ~ 61, since the

above bounds will then be obtained by taking the limit of bounds

valid for 6' < ~ Let i"

def R(s,z) = F(s,z) - G(s,z)P(s,z) .

Let us show that the function R has the desired properties.

By the definition of G(s,z ,

1 I F(o,z R(s,z) = 2~i o-s

I I=61

i - - do - 2~---i P(s,z)

I F(o,z) d~

i~j=61 P(~,z) (a-s)

l f -- roI=6

However,

F(O,z P(O,z

P(O,z)-P(s,z) m : Pj (z) ~3-sJ ~-s j~0 ~-s

[ P (~,z)-P (s,z)] G-S do .

m-1 m

j:0 i:j+l

m • , ,

P (z) (O3-1+O3-2S+...+s 3-I) j:0 J

Pi (z) oi-j-l) sJ ,

where we have used the equality Pm(Z) ~ 1. This shows that R(s,z) is

101

a polynomial in s of degree < m-1. The estimate for the coefficients

R.(z) also follows. Finally, using the maximum modulus theorem and 3

the analycity of G(s,z) , we obtain the estimate for G,

max iG(s,z) i < 1 max IG(s,z)P(s,z) i Isl_61 --~ [sl:61

1 {IF(s,z) I + IR(s,z)l}. < -- max - Isl=6

1

(Actually, here we should have taken first 81 < 61 and then the

limit for 61 ÷ 61' etc.)

Remark 5. If all ~ zeroes (counting their multiplicities) of

P(s,z) = 0, z s N, lie inside the circle Isl < 61,then the

functions R. (z) and G(s,z) are uniquely determined. Indeed, if 3

I !

F = GP + m : G m + R (Isl < 61),

then R = R(s,z) - R' (s,z) is a polynomial in s of degree at

most m-l; but R has at least m zeros (counting multiplicities)

in Isl < 61 for each z E N, hence R(w) z 0. In fact the same

reasoning shows that there is a uniqueness in the global

decomposition of Lemma I. Moreover, using Lemma 3 we could

have also deduced the global version of Lemma 1 from the

local one by taking 61 so large that all zeros of P(s,z) = 0

would be inside the circle Isl < 61 for a fixed z.

(Nevertheless the above estimates are slightly better.)

However there is another way of computing the coefficients

R. (z) as shown in the next lemma. 3

Lemma 4. Let Q(s) be a polynomial in one variable of degree m.

Let s I ..... s k be all the (distinct) roots of Q with multiplicities

ml,''-,m k, Z mj = m. Then, given arbitrary complex numbers

apq , p = 1 .... ,k, q = 0,1,...,mp_ 1 , there exists a unique polynomial

RlSm-i R(s) = + ... + Rm_IS + R m such that

d q (12) -- R(Sp) = a ds q Pq

j/ 's are linear Moreover, Rj = D D , for j = 1,2 .... ,m, where the Dj

combinations of the numbers a with coefficients which are poly- Pq

nomials in the roots s ; and, D is a polynomial in the coefficients P

of Q. The formulae for D and D depend only on the partition 3

m l,...,m k of m.

Proof: It suffices to observe that (12) is a system of m linear

equations with the coefficients R as unknowns. Let D be the square 3

of the determinant of this system. (D is similar to the Vandermonde

determinant.) Then

(13) D = c 1 I (si-s j) i~j

m,m~ ! 3

where c is the positive number

k m-i 12 c=[r fl i=l j=l

Therefore, D ~ 0, and the system (12) has a unique solution which can

be found by Cramer's rule. Hence Rj + is the quotient Dj//D. The

numerator D. is a linear combination of a with coefficients which 3 Pq

are (m-l)×(m-l) determinants involving powers of the roots Sp. Let

def ~ = Dj{[[. Then Rj = Dj/D. On the other hand, D is a symmetric Dj

function of the roots of the polynomial Q(s) ; and, as such it can be

written as a polynomial in the coefficients of Q. The rest of the

lemma is clear.

Remarks: 6. If P(s,z) is a distinguished polynomial in s of degree m,

then for every fixed z, Lemma 4 gives formulae for D(z) and

D~(Z). The expressions for these quantities change from 3

103

point to point, but it is not difficult to see that there are

only a finite number of different systems of such formulae.

In particular, D(z) = pz(Pl(Z), .... Pm(Z)) , where Pz is a

polynomial. Therefore D(~) = Dz(~) = pz(Pl(~) ..... Pm(~))

is an entire function, and we only have a finite number of

's say D ,...,D . Then, different D z , Zl zt

t (14) A(~) = I I D (~)

j=l zj

is called the discriminant of the function P. Since &(~) is

obviously a polynomial in the Pj's, A(~) must be a polynomial

in ~ whenever P is a polynomial.

7. Let us compute A(~) for some particular cases. If

P(W) = s m - Pm(Z) , then D(z) = Cl(Pm(Z)) m-I when Pm(Z) ~ 0,

m-i and D(z) = c 2 otherwise. Hence A(z) = C(Pm(Z)) for some

constant c ~ 0. In particular, for m = i, &(z) = const. ~ 0.

For P(w) = s 2 + Pl(Z)S + P2(z) we obtain A(z)

= c(P~(z)-4P2(z)), etc.

Next we have to know how the functions D , D change from 3

one point to another. The answer is given by the following lemma

due to A. Ostrowski [40].

Lemma 5. Let P(s,z) be a distinguished polynomial in s of degree m.

For each z, let Sl(Z) ,s2(z) ..... sin(z) denote the roots of P(s,z) = 0

counted according to their multiplicities. Then, for any two points

z' and z", the roots of P(s,z') = 0 and P(s,z") = 0 can be rearranged

so that for all j,

(15) Isj(z') - sj(z") I < 6m2clc21z'-z"I I/m ,

where

(16)

c I : max (I, IPj(z')I I/j J

, IPj(z") l/j)

= max {Igrad Pj (z) ll/m: z = tz + (l-t)z", 0 _< t < i}, c 2 J

Igrad Pj(z)I 2 n : [ 2 k l Pj{z) l

Proof: To begin with, let us first observe that all roots of any

polynomial

Q(s) = s m + alsm-i + ... + a m ,

lie in the disk

A = {s: Isl ~ max Imajll/J}. J

Actually, for s ~ A,

IQ(s) I >_ Isml - lal sm-I + ... + aml > o .

Let us set

(17) ~ = max {ImPj(z') [ I/j, ImPj(z") ll/J: 1 _< j _< m}

and

(is) i/m

j= l

Then clearly Isj(z') I ~ ~ and Isj(z") I j ~. Moreover, if

Pt(S) = (l-t)P(s,z') + tP(s,z") (0 _< t < i)

and Pt(So) = 0 for some fixed t, then again ISol ~ ~.

Let us consider m closed disks of radius B with centers at

sj(z'). Let CI,...,C k be the different connected components of the

union of these disks. In this way all roots s. (z') can be divided 3

into k groups of roots lying in the same set C%. Let 8' > 8 be

105

chosen so close to ~ that

(i) (2m-l)~ ' < 2m8 ; and,

(ii) the disks of radius B' with centers at s (z') define the 3

i T same number k of connected components C1,...,C k.

c C[ Let F be the boundary of C' We may also assume that C I _ l" i i" Then

F. consists of a finite number of circular arcs. We claim that for 1

all t and i,

(19) Pt(S) ~ 0 on F i

Let us assume the contrary, i.e. suppose that for some t and i,

Pt(So) = 0 for some s o s £ i. By the construction of the curves F i ,

for all j we have,

(20) ISo-Sj(Z')l >_ 8 '

On the other hand, by (17) and (19),

m

I I ISo-Sj(Z') I = IP(So,Z') I = IP(So,Z')-Pt(So) ; j = l

= tip (So,Z')-P(So,Z") < tB m •

thus, for at least one j we must have ISo-Sj (z') < tl/mB ~ 8 < 8',

which contradicts condition (20). Hence, by Rouche's theorem, the

polynomials P(s,z') = Po(S) and P(s,z") = Pl(S) have the same number

!

of zeros, say Pi ' inside the region C i. Then, however,

(21) Isi(z')-sj(z") I ~ (2Pi-l)8' < 2m8

' cl for all si(z') , sj (z") s C i (cf. the construction of the regions 1 ) .

Therefore we reorder the roots s (z') and s (z") in such a way that ] ]

the first P1 roots will lie in C 1 , etc.; and, furthermore, so that

(21) will hold with i = j for all indices. The estimate (15) then

follows.

Corollary i. For z s ~n and r > O, let

(22) M(%,r) = 6m 2 max {i, IPj({) ll/J: Iz-~l _< r, j = 1 ..... m}

x max {Igrad Pj (~)Ii/m: ]z-~ _< r, j = 1 ..... m}

( c f . ( 1 6 ) ) , a n d

23) p(z,r) = max {l+ImPj(~)Ii/J: Iz-{I r, j = 1 ..... m}.

Then by Lemma 5, for all z,z" ~ {{: Iz-61 ~ r}, all the roots si(z')

sj(z")) of P(s,z') = 0 (P(s,z") = 0 resp.) lie inside the disk

s I < p(z,r) ; and, the roots can be reordered so that, for all j ,

< M(z,r) I z'-z''I I/m 24) Isj (z')-sj (z") I _

107

§2. PROOF OF THE THEOREM

Definition i. An analytic yariety V in ~P is the set of common zeros

of a finite collection of entire functions FI,...,F r. A set V is

called a Zariski variety (Z-variety), if V is of the form V = V'-V"

where V',V" are analytic varieties. A multiplicity variety ~ is a

finite collection {Vl,dl; ..;Vr,dr}, where the Vj's are ~-varieties

(not necessarily distinct), and the d's are certain differential ]

operators in ~P with constant coefficients, i.e. each d. is a linear 3

combination of ~a/~z~' ~ = (~l'''''~r) . If G is an open (or closed)

subset of ~P, then 9~ n G denotes the collection {V 1 n G, dl; ...;

V 1 n G, dr}. If H is an entire function, then HI~ , i.e. the

restriction of H t__oo ~, is defined as the collection of functions

{Hj . is defined on V by }, where each H 3 3

.H to V. Hj = djHIV j = the restriction of d 3 3

Similarly, for H analytic in G, we can define the restriction of H to

~ G. Conversely, a collection of functions Hj, Hj defined on Vj,

is called an analytic function on 90 , if there exists an entire

function H satisfying HI~ = {Hj}. Similarly one can define analytic

functions on ~ ~ G, etc.

Let W be an AU-space with base U and an AU-structure

= {k}. Furthermore, let ~ be a fixed multiplicity variety. If

H s U , k s ~ are arbitrary, then for some constant c > 0 ,

IH(z)l < c k(z) ; and, obviously, for all j and z,

(25) IdjH(z) I ~ c' max {k(z') : Iz'-zI ~ i} (c'=const.).

By Def. i, I (cf. (iv)), the right-hand side of (25) is also a majorant

in ~ . Hence we can define the space 0(90 ) as the set of all analytic

functions {Hj} on ~ , satisfying for any k ~ ~ r

108

(26) ]Hj (z) I = ~(k(z)) (z g Vj; Vj)

It is clear that condition (26) defines not only the space U(~)) as

a set, but also it defines an l°c. topology on U(%~), under which the

natural mapping

(27)

is continuous.

I: U ÷ U(~)) where I (H) = HIT

The following notation will be used

djH(z) = djH(z) ... for z g Vj

0 ... for z ~ V 3

IIH(z)~ = IIH(z)~ = [ [djH(z) I. J

Furthermore, we shall say that an entire function H is in U(90), if

the analytic function {Hj}, defined on ~ by H, is in U(Y)), etc.

Theorem 1 (The fundamental principle). Let W be a PLAU-space of

dimension n+l with base U and an AU-structure Y<f. Let P(s,z) be a

distinguished polynomial in s of degree m. Assume that

(i) ~_ is a convolutor of W

(ii) For every k g g<~ there exists k' e ~ such that

(28) M(z,l)p(z,l)k' (s,z) < k(s,z) for all s,z

Then there exists a multiplicity variety 90 such that

(I) H g U, HI~ = 0 if and only if H = PG for some G e 6;

and,

(II) for any H ~ U(~O) there exists a function F g U such that

* i.e. the map H ~ PH of U ÷ U is continuous; cf. Def. 2, I. **cf. (22) , (23) .

109

(a) A(z)H(s,z) = F(s,z) + P(s,z)G(s,z)

for some entire function G

(b) AHI~ = FI~ ; and,

(c) the mapping ~ : U(~) + U given by H ~ F is continuous.

Moreover, suppose that, for each z O ~ ~n and b positive,

there exists a constant Q(Zo,b) such that, for each y > 0 and F

analytic in S(w O) = {w: IS-Sol ~ y, IZ-Zol ~ b},

max {IF(w) I : w E 1 S(Wo)} <_ Q(Zo,b ) max {A(z)F(z) : S(Wo)}.

Furthermore, for d > 0, let

e(zo,d) def = max {Q(z,b(z)) : IZ-Zol _< d},

and

d m

Finally we shall assume that

(ii')

Then,

(III)

Remark

for every k ~ ~ there exists a k' e ~ such that

Q(z,d)M(z,2d)p(z,2d)k' (w) < k(w)

the l.c. spaces U/PU and O(~) are isomorphic.

8. The most natural method of proving this theorem seems to be

be the following. First, the multiplicity variety 9~

should be defined in terms of pieces of the analytic

= {w: P(w) = 0}. Then given any H e U(~) variety Vp we

could apply Lemma 1 and Lemma 4 (the latter one is usually

called the Lagrange interpolation formula), and obtain the

function R(s,z) with the desired properties (in particular,

R(w) will be entire and a distinguished polynomial in s).

* A is the discriminant of P, cf. (14).

Moreover, we know how to compute the coefficients R. (s,z) 3

in R in terms of the "values" of H on ~. However, there

is one difficulty in this approach; namely, the coeffici-

ents of R(s,z), for a given (s,z) ~ V, are computed in

terms of the values of H at certain points (s',z) ~ V with

values s' far away from the original value s. Therefore we

are not able to deduce that R(s,z) 8 U. The way out of

this difficulty consists in introducing a convenient cover-

ing of ~n+l and imposing certain (local) restrictions on

the functions defined on the elements of the covering. Next

one has to show that a certain cohomology group vanishes.

The procedure of extending a function H 8 U(~0) to ~n can

be decomposed into several steps. First, we extend H

from 9 o to special rectangles. Then these extensions are

pasted together using the Cartan-Oka-Serre procedure [25]

to obtain a cocycle satisfying good bounds. Finally, one

has to repeat this procedure this time with due care to

bounds at this stage the PLAU-structure enters into the

proof). The result will be a function in U having all the

properties prescribed by the theorem.

Proof: There are several possible choices for the definition of the

multiplicity variety ~. For our purposes the simplest and the most

natural choice will be sufficient. Let us call ~ the multiplicity

variety

'Vl={W:P(w)=0}~{w:~P(w)=0}; operators d: identity;

~P 22 Z V2:{W:P(W)=-~(w)}~{w:~P(w)=O};. operators d- identity, ~--~ ;

(29)

~P ~m-ip(w) 0} ; operators: identity, Vm= {w:P (w)=~(w)=...- Dsm- 1

8 m-I - - ° 0 ,

~S ~'" ssm-i

111

Obviously, U Vj = Vp = {w: P(w) = 0} and PGI~ = 0 for any entire ]

function G. For every z fixed, the decomposition of Lemma 1 is

completely determined by the "values of the function H on the variety

above the point z" In particular, if H is entire and fIl90 = 0,

then H(w) = P(w)Q(w) for some entire Q; and, by Lemma 2, H g U => QsU.

Thus we already know that the mapping

is injective and continuous.

and ~ : H ÷ hH.

(31)

Consider the mapping z: 6 + UIPU

Then we are supposed to prove that in the diagram

U ~ ^ U/PU

0 (~p

the mapping < is continuous, ~ injective and ~o~o< = A. The theorem

does not state that U is surjective (ef. part (II) of the theorem)

unless we suppose more about the discriminant £ in part (II) which

guarantees that £ is invertible.

Let us fix a point w 0 = (s0,z 0) s {n+l and a constant a > 0.

Corollary 1 of Lemma 5 says that if

(32) b < min {a, [a/M(Zo;a) ]m},

then, for each z such that IZ-Zol _< b, the roots Sk(Z) of P(s,z) = 0

lie in the disks IS-Sk(Zo) I < a. Let T(z O) be the union of these

circles and Tk(Z o) the connected component containing Sk(Z O) . Then

either Tk(Z O) = Tj(z o) or Tk(Z o) ~ Tj(z o) = ~. Given any pair of

numbers c,d such that 0 < c < d, we can find y = Y(So,Zo,C,d) such

that

(33) d < y < 2(d + mc + ma)

Moreover, if the disk .IS-Sol < y intersects some Tk(Zo), then

Tk(Z O) _< {s: IS-Sol < y - c}.

If F is any function analytic in {w: IS-Sol < y, IZ-Zol < b}, we claim

that there exists an analytic function F' on 9{)n {w: IZ-Zol < b} such

that

(34) F' IV = FI~ for w E' {w: IS-Sol < y,

F' I~ = 0 for w s {w: IS-Sol > y,

Z-Zol < b}

Z-Zol < b}

TO verify (34), let X(S) £ Co(~) where X ~ 1 for S-Sol ~ y - c and

X ~ 0 for IS-Sol h y. Then XF can be extended as 0 to the rest of

the strip Z(Zo,b) = {w: IZ-Zol < b} and becomes a C~-function in

Z(Zo;b). Then F' is defined by F' = xF + uP, where u is the solution

of

(35) ~U = - ~(XFt P

co

Obviously the right-hand side of (35) is a C -function in Z(Zo,b)and in

theset IS-Sol _< ¥ - c and IS-Sol > ¥. By Th. 4.4.3 of [29] we know

that a solution u to (35) exists; in particular, u will be analytic

in the strip Z(Zo,b) except for the set {w: y-c < IS-Sol < y,Iz-z I <b} -- -- O --

Now it follows that conditions (34) are satisfied. To summarize,

def we have found the following: If S(Wo) = {w: IS-Sol < y,IZ-ZoI<b},

then there exists a function F' analytic in Z(Zo,b) such that

(36) FI~) ~ S(w o) = F'I~? ~ s(w o)

and

(37) F(w) = F' (w) + P(w)G(w)

where G is analytic in S(Wo).

Now we can apply the Lagrange interpolation formula (i.e.

Lemmas 1,4) to the function F' and obtain the function R(w) which is

a distinguished polynomial in s of degree m-l, defined in Z(Zo,b),

such that the coefficients Rj (z) depend linearly on the values F'I~.

113

's depend only on Since F'I~ = 0 outside the rectangle S(Wo) , the R 3

F' I~ ~ S(w ) . In other words, given H ~ U(%0) and constants a,c,d, o

for every w ° = (So,Zo) , there exist numbers b and y, satisfying

(32) and (33) such that in S(w o) we have

(38) H(W) = R w (w) + P(w)Q w (w) o o

and

(39) max IA(z)R w (w) I < CP(Zo,b) Z max IIH(w)ll wES (w o) o w~S (w ° )

where £ and C are positive numbers independent of w ° and b.

Furthermore, we claim that there are functions F w , G o Wo

d I < d in {w: IS-Sol < ~ , lw-w O 5} such that

analytic

(40) A(z)H(w) = F w (w) + P(w)G w (w) o o

and

(41) IF w (w) I < CM(Zo,2d)P(Zo,2d) o

£ max {[IH(w')ll : w' e S' (d;Wo)},

where S' (d,Wo) = {w: IS-Sol < 2(m+l)d+m, IZ-Zol < d}. (Here we

used (33) with d = a = c = 1/2). The proof of the existence of F w o

and G w follows along the same lines -namely the Cartan-Oka-Serre o

method - as the proof of (43,p) => (44,p) given below. Actually,

here this procedure becomes even simpler than in the latter case,

because it is finite; and, thus the convergence factors #,~ which

appear in the proof of (43,p) => (44,p) are not necessary. Therefore,

the proof of the existence of the functions F w and G w can be o o

omitted.

Let us write w = (Xl,X 2 ..... X2n_2) , zj = x2j_l + ix2j for

j = 1,2 ..... n and s = X2n+l + iX2n+2. For p = 1,2, .... 2n+3, e ~ ~2n+2

and 6 > 0, we set

Bp(e,6) = {w e ~n+l: ixj_~j I < 6 for j > p}.

Since the first (p-l) coordinates of ~ do not appear in the definition

of ~, they can be taken to be zero. BI(~,6) is just the "cube" of

center e and side 26, and B2n+3 = Cn+l for all e,6.

Let us choose d rather large, say d = 100n, and let L = {d}

be the set of all lattice points in ~2n+2. Then there is a system

{F } of analytic functions associated with the system of rectangles

d {B (~, ~)} as described above (cf. (40),(41)). The system {F }

eeL is a cocycle in the sense that, for any pair ~,~' g L, there is a

d function G ,~, analytic in BI(~, ~) m Bl(e', ~) such that F -F ,

= PG ,~,. Moreover the functions F satisfy good estimates.

The idea of the following proof is to extend the functions ~ to

sets B as p increases (and, of course, going from p to p+l, i.e. P

extending by one real variable at a time). Finally we will end up

= ~n+l with one function F satisfying good estimates in B2n+3

i.e. F g U.

Two lattice points e,e' will be called p-semiadjacents, if

I

(42) ~ = e. = 0 for j=l ..... p-l; ] 3

and, ei = ej for j=p+l ..... 2n+2.

!

If ep < ep , we shall write e < e'. The points ~,~' will be called 1

p-adjacent, if, in addition to (42), lap-apl = i. Moreover, the

f o l l o w i n g s p a c e s w i l l b e n e e d e d : F o r 6 > 1 , we d e f i n e

~p(6) = {F = {F }: F~ analytic in Bp(~,~)};

F~(w) Up(~) = {F = {F } g (5~p(6) : lim ~ = 0 for all k s ~ } ;

lwl ÷ ~ k(w)

Up(@ ,P) = {F={F }gUp(~ ) : m -m ,= PG ,~, on Bp(~,6)~ Bp(~' ,~ )

with G ,~, analytic for all ~,~' s L};

Up^*(6,P) = {F={F }gUp (~) : F -F ,= PG~,e, on Bp(~,~)~ Bp(~',~)

with G ,e, analytic for all

p-semiadjacent e,~' in L}.

115

All four spaces are equipped with the corresponding natural topologies.

For a small positive ~, one can define the natural maps

Ip: Up+i(6 ,P)/(PUp+I(6-s) ~ Op+l(6) P (6 ,P)/(PUp(6-c) ~ Up(6))

which are clearly continuous. Our aim is to prove:

For ~ > 0 small and F ~ U (6) ~ P O (6), there exists a P P

(43,p) G g Up(6-e) such that F = PG, and the mapping F~G is

continuous.

(44,p)

For 6 large there exist an s > 0 and a mapping

A, (PUp (6 -E) Up: Up(6 ,P)/ m Up(6 ))

+ 0p+l(6-s,P)/(PUp+l(6-2e) ~ Up+l(6-s))

such that Up is continuous and lp Up = identity.

(45,p) l is injective . P

The proof by induction will be shown as follows:

(a) (43,p) => (44,p)

(b) (45,p) holds for all p

(c) (43,p) => (43,p+i)

We know that (43,1) holds (cf. Lemma 2) . Then (a) , (b) , (c)

will imply that the mapping

: U2n+3(@ "P)/(PU2m+3(6-s) m U2n+3(6)) = U/PU

÷ UI(6 ,P)/(PUI(6-g) ~ UI(6)) ,

obtained by composing the mappings kp , Op , is an isomorphism (in

this procedure the numbers 6 and g are modified a finite number of

times). In particular we shall obtain functions F c U and G such that,

for all ],

(46)

i.e.,

(47)

116

d ' . d ° 3 (F(w)) = j (A(z)H(w)) ,

A(z)H(w) = F(w) + P(w)G(w)

and this will complete the proof of assertion (III) of the theorem.

^*

Proof of (a) : Let H e U (6 , P) . For a,B , P

we set

which are p adjacent,

F a = H B - H e

Then {F } s Up(~-l) ~ P %(6-1); and, by (43,p), F e = PNe ,

{N a} e Up(~-l-s), and {F } ~ {N } is continuous.

First let us assume that p is odd; then for p' = (p+l)/2

we consider in the plane of this p'-coordinate a square with center

ap+i~p+ 1 , side 2(6-i-e) and boundary F(a). Let F+(a) be the part of

F(a) in the halfspace {w: Re w < a } and F-(e) the other part. If -- p

#(e,Zp,) (or ~(e,s) when p' = n+l) is an entire function without

zeros inside this square or on F(a), then the Cauchy formula yields

~(~,zp,) r N (z I ..... t,zp,+l .... ,s) N a (w) = ] dt

2zi F(~) (t-Zp,)~ (a,t)

F + F-

' +is and side 2(d-2-e) If Zp is in the closed square with center ap p+l

and t e F(e), then It-Zp, I ~ i. Then for fixed ~, and ~' p-semi-

adjacent to ~, ~' > a, we get

max

[N~, (w)[ _< 86 I~(~' ,Zp) [ tcF(~')min

t~F(~')

I N ~ , ( . . . ) I

I~ (~' , t ) [ ..........

For any Zp, s 6, there are finitely many lattice points e' (bounded !

independently of Zp) such that Zp, is in the interior of F(e'). Let

117

us define

m(w) = max max I (l+ItI2)N (Zl, .,Zp,_l,t, 06' tsF(06') 06' "" ZP'+l ..... ~n'S) I'

where the max is taken only over those points ~' for which the corres- ~v

ponding is in the interior of F(~') . Since {N06}is in Up(6-l-s) , zp!

m(w)/k(w) ÷ 0 for all k 6 ]<~ . Hence we can find an m in the BAU-

structure J t of W such that m < m. By Def. 3, I we can assume in the

sequel that n = i. By condition (viii) of the same definition, for

each m E ~ , there exists m* E ~C such that for every e s L, there

is an entire function %(06,zp,) for which

m(ap+iap+l) ..... I¢(06'Zp ') I , < m (Zp,)

min l¢(~,t) I tEF (~)

in particular,

l IN~, (w) I _< 86m*(w)(l+lepI2) -1

for e' ~ ~ and Zp, in the square with ~+i~p+ 1 and side 2(6-2-s).

Similar estimates hold for Nt,, (w),a" < ~. Using the indicated

estimates we obtain the uniform convergence of the following series,

06~>06

oo

J[O 1 IP(w) I [N~, (w)I _< 86 P(w) Im*(w) "= l+j2

Z 06" <06

IP(w) llN[,,(w)l _< 86 F(w>Im*(w) oo

2 j=O l+j

for ]Xp-~pl _< ~-2-E, Ixp+l-~p+ll < 6-2-~. (The convergence estimates

are independent of ~.) Hence we can define an analytic function H 06

in B (06,~-2-s) by the formula P

06(w) -- H a(w) + sN , (w) - f sN[,,(w) ~' >~ C~" <~

It follows that {H06) s Up(d-2-s) and H -H s PUp(@-2-s). Moreover,

if B is p-adjacent to 06, say 06 < 8, then in the intersection of their

118

r e s p e c t i v e d o m a i n s we o b t a i n f r o m t h e d e f i n i t i o n o f N

~">~ ~' ~'>8 ~"< ~ 8"< 8

= H a- H~ + PN[ + pN+~ : H a- H~ + PN -- 0

Therefore H does not depend on the p-th coordinate of ~ and

{H a} s Up+l(d-2-s).

For p even, the proof is the same except that in the defini-

tion of N +~ ,N~ one has to use strips parallel to the real axis and

then apply the second part of condition (viii) of Def. 3, I.

A ^

Proof of (b) : Suppose that H ~ Up+l(6 ,P) and >,pH S PUp(d-s), then

there is N s U (d-E) such that I H = PN. if 8 is p-adjacent to ~, P P

then in B (a,d-s) ~ B (~ ,d-s), P P

P(Ns-N ) = IpH~ - IpH~ = Ip(HB-H ~) = 0

But H s 0 p+l(d ,P) implies NS- N

shows that I is injective. P

: 0. Therefore N s p+l(d-s) which

Proof of (c) : The same proof as in (a) .

To conclude the proof of the theorem, we have to prove

part (III) of the theorem, i.e., to show how to remove the

discriminant £(z) from (47). It suffices to see that under our

hypotheses, inequality (39) implies

(48) max INo(W) I < Q(Zo,b) max [p(z)ZIIH(w)II] , 2w~S (w o) w~S (w o)

where S(w o) and Q(Zo,b) are defined in part (III) of the theorem.

Then (40) and (41) become

(40') H(w) = F w (w) + P(w)G w (w) , o o

(41') IF w (w) I < CM(Zo;2d) P(Zo,2d)~ Q(Zo,d)max{HH(w')Ii : w' s s' (d,Wo) ] o

119

and the rest of the proof is the same. Hence the theorem is

completely proved.

Corollarz A. Assume that W, P and A satisfy conditions (i) and (ii)

of the theorem. Let Cp and % be the convolutors corresponding

to the multipliers P and A, respectively (cf. Def. 2, I). Then

every solution f in W of the equation ~p(f) = 0 can be written as

j (49) CA(f ) (y) = ei<y,w> dr(w) (y ~ ~n+l) k(w)

where d~ = (dr I .... ,d~ n) are Radon measures with supp vj ~ Vj

(cf. (29)) , and k is a majorant in Y~. In other words, for each

H ~ ~ (or H ~ ~(~)), /

(5o)

Proof:

f d~. (w) <H, CA(f)> = djH(w) .... J ..... j:l v k(w)

3

As we have seen above, for H s U, we have

DH = R + PG ;

and H ÷ 0 in U(%{]) implies R ÷ 0 in U. / Then, however,

< H, @A(f)> =< AH,f> = <f,R> ÷ 0

and the Hahn-Banach theorem yields the desired representation

(cf. §I, I).

Similarly we obtain,

Corollary B. Under the same hypotheses as in the foregoing corollary,

but with (ii) replaced by (ii') , we can write every solution f s W

of %(f) = 0 as

÷

r i<y,w> d~(w) (51) f(y) : ] e k(w)

In conclusion let us mention two examples of functions P to

which the fundamental principle applies:

(A) P-polynomial Then A is a polynomial of degree h. By Lemma 2,

Q(z,b) < C/b h+l

Therefore the quantities M(z,d) , Q(z,d) and p(z,d) are bounded by

Cd(l+Izl) N for some positive c d and N, and hypothesis (ii') follows.

In particular, this solves the problem mentioned in the beginning

of this chapter.

,'s are exponential polynomials Let us recall that (B) The P]

F(z) , z £ ~n, is an exponential polynomial, if F can be written as

F(z) = m <d~_, z>

ak(z)e #%

k=l

where ak(z) are polynomials and ~k are complex numbers called the

frequencies of F. If the P.'s in Def. 1 are exponential polynomials, ]

then A is also an exponential polynomial. Moreover, if all the

frequencies of the P. 's are real (or pure imaginary), the same holds ]

for A.

The following estimate, generalizing the corollary to Lemma 2,

can be derived for any exponential polynomial F. Let us set

(52) hF(Z) : max Re <~k,Z> . k

Then there exists a polynomial A(t) with positive coefficients depend-

ing only on the exponential polynomial F such that, for arbitrary

cn Zo6 ,g>0 and g entire,

hF(Z O ) 53) e Ig(Zo) I ! A(~) max IF(z)g(z) I

rZ-ZoI± From here and the definitions of the expressions M, Q and p,

it follows that all these expressions can be estimated by

121

const. (l+Izl) N eh(z) ,

where h(z) = max Re< Bk,Z >, and Bk'S are complex vectors depending on k

P. Let us observe that if all the frequencies ~k are real (or pure

imaginary), then the same holds for the vectors 8k" For instance, in

!

the case W = ~ we have to take all &k'S pure imaginary (otherwise P

would not be a multiplier in the corresponding U). In this case all

hypotheses of the fundamental principle are satisfied (cf. Theorem 2,

II).

Before concluding these notes we should mention some applica-

tions of the topics treated in this volume. However the applications

are manifold and too extensive to be covered in this short monograph.

They pertain not only to partial differential equations but also to

lacunary series, quasi-analyticity, etc. We refer the reader to

Chapters VI-XIII of [23] where several applications are discussed,

and many open problem suggested.

BIBLIOGRAPHICAL REMARKS AND OTHER COMMENTS

Chapter I

1 "this was already known to S. Bochner in 1927 (cf. [12], Chap. VI).

2 For a thorough discussion of the role distribution played in the

recent development of PDE's, the reader is referred to the beautiful

monograph of F. TrOves [52].

3 One solution of this problem has been proposed by W. SIowikowski [47].

However his conditions are not formulated in terms of standard

functional analysis. Another approach to this problem was worked out

by V. Pt~k (cf. [44,45] and the references in these papers). He

formulates the concept of a semiorthogonal subspace R of F. This

approach uses a standard framework. In certain classes of (~')-

spaces, Pt~k's conditions are necessary (and sufficient). For the

general case, the necessity of these conditions has yet to be proved,

although it is very likely that this is the case. In general, one

can say that the purpose of these works is to find an abstract

formulation (in terms of topological vector spaces) of H@rmander's

notion of strong P-convexity [26].

4 In several concrete spaces the necessary and sufficient conditions

for F to be slowly decreasing are known. Thus, e.g. Ehrenpreis

[20,21] found such conditions for the space ~ . In this case, F is

slowly decreasing if and only if there are positive numbers a, b and

c such that, for all z ~ cn,

max{IF(z')I:]z-z'l~ a(log(l+Izl)+IIm zl)} ~ b(l+Iz[)-Cexp(-clIm zl).

Let us observe that here the maximum occurs instead of the minimum

as in (7). To go from max to min one has to use the minimum modulus

theorem [33]. Similar conditions for the Gevrey classes were given by

Ch.-Ch. Chou [14].

5 Theorem 1 and its corollaries are taken from our paper [5] (for the

123

proof of Theorem I, cf. also [16]).

6 The proof is based upon an idea from [22].

CHAPTER II

1 The spaces considered in this chapter were introduced by Arne Beurl-

ing in 1961 [8]. A systematic study of Beurling spaces was later

published by G. Bj~rck [9] who, in following the program of

H~rmander's monograph [27], put the main emphasis on applications

to partial differential equations. A theorem on regularity of

solutions to elliptic partial differential equations was proved for

Beurling spaces by O. John [31] (cf. also the article of E. Magenes

[34]). Other problems concerning Beurling spaces are studied in

our papers [6,15] and in a recent paper by G. Bj~rck [I0].

2 Propositions 1 and 2 are taken from our papers [5,6,17].

Part of the proof of Proposition 1 is based on the same idea as

a theorem of B. Malgrange [36].

3 Proposition 3 and Theorem 2 appear here for the first time. It is

not without interest to observe that a different construction given

by L. Ehrenpreis yields Theorem 2 for the case (~n) ([23]

Chap. V). However his proof is different and does not seem to

generalize to Beurling spaces.

4 The proof of this fact proceeds similarly as in the classical case

of ~ = ~(~n) [46] and it is left for the interested reader.

5 An interesting characterization in terms of approximation of

Beurling test functions (and, more generally, of elements in ~)

was found by G. Bj~rck [i0]°

6 Proposition 4 and Theorem 3 are taken from our paper [6]. The proof

of Propositio~ 4 follows the proof of Theorem 5.15, [23].

t24

CHAPTER I I I

1 A c t u a l l y , i t i s n o t n e c e s s a r y t o a s s u m e t h a t t h e d i s t r i b u t i o n s L. J

h a v e c o m p a c t s u p p o r t [ 2 , 3 ] . T h e r e f o r e t h e t h e o r e m s o f 52 c a n be

g e n e r a l i z e d t o t h o s e L s a t i s f y i n g c o n d i t i o n s ( 4 2 ) .

2 T h i s t h e o r e m was o r i g i n a l l y p r o v e d by E h r e n p r e i s [23] f o r t h e c a s e

o f d i f f e r e n t i a l o p e r a t o r s . As was o b s e r v e d i n [ 2 ] , E h r e n p r e i s ' s

p r o o f c o u l d be e x t e n d e d t o c o n v o l u t o r s ( c f . T h e o r e m 1) b y u s i n g t h e

g e n e r a l i z e d f o r m o f t h e f u n d a m e n t a l p r i n c i p l e ( T h e o r e m 1 , I V ) .

The p r o o f g i v e n b e l o w c l o s e l y f o l l o w s a d i f f e r e n t a p p r o a c h due t o

B. A. T a y l o r [ 5 0 ] . B o t h m e t h o d s a r e b a s e d on t h e i d e a o f e x t e n d i n g

c e r t a i n f u n c t i o n s i n n v a r i a b l e s t o f u n c t i o n s i n n + l v a r i a b l e s . T h i s

can be d o n e by c o n s i d e r i n g t h e f u n c t i o n s i n NB(L;~ ) as d i f f e r e n t

C a u c h y d a t a o f a d i f f e r e n t i a l e q u a t i o n w h i c h i n E h r e n p r e i s ' s p r o o f

i s t h e h e a t e q u a t i o n . The p r o o f g i v e n h e r e c a n be i n t e r p r e t e d a s

t h e s t u d y o f t h e e q u a t i o n

- ~ f , ( x , t ) = L * ~ ( x , t )

i n t h e s p a c e E ( B ; ~ ) o f a l l f u n c t i o n s ~ ( x , t ) , x ¢ ~ n , t ¢ ~ , s u c h

t h a t ~ s a t i s f i e s t h e g r o w t h c o n d i t i o n s on x , a n d , a s a f u n c t i o n o f

t , ~ b e l o n g s t o t h e s p a c e S B. I t i s c l e a r t h a t f o r a n y s u c h

w h i c h s a t i s f i e s t h e a b o v e e q u a t i o n , ~ ( x , 0 ) e ~ B ( L ; ~ ) .

3 To p r o v e t h a t ~ B ( L ; ~ ) i s an A U - s p a c e ( i . e . n o t o n l y t h a t i t i s a

weak A U - s p a c e ) one s h o u l d i m p o s e a d d i t i o n a l r e s t r i c t i o n s on C. F o r

i n s t a n c e , when L i s a d i f f e r e n t i a l o p e r a t o r , ~ B ( L ; ~ ) i s o b v i o u s l y

an A U - s p a c e .

4 The c o n t e n t o f t h i s s e c t i o n s i m u l t a n e o u s l y g e n e r a l i z e s t h e u n i q u e -

n e s s t h e o r e m f o r t h e h e a t e q u a t i o n and t h e D e n j o y - C a r l e m a n t h e o r e m .

F o r t h e L a p l a c e o p e r a t o r t h i s r e s u l t c a n be t r a c e d t o S . B o c h n e r

[13] and f o r d i f f e r e n t i a l o p e r a t o r s w i t h c o n s t a n t c o e f f i c i e n t s t o

E h r e n p r e i s [ 2 3 ] . In t h i s s e c t i o n we f o l l o w [ 3 ] .

125

CHAPTER IV

As mentioned above the fundamental principle was stated first by

L. Ehrenpreis in 1960 [18]. The proof in its full generality (i.e.

for systems of linear PDE's with constant coefficients) was publish-

ed in his monograph [23] in 1970. In the meantime V. I. Palamodov

published his version of the proof in [41]. Both proofs follow

essentially the same pattern (i.e. locally extending functions from

varieties; use of the Lagrange interpolation formula; proof of the

vanishing of a certain cohomology group, etc.). Palamodov's proof

systematically uses homological methods and the H@rmander estimates

of the ~-operator [29]. A weaker version of the theorem was proved

by B. Malgrange [37]. Theorem 1 of this chapter generalizes the

fundamental principle for one equation to the case Of distinguished

polynomials (cf. Remark 3, IV). Its proof is taken from [2] and

follows the method of Ehrenpreis [23]. Although the case of

distinguished polynomials would not seem to be very different from

the case of arbitrary polynomials, it is interesting to observe that

certain "unexpected" factors appear (cf. the discriminant & in

Theorem i). Intuitively, the assumption on & says that the roots

of P do not coalesce very abruptly. It seems that in order to

generalize Theorem 1 further, one will have to impose a similar

restriction on the geometric nature of the variety Vp= {z: P(z)= 0}.

Another way of generalizing Theorem 1 is to study the case of

systems. Here the problems are of an algebraic nature, and for the

case of polynomials have been solved (cf. [Z3]). The relation of

the above mentioned theorem of Malgrange to Theorem 1 can be better

understood if we look at the problem from the point of view of real

variables (i.e. the theory of distributions). First we should prove

that P(D)T = 0 implies supp T c V (here T is taken from the space - p

which is defined as the dual of U); second, we should establish

representation (9), IV. In the case of one variable, the geometric

nature of the variety V is so simple that the second step follows P

immediately from the first one. However, it is well known (cf. [46],

Chap. IIl, §9, §I0) that the inclusion supp T ~ A does not imply

that T is a combination of derivatives of measures with supports in

A. Therefore the first step does not immediately imply representa-

tion (9). Malgrange proves essentially the first step. Ehrenpreis's

method can be viewed as a way of establishing sufficient conditions

for certain varieties to be regular in Whitney's sense [46]. Hence

it is not unexpected that for functions P which are not polynomials,

one has to impose additional restrictions as in Theorem I, II.

2 To show that D has the form D = c~ ..., where c is the constant

defined in Lemma 4, is actually quite tedious. Let us introduce m 1

different variables al,...,oml, then m 2 different variables

Tl,...,Zm2 , etc., and consider the Vandermonde determinant of

order m

V =

m - t m-2 1 a 1 01 • . . 01

m-I 0~-2 0 2 1 0 2 ...

• ° • • ° •

Then the value of V = (~2-oi) (o3-01)...(Omi-Oi)..., does not change

if we subtract the first row from the following ml-I rows containing

o's; then the row of ~I from the following rows containing T's, etc.

L e t u s d i v i d e V by ( 0 2 - a l ) ( a 3 - ~ l ) . . . ( ~ m l - a l ) ( T 2 - T 1 ) ( T 3 - T 1) . . . . T h i s

i s e q u i v a l e n t t o t h e d i v i s i o n o f t h e row o f o 2 by o2-01, e t c . Now

we s u b t r a c t t h e new row o f 02 f r o m t h e new r o w s o f o ' s t h a t f o l l o w ,

a n d d i v i d e b y ( 0 3 - 0 2 ) ( 0 4 - 0 2 ) . . . . I n t h e e n d t h e r e s u l t i n g d e t e r -

m i n a n t w i l l n o t c o n t a i n a n y t e r m o f t h e f o r m o j - o i , T j - T i , . . . ;

a n d , t h e e n t r i e s w i l l b e c e r t a i n d i v i d e d d i f f e r e n c e s • Now we c a n

t a k e o j ÷ s 1 , T k ÷ s 2 , . . . , a n d t h e l i m i t w i l l b e

mlm 2 m3m 1 (Sz-S I) (s3-s I) ....

127

Moreover, the entries in the resulting determinant are the desired

quantities divided by the corresponding factorials (for example, the

original row of ~3 appears divided by 2!, etc.). The value of D is

then the square of the previous determinant.

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