Richard D Carmichael Wake Forest University
Dragisa Mitrovic University of Zagreb
Distributions and analytic function...

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Richard D Carmichael Wake Forest University

Dragisa Mitrovic University of Zagreb

Distributions and analytic functions

m Longman

N NW

W Scientific &

- Technical

Copublished in the United States with John Wiley & Sons, Inc, New York

Longman Scientific & Technical, Longman Group UK Limited, Longman House, Burnt Mill, Harlow Essex CM20 2JE, England and Associated Companies throughout the world. Copublished in the United States with John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158

© Longman Group UK Limited 1989 All rights reserved; no part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without either the prior written permission of the Publishers or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, 33-34 Alfred Place, London, WC1E 7DP. First published 1989

AMS Subject Classification: (Main) 46F20, 46F12, 46F10 (Subsidiary) 32A10, 32A07, 32A40 ISSN 0269-3674

British Library Cataloguing in Publication Data Carmichael, Richard D. Distributions and analytic functions. 1. Calculus. Bounded analytic functions 1. Title

H. Mitrovic, Dragia

515'.223

ISBN 0-582-01856-0 Library of Congress Cataloging-in-Publication Data Carmichael, Richard D. Distributions and analytic functions/Richard D. Carmichael, Dragisa Mitrovic. p. cm.- (Pitman research notes in mathematics series, 0269-3674 206) Includes bibliographical references and index. ISBN 0-470-21398-5

1. Distributions, Theory of (Functional analysis) 2. Analytic functions. 1. Mitrovic, Dragi§a, 1922- . II. Title. III. Series. QA324.C37

515,7'82-dcl9

1989

88-34332

CIP

Printed and bound in Great Britain by Biddies Ltd, Guildford and King's Lynn

Contents

Preface

Dedication Chapter 1

1

1.1

Spaces and properties of distributions Introduction and preliminaries

1

1.2

The spaces D and D'

5

1.3

The spaces E and E'

15

1.4

The spaces S and S'

18

1.5

The spaces 0a and Oa

23

1.6

The spaces DLp and D'jjp

27

1.7

29

1.8

Convolution of distributions The Fourier transform

1.9

The spaces Z and Z'

39

Chapter 2

Distributional boundary values of analytic functions in one dimension

35

42

2.2

Introduction Distributional analytic continuation

46

2.3

Analytic representation of distributions

54

2.1

42

in E' ([R) 2.4

Analytic representation of distributions

72

in O' a(IR ) 2.5

Distributional Plemelj relations and boundary value theorems

83

2.6

Representation of half plane analytic and meromorphic functions

89

2.7

Equivalence of convergence in DI(M) and 0a- (IR)

103

2.8

Comments on Chapter 2

106

Chapter 3 3.1

3.2

Applications of distributional boundary values

Introduction Applications to boundary value problems

111

111

116

3.3

Applications to singular convolution equations

135

3.4

Comments on Chapter 3

146

Chapter 4

Analytic functions in Cn kernel functions

,

cones, and

150 150

4.3

Introduction Analytic functions of several complex variables Cones in Rn and tubes in en

4.4

Cauchy and Poisson kernel functions

157

4.5

Hp functions in tubes

167

4.6

Growth of Hp functions in tubes

174

4.7

Fourier-Laplace transform of distributions and boundary values

179

4.8

Comments on Chapter 4

201

4.1

4.2

Chapter 5

151

153

Distributional boundary values of analytic functions in n dimensions

205

Introduction Analytic representation of distributions in E': the scalar valued case

205

5.3

Analytic representation of vector valued distributions of compact support

222

5.4

Analytic representation of distributions in 0'

230

5.5

Analytic representation of distributions in D'p

232

5.1 5.2

206

LP 5.6

Comments on Chapter 5

Chapter 6

The Cauchy integral of tempered distributions and applications in n dimensions

258

260

6.1

Introduction

260

6.2

The Cauchy integral of tempered distributions:

262

the case corresponding to the quadrants in Rn 6.3

Cauchy integral representation of the analytic functions which have S' boundary values

280

6.4

The Cauchy integral of tempered distributions: the case corresponding to arbitrary regular

295

cones in Rn

6.5

6.6

6.7

Analytic functions which have S' boundary values and which are Hp functions Fourier-Laplace integral representation of Hp functions Comments on Chapter 6

301

321

327

References

333

Index

345

Preface

Analysis concerning the representation of distributions in the sense of L. Schwartz as boundary values of analytic functions in one and several variables is presented in this book. The analysis is based on the research of the authors, and the basic material presented here associated with the topic of study is not contained in any other book. Previous research books concerned with the topic under consideration here have been written by Bremermann [11], Beltrami and Wohlers [2], Roos [114], and Vladimirov ([135], [136].) The present book is a companion work to these; much of the analysis in this book has been developed since the publication of these companion works.

Research in the area

considered here finds applications in quantum field theory, partial differential equations, and convolution equations in addition to other areas; and research in this area continues with developments now occuring as well in the representation of ultradistributions in the sense of Beurling and Roumieu as boundary values of analytic functions. Basic for the study of this book are a knowledge of the rudiments of the distributions of L. Schwartz and of basic complex variable in one dimension.

For the convenience of the

reader, a review of the test spaces and distributions to be used in this book is given in Chapter 1 along with a brief discussion of the convolution and Fourier transform of distributions. The concept of analytic function of several complex variables is defined at the beginning of Chapter 4, and some basic facts of these functions have been collected No previous working experience with several complex there. variables is assumed on the part of the reader or is needed to study this book.

A review by the reader of the basic facts

concerning the Fourier transform for LP functions, 1 < p <

2,

would be helpful for the reading of some parts of Chapters 4-6.

Distributional boundary values of analytic functions in one dimension in the topologies of D'(IR) and O'(IR) will be studied Analyticity and growth properties and boundary value results of the Cauchy integral of elements in E'(IR) and in Chapter 2.

O'(6t) will be obtained; conversely, analytic and meromorphic

functions in half planes with appropriate conditions are studied with respect to distributional boundary values in the In some cases recovery of the D'(IR) and 0;(It) topologies. analytic or meromorphic functions in terms of the Cauchy integral of the boundary value is obtained.

Of particular

interest throughout the development of Chapters 2 and 3 is the construction and application of the distributional Plemelj relations concerning the distributional boundary values involving both E'(IR) and Oa(IR) distributions.

These relations

are the natural extension to distributions of the boundary value relations of half plane analytic functions introduced by the Yugoslav mathematician J. Plemelj in the early twentieth century.

The distributional Plemelj relations are used

systematically throughout Chapters 2 and 3 and in particular in generalizations of results of Bremermann and Beltrami and Wohlers. The generalization of the Plemelj relations to

distributions was given independently at approximately the same time (1966-1967) by Mitrovic and by Beltrami and Wohlers but by using different techniques.

Several new results are published and proved for the first time in Chapter 2 including the whole of section 2.7.

Revised proofs from those given in the original papers of some of the other results are given in Chapter 2.

Chapter 3 contains applications of the distributional boundary value results of Chapter 2.

In particular

generalizations of boundary value problems of Plemelj, Hilbert, Riemann-Hilbert, and Dirichlet to the setting of

Further, singular convolution equations in the distributional setting are stated distributions are stated and solved.

and solved using the distributional boundary value results and the Plemelj relations. The analysis in Chapters 4-6 is obtained in n dimensions.

After defining the concept of analytic function of several complex variables in Chapter 4 we then proceed to introduce

topics in the remainder of the chapter that are needed in 1n and tubes in Cn are defined and Cones in Chapters 5 and 6. studied. The Cauchy and Poisson kernel functions corresponding to tubes in Cn are shown to be in relevant test spaces.

Hp functions in tubes are recalled, and a pointwise

growth estimate for these functions is obtained.

The

Fourier-Laplace transform of distributions is related to analytic functions which satisfy various growth conditions; in some cases the analytic functions can be recovered by the Fourier-Laplace transform of the inverse Fourier transform of the boundary value.

Some of the results in Chapter 5 extend corresponding results in Chapter 2 to n dimensions. The Cauchy and Poisson integrals of distributions in E' = E'(Dn) and Oa = 0a(fn) ,

,

DLp = D'

(,n) are studied in Chapter 5 and are shown to have

distributional boundary value properties.

The Cauchy

integrals are analytic functions in tubes in Cn and satisfy Certain analytic functions are related to distributional boundary values in these spaces. Except for growth conditions.

section 5.3, the entire book concerns scalar valued distributions in the sense as originally defined by L.

Schwartz; in section 5.3 we present some boundary value results of vector valued distributions of compact support. H. G. Tillmann was an original investigator of the representation of distributions as boundary values of analytic functions.

In Chapter 6 we define a Cauchy integral of

tempered distributions which we use to recover the analytic functions in tubes defined by quadrants that Tillmann showed obtained S' = S'(IRn) boundary values; this analysis builds

upon that of Tillmann.

In addition, a Cauchy integral of S'

distributions that was defined by Vladimirov corresponding to arbitrary tubes in Cn is studied. The pointwise growth of Hp functions in tubes, which is obtained in Chapter 4, is a special case of the growth that characterizes boundary values in S'; necessary and sufficient conditions are given in Chapter 6 for an analytic function in a tube which has an S' Fourier-Laplace integral representations of Hp functions in tubes, 0 < p are boundary value to be an Hp function.

given at the conclusion of Chapter 6.

Examples of the topics introduced are given throughout the book.

Each chapter concludes with a section containing comments. Frequently other conclusions that can be made on the topics of the chapter are discussed, and the relations between the results presented with those of similar investigations are indicated.

Always we have attempted to give a historical

perspective of the analysis presented in this book with the development of the subject matter in general although we do not claim completeness in this effort.

The authors take great pleasure in acknowledging the contributions of three individuals in the realization of this We thank Mrs. Teresa Munnell of the Department of

book.

Mathematics and Computer Science of Wake Forest University for her expert work on the word processor at all stages in the development of the manuscript. We also thank the reviewers of the manuscript for their thoughtful and helpful suggestions.

Winston-Salem and Zagreb July 25, 1988

R.D. Carmichael and D. Mitrovic

To Jane and Mary Jane R.D.C.

To the memory of my mother Anka D.M.

1 Spaces and properties of distributions

1.1.

INTRODUCTION AND PRELIMINARIES

For the convenience of the reader, in this chapter we review the definitions, constructions, and properties of the test functions and distributions that will be considered in this book.

In addition we recall the definitions and properties of

the convolution and Fourier transform of distributions, both of which are operations on distributions that will play an important role in the analysis presented here.

We shall state important representation theorems concerning the structure of

the distributions and give references for their proofs. The analysis of Chapters 2 - 6 is built upon the theory of distributions of L.

Schwartz [117].

For a complete

development of the distributions and the topological vector space preliminaries upon which distributions are based we refer to Horvath [60] and Treves [134].

We shall use some topological vector space terminology here; we refer to [60] and [134] and to the very readable book [108] by Robertson and Robertson for definitions. The development of the theory of distributions is related to Heaviside's operational calculus, Dirac's formalism of the 6 function, contributions by J. Hadamard, M. Riesz, and S. Bochner, and S. L. Sobolev's generalized solutions of partial differential equations. A systematic and unified exposition of the distribution theory based upon topological vector spaces in functional analysis was given by L. Schwartz in 1945 - 1950.

Since the appearance of Schwartz's theory, many

mathematicians have contributed in various ways to the theory of distributions.

A very nice history of the evolution of the

mathematics leading to the theory of distributions together with an extensive bibliography has been given by Synowiec [128].

we now give a brief discussion concerning some of the ways that the theory of distributions has overcome some 1

difficulties that arise in classical analysis.

A very

familiar tool of applied mathematics is the so-called Dirac delta function 6(x) which is usually defined by

10

,

x # 0

,

x = 0

,

6(x) _

with 6(x) dx = 1 1J

fR1

and if T is a continuous function on

IR1

then

6(x) ,p(x) dx = p(0) fJ

1R1

No function exists in classical analysis which has the properties ascribed to this 6 function. The theory of distributions gives a rigorous mathematical foundation for the purely formal calculus of Dirac; the 6 function is interpreted as a continuous linear functional. Dirac's remarkable equality 6(x) = H'(x) is interpreted to mean that 6(x) is the distributional (generalized) derivative of the Heaviside function

1, x >

0

H(x) = 0

,

x < 0

.

One property of distributions which is essentially different from the situation pertaining to locally integrable functions is that distributions are infinitely differentiable.

Every locally integrable function has a distributional

derivative since it can be identified with a certain distribution. In contrast to classical analysis, a convergent sequence of distributions can always be differentiated and the resulting sequence converges to the derivative of the limit. 2

The theory of distributions treats differential equations in a qualitatively new way by introducing distributional solutions instead of the continuous function solutions of classical analysis.

This yields the proofs of the very

general theorems on the existence of solutions of partial differential equations.

In particular a linear differential

equation whose right side is a discontinuous function can not be considered in the classical sense.

Note that the equation

xy' = 1 has no classical solution on all of IR1; its

distributional solution is given by y(x) = logIxj + c1 H(x) + c2

, where H(x) is the Heaviside function and c1 and c2 are

arbitrary real constants.

The classical Fourier transform of a polynomial or the Heaviside function does not exist. In distribution theory the class of functions which are Fourier transformable is greatly enlarged and includes the polynomials and H(x). As a final contrasting idea between distributions and classical analysis we recall that the density of a mass that is distributed continuously along an interval in R1 is a continuous function. However, if the total mass is concentrated at a finite set of isolated points of R1 then the corresponding density function does not exist in classical analysis. The density of a point unit mass located at the origin is equal to the Dirac b distribution.

we complete this section by giving some important Let TVS definitions and notation for the work in this book. be an arbitrary topological vector space. A linear functional U on TVS is a mapping DEFINITION 1.1.1. from TVS into the complex numbers

C1

such that

= c1__ + c2____ for all W1 and W2 in TVS and all cl and c2 in O1; here ____ denotes the complex number obtained by__

operating with U on p E TVS.

The technical definition of continuity of a functional U on TVS is given, for example, in [108, p. 8]. DEFINITION 1.1.2.

A function f(x) defined on In is locally 3

integrable if it is measurable and

JK

If(x)I dx <

fin;

for all compact sets K C

Lloc(ln) will denote the set of

all equivalence classes of locally integrable functions on IR

Lloc(,n)

is a large class of functions; the piecewise

continuous functions on 1n are in Lloc(In) as are the integrable functions and also the measurable and locally bounded functions.

The support of a function f(x) defined on In is the closure in IRn of the set (x E 1n: f(x) x 0) and will be denoted by supp(f). DEFINITION 1.1.3.

The support of a function f(x), supp(f), is the smallest closed subset of In outside of which f(x) is identically zero. If supp(f) is a bounded set then supp(f) is compact, and we say that f has compact support.

The following containments

hold:

supp(f1f2) c supp(fl) fl supp(f2)

and

supp(fl+f2) c supp(fl) U supp(f2)

.

Let a = (al,a2,.... an) be an n-tuple of nonnegative integers, n = 1,2,3,...

For notation, the order of a is jal = a1 + a2 + ... + an and a! = al!a2!...an!. We define the .

differential operator Da = Dt

,

t = (t1,t2,...,tn) E IRn

putting

a

a

a

Dap(t) = Dtv(t) = D11 D22...Dnn ap(t) 4

,

by

where -1

_

Dj

2ai

a

= 1,...,n

j

atj

Here the inclusion of the term (-1/2zri) is simply for

convenience of notation in relation to analysis concerning the Fourier transform given in this book, and the inclusion of the subscript t in Dtp(t) emphasises that the differentiation is with respect to the variable t E t11 t22...tnn for t E IRn

Rn .

We also put to =

Throughout 0 will denote the

.

origin (0,0,...,0) in IRn

1.2. THE SPACES D AND D' DEFINITION 1.2.1.

D = D(IRn), n = 1,2,3,..., will denote the

vector space of all complex valued infinitely differentiable functions on Rn which have compact support, and the elements of D will be called test functions.

An example of a function in D is given by a2

wa(x) =

1

0 ,

Ixl < a

x12- a 2 1x1

>a

,

for any fixed real number a > 0 where x E 1x1

2 = (x1 + x2 +...+ x2)

fn

and as usual

1/2 .

Other test functions can be

constructed from wa(x) by the method of regularization. We now topologize the vector space D.

Let K be a fixed

compact subset of IRn and denote by DK = DK(IRn) the subspace of

D consisting of all functions in D which have their support in K.

We consider the locally convex topology on DK defined by

the sequence of norms

5

jalpm

IDatp(t)I

Sup

, m = 0,1,2,...

The sequence (,p,) of functions w, E DK converges in DK to the

function p E DK as X -i X0 if and only if the sequence Pa

ttpx

(t)) converges uniformly to Dap(t) on K for all n-tuples

(Here X varies over an indexing a of nonnegative integers. set which may be a discrete or continuous set.) We recall that DK is a Frechet space.

Let us choose an increasing sequence (Kj)jC1 of compact IRn

subsets of

whose union is Rn and consider the locally

convex topological vector spaces DK

,

j

= 1,2,...

We have

.

J CO

D = JU1

,

and the topology of DKJ is

DKJ C DKj+1 ,

DKJ

identical to the one induced on it from DK

Since the

J+1 conditions are satisfied for applying the inductive limit construction, we define on D = D(Itn) the inductive limit topology of the spaces DK

,

j

= 1,2,...

.

This topology is

J

independent of the choice of the (K1), and the topology of DK J

is identical to the one induced on it by that of D.

We

LRn

further have that for any compact set K C the topology of DK is identical to the one induced on it by the topology of D. ,

It is interesting to note that the space D is not metrizable or normable. REMARK 1.2.1. From the topology defined on D, we have the following criterion for convergence in the space D: a sequence (,p,) of functions in D converges in the topology of D to a

function tp e D as X --, X0 if and only if there is a compact subset K C n such that supp (,px) c K for each X

,

supp (gyp)

c K,

and for every n-tuple a of nonnegative integers the sequence converges to

6

uniformly on K as A -i Xo

.

It is easy to prove that the operation of differentiation Also multiplication of elements

is continuous from D into D.

in D by a function g E Cw(IRn), the set of all infinitely

differentiable functions on Mn is continuous from D into D. DEFINITION 1.2.2. The dual space D' = D'(IRn) of D equipped ,

with the previously stated topology is the space of distributions; D' is the set of all continuous linear functionals on D equipped with this previously stated topology.

Let T E D' and p E D.

The value of T at c will be denoted

The vector space structure of D' is defined in the and usual way as follows: for T, T1, and T2 in D' w E D.

,

,

c E C1 we put

= c and

= + .

Among the topologies on D' that are compatible with the vector space structure, the most important are the weak topology (convergence on finite subsets of D) and the strong topology (convergence on bounded subsets of D.) The weak dual topology is defined by the family of seminorms (p

p E D)

where

pw(T) = Jj, T E D'

.

This implies the following criterion of convergence for sequences of distributions: a sequence {T.} of elements of D' converges weakly to zero in D' as X -> X0 if and only if for every W E D the sequence of complex numbers {} converges to zero as X

X0

.

7

A subset B of D is bounded in D if and only if the supports of all functions p E B lie in some fixed compact subset K C Otn

and for every a there is a number Ma such that

SUP JDap(t)j tUK

< Ma

E B

The strong topology is introduced on D' by means of the seminorms pB(T) = SUB Il

Thus in the strong topology of D' the sequence (T.) of elements in D' converges as B varies over all bounded sets in D.

to zero as X --> X0 if and only if

lim X)X O sEB = 0 qp

That is, a sequence (T.) of elements in D' converges strongly to zero in D' as X -. X0 if the sequence of complex numbers

converges to zero uniformly on every bounded subset

of D as X

X

0

From the above two definitions it is clear that strong convergence in D' implies weak convergence in D' .

Conversely, if TX -' T weakly in D' as X -, X0 for a sequence (TX) in D' then T E D' and T. --+ T strongly in D' as A

A0

the fact that weak convergence implies strong convergence in D' follows from the fact that D is a Montel space [60, p. 241 and p. 314] and the result [46, Corollary 8.4.9, p. 510]. (In Chapters 2 and 3 we restrict ourselves to the weak topology of D'.)

The following theorem gives equivalent conditions for a linear functional on D to be a distribution. 8

Let T be a linear functional on D.

THEOREM 1.2.1.

Then the

following assertions are equivalent: (i)

T is a distribution;

(ii)

T is sequentially continuous on D; that is, for every

sequence (pA) which converges to zero in D as A --+ X0 then

converges to zero as A --b Ao ; (iii) for every compact subset K C Rn the restriction of T on DK is continuous on DK ;

(iv)

for every compact subset K C Rn there exist a

positive real number M and a nonnegative integer m depending only on K such that Il

< M Is al

s sup p IDaw(t)I, w e DK

One of the most important examples of a distribution is the Dirac b functional defined by w E D. The functional T from D to O1 defined by

EXAMPLE 1.2.1.

00

=

j=0

p(1)

is a distribution.

The functional T from D to O1 defined by = Iw(0)I is not a distribution. The simplest distributions, but among the most useful, are those generated by locally integrable functions. integrable function f on IRn

,

For every fixed locally

the functional Tf from D to O1

defined by

=

fn

f( t) p(t) dt

,

pED

is a distribution. REMARK 1.2.2.

A distribution T is called a regular

distribution if there exists a locally integrable function such that the equality (1.1) holds. All other distributions 9

are said to be singular.

For example the Dirac 6 distribtuion

is a singular distribution.

The function 1/x, x E IR1

,

is not

locally integrable on IR1; hence it does not define a regular

But let us set

distribution.

f

p> = vp

°xx

dx = lim

Wxx

a D.

dx,

IX >£ The limit on the right side is called the Cauchy principal value of the integral

and vp

x

is a distribution.

Two distributions T and U are said to be

DEFINITION 1.2.3.

equal if

__ for all qp E D__

.

The basic identification between regular distributions and locally integrable functions is contained in the following theorem.

THEOREM 1.2.2.

Let f and g be two locally integrable The regular distributions Tf and Tg are

functions on stn.

equal in D' if and only if f(x) = g(x) almost everywhere on n IR

Now denote by D'(Otn;r) the space of all regular

distributions generated by locally integrable functions. Consider the map A:

D' (IRn,r)

Lloc(IRn)

given by A(f) = Tf

.

We recall that an element of Lloc(IRn) is

the class formed by all functions that are equal almost everywhere to a given locally integrable function on IRn is easy to show that the map A is linear.

10

.

It

Since A(f) = A(g)

implies Tf = T9 and this implies f(x) = g(x) almost everywhere, the map A is an injection from Lloc(IR') to D'(IR n;r).

By definition A(Lloc(IR n))

map A is a bijection.

=

D'(IR n;r) so that the

The map A is thus an algebraic

isomorphism from Lloc(Rn) onto D'(IRn;r).

Since these spaces

are isomorphic, we can identify an equivalence class of locally integrable functions with the distribution generated by one of its representatives. In particular two continuous functions on

IRn

are identical.

which generate the same regular distribution Hence we can identify a continuous function f

with the regular distribution Tf and write for . Since D'(Mn;r) C D'(IRn) the notion of distribution generalizes that of continuous function. Therefore a continuous function In can be interpreted in two different ways, first as an f on In ordinary function f: -, C1 and secondly as a distribution

f: p -+ In f( x) p(x) dx from D to

O1 .

Similarly a constant M has three meanings,

first as a complex number, secondly as a constant function, and thirdly as a constant distribution

1

In

M p (x) dx

A distribution T E D' equals zero on an In if = 0 for every p E D with support open set 0 C DEFINITION 1.2.4. in 0

.

For example the Dirac 6 distribution equals zero on 0 = Rn\{0}.

The regular distribution which corresponds to the Heaviside function H(t) equals zero on the set 0 = {t a 1R1: - < t < 0). This is a particular case of the result that if a locally integrable function equals zero on an open subset 0 of IR

n then the corresponding regular distribution equals zero 11

on 0 also.

Two distributions T and U in D' are equal fn that is if T - U equals zero on R on an open subset 0 C if = __ for every p E D with support in 0 are For example the distributions T and T + 6, T E D' Rn that does not include the equal on every open subset of DEFINITION 1.2.5.__

,

,

origin.

The support of T E D' is the complement in DEFINITION 1.2.6. n of the largest open subset of IR n where T equals zero and is IR denoted by supp(T).

Equivalently, a point belongs to supp(T) if an only if there is no open neighborhood of the point on which T equals zero.

EXAMPLE 1.2.2.

We have supp(8) _ (0) and supp(H(t)) _

(t E f1: 0 < t < w).

Often one uses the following result: if p E D and T E D' are such that supp(T) O supp(p) = 0 then = 0. If T and U belong to D' we have supp(T+U) c supp(T) U supp(U). We now discuss and define several important operations on distributions, the first being multiplication by an infinitely differentiable function.

To motivate a general definition

first let us consider the product of a regular distribution with a function in Cw(IRn).

Let f E Lloc(,n) and g E Cw(ln).

For all p E D we have the equality = fn (g (x) f(x)) w(x) dx

=

f f(x) (g(x) w(x)) dx =

,

g(x) p(x)>.

Recalling that gp E D we are led to the following definition. The product of a distribution T E D' with a DEFINITION 1.2.7. function g E Ct(IRn) is the functional gT defined by = , T E D. 12

It is easy to prove that gT as given in Definition 1.2.7 is a distribution for T E D' and g E C*(Rn).

The linearity is

obvious and = -+ 0 for V. -+ 0 in D as X --> X0 . Note, for example, that if g E CO'(IRn) then _ ** = (g(0) p(0)) = g(0) ****, E D, which yields tip**

g5 = (g(0) 0) in D'

Note also that supp(gT) c supp(g) fl

.

supp(T).

In general it is not possible to define multiplicaton of two arbitrary distributions; it may not be possible to do so in D' even if the two distributions are regular.

However,

this disadvantage of distribution theory does not exclude the possibility of defining a product of distributions in certain Progress on multiplication of distributions during the last three decades appears to have significant subspaces of D'.

application in quantum field theory.

Differentiation of distributions is a basic operation which has significant applications in pure and applied mathematics. The definition of distributional differentiation is motivated by considering the situation for regular distributions. Let f Then f and all of its partial

be a function in C1(Dtn).

derivatives ef(t)/atj

= 1,2,...,n, define the following

j

,

regular distributions:

= in f( t) tip(t) dt,

<

'P> =

of J

af(t) I

tip

ED

4p(t) dt

,

E D

j

J

in

Integrating by parts in the last integral we obtain

Of <

p> _ -

)>

a

pED

.

(1.2)

J

13

The equality (1.2) suggests the definition of distributional differentiation.

The partial derivative aT/atj

DEFINITION 1.2.8.

j

= 1,...,n,

of T E D' is the functional given by < aT

atj

.p> _ -

ap(t)

atj

The functional aT/atj

,

j

>, j = 1,...,n

= 1,...,n

,

w E D

.

(1.3)

is a distribution.

,

Linearity is obvious; continuity follows from the fact that a'px(t)/atj converges to zero in D as A -+ A 0 for each j

= 1,...,n when wX --> 0 in D as A -> X0

.

By iteration we

have

= (-1) lal , f E D

(1.4)

,

for any n-tuple a of nonnegative integers, and DaT E D' for T E D'

.

For example,

= - _ -

J 0

%p '(x) dx = %p(0)

for p E D and H(t) being the Heaviside function; thus H' = b in D'.

Let f be a function of

Let x0 be a fixed point in 1t1.

class C1 on IR 1\(x0) with a discontinuity of the first kind at

the point x0

Suppose also that the classical derivative f'

.

of the function f is locally integrable on R1

.

Then the

distributional derivative Df of the function f is given by

Df = f' + bx in D' where b

X

0

is defined by

- f (x0-e) ),

[e--+O+ ( f (x0+e ) ,

w> = p(x0), w E D

0

We see that in contrast to classical differentiation of 14

functions, every distribution has derivatives of all order which are also distributions.

In particular every locally

integrable function has distributional derivatives of all order; these derivatives, in general, are not regular distributions. In the case of a continuous function possessing continuous derivatives, the distributional derivatives coincide with the classical derivatives. Moreover, in contrast to classical differential calculus, we have the following result: if (T.) is a sequence of distributions which converges to T in D' as A -+ X0 and a is any n-tuple of nonnegative integers then DaTx X -4 X0

.

This follows directly from (1.4)

.

DaT in D' as

We conclude

that distributional differentiation is a continuous linear operator from D' to D' .

1.3. THE SPACES E and E'

There are a number of important subspaces of the vector space of distributions D' such as the space DK = DK(IRn) which

consists of the continuous linear functions on DK functions that was introduced in section 1.2. section deals with another subspace of D'

,

,

a space of

The present

the distributions

with compact support.

We denote by E = E(on) the vector space of all infinitely differentiable complex valued functions on IRn In order to topologize E let (Kj) 1 be an increasing DEFINITION 1.3.1.

,

sequence of compact sets in Rn whose union is In j

.

For each

= 1,2,... and each m = 0,1,2,... define the seminorm pm,j by

pm,JOP

DaP (t)I

ialPm

tEK.

E E

J

The family of seminorms (pm,j) defines a locally convex topology on E.

This topology does not change if we replace

the sequence (Kj)

1

by another increasing sequence of compact

subsets of IRn whose union is Rn

.

The topology of E

,

often 15

called the natural topology, is the topology of uniform convergence on compact subsets of Un for sequences of functions in E and for the corresponding sequences of derivatives of all order. we make this explicit in the following criterion for convergence in E. A sequence (gyp.) of functions V. E E converges

REMARK 1.3.1.

to a function ,p in E as A -> X0 if and only if for each

n-tuple a of nonnegative integers the sequence (Dt4px(t))

converges to Dtp(t) uniformly on every compact subset of

Rn

as

Since the family of seminorms (pm'j ) is countable then E is metrizable.

Additionally E is complete.

Thus the topological

vector space E is a Frechet space. > -' 0 as

A linear functional T on E is continuous if

---> 0 in E as X -> No

-4 X0 when q,

.

The set of all

continuous linear functionals on E is denoted by E'

= E'(IRn).

We have the following characterization of elements in E': a linear functional T on E belongs to E' if and only if there is a constant M > 0

,

an integer m > 0, and a compact subset K of

IRn such that

M IaIPm

for all 'p

EE

tsup uK

IDa'p(t)

.

Every element T of E' is an element of D'; that is, T is a distribution. To see this first note that if T E E' then Further, if

is well defined for all 'p E D since D C E.

the sequence {gyp.} converges to zero in D as X -- X0 then

YX -* 0 in E also; hence -' 0 as A -* X0 and T E D' We conclude that E' C D'

.

Let f be a locally integrable function on support.

16

IRn

The linear functional Tf from E into

with compact C1

defined by

= fn f(t) p(t) dt

p e E

,

is a regular distribution in E'

.

One of the most important distributions in E' is the Dirac We recall that supp(b) = (0).

6 distribution.

As in the case of D' we shall describe the weak and strong topologies on E'. A sequence (TX) of distributions T. E E' is

said to converge weakly to zero in E' as X -1 X0 if for every p E E the numerical sequence () converges to zero. sequence (T.), TX E E'

,

A

converges strongly to zero in E' as

X -' X0 if the numerical sequence () converges to zero uniformly on bounded sets of E

We recall that a subset B of

.

IRn

E is bounded if for every compact set K in

and every a

there is a constant M depending on K and a such that IDap(t)I < M for all t E K and p E B.

The space D is dense in E; that is, for each p E E there is a sequence of functions in D which converges to p in E. Indeed, let (Kj)1

be an increasing sequence of compact

1

subsets of Dn such that their union is In

,

and let (pj}1W1 be

a sequence of functions in D such that yj(t) = 1 on a neighborhood of Kj j

= 1,2,...

.

,

j = 1,2,...

We have that p

.

For p E E

E D for each j.

show that pj -> V in E as j - w

,

put Vj = wyj It is easy to

.

This result implies the following useful fact: if T and U are in E' and if = __ for all V E D then __ _ Another useful fact concerns a __ for all v E E. modification of test functions outside the support of a distribution.__

If T E D' and g(t) E Co(LRn) such that g(t) = 1

on a neighborhood of the support of T then _ for all V E D.

This result also holds for T E E if T E E'

Additionally we have that E' is dense in D' and the The elements of

canonical injection E' -- D' is continuous.

17

,

E' are distributions with compact support. We shall state two results of L. Schwartz [117] that give additional structure for E' distributions, and we shall use these results in the succeeding material of this book. Every distribution T E E' with [117, p. 91] THEOREM 1.3.1. Rn Rn in infinitely can be represented in compact support K C

many ways as a sum of a finite number of distributional derivatives of continuous functions which have their support contained in an arbitrary neighborhood of K. Every distribution T E E' whose [117, p. 100] THEOREM 1.3.2. support is the origin can be represented in a unique way as a finite linear combination of distributional derivatives of the Dirac 6 distribution. 1.4. THE SPACES S AND S'

In order to define a distributional Fourier transform, L. Schwartz introduced the space of functions of rapid decrease S = S(IR n)

and the corresponding dual space of tempered distributions S' = S'M n). DEFINITION 1.4.1. S = S(Rn) is the vector space of all COO(Un) functions 'p such that

ItI

Ito Da,p (t)I

= 0

for all n-tuples a and p of nonnegative integers; equivalently

S is the vector space of all CW(Rn) functions p such that for

each a and p there is a constant Map for which sup

It'3

Da,P(t)I

<_ Ma,p

tCIR

An example of a function in S is exp(-IxI2).

For n-tuples a and p of nonnegative integers, define the seminorms Pa'p(W) = Sup

Ito

Da.p(t)I

,

(p E S

tEIRn

The space S equipped with the topology generated by the 18

countable family of seminorms (p a,

is metrizable and

complete; hence S is a Frechet space. REMARK 1.4.1. From the topology of S we have that a sequence (yp1) of functions in S converges to a function W E S in S as

) X0 if and only if for all a and p

A

limy

supra

O tEIR

Ito Dt(wx(t) - w(t))I = 0

We have D C S and convergence in D implies convergence in

S.

If Ox -i p in D as X --1 AO for the sequence (c1) in D and

p E D then all supports supp(p1) and supp(p) are contained in a fixed compact subset of Rn

p in S as X -> XO

wX

Hence (1.5) will hold, and

.

.

The product of the function p E S with an arbitrary infinitely differentiable function may not be a function of S. For example (exp(Ix12) exp(-1x12)) = 1 which is not in S.

A

set of CO(Mn) functions whose elements are multipliers of S is the space OM = OM(IRn) of Schwartz [117, p. 243]. DEFINITION 1.4.2.

OM = 0M(IRn) is the set of all functions W E

CW(Mn) such that for every a there is a polynomial Pa(t) for which

I Da'p (t) I

< Pa (t)

,

tE

OM is called the space of

CCO

IR n

;

(IRn) functions which are slowly

increasing at infinity.

We collect several important facts concerning S in the following result. THEOREM 1.4.1. (i)

The operation of multiplication by a function in OM

is continuous from S into S; the operation of differentiation is continuous from S (ii) 19

into S;

(iii) D C S C E with continuous injections; (iv)

D is dense in S and S is dense in E;

(v)

S C L1 with continuous injection.

The dual S'

S'(Pn) of S is the set of all continuous The elements of S' are called

=

linear functionals on S.

It is known that a linear functional

tempered distributions.

T on S is continuous if and only if there is a constant M > 0 and a nonnegative integer m such that II

sup

< M

IaI<m

Ito Daw(t)I, w C S

tEIRn

I1I<m

From this result it follows that every tempered distribution is of finite order.

(The order of an element T C S' is

explicitly defined in the alternative definition for S and S' which is given in Remark 1.4.2 below.) EXAMPLE 1.4.1. Every continuous function f(t) from IRn to C1 which is slowly increasing at infinity, that is which is bounded by a polynomial, generates a regular tempered distribution Tf

.

As in the case of D' and E' we shall consider the weak and strong topologies on S' A sequence (T.) of distributions in .

S' converges weakly to zero in S' as X --+ X0 if

0

X

To introduce the strong topology on S' we must define the boundedness of a set in S. A set B C S is said to be bounded in S if for each pair of n-tuples a and p of nonnegative integers the products (to Dap(t)) are uniformly bounded on IRn as p varies over B. A set B' C S' is bounded if 'up II < MB

for all T C B' and all bounded sets B C S. 20

Now we have the

following condition for strong convergence in S': a sequence (T.) of distributions in S' converges strongly to zero in S'

asl' -->A 0

if

lim

N-'N0

sup pEB

, p> = 0

for all bounded sets B C S

.

Obviously strong convergence in S' implies weak The converse is also true; the fact that weak

convergence.

convergence implies strong convergence in S' follows from the fact that S is a Montel space [135, p. 21] and the result [46, Corollary 8.4.9, p. 510]. Let T E S'

Since D C S and convergence in D implies

.

convergence in S then T is continuous on D.

Hence T E D' and

S' C D' with proper containment since there are distributions in D' that are not in S'.

Further, convergence of sequences

in S' implies convergence in D'; for if Tx -p 0 weakly in S'

as X

--i 0 for all rp E D and hence T. -p 0

A0 then

weakly in D' as X --b N0

From Theorem 1.4.1(iii) we have E' C S' C D' with continuous injections.

Let T E S' and let g E OM _

ES

,

.

The product gT given by

is an element of S'

,

and the map T --> gT is

continuous in S'.

The differentiation of a tempered distribution T is given by the relation

= (-1) Ia I

Dap>, 4p E S

,

for every n-tuple a of nonnegative integers. Following V. S. Vladimirov [135, pp. 20-22] we REMARK 1.4.2. give an alternative definition for S and its topology.

Consider the countably many norms

21

sup

=

II,P II

(1 + ItI)m IDaw(t)I

, m = 0,1,2,...,

t E IRn

m

IaI<m

for tip

A sequence {'h} converges to zero in S if and only

E S.

-i 0 for all m as X

if IIpxII

AO

.

Denote by S(m) the

m

completion of S with respect to the mth norm. The spaces S(m) are separable Banach spaces, and D is dense in S(m) A .

function p belongs to S(m) if and only if ,p has continuous

derivatives up to and including order m and (Itlm as Iti -i - for all a such that IaI < m. S(0) D S(1) D S(2) D

-i 0

Now we have

...

and

s=

fl

s (m)

m>0

This construction of S as the intersection of the S(m) spaces also yields an alternative way to construct S' The dual spaces S(m)' of S(m) form an increasing chain .

S(0), C S(1)I C S(2)' c

...

and we have U S'

_

S(m)

m>0

Thus if U E S' then U E S(m)' for some m > 0

the smallest M for which U E S(M)' will be called the order of U. For each U E S' we introduce the decreasing sequence of norms

22

;

-m

sup

=

11M

II'

m = M, M + 1,...

=1

where M is the order of U

We shall use this alternative construction of S and S' in terms of S(m) and S(m)' in Chapter .

5.

We conclude this section by stating the following characterization theorem of S' which has been given by L. Schwartz.

THEOREM 1.4.2.

[117, p. 239] A distribution U E D' is an element of S' if and only if U has the form U = Dot((1+It12)k/2 g(t))

for some n-tuple p of nonnegative integers and some real number k > 0 where g(t) is a continuous bounded function on It easily follows from this characterization theorem that if U E S' then there exists a function h(t) E L2 and n-tuples p and -r

of nonnegative integers such that n

U = DR((

II

(1+t2 )ry3) h(t))

j=1 1.5.

THE SPACES 0a AND 0' a

In order to represent as many distributions as possible with the Cauchy integral, H. Bremermann [11] introduced the test spaces 0a = Oa(Rn) and distribution spaces 0' = 0I(Rn) give two definitions of spaces of this type in Otn

,

.

We

one for a

being a real number and another for a being a n-tuple of real numbers.

DEFINITION 1.5.1.

For a being a given real number, we say

that a function N E 0a = Oa(,n) if p is infinitely 23

differentiable and_ if for each n-tuple p of nonnegative

integers there exists a constant MP such that

Mp (1+ItI)a

t e n .

,

Convergence in the vector space 0a is defined as follows:

Xo if

converges to p in 0a as X

a sequence (i)

each pX E Oa

(ii)

for each p the sequence (Dpw,) converges uniformly

;

on every compact subset of

IRn

to Dpw as X -+ X0 ;

and (iii)

for each p there exists a constant MP , which is

independent of X

ID13

,

such that

< MP (1+ItI)a

(t)I

,

t E In

(1.6)

.

Note that the space 0a is complete; that is, the limit function w belongs to 0a

.

Also observe that (w.) converges

in 0a if it converges in E and (1.6) holds.

The vector space 0a is the space of all continuous linear functionals on 0a with continuity having the usual meaning

that__ -> ____ if P -p w in 0a for U E 0' We have D C 0a for every a E R1__

;

and if a sequence

converges to 'p in D then the sequence also converges to 'p in Oa

.

Accordingly every continuous linear functional in 0a is

also a distribution; that is, 0a C D' Let U E 0a

.

We define the derivative A by

= (-1)I1I __, 'p E 0 a__

24

.

The right side in (1.7) is well defined since qp E 0a implies DRv E 0a.

For U E 0a the functional DOU is linear; it is also

continuous since

X - X0

0 when V. - 0 in 0a as

Therefore DRU E 0' if U E 0' a

If al < a2 then 0a

C 0a

a

Also we have

c 0'

and 0' 2

1

a

2

a

1

D C 0a c E and E' C 0' C D' with continuous imbedding.

D is

a

dense in Oa for all a E 6t1

.

As in the case of D' we define the convergence in 0' as

a

weak convergence: the sequence (U.) of distributions in Oa

converges to U if__ ---+__

for every p E 0a as X --' X0.

The space Oa is closed under convergence.

Following Bremermann [11, p. 53] we give the following definition of asymptotic bound of a distribution. A distribution U E D' is said to have the asymptotic bound g(t) Z 0, and we write U = 0(g(t)), if there

DEFINITION 1.5.2.

exist constants R and M such that for all qp E D with support in (t E IRn:

J__j__

Itl

> R) we have

< M J In

THEOREM 1.5.1.

g(t)

Iw(t)I dt

[11, p. 54] Let U E D' and U = 0(Itlr).

Then

U can be extended to 0' for any a such that a+r+n < 0 where n

a

further, the extension is unique. to the case that We now extend the definition of 0a and 0' a

is the dimension of Rn ;

a = (al,a2,...,an) is an arbitrary n-tuple of real numbers. DEFINITION 1.5.3.

A function p belongs to 0a = 0a(IRn) with

a = (al,a2,...,an) if p is infinitely differentiable and if for every n-tuple p of nonnegative integers there exists a constant MP such that

25

a

n

3

I )

I

j=1

,

t E IRn

A sequence (gyp.) converges to p in 0a

(1.8)

.

a = (al,a2,...,an)I

,

if

--

(i)

each pX E 0a

(ii)

for each n-tuple p of nonnegative integers, DP-px(t)

Dpp (t) uniformly on every compact subset of R n as X -b X0 (iii)

for each p there exists a constant MR

independent of X

tE

,

,

which is

such that (1.8) holds for DRp,(t) for all

IR n

We then denote Oa

,

a = (al,a2,...,an)

all continuous linear functionals on 0a

,

as the space of

a = (al,a2,...,an).

,

The following two results are due to Carmichael [19]. THEOREM 1.5.2. Let U E 0Q a = (al,a2,...,an). Then there ,

exist constants M and m depending only on U such that

I__I__

<_

M

tE[R

n IDap(t)

IPI<<m

for all w E Oa

U

(1+It.1)-ail

j=1

.

THEOREM 1.5.3.

Let m be a fixed positive real number and let p be an n-tuple of nonnegative integers. Let U = 13fP(t) where for each p fp(t) is a Lebesgue IPI < m ,

,

Ipkm measurable function which satisfies

n If p(t)I

e > 0

26

,

-a . -1-E

< Rp ]nl (1+It1I)

for all t c IRn with R0 being a constant depending on

Then U E 0'

p.

a

In Chapters 2-6 we shall always state explicitly whether a is a fixed real number or an n-tuple of real numbers in Oa and 0'. a

THE SPACES DLp AND DLp

1.6.

The definitions and results of this section are taken from L. Schwartz [117, pp. 199 - 203]. DEFINITION 1.6.1. DLp = DLp(,n)

1 < p <

,

,

is the space of

all infinitely differentiable functions 'p for which Dp'p(t) E

LP for each n-tuple p of nonnegative integers.

B = D

_

L

(,n) is the space of all infinitely differentiable

D L

functions which are bounded on Rn B is the subspace of B consisting of all functions which vanish at infinity together .

with each of their derivatives.

The topology of D p is given in terms of the norms L

11`-11

(f IDpw(t)IP `Jn

m.P

A sequence of functions the topology of DLp

pED

,

< p <

1

dt]1/p

.

I3

< m , m = 0,1,2,...

converges to a function w in ,

as X -> X0 if each V. E DLp

and

L

0

lim for every p

.

A sequence of functions (gyp.) converges to a function w in

if each w, E B

,

p E B

,

and (1.9) holds for p = -

.

27

.

We have that D is dense in D Lp

not in B = D L IRn

in

If W E D

.

w

,

Dragisa Mitrovic University of Zagreb

Distributions and analytic functions

m Longman

N NW

W Scientific &

- Technical

Copublished in the United States with John Wiley & Sons, Inc, New York

Longman Scientific & Technical, Longman Group UK Limited, Longman House, Burnt Mill, Harlow Essex CM20 2JE, England and Associated Companies throughout the world. Copublished in the United States with John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158

© Longman Group UK Limited 1989 All rights reserved; no part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without either the prior written permission of the Publishers or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, 33-34 Alfred Place, London, WC1E 7DP. First published 1989

AMS Subject Classification: (Main) 46F20, 46F12, 46F10 (Subsidiary) 32A10, 32A07, 32A40 ISSN 0269-3674

British Library Cataloguing in Publication Data Carmichael, Richard D. Distributions and analytic functions. 1. Calculus. Bounded analytic functions 1. Title

H. Mitrovic, Dragia

515'.223

ISBN 0-582-01856-0 Library of Congress Cataloging-in-Publication Data Carmichael, Richard D. Distributions and analytic functions/Richard D. Carmichael, Dragisa Mitrovic. p. cm.- (Pitman research notes in mathematics series, 0269-3674 206) Includes bibliographical references and index. ISBN 0-470-21398-5

1. Distributions, Theory of (Functional analysis) 2. Analytic functions. 1. Mitrovic, Dragi§a, 1922- . II. Title. III. Series. QA324.C37

515,7'82-dcl9

1989

88-34332

CIP

Printed and bound in Great Britain by Biddies Ltd, Guildford and King's Lynn

Contents

Preface

Dedication Chapter 1

1

1.1

Spaces and properties of distributions Introduction and preliminaries

1

1.2

The spaces D and D'

5

1.3

The spaces E and E'

15

1.4

The spaces S and S'

18

1.5

The spaces 0a and Oa

23

1.6

The spaces DLp and D'jjp

27

1.7

29

1.8

Convolution of distributions The Fourier transform

1.9

The spaces Z and Z'

39

Chapter 2

Distributional boundary values of analytic functions in one dimension

35

42

2.2

Introduction Distributional analytic continuation

46

2.3

Analytic representation of distributions

54

2.1

42

in E' ([R) 2.4

Analytic representation of distributions

72

in O' a(IR ) 2.5

Distributional Plemelj relations and boundary value theorems

83

2.6

Representation of half plane analytic and meromorphic functions

89

2.7

Equivalence of convergence in DI(M) and 0a- (IR)

103

2.8

Comments on Chapter 2

106

Chapter 3 3.1

3.2

Applications of distributional boundary values

Introduction Applications to boundary value problems

111

111

116

3.3

Applications to singular convolution equations

135

3.4

Comments on Chapter 3

146

Chapter 4

Analytic functions in Cn kernel functions

,

cones, and

150 150

4.3

Introduction Analytic functions of several complex variables Cones in Rn and tubes in en

4.4

Cauchy and Poisson kernel functions

157

4.5

Hp functions in tubes

167

4.6

Growth of Hp functions in tubes

174

4.7

Fourier-Laplace transform of distributions and boundary values

179

4.8

Comments on Chapter 4

201

4.1

4.2

Chapter 5

151

153

Distributional boundary values of analytic functions in n dimensions

205

Introduction Analytic representation of distributions in E': the scalar valued case

205

5.3

Analytic representation of vector valued distributions of compact support

222

5.4

Analytic representation of distributions in 0'

230

5.5

Analytic representation of distributions in D'p

232

5.1 5.2

206

LP 5.6

Comments on Chapter 5

Chapter 6

The Cauchy integral of tempered distributions and applications in n dimensions

258

260

6.1

Introduction

260

6.2

The Cauchy integral of tempered distributions:

262

the case corresponding to the quadrants in Rn 6.3

Cauchy integral representation of the analytic functions which have S' boundary values

280

6.4

The Cauchy integral of tempered distributions: the case corresponding to arbitrary regular

295

cones in Rn

6.5

6.6

6.7

Analytic functions which have S' boundary values and which are Hp functions Fourier-Laplace integral representation of Hp functions Comments on Chapter 6

301

321

327

References

333

Index

345

Preface

Analysis concerning the representation of distributions in the sense of L. Schwartz as boundary values of analytic functions in one and several variables is presented in this book. The analysis is based on the research of the authors, and the basic material presented here associated with the topic of study is not contained in any other book. Previous research books concerned with the topic under consideration here have been written by Bremermann [11], Beltrami and Wohlers [2], Roos [114], and Vladimirov ([135], [136].) The present book is a companion work to these; much of the analysis in this book has been developed since the publication of these companion works.

Research in the area

considered here finds applications in quantum field theory, partial differential equations, and convolution equations in addition to other areas; and research in this area continues with developments now occuring as well in the representation of ultradistributions in the sense of Beurling and Roumieu as boundary values of analytic functions. Basic for the study of this book are a knowledge of the rudiments of the distributions of L. Schwartz and of basic complex variable in one dimension.

For the convenience of the

reader, a review of the test spaces and distributions to be used in this book is given in Chapter 1 along with a brief discussion of the convolution and Fourier transform of distributions. The concept of analytic function of several complex variables is defined at the beginning of Chapter 4, and some basic facts of these functions have been collected No previous working experience with several complex there. variables is assumed on the part of the reader or is needed to study this book.

A review by the reader of the basic facts

concerning the Fourier transform for LP functions, 1 < p <

2,

would be helpful for the reading of some parts of Chapters 4-6.

Distributional boundary values of analytic functions in one dimension in the topologies of D'(IR) and O'(IR) will be studied Analyticity and growth properties and boundary value results of the Cauchy integral of elements in E'(IR) and in Chapter 2.

O'(6t) will be obtained; conversely, analytic and meromorphic

functions in half planes with appropriate conditions are studied with respect to distributional boundary values in the In some cases recovery of the D'(IR) and 0;(It) topologies. analytic or meromorphic functions in terms of the Cauchy integral of the boundary value is obtained.

Of particular

interest throughout the development of Chapters 2 and 3 is the construction and application of the distributional Plemelj relations concerning the distributional boundary values involving both E'(IR) and Oa(IR) distributions.

These relations

are the natural extension to distributions of the boundary value relations of half plane analytic functions introduced by the Yugoslav mathematician J. Plemelj in the early twentieth century.

The distributional Plemelj relations are used

systematically throughout Chapters 2 and 3 and in particular in generalizations of results of Bremermann and Beltrami and Wohlers. The generalization of the Plemelj relations to

distributions was given independently at approximately the same time (1966-1967) by Mitrovic and by Beltrami and Wohlers but by using different techniques.

Several new results are published and proved for the first time in Chapter 2 including the whole of section 2.7.

Revised proofs from those given in the original papers of some of the other results are given in Chapter 2.

Chapter 3 contains applications of the distributional boundary value results of Chapter 2.

In particular

generalizations of boundary value problems of Plemelj, Hilbert, Riemann-Hilbert, and Dirichlet to the setting of

Further, singular convolution equations in the distributional setting are stated distributions are stated and solved.

and solved using the distributional boundary value results and the Plemelj relations. The analysis in Chapters 4-6 is obtained in n dimensions.

After defining the concept of analytic function of several complex variables in Chapter 4 we then proceed to introduce

topics in the remainder of the chapter that are needed in 1n and tubes in Cn are defined and Cones in Chapters 5 and 6. studied. The Cauchy and Poisson kernel functions corresponding to tubes in Cn are shown to be in relevant test spaces.

Hp functions in tubes are recalled, and a pointwise

growth estimate for these functions is obtained.

The

Fourier-Laplace transform of distributions is related to analytic functions which satisfy various growth conditions; in some cases the analytic functions can be recovered by the Fourier-Laplace transform of the inverse Fourier transform of the boundary value.

Some of the results in Chapter 5 extend corresponding results in Chapter 2 to n dimensions. The Cauchy and Poisson integrals of distributions in E' = E'(Dn) and Oa = 0a(fn) ,

,

DLp = D'

(,n) are studied in Chapter 5 and are shown to have

distributional boundary value properties.

The Cauchy

integrals are analytic functions in tubes in Cn and satisfy Certain analytic functions are related to distributional boundary values in these spaces. Except for growth conditions.

section 5.3, the entire book concerns scalar valued distributions in the sense as originally defined by L.

Schwartz; in section 5.3 we present some boundary value results of vector valued distributions of compact support. H. G. Tillmann was an original investigator of the representation of distributions as boundary values of analytic functions.

In Chapter 6 we define a Cauchy integral of

tempered distributions which we use to recover the analytic functions in tubes defined by quadrants that Tillmann showed obtained S' = S'(IRn) boundary values; this analysis builds

upon that of Tillmann.

In addition, a Cauchy integral of S'

distributions that was defined by Vladimirov corresponding to arbitrary tubes in Cn is studied. The pointwise growth of Hp functions in tubes, which is obtained in Chapter 4, is a special case of the growth that characterizes boundary values in S'; necessary and sufficient conditions are given in Chapter 6 for an analytic function in a tube which has an S' Fourier-Laplace integral representations of Hp functions in tubes, 0 < p are boundary value to be an Hp function.

given at the conclusion of Chapter 6.

Examples of the topics introduced are given throughout the book.

Each chapter concludes with a section containing comments. Frequently other conclusions that can be made on the topics of the chapter are discussed, and the relations between the results presented with those of similar investigations are indicated.

Always we have attempted to give a historical

perspective of the analysis presented in this book with the development of the subject matter in general although we do not claim completeness in this effort.

The authors take great pleasure in acknowledging the contributions of three individuals in the realization of this We thank Mrs. Teresa Munnell of the Department of

book.

Mathematics and Computer Science of Wake Forest University for her expert work on the word processor at all stages in the development of the manuscript. We also thank the reviewers of the manuscript for their thoughtful and helpful suggestions.

Winston-Salem and Zagreb July 25, 1988

R.D. Carmichael and D. Mitrovic

To Jane and Mary Jane R.D.C.

To the memory of my mother Anka D.M.

1 Spaces and properties of distributions

1.1.

INTRODUCTION AND PRELIMINARIES

For the convenience of the reader, in this chapter we review the definitions, constructions, and properties of the test functions and distributions that will be considered in this book.

In addition we recall the definitions and properties of

the convolution and Fourier transform of distributions, both of which are operations on distributions that will play an important role in the analysis presented here.

We shall state important representation theorems concerning the structure of

the distributions and give references for their proofs. The analysis of Chapters 2 - 6 is built upon the theory of distributions of L.

Schwartz [117].

For a complete

development of the distributions and the topological vector space preliminaries upon which distributions are based we refer to Horvath [60] and Treves [134].

We shall use some topological vector space terminology here; we refer to [60] and [134] and to the very readable book [108] by Robertson and Robertson for definitions. The development of the theory of distributions is related to Heaviside's operational calculus, Dirac's formalism of the 6 function, contributions by J. Hadamard, M. Riesz, and S. Bochner, and S. L. Sobolev's generalized solutions of partial differential equations. A systematic and unified exposition of the distribution theory based upon topological vector spaces in functional analysis was given by L. Schwartz in 1945 - 1950.

Since the appearance of Schwartz's theory, many

mathematicians have contributed in various ways to the theory of distributions.

A very nice history of the evolution of the

mathematics leading to the theory of distributions together with an extensive bibliography has been given by Synowiec [128].

we now give a brief discussion concerning some of the ways that the theory of distributions has overcome some 1

difficulties that arise in classical analysis.

A very

familiar tool of applied mathematics is the so-called Dirac delta function 6(x) which is usually defined by

10

,

x # 0

,

x = 0

,

6(x) _

with 6(x) dx = 1 1J

fR1

and if T is a continuous function on

IR1

then

6(x) ,p(x) dx = p(0) fJ

1R1

No function exists in classical analysis which has the properties ascribed to this 6 function. The theory of distributions gives a rigorous mathematical foundation for the purely formal calculus of Dirac; the 6 function is interpreted as a continuous linear functional. Dirac's remarkable equality 6(x) = H'(x) is interpreted to mean that 6(x) is the distributional (generalized) derivative of the Heaviside function

1, x >

0

H(x) = 0

,

x < 0

.

One property of distributions which is essentially different from the situation pertaining to locally integrable functions is that distributions are infinitely differentiable.

Every locally integrable function has a distributional

derivative since it can be identified with a certain distribution. In contrast to classical analysis, a convergent sequence of distributions can always be differentiated and the resulting sequence converges to the derivative of the limit. 2

The theory of distributions treats differential equations in a qualitatively new way by introducing distributional solutions instead of the continuous function solutions of classical analysis.

This yields the proofs of the very

general theorems on the existence of solutions of partial differential equations.

In particular a linear differential

equation whose right side is a discontinuous function can not be considered in the classical sense.

Note that the equation

xy' = 1 has no classical solution on all of IR1; its

distributional solution is given by y(x) = logIxj + c1 H(x) + c2

, where H(x) is the Heaviside function and c1 and c2 are

arbitrary real constants.

The classical Fourier transform of a polynomial or the Heaviside function does not exist. In distribution theory the class of functions which are Fourier transformable is greatly enlarged and includes the polynomials and H(x). As a final contrasting idea between distributions and classical analysis we recall that the density of a mass that is distributed continuously along an interval in R1 is a continuous function. However, if the total mass is concentrated at a finite set of isolated points of R1 then the corresponding density function does not exist in classical analysis. The density of a point unit mass located at the origin is equal to the Dirac b distribution.

we complete this section by giving some important Let TVS definitions and notation for the work in this book. be an arbitrary topological vector space. A linear functional U on TVS is a mapping DEFINITION 1.1.1. from TVS into the complex numbers

C1

such that

= c1

operating with U on p E TVS.

The technical definition of continuity of a functional U on TVS is given, for example, in [108, p. 8]. DEFINITION 1.1.2.

A function f(x) defined on In is locally 3

integrable if it is measurable and

JK

If(x)I dx <

fin;

for all compact sets K C

Lloc(ln) will denote the set of

all equivalence classes of locally integrable functions on IR

Lloc(,n)

is a large class of functions; the piecewise

continuous functions on 1n are in Lloc(In) as are the integrable functions and also the measurable and locally bounded functions.

The support of a function f(x) defined on In is the closure in IRn of the set (x E 1n: f(x) x 0) and will be denoted by supp(f). DEFINITION 1.1.3.

The support of a function f(x), supp(f), is the smallest closed subset of In outside of which f(x) is identically zero. If supp(f) is a bounded set then supp(f) is compact, and we say that f has compact support.

The following containments

hold:

supp(f1f2) c supp(fl) fl supp(f2)

and

supp(fl+f2) c supp(fl) U supp(f2)

.

Let a = (al,a2,.... an) be an n-tuple of nonnegative integers, n = 1,2,3,...

For notation, the order of a is jal = a1 + a2 + ... + an and a! = al!a2!...an!. We define the .

differential operator Da = Dt

,

t = (t1,t2,...,tn) E IRn

putting

a

a

a

Dap(t) = Dtv(t) = D11 D22...Dnn ap(t) 4

,

by

where -1

_

Dj

2ai

a

= 1,...,n

j

atj

Here the inclusion of the term (-1/2zri) is simply for

convenience of notation in relation to analysis concerning the Fourier transform given in this book, and the inclusion of the subscript t in Dtp(t) emphasises that the differentiation is with respect to the variable t E t11 t22...tnn for t E IRn

Rn .

We also put to =

Throughout 0 will denote the

.

origin (0,0,...,0) in IRn

1.2. THE SPACES D AND D' DEFINITION 1.2.1.

D = D(IRn), n = 1,2,3,..., will denote the

vector space of all complex valued infinitely differentiable functions on Rn which have compact support, and the elements of D will be called test functions.

An example of a function in D is given by a2

wa(x) =

1

0 ,

Ixl < a

x12- a 2 1x1

>a

,

for any fixed real number a > 0 where x E 1x1

2 = (x1 + x2 +...+ x2)

fn

and as usual

1/2 .

Other test functions can be

constructed from wa(x) by the method of regularization. We now topologize the vector space D.

Let K be a fixed

compact subset of IRn and denote by DK = DK(IRn) the subspace of

D consisting of all functions in D which have their support in K.

We consider the locally convex topology on DK defined by

the sequence of norms

5

jalpm

IDatp(t)I

Sup

, m = 0,1,2,...

The sequence (,p,) of functions w, E DK converges in DK to the

function p E DK as X -i X0 if and only if the sequence Pa

ttpx

(t)) converges uniformly to Dap(t) on K for all n-tuples

(Here X varies over an indexing a of nonnegative integers. set which may be a discrete or continuous set.) We recall that DK is a Frechet space.

Let us choose an increasing sequence (Kj)jC1 of compact IRn

subsets of

whose union is Rn and consider the locally

convex topological vector spaces DK

,

j

= 1,2,...

We have

.

J CO

D = JU1

,

and the topology of DKJ is

DKJ C DKj+1 ,

DKJ

identical to the one induced on it from DK

Since the

J+1 conditions are satisfied for applying the inductive limit construction, we define on D = D(Itn) the inductive limit topology of the spaces DK

,

j

= 1,2,...

.

This topology is

J

independent of the choice of the (K1), and the topology of DK J

is identical to the one induced on it by that of D.

We

LRn

further have that for any compact set K C the topology of DK is identical to the one induced on it by the topology of D. ,

It is interesting to note that the space D is not metrizable or normable. REMARK 1.2.1. From the topology defined on D, we have the following criterion for convergence in the space D: a sequence (,p,) of functions in D converges in the topology of D to a

function tp e D as X --, X0 if and only if there is a compact subset K C n such that supp (,px) c K for each X

,

supp (gyp)

c K,

and for every n-tuple a of nonnegative integers the sequence converges to

6

uniformly on K as A -i Xo

.

It is easy to prove that the operation of differentiation Also multiplication of elements

is continuous from D into D.

in D by a function g E Cw(IRn), the set of all infinitely

differentiable functions on Mn is continuous from D into D. DEFINITION 1.2.2. The dual space D' = D'(IRn) of D equipped ,

with the previously stated topology is the space of distributions; D' is the set of all continuous linear functionals on D equipped with this previously stated topology.

Let T E D' and p E D.

The value of T at c will be denoted

The vector space structure of D' is defined in the and usual way as follows: for T, T1, and T2 in D' w E D

,

,

c E C1 we put

Among the topologies on D' that are compatible with the vector space structure, the most important are the weak topology (convergence on finite subsets of D) and the strong topology (convergence on bounded subsets of D.) The weak dual topology is defined by the family of seminorms (p

p E D)

where

pw(T) = J

.

This implies the following criterion of convergence for sequences of distributions: a sequence {T.} of elements of D' converges weakly to zero in D' as X -> X0 if and only if for every W E D the sequence of complex numbers {

X0

.

7

A subset B of D is bounded in D if and only if the supports of all functions p E B lie in some fixed compact subset K C Otn

and for every a there is a number Ma such that

SUP JDap(t)j tUK

< Ma

E B

The strong topology is introduced on D' by means of the seminorms pB(T) = SUB I

Thus in the strong topology of D' the sequence (T.) of elements in D' converges as B varies over all bounded sets in D.

to zero as X --> X0 if and only if

lim X)X O sEB

That is, a sequence (T.) of elements in D' converges strongly to zero in D' as X -. X0 if the sequence of complex numbers

converges to zero uniformly on every bounded subset

of D as X

X

0

From the above two definitions it is clear that strong convergence in D' implies weak convergence in D' .

Conversely, if TX -' T weakly in D' as X -, X0 for a sequence (TX) in D' then T E D' and T. --+ T strongly in D' as A

A0

the fact that weak convergence implies strong convergence in D' follows from the fact that D is a Montel space [60, p. 241 and p. 314] and the result [46, Corollary 8.4.9, p. 510]. (In Chapters 2 and 3 we restrict ourselves to the weak topology of D'.)

The following theorem gives equivalent conditions for a linear functional on D to be a distribution. 8

Let T be a linear functional on D.

THEOREM 1.2.1.

Then the

following assertions are equivalent: (i)

T is a distribution;

(ii)

T is sequentially continuous on D; that is, for every

sequence (pA) which converges to zero in D as A --+ X0 then

converges to zero as A --b Ao ; (iii) for every compact subset K C Rn the restriction of T on DK is continuous on DK ;

(iv)

for every compact subset K C Rn there exist a

positive real number M and a nonnegative integer m depending only on K such that I

< M Is al

s sup p IDaw(t)I, w e DK

One of the most important examples of a distribution is the Dirac b functional defined by w E D. The functional T from D to O1 defined by

EXAMPLE 1.2.1.

00

j=0

p(1)

is a distribution.

The functional T from D to O1 defined by

,

For every fixed locally

the functional Tf from D to O1

defined by

fn

f( t) p(t) dt

,

pED

is a distribution. REMARK 1.2.2.

A distribution T is called a regular

distribution if there exists a locally integrable function such that the equality (1.1) holds. All other distributions 9

are said to be singular.

For example the Dirac 6 distribtuion

is a singular distribution.

The function 1/x, x E IR1

,

is not

locally integrable on IR1; hence it does not define a regular

But let us set

distribution.

f

p> = vp

°xx

dx = lim

Wxx

a D.

dx,

IX >£ The limit on the right side is called the Cauchy principal value of the integral

and vp

x

is a distribution.

Two distributions T and U are said to be

DEFINITION 1.2.3.

equal if

.

The basic identification between regular distributions and locally integrable functions is contained in the following theorem.

THEOREM 1.2.2.

Let f and g be two locally integrable The regular distributions Tf and Tg are

functions on stn.

equal in D' if and only if f(x) = g(x) almost everywhere on n IR

Now denote by D'(Otn;r) the space of all regular

distributions generated by locally integrable functions. Consider the map A:

D' (IRn,r)

Lloc(IRn)

given by A(f) = Tf

.

We recall that an element of Lloc(IRn) is

the class formed by all functions that are equal almost everywhere to a given locally integrable function on IRn is easy to show that the map A is linear.

10

.

It

Since A(f) = A(g)

implies Tf = T9 and this implies f(x) = g(x) almost everywhere, the map A is an injection from Lloc(IR') to D'(IR n;r).

By definition A(Lloc(IR n))

map A is a bijection.

=

D'(IR n;r) so that the

The map A is thus an algebraic

isomorphism from Lloc(Rn) onto D'(IRn;r).

Since these spaces

are isomorphic, we can identify an equivalence class of locally integrable functions with the distribution generated by one of its representatives. In particular two continuous functions on

IRn

are identical.

which generate the same regular distribution Hence we can identify a continuous function f

with the regular distribution Tf and write

f: p -+ In f( x) p(x) dx from D to

O1 .

Similarly a constant M has three meanings,

first as a complex number, secondly as a constant function, and thirdly as a constant distribution

1

In

M p (x) dx

A distribution T E D' equals zero on an In if

.

For example the Dirac 6 distribution equals zero on 0 = Rn\{0}.

The regular distribution which corresponds to the Heaviside function H(t) equals zero on the set 0 = {t a 1R1: - < t < 0). This is a particular case of the result that if a locally integrable function equals zero on an open subset 0 of IR

n then the corresponding regular distribution equals zero 11

on 0 also.

Two distributions T and U in D' are equal fn that is if T - U equals zero on R on an open subset 0 C if

,

,

origin.

The support of T E D' is the complement in DEFINITION 1.2.6. n of the largest open subset of IR n where T equals zero and is IR denoted by supp(T).

Equivalently, a point belongs to supp(T) if an only if there is no open neighborhood of the point on which T equals zero.

EXAMPLE 1.2.2.

We have supp(8) _ (0) and supp(H(t)) _

(t E f1: 0 < t < w).

Often one uses the following result: if p E D and T E D' are such that supp(T) O supp(p) = 0 then

To motivate a general definition

first let us consider the product of a regular distribution with a function in Cw(IRn).

Let f E Lloc(,n) and g E Cw(ln).

For all p E D we have the equality

=

f f(x) (g(x) w(x)) dx =

,

g(x) p(x)>.

Recalling that gp E D we are led to the following definition. The product of a distribution T E D' with a DEFINITION 1.2.7. function g E Ct(IRn) is the functional gT defined by

It is easy to prove that gT as given in Definition 1.2.7 is a distribution for T E D' and g E C*(Rn).

The linearity is

obvious and

g5 = (g(0) 0) in D'

Note also that supp(gT) c supp(g) fl

.

supp(T).

In general it is not possible to define multiplicaton of two arbitrary distributions; it may not be possible to do so in D' even if the two distributions are regular.

However,

this disadvantage of distribution theory does not exclude the possibility of defining a product of distributions in certain Progress on multiplication of distributions during the last three decades appears to have significant subspaces of D'.

application in quantum field theory.

Differentiation of distributions is a basic operation which has significant applications in pure and applied mathematics. The definition of distributional differentiation is motivated by considering the situation for regular distributions. Let f Then f and all of its partial

be a function in C1(Dtn).

derivatives ef(t)/atj

= 1,2,...,n, define the following

j

,

regular distributions:

<

'P> =

of J

af(t) I

tip

ED

4p(t) dt

,

E D

j

J

in

Integrating by parts in the last integral we obtain

Of <

p> _ -

)>

a

pED

.

(1.2)

J

13

The equality (1.2) suggests the definition of distributional differentiation.

The partial derivative aT/atj

DEFINITION 1.2.8.

j

= 1,...,n,

of T E D' is the functional given by < aT

atj

.p> _ -

ap(t)

atj

The functional aT/atj

,

j

>, j = 1,...,n

= 1,...,n

,

w E D

.

(1.3)

is a distribution.

,

Linearity is obvious; continuity follows from the fact that a'px(t)/atj converges to zero in D as A -+ A 0 for each j

= 1,...,n when wX --> 0 in D as A -> X0

.

By iteration we

have

(1.4)

,

for any n-tuple a of nonnegative integers, and DaT E D' for T E D'

.

For example,

J 0

%p '(x) dx = %p(0)

for p E D and H(t) being the Heaviside function; thus H' = b in D'.

Let f be a function of

Let x0 be a fixed point in 1t1.

class C1 on IR 1\(x0) with a discontinuity of the first kind at

the point x0

Suppose also that the classical derivative f'

.

of the function f is locally integrable on R1

.

Then the

distributional derivative Df of the function f is given by

Df = f' + bx in D' where b

X

0

is defined by

- f (x0-e) ),

[e--+O+ ( f (x0+e ) ,

w> = p(x0), w E D

0

We see that in contrast to classical differentiation of 14

functions, every distribution has derivatives of all order which are also distributions.

In particular every locally

integrable function has distributional derivatives of all order; these derivatives, in general, are not regular distributions. In the case of a continuous function possessing continuous derivatives, the distributional derivatives coincide with the classical derivatives. Moreover, in contrast to classical differential calculus, we have the following result: if (T.) is a sequence of distributions which converges to T in D' as A -+ X0 and a is any n-tuple of nonnegative integers then DaTx X -4 X0

.

This follows directly from (1.4)

.

DaT in D' as

We conclude

that distributional differentiation is a continuous linear operator from D' to D' .

1.3. THE SPACES E and E'

There are a number of important subspaces of the vector space of distributions D' such as the space DK = DK(IRn) which

consists of the continuous linear functions on DK functions that was introduced in section 1.2. section deals with another subspace of D'

,

,

a space of

The present

the distributions

with compact support.

We denote by E = E(on) the vector space of all infinitely differentiable complex valued functions on IRn In order to topologize E let (Kj) 1 be an increasing DEFINITION 1.3.1.

,

sequence of compact sets in Rn whose union is In j

.

For each

= 1,2,... and each m = 0,1,2,... define the seminorm pm,j by

pm,JOP

DaP (t)I

ialPm

tEK.

E E

J

The family of seminorms (pm,j) defines a locally convex topology on E.

This topology does not change if we replace

the sequence (Kj)

1

by another increasing sequence of compact

subsets of IRn whose union is Rn

.

The topology of E

,

often 15

called the natural topology, is the topology of uniform convergence on compact subsets of Un for sequences of functions in E and for the corresponding sequences of derivatives of all order. we make this explicit in the following criterion for convergence in E. A sequence (gyp.) of functions V. E E converges

REMARK 1.3.1.

to a function ,p in E as A -> X0 if and only if for each

n-tuple a of nonnegative integers the sequence (Dt4px(t))

converges to Dtp(t) uniformly on every compact subset of

Rn

as

Since the family of seminorms (pm'j ) is countable then E is metrizable.

Additionally E is complete.

Thus the topological

vector space E is a Frechet space. > -' 0 as

A linear functional T on E is continuous if

---> 0 in E as X -> No

-4 X0 when q,

.

The set of all

continuous linear functionals on E is denoted by E'

= E'(IRn).

We have the following characterization of elements in E': a linear functional T on E belongs to E' if and only if there is a constant M > 0

,

an integer m > 0, and a compact subset K of

IRn such that

M IaIPm

for all 'p

EE

tsup uK

IDa'p(t)

.

Every element T of E' is an element of D'; that is, T is a distribution. To see this first note that if T E E' then Further, if

the sequence {gyp.} converges to zero in D as X -- X0 then

YX -* 0 in E also; hence

.

Let f be a locally integrable function on support.

16

IRn

The linear functional Tf from E into

with compact C1

defined by

p e E

,

is a regular distribution in E'

.

One of the most important distributions in E' is the Dirac We recall that supp(b) = (0).

6 distribution.

As in the case of D' we shall describe the weak and strong topologies on E'. A sequence (TX) of distributions T. E E' is

said to converge weakly to zero in E' as X -1 X0 if for every p E E the numerical sequence (

,

A

converges strongly to zero in E' as

X -' X0 if the numerical sequence (

We recall that a subset B of

.

IRn

E is bounded if for every compact set K in

and every a

there is a constant M depending on K and a such that IDap(t)I < M for all t E K and p E B.

The space D is dense in E; that is, for each p E E there is a sequence of functions in D which converges to p in E. Indeed, let (Kj)1

be an increasing sequence of compact

1

subsets of Dn such that their union is In

,

and let (pj}1W1 be

a sequence of functions in D such that yj(t) = 1 on a neighborhood of Kj j

= 1,2,...

.

,

j = 1,2,...

We have that p

.

For p E E

E D for each j.

show that pj -> V in E as j - w

,

put Vj = wyj It is easy to

.

This result implies the following useful fact: if T and U are in E' and if

If T E D' and g(t) E Co(LRn) such that g(t) = 1

on a neighborhood of the support of T then

This result also holds for T E E if T E E'

Additionally we have that E' is dense in D' and the The elements of

canonical injection E' -- D' is continuous.

17

,

E' are distributions with compact support. We shall state two results of L. Schwartz [117] that give additional structure for E' distributions, and we shall use these results in the succeeding material of this book. Every distribution T E E' with [117, p. 91] THEOREM 1.3.1. Rn Rn in infinitely can be represented in compact support K C

many ways as a sum of a finite number of distributional derivatives of continuous functions which have their support contained in an arbitrary neighborhood of K. Every distribution T E E' whose [117, p. 100] THEOREM 1.3.2. support is the origin can be represented in a unique way as a finite linear combination of distributional derivatives of the Dirac 6 distribution. 1.4. THE SPACES S AND S'

In order to define a distributional Fourier transform, L. Schwartz introduced the space of functions of rapid decrease S = S(IR n)

and the corresponding dual space of tempered distributions S' = S'M n). DEFINITION 1.4.1. S = S(Rn) is the vector space of all COO(Un) functions 'p such that

ItI

Ito Da,p (t)I

= 0

for all n-tuples a and p of nonnegative integers; equivalently

S is the vector space of all CW(Rn) functions p such that for

each a and p there is a constant Map for which sup

It'3

Da,P(t)I

<_ Ma,p

tCIR

An example of a function in S is exp(-IxI2).

For n-tuples a and p of nonnegative integers, define the seminorms Pa'p(W) = Sup

Ito

Da.p(t)I

,

(p E S

tEIRn

The space S equipped with the topology generated by the 18

countable family of seminorms (p a,

is metrizable and

complete; hence S is a Frechet space. REMARK 1.4.1. From the topology of S we have that a sequence (yp1) of functions in S converges to a function W E S in S as

) X0 if and only if for all a and p

A

limy

supra

O tEIR

Ito Dt(wx(t) - w(t))I = 0

We have D C S and convergence in D implies convergence in

S.

If Ox -i p in D as X --1 AO for the sequence (c1) in D and

p E D then all supports supp(p1) and supp(p) are contained in a fixed compact subset of Rn

p in S as X -> XO

wX

Hence (1.5) will hold, and

.

.

The product of the function p E S with an arbitrary infinitely differentiable function may not be a function of S. For example (exp(Ix12) exp(-1x12)) = 1 which is not in S.

A

set of CO(Mn) functions whose elements are multipliers of S is the space OM = OM(IRn) of Schwartz [117, p. 243]. DEFINITION 1.4.2.

OM = 0M(IRn) is the set of all functions W E

CW(Mn) such that for every a there is a polynomial Pa(t) for which

I Da'p (t) I

< Pa (t)

,

tE

OM is called the space of

CCO

IR n

;

(IRn) functions which are slowly

increasing at infinity.

We collect several important facts concerning S in the following result. THEOREM 1.4.1. (i)

The operation of multiplication by a function in OM

is continuous from S into S; the operation of differentiation is continuous from S (ii) 19

into S;

(iii) D C S C E with continuous injections; (iv)

D is dense in S and S is dense in E;

(v)

S C L1 with continuous injection.

The dual S'

S'(Pn) of S is the set of all continuous The elements of S' are called

=

linear functionals on S.

It is known that a linear functional

tempered distributions.

T on S is continuous if and only if there is a constant M > 0 and a nonnegative integer m such that I

sup

< M

IaI<m

Ito Daw(t)I, w C S

tEIRn

I1I<m

From this result it follows that every tempered distribution is of finite order.

(The order of an element T C S' is

explicitly defined in the alternative definition for S and S' which is given in Remark 1.4.2 below.) EXAMPLE 1.4.1. Every continuous function f(t) from IRn to C1 which is slowly increasing at infinity, that is which is bounded by a polynomial, generates a regular tempered distribution Tf

.

As in the case of D' and E' we shall consider the weak and strong topologies on S' A sequence (T.) of distributions in .

S' converges weakly to zero in S' as X --+ X0 if

0

X

To introduce the strong topology on S' we must define the boundedness of a set in S. A set B C S is said to be bounded in S if for each pair of n-tuples a and p of nonnegative integers the products (to Dap(t)) are uniformly bounded on IRn as p varies over B. A set B' C S' is bounded if 'up I

for all T C B' and all bounded sets B C S. 20

Now we have the

following condition for strong convergence in S': a sequence (T.) of distributions in S' converges strongly to zero in S'

asl' -->A 0

if

lim

N-'N0

sup pEB

, p> = 0

for all bounded sets B C S

.

Obviously strong convergence in S' implies weak The converse is also true; the fact that weak

convergence.

convergence implies strong convergence in S' follows from the fact that S is a Montel space [135, p. 21] and the result [46, Corollary 8.4.9, p. 510]. Let T E S'

Since D C S and convergence in D implies

.

convergence in S then T is continuous on D.

Hence T E D' and

S' C D' with proper containment since there are distributions in D' that are not in S'.

Further, convergence of sequences

in S' implies convergence in D'; for if Tx -p 0 weakly in S'

as X

--i 0 for all rp E D and hence T. -p 0

A0 then

weakly in D' as X --b N0

From Theorem 1.4.1(iii) we have E' C S' C D' with continuous injections.

Let T E S' and let g E OM _

ES

,

.

The product gT given by

is an element of S'

,

and the map T --> gT is

continuous in S'.

The differentiation of a tempered distribution T is given by the relation

= (-1) Ia I

Dap>, 4p E S

,

for every n-tuple a of nonnegative integers. Following V. S. Vladimirov [135, pp. 20-22] we REMARK 1.4.2. give an alternative definition for S and its topology.

Consider the countably many norms

21

sup

=

II,P II

(1 + ItI)m IDaw(t)I

, m = 0,1,2,...,

t E IRn

m

IaI<m

for tip

A sequence {'h} converges to zero in S if and only

E S.

-i 0 for all m as X

if IIpxII

AO

.

Denote by S(m) the

m

completion of S with respect to the mth norm. The spaces S(m) are separable Banach spaces, and D is dense in S(m) A .

function p belongs to S(m) if and only if ,p has continuous

derivatives up to and including order m and (Itlm as Iti -i - for all a such that IaI < m. S(0) D S(1) D S(2) D

-i 0

Now we have

...

and

s=

fl

s (m)

m>0

This construction of S as the intersection of the S(m) spaces also yields an alternative way to construct S' The dual spaces S(m)' of S(m) form an increasing chain .

S(0), C S(1)I C S(2)' c

...

and we have U S'

_

S(m)

m>0

Thus if U E S' then U E S(m)' for some m > 0

the smallest M for which U E S(M)' will be called the order of U. For each U E S' we introduce the decreasing sequence of norms

22

;

-m

sup

=

11M

II'

m = M, M + 1,...

=1

where M is the order of U

We shall use this alternative construction of S and S' in terms of S(m) and S(m)' in Chapter .

5.

We conclude this section by stating the following characterization theorem of S' which has been given by L. Schwartz.

THEOREM 1.4.2.

[117, p. 239] A distribution U E D' is an element of S' if and only if U has the form U = Dot((1+It12)k/2 g(t))

for some n-tuple p of nonnegative integers and some real number k > 0 where g(t) is a continuous bounded function on It easily follows from this characterization theorem that if U E S' then there exists a function h(t) E L2 and n-tuples p and -r

of nonnegative integers such that n

U = DR((

II

(1+t2 )ry3) h(t))

j=1 1.5.

THE SPACES 0a AND 0' a

In order to represent as many distributions as possible with the Cauchy integral, H. Bremermann [11] introduced the test spaces 0a = Oa(Rn) and distribution spaces 0' = 0I(Rn) give two definitions of spaces of this type in Otn

,

.

We

one for a

being a real number and another for a being a n-tuple of real numbers.

DEFINITION 1.5.1.

For a being a given real number, we say

that a function N E 0a = Oa(,n) if p is infinitely 23

differentiable and_ if for each n-tuple p of nonnegative

integers there exists a constant MP such that

Mp (1+ItI)a

t e n .

,

Convergence in the vector space 0a is defined as follows:

Xo if

converges to p in 0a as X

a sequence (i)

each pX E Oa

(ii)

for each p the sequence (Dpw,) converges uniformly

;

on every compact subset of

IRn

to Dpw as X -+ X0 ;

and (iii)

for each p there exists a constant MP , which is

independent of X

ID13

,

such that

< MP (1+ItI)a

(t)I

,

t E In

(1.6)

.

Note that the space 0a is complete; that is, the limit function w belongs to 0a

.

Also observe that (w.) converges

in 0a if it converges in E and (1.6) holds.

The vector space 0a is the space of all continuous linear functionals on 0a with continuity having the usual meaning

that

;

and if a sequence

converges to 'p in D then the sequence also converges to 'p in Oa

.

Accordingly every continuous linear functional in 0a is

also a distribution; that is, 0a C D' Let U E 0a

.

We define the derivative A by

24

.

The right side in (1.7) is well defined since qp E 0a implies DRv E 0a.

For U E 0a the functional DOU is linear; it is also

continuous since

X - X0

0 when V. - 0 in 0a as

Therefore DRU E 0' if U E 0' a

If al < a2 then 0a

C 0a

a

Also we have

c 0'

and 0' 2

1

a

2

a

1

D C 0a c E and E' C 0' C D' with continuous imbedding.

D is

a

dense in Oa for all a E 6t1

.

As in the case of D' we define the convergence in 0' as

a

weak convergence: the sequence (U.) of distributions in Oa

converges to U if

for every p E 0a as X --' X0.

The space Oa is closed under convergence.

Following Bremermann [11, p. 53] we give the following definition of asymptotic bound of a distribution. A distribution U E D' is said to have the asymptotic bound g(t) Z 0, and we write U = 0(g(t)), if there

DEFINITION 1.5.2.

exist constants R and M such that for all qp E D with support in (t E IRn:

J

Itl

> R) we have

< M J In

THEOREM 1.5.1.

g(t)

Iw(t)I dt

[11, p. 54] Let U E D' and U = 0(Itlr).

Then

U can be extended to 0' for any a such that a+r+n < 0 where n

a

further, the extension is unique. to the case that We now extend the definition of 0a and 0' a

is the dimension of Rn ;

a = (al,a2,...,an) is an arbitrary n-tuple of real numbers. DEFINITION 1.5.3.

A function p belongs to 0a = 0a(IRn) with

a = (al,a2,...,an) if p is infinitely differentiable and if for every n-tuple p of nonnegative integers there exists a constant MP such that

25

a

n

3

I )

I

j=1

,

t E IRn

A sequence (gyp.) converges to p in 0a

(1.8)

.

a = (al,a2,...,an)I

,

if

--

(i)

each pX E 0a

(ii)

for each n-tuple p of nonnegative integers, DP-px(t)

Dpp (t) uniformly on every compact subset of R n as X -b X0 (iii)

for each p there exists a constant MR

independent of X

tE

,

,

which is

such that (1.8) holds for DRp,(t) for all

IR n

We then denote Oa

,

a = (al,a2,...,an)

all continuous linear functionals on 0a

,

as the space of

a = (al,a2,...,an).

,

The following two results are due to Carmichael [19]. THEOREM 1.5.2. Let U E 0Q a = (al,a2,...,an). Then there ,

exist constants M and m depending only on U such that

I

<_

M

tE[R

n IDap(t)

IPI<<m

for all w E Oa

U

(1+It.1)-ail

j=1

.

THEOREM 1.5.3.

Let m be a fixed positive real number and let p be an n-tuple of nonnegative integers. Let U = 13fP(t) where for each p fp(t) is a Lebesgue IPI < m ,

,

Ipkm measurable function which satisfies

n If p(t)I

e > 0

26

,

-a . -1-E

< Rp ]nl (1+It1I)

for all t c IRn with R0 being a constant depending on

Then U E 0'

p.

a

In Chapters 2-6 we shall always state explicitly whether a is a fixed real number or an n-tuple of real numbers in Oa and 0'. a

THE SPACES DLp AND DLp

1.6.

The definitions and results of this section are taken from L. Schwartz [117, pp. 199 - 203]. DEFINITION 1.6.1. DLp = DLp(,n)

1 < p <

,

,

is the space of

all infinitely differentiable functions 'p for which Dp'p(t) E

LP for each n-tuple p of nonnegative integers.

B = D

_

L

(,n) is the space of all infinitely differentiable

D L

functions which are bounded on Rn B is the subspace of B consisting of all functions which vanish at infinity together .

with each of their derivatives.

The topology of D p is given in terms of the norms L

11`-11

(f IDpw(t)IP `Jn

m.P

A sequence of functions the topology of DLp

pED

,

< p <

1

dt]1/p

.

I3

< m , m = 0,1,2,...

converges to a function w in ,

as X -> X0 if each V. E DLp

and

L

0

lim for every p

.

A sequence of functions (gyp.) converges to a function w in

if each w, E B

,

p E B

,

and (1.9) holds for p = -

.

27

.

We have that D is dense in D Lp

not in B = D L IRn

in

If W E D

.

w

,

1

,

and in B but

,

then p is bounded

Lp

and converges to zero at infinity with the same being

true for all derivatives of W

1

1

,

We have D C DLp C DLq c B for

.

.

DEFINITION 1.6.2.

DLp = D'

(IRn),

is the space of

1 < p

continuous linear functionals on D

,

1/p + 1/q = 1; D'

Lq

_

L

D'1(IRn) is the space of continuous linear functionals on L

The space DLp is a subspace of D' since D C D q

,

.

Indeed, let U E DLp

__ is defined for all W E D
.
Clearly U is
L
Since convergence in D implies convergence in D q then __