Distributions and Analytic Functions (Pitman Research Notes in Mathematics Series)

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


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
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£ 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 = .

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,


_ -, 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 = 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

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

, '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:

Jj

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

II

0 depending only on U such that

U = 1131<m

where f13 E Lp for each p 28

,

IRI

< m

.

1.7.

CONVOLUTION OF DISTRIBUTIONS

We have already seen that, in general, it is not possible to define the product of distributions.

However, we can define

another type of product, the direct or tensor product of distributions. We begin by defining this type of product for functions. The direct product f(x) 0 g(y) of the functions f(x) and g(y), x E IRn and y E IRm

,

is the function h(x,y) = f(x)g(y) defined

Thus the direct product of two real valued functions coincides with their usual product. Now let us consider the direct product of two regular distributions. Let us assume that f(x) and g(y) are locally integrable functions on IRn and 1m respectively; and let p(x,y) be a test function in D(IRnxIm). The functions f(x) and g(y) induce the regular distributions Tf(x) and Tg(y) defined on IR n x IRm

,

by

= In f( x) p(x,y) dx

= In g( Y) v(x,y) dy

and

9(y)

Since the direct product of two locally integrable functions is also a locally integrable function, we can define the regular distribution fog by = f InxIR

f(x) g(y) p(x,y) dx dy, w E D(RnxRm).

Using Fubini's theorem we obtain g(Y) w(x,y) dy dx

= in f(x) I m

29

= fn f(x) dx = .

In the same manner we obtain =

.

Consequently the direct product of two regular distributions f E D'(R1) and g E D'(Rm) is a new distribution f®g in D'(RnxRm) that satisfies the equalities = .

In particular if p(x,y) = y(x)x(y) where y E D(Mn) and X E D(IRm) we have = X0 then Tx 0 Ux -> T ® U in D'(UnxIRm).

If a and p are an n-tuple and m-tuple, respectively, of nonnegative integers then DX DP (T®U) = DXT 0 DyU y

Multiplication of the direct product by a function g(x) e CCO(Rn)

is defined by

g(x) (T®U) = (g(x)T) 0 U

.

with the use of the direct product for m = n, we can now define one of the most important operations in the space of distributions, that of convolution. Let f and g be two locally integrable functions on 0n with one having compact support; their convolution is the function h = f*g defined by

h(x) = (f*g)(x) = in f(x-y) g(y) dy

=

(1.10)

f f(y) g(x-y) dy In

Let q(x) be a test function in D

.

By taking (1.10) into

account and using Fubini's theorem we have IRn

in

f(x) g(y) p(x+y) dx dy

,

w E D

,

or

=

,

v E D

.

(1.11)

For arbitrary T and U in D' = D'(Dn) the equation (1.11) can be written formally as 31

= ,

'p E D

(1.12)

.

When both T and U have compact support then so has T®U by a property of the direct product, and the second term in (1.12) In general the right side of (1.12) may not is well defined. exist for arbitrary T and U in D' since p(x+y) as a function of (x,y) does not have compact support in Rn x IRn

.

Schwartz

[117, pp. 153 - 156] has given conditions on the distributions T and U such that the right side of (1.12) is well defined and T*U is a distribution; see also Horvath [60, pp. 381 - 401].

We use equation (1.12) to define convolution of two distributions. DEFINITION 1.7.2.

Let T and U be two distributions in D' of T and U is defined by (1.12).

The convolution T*U

Under conditions that ensure T*U E D' and under which the right side of (1.12) is well defined the convolution can be computed as =

= ,

(1.13)

pED

.

In certain cases the convolution is a continuous operator. Let T E E' and U E D'

If a sequence UX --. U in D' as

.

A -- A0 then T*UX --> T*U in D'.

If T E D' and a sequence

U -. U in D' as X - X0 such that all supports of UA are contained in a fixed compact set K C Rn then T*UX -' T*U in D'.

The first result follows from the definition of the

convolution and the fact that X0 Let us note that if T E S' and W E E' then T*W E S'

.

Also

the bilinear map (T,W) - T*W from S'xE' to S' is separately continuous. Let T and U be two distributions on Rn with at least one of them having compact support. Then we have the following properties: (i)

commutativity: T*U = U*T;

(ii)

associativity: for R E E' we have

(iii)

R*T*U = R*(T*U) = (R*T)*U; differentiation: for every n-tuple R of nonnegative

integers we have 34

D13 (T*U)

=

(D)3T)*U = T*(DRU);

translation: if T is a given point in Mn

(iv)

the

,

translation of T by T is the distribution T(T) defined by

=

0

1/r = 1/p + 1/q - 1; the bilinear map (T,U)

,

then T*U T*U

L

from D'

LP

x D'

Lq

is continuous. into D' Lr

1.8. THE FOURIER TRANSFORM Let denote the usual dot (inner) product on

Rn

given by

35

= t1x1+t2x2+...+tnxn where t = (t1,t2,...,tn) E In and x = (x1,x2,...,xn) E DEFINITION 1.8.1.

[Rn .

Let pp E L1 = L1(IRn)

The Fourier

.

transform of p is the function

w(x) = ?[p(t);x] = '11n v(t) e2,ri dt

,

x E IRn

T he Fourier transform is a linear map from L1 into LW _ LCO

(IRn)

For V E L1

.

(x)

II

1= L

p exists, is continuous, and satisfies

,

fIn

dt
W DEFINITION 1.8.2.

.

The inverse Fourier transform of p E L1 is

the function

-1[,P(t);x] = In ,p (t) e-2ni dt

,

x E IRn

the Fourier transform pp(x) _ is constructed by the usual limit in the mean process in the Lq norm 1/p + 1/q = 1 We know that pp E Lq 5-1 and p can be recovered from [p (x) ; t ] , the If

E Lp = Lp(,n)

1 < p < 2

,

,

,

.

as p (t) _

inverse Fourier transform of w by a similar limit in the mean process in the LP norm. The Fourier transform is a one-one mapping of L2 onto L2 and for any p C LP ,

;

1 < p
0,

Iz13

41(z)l

j

= 1,...,n, for which

< MR exp(allim(zl)I + ... + anllm(zn)l), z e

en ,

for all n-tuples p of nonnegative integers where MR depends on p and possibly on y and (al,a21.... an) depends on '

.

In this definition we of course have z = (zl,z2.... .zn) with zj = xj+iyj

,

j

= 1,...,n.

We have defined the concept

of analytic function of several complex variables in section 4.2.

A sequence (4,) converges in Z if (i)

each y, C Z

(ii)

there exist constants Ma and (al,a2,...,an), which

are independent of X Iz13

41x(z)l

;

,

such that for all X

< MP exp(a1IIm(z1)I + ... + anllm(z )I), z e

en ,

for each p;

(iii) (ipx(z)) converges uniformly on every bounded set in

On DEFINITION 1.9.2. Z' = Z'(ln) is the set of all continuous linear functionals on Z .

Ehrenpreis and Gel'fand and Shilov proved that the Fourier transform is a topological isomorphism from D onto Z. Using this fact we can define a Fourier transform on D' .

The Fourier transform of V is the element U = 9[V] = V such that DEFINITION 1.9.3.

= ,

with supp(T) = K such that C(T;z) = f(z)?

In general the

The function f(z) _ (el/z - 1) is analytic for

answer is no.

z # 0 and tends to zero as Izi -- co

,

but it constitutes a

In fact, suppose the contrary; suppose that the desired distribution T exists such that C(T;z) = (e1/z counterexample.

Since (el/z - 1) is analytic except at z = 0 then T must be concentrated at the origin, that is supp(T) _ (0); hence T is a finite linear combination of Dirac's 6 distribution and 1).

This implies that the Cauchy integral of T

its derivatives. is

N

a

n

C(T;z) _ n=1

0

z

where the an are complex constants and N is a positive integer.

But we have

el/z _ 1 =

C

1

n==1

n! zn

, z# 0.

Hence there does not exist a distribution T E E'(IR) such that C(T;z) = (el/z - 1). Under some additional conditions on the 45

analytic functions we do obtain a converse result which gives

an interesting correspondence between the space E'(R) and the considered class of analytic functions. The previous considerations of this paragraph are all contained in section 2.6 as is the representation of a given function analytic in a strip, which converges in the DI(R) topology to certain boundary values from the interior of the strip, as the difference of two generalized Cauchy integrals. In section 2.7 we show that under certain conditions, convergence in DI(R) implies convergence in Oa(R).

The

Due to this equivalence of topologies, the well known formulas of quantum mechanics converse is always true.

stated by Bremermann in 0a1(R) for a < 0

[11, pp. 60 - 66] are deduced from the same formulas stated originally in D'(R).

Section 2.8, where comments concerning the analysis of this chapter and related analysis in other works are contained, concludes Chapter 2.

p = 0,1,2,..., will denote the pth derivative of the function f with respect to Throughout Chapters 2 and 3, f(P)

,

its variable unless specifically stated otherwise. f(O)

As usual

= f.

DISTRIBUTIONAL ANALYTIC CONTINUATION As is mentioned in section 2.1, we shall consider here a 2.2.

distributional extension of the Painleve theorem.

The

condition of continuity is replaced by a weak distributional convergence, and the usual equality of the boundary values on a is to be understood in a distributional sense. To obtain this result and some others throughout Chapter 2 we need the following lemma. LEMMA 2.2.1.

Let h+(z) be analytic in A+ with h+(z) _

0(1/lzI) as Izi -- w in A+ Let h+(x+ie) converge to the boundary value h+ E D'(IR) in the DI(R) topology as e ---).0+ .

that is, let

46

JCO

= lim



for all V E D(R).

=

E-

h+(x+ie) w(x) dx

+

(2.1)

Then

h+ E O'(lt) for all a < 0;

(i)

and

h+(x+ie) converges to h+ in the Oa(ft) topology, a < 0,

(ii)

as a-+0+, that is (2.1) holds for all w E 0a(9t), a < 0.

Additionally, if -1 < a < 0 we have

(iii)

1

2,ri

PROOF.

+

1

t-z



=

jh(z) 0

zE

.

z E e-

,

(2.2)

For each e > 0, h+(x+ie) is continuous as a function Therefore for each e > 0 the linear functional

of x c R.

h+(x+ie) on D(IR) to C defined by the integral



=

f'

h+(x+ie) w(x) dx

V E D(IR) ,

,

By the hypothesis on the

is a regular distribution in D'(IR).

behavior of h+(z) there exist constants R > 0 and A > 0 such that for each e > 0 and all IxI > R the inequality lh+(x+ie)l S

holds.


R

Then for all V E D(IR) with support contained in the

set A = (x E M

il

(x2+e2)1/2

=

:

IxI

> r > R) it follows that

u rn 1h o0

p(x) dxl < A

f.

IxI-1 lp(x)I dx. W

Thus the distribution h+ has IxI-1 as asymptotic bound; and by the theorem in [11, p. 54] (Theorem 1.5.1), h+ can be extended

from D' (It) to O' (lt) , a < 0.

That is, h+ E 0;(R), a < 0, and

the proof of (i) is complete. We now prove (ii). First we must show that for each e > 0 47

the linear functional h+(x+ie) on Oa(IR) to C defined by

=

J h+(x+ie)

pp(x) dx

, 'p E Oa(QR)

,

(2.3)

Clearly this is implied by

is a distribution in Oa(IR), a < 0.

the cited theorem in [11, p. 54] (Theorem 1.5.1) since

f

ll < A

lxl-1 l,p(x)l dx

lxl>r

for all p E D(IR) with support in the set A C R

.

In addition

it is of interest to prove this fact directly which we do now.

For each e > 0 the integral in (2.3) exists because the integrand is O [ixi_1] a < 0. Let be any sequence ,

which convergesto zero in Oa(IR) as n

-- 0 as n -> -

We must show that

Let r > 0. Then we can

.

write

h+(x+ie) pn(x) dxI

f


0 be as in the proof of part (i). To consider the limit in part (ii) we write

h+(x+ie) %p(x) dx

=

h+(x+ie) p(x) dx +

f

IxIr where %p

E 0a (IR) and r > 0.

Since each 9 C Oa (1R) C CO (IR) , for

any given compact set in U there exists a function in D(IR)

that is identical to p over this compact set [144, p. 41]. Thus by the hypothesis

lim

e-O+

f

h+(x+ie) %p(x) dx

=



Since h+(z) is analytic and bounded in the domain (z a d+: IRe(z) I

>

r > R) it follows that h+(x+ie) - H+(x) E L- for

almost all x with lx I > r as a lh+(x+ie)I
0.

,

A

IxI

---> 0+

.

Also

> r > R

Using the Lebesgue dominated convergence

theorem we obtain

49

Elim -

h+(x+ie) p(x) dx

+

H+(x) p(x) dx

= J

J

IxI>r

IxI>r

Combining these facts, there exists an element U E O'(IR) a < 0

,

such that

lim This implies h

=

,

P E 0a(R)

,

a 0.) Let z be any point in A+. We apply the Cauchy integral formula to the function

h(c+i6) S-z

Re(C) = t

as a function of c along the closed path consisting of a sufficiently large semicircle in A+ of radius r and the segment [-r,r] on the real axis and obtain

50

h+(z+ie) =

21

i

r°'

,f _00

h(t+ie) dt t-z

this integral vanishes.

For z c A

z E A+

,

.

Thus, letting a --> 0+ we

have

lim e-40+

1

h+(z), z E A+

1

-1, as a function of t e R for Im(z) # 0.

The proof of Lemma 2.2.1 is

complete.

Obviously, corresponding results to (i) and

REMARK 2.2.1.

(ii) of Lemma 2.2.1 hold for a given analytic function h (z) in A

with h -(z) = 0(1/Izl) as Izi -->

exists h

and for which there

E D'(IR) such that

0+ such that

E D' (61) in D' (61)

- f-, p> = 0 for all 'P E D(R) whose support lies in some

Then there exists a unique function f(z) and is that is equal to f(z) in A+ and to f (z) in A open set 0 C 61

.

,

analytic in A+ U 0 U A

,

.

Since every open set on the real line can be expressed as a countable union of disjoint open intervals, if suffices

PROOF.

to consider 0 as an open interval in R. arbitrary open set follows immediately.

2.2.1 that the distributions f+ and f

The extension to an We know from Lemma belong to 0'1(61).

Using the Cauchy integral of these distributions given by (2.2) and (2.7) we introduce a new function f(z) defined by

f(z) =

1

2vi

_

1



1

2ni

t-z >

w in A+

.

If f+(x+ie) -> f+ E

in D' (It) as a -> 0+ where f+ is such that = 0 for all p E D (M) with supp ( gyp) in some open interval 0 C l then f+(z) = 0 in A+ and 0.

D' (It)

PROOF.

Let us introduce the auxiliary function f (z) such Evidently

and

with f+(z) being the analytic continuation of f -(z) into A+ the function f(z) is However, since f (z) = 0 in A ,

identically zero at all points of its domain of analyticity. and its D'(ll) boundary value f+ must Thus f+(z) = 0, z E A+ ,

53

be the zero distribution on R.

The proof is complete.

Let the functions f+(z) and f -(z) satisfy

CONSEQUENCES 2.2.1.

and let f(z) be the the conditions of Theorem 2.2.2 constructed function in Theorem 2.2.2 which is analytic in A+ ,

We have the following additional consequences of

U 0 U A

Theorem 2.2.2. (i)

Hence

If 0 = R then f(z) is analytic in C with f(o3) = 0.

f(z) = 0 in C If 0 = l and if f+(z) or f -(z) has a pole of order m at (ii) .

some point z = a, a E A+ or a E A

then by Theorem 2.2.2 and

,

a Liouville theorem we have

f(z) =

Pm-1(z) (z-a)m

where Pm_1(z) is a polynomial of degree less than or equal to m-1

The assertions of Theorem 2.2.2 and Corollary 2.2.1

(iii)

hold if the D'(IR) topology is replaced by the Oa(IR) topology,

-1 < a < 0. 2.3

ANALYTIC REPRESENTATION OF DISTRIBUTIONS IN E'((R)

For completeness we first present a brief survey of some major results concerning the analytic representation of distributions, a phrase which we formalize below in Definition 2.3.2.

These results have been known for some time, and we

Later in this section the formulas of Plemelj concerning boundary values of analytic functions are extended to the distributional E'(f) setting.

note sources of proof for them.

The starting point for our review survey is the simple assertion that an arbitrary continuous complex valued function on 6t cannot be analytically continued (extended) into the complex plane C

.

In fact let f(x)

,

which maps IR to C

,

be a

continuous function with compact support, and assume that a function h(z) is its analytic continuation into C. We have f(x) = h(x), x E R.

This implies h(x) = 0 on IR\supp(f), and

by the uniqueness theorem we have h(z) = 0, z E C. 54

Although it is impossible to represent an arbitrary f(x) as

the restriction of an analytic function, it is possible to find a function f(z) which is analytic in a subset of C and which represents f(x) by a jump (f(x+ie) - f(x-ie)) arbitrarily close to the real axis.

This significant and

inspiring property is given in the following theorem. Let f(t) map C to C and be a continuous

THEOREM 2.3.1.

function with f(t) = 0(1/ItIa) for some a > 0 as Itl -> Let ?(z) be the function defined by

i(z) =

211

i

f(t)

J

dt

,

.

z E A = (z: Im(z) s 0)

Then

lim e->0+

?(x-ie)) = f(x)

(2.8)

uniformly on every compact subset of R. We call r(z) the Cauchy integral (representation) of f and the limit (2.8) the analytic representation of f. For the proof see [11, p. 47]. REMARK 2.3.1.

Let h(z) be a function which is analytic in C

for all z outside a closed set K C R.

We shall refer

occasionally to this assumption by saying that the function h(z) is sectionally (locally) analytic in the complex plane C with boundary on the real axis consisting of the set K.

(The

set K does not belong to the domain of analyticity of h(z).) In this situation the function h(z) can be decomposed into two functions:

1.

kz)

=

Jh+(z)

h(z)

z E A+ , zEA ,

,

In general the functions h+(z) and h -(z) are not the analytic

continuation of each other.

In the case of the Cauchy

integral ?(z) in Theorem 2.3.1 the decomposition becomes 55

z E A+

+(z)

(z) _ where

2,1

J

co

t(z) dt

,

z E A+

,

z E A

zeA

0

and

0

,

z e A+ f(t)

2ai

f_c

t-z

dt

-

In particular, if f(t) is a continuous function with compact support then the functior.+(z) is the analytic continuation of the function ?-(z) across the set IR\supp(f).

Theorem 2.3.1 with added definitions is the foundation of the development of the theory presented here which begins with Definitions 2.3.1 and 2.3.2 below.

Let z denote a point located in the half plane A+ or the half planed but not on the real axis R. The function 1/(t-z) is continuous as a function of t e U and has continuous derivatives of all order for all values of t e U; hence 1/(t-z) a E(IR) as a function of t E R.

If T C E'(IR)

acts on this function, a function of the complex variable z is obtained as alluded to in section 2.1. DEFINITION 2.3.1.

Let T E E'(IR).

The function (2.9)

for z varying over an appropriate subset of C is called the

56

Cauchy integral of T.

Other terminologies used for C(T;z) are the Cauchy representation or the analytic representation of T by means of the Cauchy kernel.

In particular if T is a regular distribution corresponding to a locally integrable function T(t), t C IR, with compact support K C IR then C(T;z) is reduced to the ordinary Cauchy integral

C(T;z)

2Tri

EXAMPLE 2.3.1.

t-z

J-00

dt

JK

2ai

t-z

)

dt

Consider the Dirac 6 distribution defined on

E(IR) by = T(0), T E E(IR).

b E E' (R) with supp(b) = (0).

Let A(t) E C°(IR) with X(O) 0 0.

The Cauchy integral of

(X(t) 6) is the function

C(X(t) 6 ;z) =

27r1

<X(t) bt

i

tlz > _

.

-X(O)

To find the Cauchy integral (representation)

EXAMPLE 2.3.2.

of Heisenberg's delta distribution b+

_

lim

-1

1

X+i

2'rri

with convergence in D'(IR) let us consider the function

h+(z) =

z C A+

2,riz

By part (iii) of Lemma 2.2.1 we obtain

C(b+;z) =

+

1

2rri

1

t-z

=

z E A+

2iriz 0

,

zEA

Also the Cauchy integral (representation) of the distribution

-1 2iri

lim

6-O+

1

X-i 57

with convergence in D'(IR) can be obtained in a similar manner. Let h (z)

2niz

'

z E A

Then by Remark 2.2.1 and the formula (2.7) we have 0

1

C(b ;z)

=



-

,

1

1

2niz

z E e+

,

z E e-

Note that the evaluation of the Cauchy integral of the distribution S+ given in [114, pp. 330 - 331] depends essentially on a definition of a limit that is now justified by Lemma 2.2.1.

The basic properties of C(T;z) are described in the following result. THEOREM 2.3.2.

Let T E E'(IR).

The Cauchy integral C(T;z) is defined and analytic in the

(i)

domain C\supp(T); C(T;z)

(ii)

dzn

=

C(n)(T;z)

=



where n

C(n)(T;z)

2ri

< d

dt

n

T,

t1

>.

(Thus the nth derivative of the Cauchy integral of T is the Cauchy integral of the nth derivative of T.) (iii)

C(T;z) = 0(1/Izl) as Izi-> -.

A complete proof of Theorem 2.3.2 is given in [11, pp. 43 - 45].

The definition of the distributional version of the limit (2.8) is now given.

DEFINITION 2.3.2.

Let T be a given distribution in D'(IR).

Any function f(z) which is defined and analytic in the domain C\supp(T) such that

58

J00

lim -w

e-*0+

(f(x+ie) - f(x-ie)) qp(x) dx =

(2.10)

for all p a D(R) is called an analytic representation of T. Sometimes the limit (2.10) is written in the form

(f (x+ie) - f(x-ie)) --1 T in DI(R) as a --i 0+. We shall use the terminology "analytic representation" defined in Definition 2.3.2 for other spaces of distributions as well as for those in D, (R) when we represent these distributions as in

(2.10) for p in the corresponding test space. Let EB(R) denote the subspace of E(Ot) consisting of all

bounded functions in E(R).

We now state an analytic

representation theorem for distributions in E'(IR). THEOREM 2.3.3.

If T e E'(Ot) then

M

lim

(C(T;x+ie) - C(T;x-ie)) p(x) dx = J

for all pp

(2.11)

-w

a EB (Ot)

.

A proof can be found in [ll,p. 48]. COROLLARY 2.3.1.

If T E E'(Ot) then (2.11) holds for all p E

D(IR). The construction of an analytic representation of an arbitrary distribution in DI(R) or S'(Ot) by means of its Cauchy integral is not always possible. An arbitrary element of D'(IR) or S'(IR) does not have a Cauchy integral since the Yet Cauchy kernel 1/(t-z) is not an element of D(R) or S(R). we do have analytic representation results for both D'(IR) and

S'(Ot). THEOREM 2.3.4.

Every distribution T E S'(IR) has an analytic

representation.

For a proof see [131].

We shall give much more information

concerning the representation of tempered distributions as boundary values of analytic functions in one and many dimensions in Chapters 4-6 of this book. THEOREM 2.3.5. Every distribution T E D'(IR) has an analytic 59

representation.

A complete proof is found in [11, p. 50]. Note that an analytic representation f(z) of a distribution In fact if T is not unique if such a representation exists. H(z) is an entire function then the function (f(z) + H(z)) is

also an analytic representation of T because every entire function is an analytic representation of the zero distribution.

Our brief review of some major results concerning the analytic representation of distributions is complete.

We now

desire to formulate the distributional version of the following theorem which is of great interest in analytic function theory and diverse applications. The following theorem yields the formulas of Plemelj [104], and we shall extend these formulas to the distributional setting for E'(IR) distributions in this section and for O'(IR) distributions in a section 2.4.

Let f(t), t E U, be a complex valued function which is Holder continuous on every compact

THEOREM 2.3.6.

(Plemelj [104])

subset of IR, and let f(t) = 0(1/Itla) for some a > 0 as Iti -a W

.

Let

?(z)

=

2ni

JW

dt

t(z)

,

z e A

The boundary values of ?(z) from A+ and A

on Ut exist, and we

have

lim ->0+

(X+16) = 2 f( x) +

1

2rri

t-x iTco_()

dt =

+(x)

(2.12)

and

lim

1

2

60

f(x) +

tai

JW

f(t)

dt =

_(x) (2.13)

with the convergence being uniform on every compact subset of O2 and where the singular integral

(

x)

=

f

2n1 i

t- x

J _oo

dt

(2.14)

is taken as the Cauchy principal value.

The symbols ?+(x) and ?_(x) on the right of (2.12) and (2.13), respectively, denote the boundary values (limits) obtained in these two equations.

Subtracting and adding these

limits we obtain the famous formulas of Plemelj which are ?+(x) - ?_(x) = f(x)

and

Under the hypotheses of Theorem 2.3.6 the function f(t)/(t-z) is continuous on Ut x A and is analytic in z e A for

Further, the Cauchy integral ?(z) converges uniformly on every compact subset of A. These facts imply that the function ?(z) is analytic in A, and we have each t e R.

(P) (

z) = ---

2iri !

f -CO

f(t)

(t-z)

+1 dt, p = 0,1,2,...,

where as usual ?(P)(z) means the pth derivative of ?(z) with In addition if f respect to its variable and f(0)(z) = f(z). E CW(JR) with f(P)(t) = 0(1/Itla) for all p = 0,1,2,... as Iti -. -

,

integration by parts p times yields

61

f(P) (z) =

°°

f_

2,ri

f (Pt)

-zt) - dt.

OD

Since for each p = 0,1,2,... the function f(P)(t) is Holder continuous on every compact subset of ER then by Theorem 2.3.6 applied to f(P)(t) we also have the relations lim

f(P)(x+ie)

_

f(P)(x) +

2

+

2ni

1

f(P)(x) +

-CO

(2.15)

f(P)(t) dt t-x

=

f(P)( + x)

and

lim f(P)(x-ie) e-40+

2

+

1

(2.16)

°° f(P)(t)

2ni

t-x

dt = f(P)(x)

for p = 0,1,2,... which extend (2.12) and (2.13) and where convergence is uniform on every compact subset of R. As in (2.12) and (2.13), the symbols f(P)(x) and f(P)(x) in (2.15) and (2.16), respectively, denote the limits obtained in these

two equations, and the integral in each equation is a Cauchy principal value as in (2.14). In addition to Theorem 2.3.6 we also need the following lemma.

Lemma 2.3.1.

w(z) =

Let

2ni F

dt, z E A, )

be the Cauchy integral of p E D(R).

62

Let

n

1

p(tk)

vn(z) = k=1

tk-z

(tk - tk-1)

(2.17)

Then for each p = 0,1,2,..., the

be the Riemann sum of (z). sequence of pth derivatives

converges uniformly to

the pth derivative ;(P)(z) on every compact subset of A as n 1

00

First observe that for a given positive integer n the

PROOF.

Riemann sum (2.17) of 4p(z) corresponds to a partition which

(a = t0,t1,...,tn = b) of the interval [a,b] = (;nP)(z)) is the sequence of functions

depends on gyp.

n

n (z)

2vi

=

k=1

w(tk) p+1 (tk-z)

(tk - tk-1)

each of which is analytic in A for p = 0,1,2,...

.

Now let K

be an arbitrary but fixed compact subset of A, and let (Both of d = d(K,supp(4p)) be the distance from K to supp(,p). these sets being compact, there are points z' E K and t' E supp(rp) such that d =

Iz'

n

;(P)(z) n

-

w(tk)

tk

p!

2ni

k=1

Hence from

- t'I.)

dt

tk-1

(tk-z)

p+1

we get for all n and all z E K the inequality

n

`pnp) (z) k=1 (b-a) p!

2,rdP+1 for p = 0,1,2,...

.

rtk

'v (tk )

ft k-1

t k-z

I

P+1

dt

max 1w(t)I

l

Thus the sequence (,p1

(z) )

is uniformly 63

bounded on every compact subset K of A. subset of A with accumulation point in A.

Let r be an infinite By the theory of

integration, the limit

Jim ;p(P) (z)

=

n-

;(P) (Z)

=

2!

00

p(t)±1 dt

(t-z)P

f-W 1

2iri exists for every z E r

00

I

(P)(t)

t-z

dt

In view of the Stieltjes-Vitali theorem [42, p.309], for every p = 0,1,2,... the sequence (;(P)(z)) converges uniformly to -P(P)(z) on every compact .

subset K of A as n The proof is complete. Using the Plemelj formulas derived from Theorem 2.3.6 we add to the results of Theorem 2.3.3 and Corollary 2.3.1 in the following theorem.

If T E E'(R) then

THEOREM 2.3.7. CO

lim e-*0+

--

(C(T;x+ie) - C(T;x-ie)) v(x) dx =

and

lim

(C(T;x+ie) + C(T;x-ie)) .p(x) dx = -2

JW

e-90+

-w

for all p E D(R) where ap(t) is the principal value integral

w(t)

1

=

2,i

J -00

P(X)

x-t

dx.

For each E > 0 the function C(T;x+iE) is a continuous function of x E R. Therefore for each E > 0 the linear PROOF.

functional C(T;x+ie) on D(D) to C defined by the integral

64

J

=

C(T;x+ie) %p(x) dx

(2.18)

-CO

The integral (2.18),

is a regular distribution in D'(ll).

being a Riemann integral over the support of gyp, can be

approximated by the Riemann sum n

C(T;xk+ie) %O(xk)

xk-1)

(xk

k=1

corresponding to a partition (a = x0,x1,x2,...,xn = b) of the

We have

interval [a,b] =

JCO

=

-00 `p(xk)

(xk

xk-1)'

Because T is a linear functional on E(R) we can write

= lim .

k=1

By Lemma 2.3.1, for each e > 0 the sequence of functions

1

IPn(t-ie) _

n

2ai

P(xk) xk- (t- ie)

(xk

xk-1)

k=1

in E(IR) as functions of t E IR, upon which T acts, converges in E(IR) to the Cauchy integral

65

as n -> co

.

f

1

W(t-ie) =

2ir i

w(x) x- (t-ie)

J _0,

dx

(2.19)

By continuity of T E E'(IR) we obtain

=

t'

.

Now for each p = 0,1,2,..., the function V(P)(x) is Holder continuous on every compact subset of Qt; thus, according to

the Plemelj formula (2.16), the corresponding Cauchy integral

(P) (t-ie

IP(P)(x) )

1

=

dx

i_ x-(t-ie)

2,ri

converges uniformly to ;(P)(t) defined by (2.16) on every compact subset of 1k as e 3 0+

t e 6t

,

.

Hence as a function of

(t-ie) converges in E(IR) to cp_(t) as e - 0+. This

permits us to write 0+ for T E E'(D). D'(IR) is closed with respect to convergence of sequences of distributions in D'(IR), there exist unique distributions T+ E D'(IR) and T

lim e-*0+

E D'(IR) such that

C(T;x+ie) = T+

and 67

lim e-+0+

in D'(IR).

C(T;x-ie) = T

In addition, by the definition of convolution and

the notation (2.14) we have 1 ai __

1

Vi



0 as

ti

-) -

(,- (0) = T .

.

convergence in Ea(IR) as that of E(IR).

)

Define the

We obtain now that the

results of Theorem 2.3.7, and hence of Theorem 2.3.8, can be generalized somewhat by allowing the test space in these results to be Ea(IR) instead of D(IR). To do this we need some preliminary results. LEMMA 2.3.2.

w(z) =

Let

2ni

F.

t(z) dt

,

z E A

be the Cauchy integral of v E Ea(IR) and let

70

(2.26)

px(z) =

z E A

t(z) dt

2ai

where A is a positive real number. uniformly to ;(P)(z)

K C A as X -->-

,

(2.27)

Then -(P)(z) converges

p = 0,1,2,..., on every compact subset

.

The integral in (2.26) exists due to the fact that the integrand behaves like 1/Itll+a as Iti -+ Since P(P)(t) ---' 0 as Iti - p = 0,1,2,..., we integrate by parts in (2.26) and obtain PROOF.

.

,

(P)(z) _ _

i -w

dt =

w(t)

(t-z) p+1

,P(P)(t)

°°

1

dt

t-z

2,ri

From the definition of a convergent improper integral ,P

271

(P)

t-z

J

(t) dt = lim 1 A-4- 2ni

('A A

4P(P) (t)

dt

t-z

Starting from (2.27) and integrating by parts we get

(P) (z)

=

(P)

1 27 1

Ex

t-z(t) dt +

(2.28)

P-1 +

p(k)(A)

1

2ni

-

1p(k)(-A)

(A-z)P-k

k=0

(-A-z)P-k

Now we form the difference (P)

(Z) -

1

2ni

(P)(z) A

J-W

_

,P(P)(t)

t-z

dt +

,P

1

2ni

rX

(P)(t) t-z

dt

71

P-1 2ni

k=O

(X-z)P-k (-X-z)P-k

which can be estimated by

I,p(P) (Z)

- c(P) (Z) I AP

ItI-1-a

f_ J

dt + - I

0,

ItI-1-a dt X

1

A

+

(2.29)

[ixi__a I?-zlk + IXI-1-a

pC

I+zl-kl J

k=O

P

A

P

airXa

+

AP 2wX1+a

[i_i-P+k

+

IX+zl-p+kl

k=0

J

It is easy to see that for z varying over any compact set K C A the right side in (2.29) can be made arbitrarily small for sufficiently large X. The proof is complete. COROLLARY 2.3.2. functions

For each fixed a > 0 the sequence of converges to p(xfie), respectively,

x E R. in E(IR) as X -. -

.

Now we can show that Theorem 2.3.7 remains valid for W E The proof can be obtained very similarly to that of

Ea(IR).

Theorem 2.3.7 and hence will not be repeated. 2.4.

ANALYTIC REPRESENTATION OF DISTRIBUTIONS IN O'(IR)

The object of this section is to introduce a broader extension of some of Bremermann's results on analytic representations of distributions in the space O'(R) [11, pp. 56 - 59] from the viewpoint of the distributional Plemelj relations. In order to discuss the Cauchy integral of distributions in Oa(t) we first note the following. Consider 1/(t-z) as a

72

function of t E l for any z E A; it is clear that 1/(t-z) _ 0((l+ltl)-1) as Iti -> dP

Also

.

(-1) P p'

(1/(t-z)) _

dtp

=

0((l+Itl)-p-1)

=

0((1+Itl)-1)

(t-z)P+i

as Iti - - for p = 0,1,2,... function of t E R for z E A

Thus 1/(t-z) E O_1(C) as a

.

Since -1 < a implies 0_1(m) C

.

0 (R) we have that 1/(t-z) E 0 (l) for all a > -1. every T E O'(C)

C(T;z)

,

Hence for

a > -1, the Cauchy integral

2ni



(2.30)

is well defined for z E A; in fact we know that C(T;z) is an analytic function of z in C\supp(T) [11, p.56]. Let us now obtain the analytic representation of elements T E O'(C), a > -1, in terms of the Cauchy integral.

If T E O'(C), a > -1, then

THEOREM 2.4.1.

lim F- -4

(C(T;x+ie) - C(T;x-ie)) v(x) dx =

(2.31)

J,--Co

and

lim e-*0+

CO

f-m(C(T;x+ie) + C(T;x-ie)) v(x) dx = -2

(2.32)

for all w E D(R) where fi(t) is the principal value integral x

1

w(t)

2ni

F-.

x-t

dx

It suffices to prove the theorem for a = -1 because of the inclusion 0'(C) C 0'1(C) for a > -1. As in the proof of PROOF.

73

Theorem 2.3.7 we can approximate the integral

C(T;x+ie) p(x) dx,

= J

E D(M), e > 0

4p

by Riemann sums and exchange summation and application of T.

We obtain n

= nlim 0 fixed.

Every function Vn(t-ie)

has derivatives of all order on IR with respect to t E IR which

are given by

nP)(t-ie)

1

=

n

p! p(xk)

2ni

(xk - xk-1

(xk-(t-ie))P+1

k=1

CCO

p = 0,1,2,...

.

This shows that ;n (t-ie) E

function of t E U for each n.

(P)(t-ie) n

=

In addition we see that

0((l+Itl)-1) for each fixed n and all p

0,1,2,... as Itl - w of t E IR for n = 1,2,...

I(1+Itl)

=

Thus Pn(t-ie) E 0-1(IR) as a function .

Also the inequalities

nP)(t-ie)I < Mp,e

,

hold for n = 0,1,2,... and t E IR 74

(l) as a

p = 0,1,2,...,

.

By Lemma 2.3.1, for each

fixed e > 0 the sequence (Ppn(t-iE)) converges in 0_1(IR) to

'P (t-i6 )

f

1

=

27ri

Px

x- (

-ro

-i

)

dX

By continuity of T on 0_1(R) we get

= 0, the continuity of the function (t VP) (t)) on lt, and the maximum modulus

principle for a horizontal strip together imply the inequalities 75

I(l+Itl) w(P)(t-ie)I ( MP

0 < e < rj for t E IR positive real number. ,

p_(t) as e - 0+

,

and p = 0,1,2,... with n being a Thus .p(t-ie) converges in 0_1(IR) to

This permits us to write

.

lim = 0+

e->0+



.

Repeating the previous process for the integral

= J C(T;x-ie) p(x) dx, v E D(Qt) -00

we obtain

u rn

=

.

Applying once more the Plemelj formulas (2.12) and (2.13) to the Cauchy integrals

w(tfie) =

2Tr1

i

00

x

(tfie)

dx

as in the proof of Theorem 2.3.7 we obtain the relations (2.31) and (2.32). The proof is complete. Sometimes it is useful to have Theorem 2.4.1 formulated without the integral forms in (2.31) and (2.32) as in the following restatement of this result. THEOREM 2.4.2.

Let T E Oa(IR), a > -1

lim C (T; xf ie ) 6-40+ 76

=

T}

exist in D'(IR) with

T+

-T =T

and

T+ + T

(T * vp

iri

=

1

)

.

By replacing the space D(IR) in Theorems 2.4.1 and 2.4.2

with certain of the spaces 0a(R) we obtain new theorems in which we need the following lemma whose proof is similar to that of Lemma 2.3.2. LEMMA 2.4.1.

(z) =

Let

2ni

t(t) dt, z E A

E O' (IR), a < 0

I

and let A

VX(z)

- 2,ri

J-

x

t(t) -z dt

The sequence 0,1,2,...

,

zEA

, l

>0

converges uniformly to p(p)(z), p = on every compact subset K C A as T

COROLLARY 2.4.1.

O

For every fixed e > 0 the sequence

(pA(xiie)) converges in 0a(IR), -1 < a < 0, to !p(xfie), respectively, as A --> m PROOF.

.

It suffices to prove this corollary for a = -1 since

the convergence in 0_1(IR) implies the convergence in 0a(R) for a > -1.

Let e > 0 be fixed.

For each nonnegative integer p

there exists a constant Mp, which is independent of X, such that

77

_

where

M

(xfie) l

gy(P)

xEf

here means the pth derivative of p,(xfie) with

(This follows from respect to x E IR for e > 0 being fixed. 0(1/Izl) as Izi -i - that the order relation ;p)(z)

characterizes the behavior of the Cauchy integral when the integration is defined on a compact subset of I2, in our case For a < 0 Lemma or on a simple closed curve in C.) converges uniformly 2.4.1 shows that the sequence for each to (p)(xfie) on every compact subset of C as X -> Consequently all conditions for convergence in 0_1(R) are p.

The proof is complete.

satisfied.

If the Cauchy integral in Theorem 2.4.1 satisfies a certain growth, then 0a(IR), -1 < a < 0, can replace D(IR) in this

result as we now show. Let T E Oa(li), -1 < a < 0, and C(T;z) _

THEOREM 2.4.3.

we have

0(1/Izl) as Izi -->

lim

m

(C(T;x+ie) - C(T;x-ie)) p(x) dx = -oo

and

lim

e-0+

(C(T;x+ie) + C(T;x-ie)) p(x) dx = -2 -00

for all p c 0a(U2) where ap(t) is the principal value integral

1

2ni

PROOF.

( ,J

xP

x-tx

dx

The hypothesis C(T;z) = 0(1/JzI) ensures the

convergence of the improper integrals

78

F J_f ao

C(T;xfie) p(x) dx

,

pp E 0a(IR), -1 < a < 0

In fact, because the integrands behave as

lxl-1+a

these

integrals exist and C(T;xtie) are regular distributions in Oa(IR). The remainder of the proof follows along the same lines as that given in Theorem 2.3.7 with an appeal to Lemma 2.4.1 and Corollary 2.4.1.

The following result is equivalent to Theorem 2.4.3 in the same way that Theorem 2.4.2 is equivalent to Theorem 2.4.1. THEOREM 2.4.4.

Let T E 0'(IR), -1 < a < 0, and C(T;z) =

0(1/lzl) as IzI

---4 W

.

C(T;xf1e) = T

exist in 0'(IR) with

T+

-T =T

and

T+ + T

=

(T * vp

it i

R

The Plemelj relations for the distribution vp

are given

x

in [114, p. 359] by definition.

Using Theorems 2.4.2 and 2.4.4 these relations are now proved in the following example. EXAMPLE 2.4.1. The Cauchy integral C(vp t ;z) of the

principal value distribution vp

t

, which is an element of

0a(IR), a < 0, is C(vp

1

t ;z

_

1

2Tri

-1, the case for arbitrary a E

80

Ot

is treated in the following two theorems. THEOREM 2.4.5.

Let T E O'(IR), a E M.

chosen such that for z E A Iti -> m The function

Let k = 0,1,2,... be

1/(t-z)k+l = 0(Itla) as

,

.

F (z )

k'

=

i 0 we have 1/(t-z) = 0(ItIa) as Itl --> -

.

For

Thus for

arbitrary a E IR there exists a nonnegative integer k such that 1/(t-z)k+l E 0a(1k) as a function of t E R This shows that .

the function F(z) is well defined for z E C\supp(T).

One

proves the analyticity of F(z) as for the Cauchy integral of a distribution in E'(IR).

The proof is completed by the use of

Lemma 2.3.1 and Theorem 2.3.7 essentially in the same way as 81

in the proof of Theorem 2.4.1.

We obtain

=

= 0.

equalities converge to

- _(t)>

=

-k-1.

We have 0-k-1(IR) c

a > -k-1, and the function [4_}(z) is well defined.

0a(IR),

Now

proceeding as in the proof of Theorem 2.4.1 we obtain

=

< P I (x-ie), P(x)> = w and if this boundary value is equal to the zero distribution on an open subset of R PROOF.

,

then T = 0

.

The proof is obtained by a slight modification of that

Let f+(z) satisfy the hypotheses in used for Corollary 2.2.1. In this case we know that (2.33) holds. By the A hypotheses and (2.33) we have T = f+ = T open subset R C R where T C(T;z) = 0, z E A

is the boundary value for

from (2.33).

,

= 0 in DI(R) on an

We conclude that C(T;z) is

analytic in R and that f+(z) is the analytic continuation in This implies A+ of the function C(T;z) = 0, z E A The proof is the same f+(z) = 0, z E A+ ; hence T = f+ = 0. .

if a function f (z) which is analytic in A

is considered by

using (2.37).

Using Theorem 2.4.3 we can establish three new theorems in the Oa(R) topology for -1 < a < 0 which are REMARK 2.5.1.

similar to Theorems 2.5.1, 2.5.2, and 2.5.3. As examples of the analysis of Theorems 2.5.1 and 2.5.2 and

Remark 2.5.1 let us consider the Heisenberg delta distributions [114, p. 331] lim s+

E-40+

-1

2ai(x+iE)

and

b

=

lim E-40+

1

2ai(x-iE)

These two distributions, which are in 0'1(R) and are the 86

boundary values of f+(z) _ -1/2,riz

f (z) = 1/27iz

z E A

,

1

1

x+10

+

,

* VP

1

x+i0

,ri

,

z E A+

,

and

respectively, satisfy the formulas 1

= 0

x

and _

1

x-i0

1

1

1

1

1

* vp x - 0

x-i0

Ti

In the remainder of this section we shall describe the behavior of the derivatives of the Cauchy integral of

distributions as Im(z) -> Of C(T;z)

211

0, as Its --1 and if the Cauchy integral of T(t) vanishes in A+ or A then .

T(t) vanishes on R

.

91

THEOREM 2.6.4.

Let f(z) be sectionally analytic in C except

for a finite number of poles at ak, k = 1,2,...,n, of order ak located in A+ U A closed set K C IR

and with a boundary on IR consisting of a

Let f(z) = 0(1/Izj) as IzI

.

lim f (x+ie ) e-40+

=

--> w

.

Let

f+

and

lim

f(x-ie) = f

in D'(IR)

Then for z f K and different from the ak,

.

k = 1,...,n, the function f(z) has the representation ak

n f( z) z)

+

+

k=1 p=1

(z-ak)

D

(2.50)

where a

m

Ak,ak-m =

PROOF.

m!

z->ak

((z-ak)

k

f(z)), m =

k-1.

dzm

Put T = f+ - f

First we shall prove that supp(T) C To do this, first consider K to be a closed proper subset

K. of IR

.

.

Since the function f(z) is analytic on the open set

IR\K we have



=

lim0+

= lim

for all N E D(IR) with support in IR\K

0 and yl > 0) and

.

yl - y2-

It

= (y: y2 > 0 and y1 < 0) are the forward light cone and the

We associate with the

backward light cone, respectively.

light cones the following domains in n dimensional complex space Cn: TG

utn+iG+ is the forward tube domain and TG

=

_ Otn+iG

=

+

Here z E TG

is the backward tube domain.

example, means z = x+iy with x = (x1,x2,...,xn) E (y1,y2,...,yn) E G+ x2+iy2,...,xn+iyn).

,

for

,

n

y =

,

and z = (z1,z2,...,zn) = (xl+iyl,

Now let f+(z) be analytic in TG

f -(z) be analytic in TG

and

(For the definition of an analytic

.

function of several complex variables see section 4.2.) 1n we have Assume that on an open set 0 C lim

lim

f+ (x+iy) = y-+0_ yEG

f_(x+iy) ,

x E ft C

Mn

yEG

in D'(IR) where 0 = (0,0,...,0); then there exists a function

f(z) which is analytic in TG

U ft U TG

and satisfies f(z) _

_

+

This is the and f(z) = f (z), z E TG f+(z), z E TG famous "edge of the wedge" theorem; a sketch of the proof for ,

.

a variant of it is given in [2, pp. 110 - 111] where further references on this problem can be obtained. See also [75, pp. 244 and 256] and [115]. n = 1 the set TG

U c U TG

Let us observe that for

is a domain; for n > 1 it is not.

Hence it is perhaps not desirable to regard Theorem 2.2.2 as we give an analogue of the edge of the wedge theorem. consideration to analytic functions of several complex 107

variables and distributional boundary values in Chapters

4 - 6. The distributional relations presented in sections 2.3 and 2.4 have been developed gradually over a period of time.

It

should be noted that two different analytic representations in the sense of Definition 2.3.2 are given by C(T;z) and C(U;z) where U = (-1/,ri)(T * vp(1/t)). Hence we may speak of the

first Plemelj relation as an analytic representation of the distributions T and U. To our knowledge the first results which extend the classical Plemelj relations in a general way to the setting of

Schwartz distributions are found in the references [2] and [3] of Beltrami and Wohlers, in the reference [114] of Roos, and in the references [79], [80], and [81] of Mitrovic.

The

overlap of the results in [2], [3], and [114] with our results here is small.

Among other contributions in this area we

select the following result of Orton [98].

Let U E D'(IR). As

we know, a function ?(z) which is analytic in C\supp(U) such that lim e-*C)+ f-CO

?(x-ie)) v(x) dx = , w E D(IR),

For such an analytic representation, Orton has shown in [98, Lemma 2.1] is called an analytic representation of U

.

that

lim

i(?(x+ie) +(x-ie))

exists in D'(IR) and defines a continuous linear functional on D(IR).

This limit is defined to be the Hilbert transform of U

relative to ?(z) and is written YF?(x).

distributional limits lim e-40+

108

?(x+ie) _ ?+(x)

Consequently the

and

lim e)0+

(x-ie) __(x)

exist and satisfy the relations of Plemelj type ?+(x) =

*?(x)

1

U +

2

2i

and

U +

2

1

2i

*?(x)

as can be seen in [98, Lemma 3.1].

We conjecture that distributional Plemelj relations exist corresponding to the Cauchy integral n

C(T;z) =

1

n

1.

> j

Plemelj relations in the

n dimensional setting should be considered in future research. Let h(t) be a given Holder continuous function on a simple closed smooth curve L.

Theorems 2.5.1 and 2.5.2 are motivated

by the following question [52, pp. 40 - 41]: what conditions must be satisfied by the function h(t) so that it will be the boundary value of some function h(z) which is analytic in the interior domain G of L and continuous on G U L? The classical version of the formulas (2.45) and (2.46) derived in section 2.5 is treated in Gakhov [52, pp. 42 - 43].

A theorem similar to Theorems 2.6.1 and 2.6.2 is found in [2, p. 66] but with conditions which differ from those involved in our theorems.

Theorems 2.6.5 and 2.6.6 are motivated by the following result from the theory of the Cauchy integral. Let h(t), For h(t) being Holder be a complex valued function. t E f ,

109

continuous with compact support K

fi(z)

=

We have (i)

21 i

J

t)

dt

,

,

consider the function

z E C\K

-CO

Fi(z) is an analytic function in C\K; (ii)

h(z)

has boundary values H+(x) and H -(x) on IR as Im(z) - Of, respectively; and (iii)

h(z) = 0(1/Izl) as Izi ->

.

Conversely, any function E(z) which satisfies the conditions (i), (ii), and (iii) is the Cauchy integral of some function See also [2, Theorem 3.3] and h(t) with supp(h) = K C IR .

[95].

Theorem 2.6.7 is influenced by [133, Theorem 97, p. 130] In [2, Theorem 3.6] a decomposition of and [2, Theorem 3.6]. strip analytic functions into the difference of two distributional Cauchy integrals involving the S'(IR) topology

Theorem 2.6.7 is a version of this boundary value theorem which is proved using the D'(IR) topology. is established.

The basis of the results presented in section 2.7 is the note [94] where some additional comments on the topic of section 2.7 can be found.

110

3 Applications of distributional boundary values INTRODUCTION

3.1.

The generalization of the boundary value problems of Plemelj,

Hilbert, Riemann-Hilbert, and Dirichlet to the setting of Schwartz distributions is the subject of section 3.2. In the first two problems we desire to find a function which is sectionally analytic in the complex plane C cut along the real

axis M, which has boundary values on Ut in the DI (R) topology from A+ and A and which satisfies a given boundary In the last two problems of section 3.2 we ,

condition.

construct an analytic function in A+ which has boundary values on Ut in D'(Ut) from A+ and which satisfies a given boundary

The distributional Plemelj relations and distributional principle of analytic continuation will provide condition.

the main tool in solving these problems. For completeness we briefly describe the classical version of these problems following the nomenclature of Muskhelishvili [97].

(See also Gakhov [52].)

Plemelj.

We begin with the problem of

Let G(x) be a given complex valued function on Ut

which is Holder continuous on compact subsets in U2 and

satisfies a Holder condition at infinity.

(Later in this

section we recall the definitions of Holder continuity and of The problem is to find a function F(z) vanishes at which is sectionally analytic in the domain C\R a Holder condition.)

,

infinity, and has boundary values F+(x) and F -(x) on Ut which

satisfy the condition F+(x) - F -(x) = G(x) on R.

We now state the Hilbert problem.

Let G(x) have the

properties in the previous paragraph and assume that G(x) - 0 on Ut including the points at infinity. Let g(x) be a given function on Ut which is Holder continuous on compact subsets of Ut and satisfies a Holder condition at infinity. Find a

function F(z) which is sectionally analytic in the domain C\Ut

,

vanishes at infinity, and has boundary values F+(x) and

F-(x) on Ut which satisfy the boundary condition F+(x) = 111

(G(x) F (x)) + g(x) on Ot

.

In the Riemann-Hilbert problem a function F+(z) is desired and bounded which is analytic in A+ , continuous on A U R ,

at infinity, and which satisfies the boundary condition Re((a(x)+ib(x)) F+(x)) = c(x)

,

x E IR

,

(3.1)

where a(x), b(x), and c(x) are given real valued functions on These functions are assumed to be Holder continuous on R. compact subsets of R and to satisfy a Holder condition at infinity; we also assume that (a(x))2 + (b(x))2 # 0 on l and In particular by putting a(x) = 1 at the points at infinity. and b(x) = 0 in (3.1), the boundary condition of the Dirichlet is obtained. As previously noted problem, Re(F+(x)) = c(x) ,

these classical problems of Plemelj, Hilbert, Riemann-Hilbert, and Dirichlet will be stated and solved in the context of distributions in section 3.2.

In section 3.3 we study the distributional version of equations of the type

a(x) (x) +

Vi

bb (x)

J

t(t) -x

dt = f(x)

(3

.

3

.

2)

L

and

a x (

)

y x (

)

-

r

1

L

b(t) V(t)

dt

=

h (x).

(

3

)

t-x

Here p(x) and y(x) are unknown functions; while a(x), b(x), f(x), and h(x) are given functions which satisfy certain conditions on the particular choices of the smooth curve L. Equations (3.2) and (3.3) have great importance in the general theory of singular equations with Cauchy type kernels. The key elements for the extension of equations (3.2) and (3.3) to distributions are the distributional Plemelj relations.

Now we state some notation and results involved in our discussion in this chapter. We encountered Holder continuous 112

functions and Holder conditions in Chapter 2, but now we need some technical notation concerning such functions.

We first

recall that a function f: C --> C is said to be Holder

continuous (or to satisfy a Holder condition) on a bounded closed interval [a,b] C C if for any two points x1 and x2 in [a,b] there exist two positive numbers M and v such that If(x2) - f(xl)I

where 0 < v




which satisfies the

order relation f(x) - f(OD)

= 0(1/Ixla)

for some a > 0 as Ixi - If(x) - f(o)I
0+ F (x-ie ) = T

in the D'(O) topology.

-F

In addition the function F(z), which

is analytic in C\supp(T), has a Laurent expansion at infinity.

Since F(-) = 0 it follows that F(z) = 0(1/Izl) as Izj -i Hence H(z) = 0(1/Izl) as Izi - m On the other hand we have .

.

= -

= - = 0

Hence = for all p E D(IR).

This implies, according to the principle of analytic continuation given in Theorem 2.2.2, that the function H(z) is analytic in C; that is, H(z) is an entire function. Since = 0, by Liouville's theorem we have H(z) = 0 in C Consequently C(T;z) = F(z), and the solution of the problem

H(co)

.

given by (3.7) is unique. CONSEQUENCE 3.2.1.

Every T E E'(IR) can be represented

uniquely as the difference given in (3.6) where the distributions T+ and T are boundary values in the D'(IR) topology of the Cauchy integral C(T;z). A meromorphic version of Problem 3.2.1 can be stated as follows.

PROBLEM 3.2.2.

Let T E El (R) be given.

Find a function F(z)

which is sectionally analytic in C cut along supp(T), has a pole of order m at infinity, and has boundary values F+ and Fin the D'(IR) topology which satisfy the boundary condition F+ - F

SOLUTION.

= T

.

(3.8)

Let us introduce the function

117

H(z) = F(z) - C(T;z)

.

Using the boundary conditions T+ - T

that

= T and (3.8) we get

By the generalization

for all %p E D(IR).

of Theorem 2.2.2 indicated in the third paragraph of section 2.8 with n = 1, the function H(z) is analytic in the whole of Since this point is a pole of C except at the point z = .

order m for H(z), by the generalized Liouville theorem H(z) is Therefore the general solution of a polynomial of degree M. the problem is given by 1

1

F(z) =

2 7ri

+ Pm(z)

where Pm(z) is an arbitrary polynomial of degree m. PROBLEM 3.2.3.

Let T E E'(IR) be given.

Find a function F(z)

which is sectionally analytic in the domain C\supp(T) except for a finite number of poles ak, k = 1,2,...,n, of order ak respectively, that are located in A+ U A

,

which vanishes at

infinity, and which has boundary values F+ and F

in the D'(IR)

topology that satisfy the condition

SOLUTION.

Let us define

H(z) = F(z) - C(T;z)

.

By means of Theorem 2.2.2 and the generalized Liouville theorem we obtain that H(z) is a rational function in C with poles at ak

,

k = 1,...,n, which vanishes at infinity with

Hence the general solution of the problem has the representation order 1/jzj.

118

n F(z)

+

ak

k=1 p=1

k.p (z-ak)p

where the Bk,p are arbitrary complex coefficients. Let T E O'(IR), a Z -1, be given.

PROBLEM 3.2.4.

Find a

function which is sectionally analytic in C\supp(T), vanishes at infinity, and has boundary values T+ and T topology that satisfy the condition

in the DI(R)

SOLUTION. Following the solution of Problem 3.2.1 and using Theorem 2.4.2 and the principle of analytic continuation

indicated in the third paragraph of section 2.8 with n = 1, we

obtain the unique solution

Observe that supp(T) of T C O'(R) can be any

to this problem.

closed set contained in O PROBLEM 3.2.5.

Let T E O'(IR), -1 S a < 0, be given.

Find a

function which is sectionally analytic in C\supp(T), vanishes at infinity as 1/IzI when I z I , -

,

and has boundary values

T+ and T- in the Oa(6t) topology which satisfy the condition

(3.9)

SOLUTION.

Theorem 2.4.4 together with Consequences 2.2.1(iii)

stated at the end of section 2.2 yield the unique solution C(T;z) =

2,ri

, z E C\supp(T)

Let us observe that the assumed order relation 1/IzI of the 119

unknown function ensures the existence of the boundary values T+ and T

in Oa(R) according to Theorem 2.4.4. Every T E O'(R), -1 < a < 0

CONSEQUENCE 3.2.2.

,

can be

represented uniquely as the difference (3.9) where the distributions T+ and T

are boundary values in the Oa(l)

topology of the Cauchy integral C(T;z) which vanishes as 1/1zI when IzI -> w

.

We now solve the distributional Dirichlet boundary value problem for the half plane by reducing it to a Plemelj problem.

PROBLEM 3.2.6. analytic in A

Let U be a given real

(Dirichlet)

distribution in E'(IR)

.

Find a function f+(z) which is

vanishes at infinity, and satisfies the boundary value condition ,

=

(3.10)

for all real valued p E D(IR) where f+ is the DI(R) boundary value of f+(z) from A+ on f SOLUTION.

=

lim

for all p E D(l).



f+(x+ie) p(x) dx

l -40+

Further we put

=

,--lim

.0+

Re(f+(x+ie)) p(x) dx J'0 -00

for all real valued w E D(l).

First we prove that the

boundary condition (3.10) is equivalent to the boundary 120

condition

f +f

= 2U

(3.11)

We have



=

+ i

and

< f+, p> = - i for all real valued p E D(IR).

Adding these equalities we

obtain

= = Obviously the converse also

Thus (3.10) implies (3.11). holds.

Now let us introduce the sectionally analytic function

F(z) defined by means of f+(z) as follows: f+(z)

,

z E A+

-f+(z)

,

z E A

F(z) =

Since

and

F

in D'(IR)

lim

=

,

F(X-iE) _

+

the boundary condition (3.11) may be written in the

form 121

F+

-F

= 2U

(3.12)

.

Now we find a solution to the Plemelj problem (3.12); from Problem 3.2.1 the solution is

F(z) =

27ri

= , * E D(IR), where X+ and X

(3.18)

are regular distributions which are the

In fact boundary values of X}(xfie), respectively, in DI(R) since X±(xfi6) converges to the function X±(x) pointwise .

everywhere on O

,

then X}(xfie) converges to the regular

distributions X±(x) in D'(IR) 124

,

respectively, by the Lebesgue

dominated convergence theorem; that is

<X+,

jGo

elim V> = -0+

x+(x+ie) V(x) dx = f

x+ (x) v(x) dx w

0o

and

,p> = lim

<X

F--*O+

J X (x-ie) v(x) dx = rw -W

X- (x) %p (x) dx

for all p E D(Ot).

It is easy to verify that the function X(z)

satisfies (3.18).

Indeed, from the definition of X(z) and

using the first classical Plemelj formulas (2.12) and (2.13)

we obtain

X+(x) = exp(T+(x)) = exp[T-(x) + log[[ xX-i +i

x-i

[ x+i

Y

A

,

G(x)

X_(x)

exp(T_(x))

= G(x)

G(x)

Using this equality we proceed to solve the problem (3.16). Let X = Ind(G(x)) as noted before.

Setting G(x) = X+(x)/X (x)

in (3.16) we get the boundary condition +

F


_


E D (R)

.

(3.19)

Using the original definition of the function X(z) and properties concerning the function G(x), we conclude that X+(x) and X -(x) differ from zero and belong to the space Cw(Ot)

.

This implies that 1/X+(x) and 1/X (x) are multipliers

Therefore the two distributions in (3.19) According to the limits of the are members of D' (R) for D(R) and E(Ot).

.

products in (3.13) and (3.14), we have

125

+

F

P(x)> =

x (x)

lim

F(x+ie)

J -00

e-4

X+ (x+ie)

W

lim

w(x) dx

F(x+ie) X+(x)

e--0+ J_'M

and

F


=

X - (X)

F(x- ie)

lira

e-+0+

X-(X-ie)

lim

F(x- ie)

e40+

X-

p(x) dx

w (x) dx

(X)

for all w E D(IR). Now introduce the auxiliary sectionally analytic function F(z) X(z)

H(z7 =

z E A

,

The function H+(z) = H(z),

and assume X = Ind(G(x)) > 0. z E A+

,

is analytic in A+

,

and H (z) = H(z), Z E A

is

,

analytic in A

everywhere except at the point z = -i where it has a pole of order X w by Theorem 2.2.2 and the generalized Liouville theorem we have that H(z) is a .

,

rational function in C which vanishes at infinity.

F(z) X(z)

=

Hence

P(z) (z+i)X

where P(z) is an arbitrary polynomial of degree m < X-1. F(z) =

X(z) P(z)

,

z E A

.

Thus

(3.20)

(z+i)A

For X = Ind(G(x)) = 0 the auxiliary function takes the form

126

H+(z)

F (z) exp(T+(z))

=

,

z E A+

,

H(z) = F (z)

H (z)

zEA

exp(T (z))

For A = Ind(G(x)) < 0 we have

+' z) =

F+(z) exp(T+(z))

,

z

E A+

H(z)

[z+ii'

z-i Y

F- (Z)

In both cases F+(z) = F(z), Z E A+ z E A

;

,

exp(T (z) ) ,

z E A-

and F -(Z) = F(z),

further, in both cases H+(z) and H -(z) are analytic

functions in A

and A

,

respectively.

Also =

Using

A > 0 we obtain in both cases Hence F(z) = 0 in C in C the general solution to the For A S 0 the given by (3.20). .

problem has the trivial solution F(z) problem (3.16) has only the classical We can now turn to the solution of Hilbert problem (3.15). Suppose that

Thus the = 0 in C. solutions. the nonhomogeneous A = Ind(G(x)) > 0.

Proceeding as before, the substitution G(x) = X+(x)/X (x) in the boundary condition (3.15) leads to the boundary condition

F+

=

X +(x)

in D'(IR)

.

F

+

X (x)

Ti

X+

(3.21)

(x)

We know already that the three quotients in (3.21)

are members of D'(IR)

.

In particular the distribution U/X+(x)

is a element of E'(IR) and its support is precisely supp(U).

Put T = U/X+(t) and introduce the Cauchy integral

127

C(T;z)

tlz >, z E C\supp(T)

0 given by (3.23) is also the solution for X = 0 when P(z) = 0; that is, F(z) given in (3.24) is the general solution to Problem 3.2.7 when X = 0. For X < 0 the solution is given by (3.24) if and only if the equalities in (3.25) hold. This concludes the solution to Problem 3.2.7.

Perhaps it will be helpful to consider the following problem which is a variant of Problem 3.2.7. PROBLEM 3.2.8. (Hilbert) Assume that the function G(x) and its derivatives and the distribution U have the same properties as in Problem 3.2.7. sectionally analtyic in 0\IR

IzI - w

,

,

Find a function F(z) which is is of order 0(1/Izl) as

has boundary values F+ and F

for -1 < a < 0

,

in the Oa(IR) topology

and satisfies the boundary condition (3.15).

The solution is the function F(z) given by (3.23); since F(z) satisfies the boundary condition (3.15) in D'(IR)

SOLUTION.

and has order 0(1/Izl) as IzI --- w then by Lemma 2.2.1 F(z) satisfies (3.15) in Oa(IR) for any a

,

-1 < a < 0.

Although

the direct solution is almost the same as the one we have given for Problem 3.2.7, some comments are necessary here. First, the boundary condition (3.15) is well defined in Oa(IR) since the function G(x) is a multiplier for Oa(IR) and U E E'(l) C O (IR) for all a E R.

Secondly, the boundary condition

(3.21) is well defined in O'(IR) since the functions 1/X+(x)

and 1/X (x) are multipliers for OQ(U).

Lastly, the boundary

condition (3.22) is implied by the first Plemelj relation from Theorem 2.4.4. From this point to the end of this section Dr(IR) will denote those elements in D(IR) that are real valued, and D.(IR) 130

will denote the elements in D'(R) that are real valued on Dr(R).

Let a(x) and b(x) be given PROBLEM 3.2.9. (Riemann-Hilbert) real valued CW(R) functions which satisfy, together with all of their derivatives, a Holder condition at infinity that depends on the order p of the derivative, p = 0,1,2,... In addition assume that the function (a(x)) 2 + (b(x))2 does not .

Further, let U be a given real distribution in

vanish on

Find a function F+(z) which is analytic in A+ bounded at infinity, and has a boundary value F+ in the Dr'(R)

E'(R)

.

topology which satisfies the boundary condition Re((a(x)+ib(x)) F+) = U SOLUTION.

(3.26)

.

Put

F+ = Re(F+) + iIm(F+) and

F+ = Re(F+) - iIm(F

where f00



e-0+

F+(x+ie) p(x) dx _W

and



P-lim =

f-COF+(x+ie)

fi(x) dx

-.)O+

for all p E Dr(R).

Using familiar operations with

distributions it is easy to show that the boundary condition (3.26) is equivalent to the boundary condition

131

(a(x) Re(F+)) - (b(x) Im(F+)) = U in Dr(M)

(3.27)

Also this later condition is equivalent to the

.

boundary condition

+ (a(x)-ib(x)) F

(a(x)+ib(x)) F in Dr(IR)

= 2U

(3.28)

Thus the condition (3.26) is equivalent to the

.

condition (3.28).

Now let us introduce the function F(z)

defined by F+(z)

,

z E A+ (3.29)

F(z) _

F (z) = F+(z)

zEA

From the definition of F(z) we have for all p E Dr(R) that

= lim

F (x-ie) p(x) dx

lim J

hence F

=

F+ in Dr(IR)

F+(x+ie) p(x) dx

.

=

In this notation the boundary

condition (3.28) takes the form (a(x)+ib(x)) F+ + (a(x)-ib(x)) F

= 2U

,

or

F+

= G(x) F + T

in Dr (lt) where

132

(3.30)

-a(x)+ib(x) a(x)+ib(x) and

T =

2

a(x)+ib(x)

U

Thus by introducing the function F(z), the Riemann-Hilbert problem is reduced to the Hilbert problem (3.30); now we desire to find a solution to (3.30) which is bounded at For simplicity it will be assumed that X = Ind(G(x)) = 0. It is not hard to show the following three infinity.

facts which are needed in our solution. Fact 1.

If a sectionally analytic function F+(z), z E A+

F(z) =

(3.31)

F (z), z E A

,

can be represented in the form (3.29) then

F(z) = F(z)

,

z E A = A

U A

.

(3.32)

Conversely, if the sectionally analytic function defined in (3.31) has the property (3.32) then it has the representation (3.29).

Fact 2.

A solution F(z) of the Hilbert problem is a

solution of the Riemann-Hilbert problem if and only if

F-(z) = F+(z), Z E A-

.

Combining Fact 1 with Fact 2 we obtain a further result as follows.

Fact 3. A solution F(z) of the Hilbert problem (3.30) is a solution of the Riemann-Hilbert problem (3.26) if and only if

F(z) satisfies (3.32). 133

According to the discussion of the Hilbert problem (3.15) (see (3.23)), the general solution of the problem (3.30) which is bounded at infinity will be given by U

F(z) = X(z)

a1


+ K

(3.33)

(t)

Let us mention that X(z) _

exp(r(z)) here with r(z) =

1

2wi

I.

log(G(t))

t-z

dt =

2r 1

-W

arg(G(t)) t-z

dt

Observe that logIG(t)l = log(1) = 0. For F+(z) = F(z), z E A+ to be a solution of the original Riemann-Hilbert ,

problem, we must prove by Fact 3 that F(z) satisfies (3.32). To this end note that the integral r(z) is real for z real and r(z) = r(z) for z E A

This implies that X(z) = X(z) for Z E A Thus the function X(z) satisfies the homogeneous Riemann-Hilbert boundary problem .

.

(a(x)+ib(x)) X+(x) + (a(x)-ib(x)) X+(x) = 0

in Dr(IR)

; or

(a(x)-ib(x)) X+(x) = -(a(x)+ib(x)) X+(x) in Dr(QR)

.

Using (3.33) we have U

F(z) = X(z) 'ri [_1

Thus

134

(3.34)

t

1

(a(t)+ib(t))X+(t)

t-z

U

-1

F(z) = X(z)

ni

> + K

t

1

(a(t)+ib(t))X+(t)

t-z


0. X+(z)

X

=

exp(r+(z))

Z-i

(z)

,

Let

z e A

exp (r (z)), z E A

,

where

r(z)

2a1

=

i

F

log[[ t+ii

J

K(t)J

t-z dt, z E A

with K(t) defined by (3.41), f(z) = r(z) for z e A The substitution r (z) = r(z) for z e A

,

and

.

137

X+(x)

a(x)+b x a(x)- (x)

=

X(x) in (3.40) gives


0 that

F+ = X+(x)

U

2 X+ (x)(a(x)-b(x))

138

-

U

1

2ni

* VP

x+(x)(a(x)-b(x))

x

+

P(x)

(x+i)X

and

F

-U

= X- (X)

2 X+(x)(a(x)-b(x)) U

1

2ai

in D'(IR)

.

1

v P -x

+

x+(x)(a(x)-b(x))

P(x)

(x+i)x

Consequently from (3.38) X+(x) + X -(X)

T =

*

U

(3.45)

2X(x)(a(x)-b(x)) X+(x) - X-(x) 2wi

U

X+(X)(a

+ (X+(x) - X -M)

* VP (x)-o(x) )

P(x) (x+i)1'

The fact that T E O'(IR), -1 < a < 0

remains to be shown.

,

The first term on the right side in (3.45) is a distribution in O'(IR) for all a E R. The second and third terms are

a

distributions in 0; (IR) for all a < 0

for all a < 0.

.

This implies T e

Since T generates the Cauchy integral (3.36) Note -1 a < 0.

for all a > -1 it follows that T E O'(R)

,

that supp(T) C I The formula (3.45) giving the general solution of (3.35) .

for X > 0 also gives the solution for X < 0 if we put P(x) _ and assume that the necessary and sufficient conditions (3.44) are satisfied when X < 0. The solution of 0, x e IR,

Problem 3.3.1 is complete. REMARK 3.3.1. Assume that the hypothesis in Problem 3.3.1 on the functions a(x), b(x), and their derivatives, and on U 139

If the equation (3.35) is given in 0a1(R) for an a with

hold.

-1 < a < 0, we remark that the solution (3.45) remains valid. Now let a*

(x) =

acx) (a(x))2-(b(x))2

b*(x) =

b(x)

(a(x))2-(b(x))2 and

Z(x) = X+(x)(a(x) - b(x)) = X-(x)(a(x) + b(x))

.

After a simple transformation the solution (3.45) may be written as *

*

T = a (x) U -

Z(x) b (x) ni

U Z(x)

*

VP

1

x

+ 2 Z(x) b*(x)

P(x) (x+i)

For example if U = 6 and X = 0

,

the solution of (3.35) has

the form *

T = a * (0)

EXAMPLE 3.3.1.

6-

Z(x) b (x) aiZ(0)

vp

(Hilbert transform).

1

x

If U is a known

distribution in E'(IR) then the solution of the equation W1

IT * in D' (IR) T 140

x ] =U

(3.46)

{U*VP_1_j

(3.47)

vp

is ni

For a(x) = 0 and b(x) = 1 on R the equation (3.35)

PROOF.

In this case K(x) = -1 on ut

takes the form (3.46). Ind(K(x)) = 0

r (z) =

and we may assume log(K(x)) = ,ri

,

ni

1

27ri

t-z dt

-CO

=

_

.

n2

z E A+

ni

zEA

2

,

X =

Thus

Further, by definition X+(z) = exp(r+(z)) = i

z E A+

,

and

X (z) = exp(F (z)) = -i

z E A

,

From (3.45) we now obtain (3.47). To verify by calculation that T given by (3.47) is a

solution to (3.46) we substitute this T into (3.46) and use Example 2.4.1 to obtain

n1

f

(U * vp

ni

x

)

1

n2 1

n

in D' (Ot)

x ]

[u* [vp__*vp__.J} IU *

I- n2

S)

J

=U*b=U

.

EXAMPLE 3.3.2. 2T +

in O' (IR)

2

* vp

n1 ,

Find the solution of the equation

[T*vp_-_]

= 6

-1 < a < 0 141

In (3.35) we have a(x) = 2 and b(x) = 1 on IR

SOLUTION.

Hence K(x) = 3 on IR and A = Ind(K(x)) = 0. X +(z)

= 3 1/2 and X (z) = 1/31/2

T=

3

b-

1 vp 3ai

.

.

These facts imply

From (3.45) we obtain

x

Of course the given

Observe that T E O'(IR) for all a < 0.

distribution U E E'(IR) in (3.35) is U = b here.

In the remaining problem to be considered in this section we solve an equation which is adjoint to (3.35).

Assume that all hypotheses concerning the given functions a(x) and b(x) in Problem 3.3.1 hold. Let B be Find a distribution solution a given distribution in E'(l2) PROBLEM 3.3.2.

.

A of the equation a(x) A -

in D' (IR)

71

I(b(x) A) * vp

= B

x

(3.48)

J

.

SOLUTION. -1 < a < 0.

We seek a solution A that is an element of O'(IR)

First let us observe that the functions a(x) and

The detailed discussion of the

b(x) are multipliers in 0a(IR).

solution of Problem 3.3.1 suggests that we introduce at once the locally analytic function 1

A(z) _



,

z E A

By the Plemelj relations

and

142

a1

[(b(x)

A) * vp x J

in D'(IR), we can show that the equation (3.48) is equivalent

to the following boundary value problem: find a distribution A and a locally analytic function A(z), which vanishes at infinity, that satisfy the conditions

b(x) A = A+ - A and

a(x) A = - A+ - A

+B

Adding and subtracting these two conditions, we obtain the equivalent boundary conditions in D'(IR)

.

(a(x)+b(x)) A = -2A

+ B

and

(a(x)-b(x)) A = -2A + + B

in D' (IR); or A

_

=

2A a(x)+b(x)

+

a(x)+b(x)

2A+ a(x)-b(x)

+

a(x)-b(x)

B

(3.49)

and

A

_ =

B

(3.50)

A comparison of the right side of (3.49) and (3.50) leads to the Hilbert problem A+

=

a(x)-b(x) a(x)+b(x)

A

+

b(x) a(x)+b(x)

B

(3.51)

The coefficient of the problem (3.51) is equal to the reciprocal coefficient of the Hilbert problem corresponding to 143

Hence

the equation (3.35).

X

2iri 1

[[log

l

a(x)+b(x) a(x)-b(x)

JJ-.

Let

Suppose x = -X > 0.

Y+(z) = exp(rX())

and

zZ-i +i

Y -(Z)

exp(FX(z))

)

be canonical functions of the problem (3.51) where rX(z) =

1 2iri

(3.52)

Iw

log[[ t-i 1-X

a(t)-b(t) 1

1 a(t)+b(t) J t-z

t+i j

Since IX(z) = -rx(z)

,

dt, z E A

comparing the present canonical

function, denoted Y(z), with the former one X(z) defined by

r,(z) in the solution of Problem 3.3.1 we obtain

Y(z) =

X(Z1

)

,

z E A

,

z E A+

,

where Y+(z)

Y(z) =

Now according to the relation

144

(3.53)

Y+(x)

=

Y (x)

-

a(x)-b(x) a(x)+b(x)

the boundary condition (3.51) becomes A+

A

=

Y+ (X)

b(x) B

+

Y -(X)

(3.54)

Y+(x)(a(x)+b(x))

Since the logarithm in (3.52) together with all of its derivatives satisfy a Holder condition on compact sets in IR

and a Holder condition at infinity, the Plemelj relations for each derivative of rX(z) hold.

This implies that Y+(x) and

Y (x) belong to the subspace of Ce(lt) functions whose elements are bounded on It and are different from zero; therefore these functions are multipliers in D(R) and in 0a(lt).

Thus (3.54)

is well defined in DI (R) and in O' (lt) . By an argument like that used in the solution of Problem 3.3.1 and using (3.53) we obtain

A(z)

_

1

X(z)

1

X+(t) b(t) < a(t)+b(t) Bt,

1

l 2iri

1

t-z > +

(3.55)

+

Q(z) , (z+i)X

where Q(z) is an arbitrary polynomial of degree less than or equal to X-1.

If X = 0 we put Q(z) = 0.

Now consider x = -X < 0. In this case the function A(z) is also given by (3.55) with Q(z) = 0 in C if the following necessary and sufficient conditions are satisfied:

ax


= 0 1

,

k =

1,2,...,-X

(3.56)

(t+i)k

Finally, let us compute the boundary values A+ and A

of

the function A(z) by the distributional Plemelj relations. 145

Recall the functions a*(x), b*(x), and Z(x) from the paragraph From the equalities preceding Example 3.3.1.

a(x)-b(x) 1

B -

b(x)

(a(x))2-(b(x))2

B = a*(x) B

and +(x)

b(x) -X a(x)+b(x)

= Z(x) b*(x)

and either of the formulas (3.49) or (3.50), we obtain the solution of the equation (3.48) for X > 0 to be A = a*(x) B +

,riZ(1

x)

[(z(x) b*(x) B) * vp x 1 ,

1

Z(x)

2 Q(x) (x+i)'X

(3.57)

Here for K = 0 it is necessary to put Q(x) = 0 on M.

If

< 0

the solution is also given by (3.57) with Q(x) = 0 if and only if the conditions (3.56) hold.

COMMENTS ON CHAPTER 3 The problem of finding a function which is analytic in the 3.4.

plane C with a boundary path L and which has boundary values that satisfy a given boundary condition on L is called a boundary value problem. The classical boundary value problems with a boundary L consisting of a line with infinite length that is parallel to the real or imaginary axis are of special importance in mathematical physics as are similar problems with the boundary values being distributions.

In our

discussion in this chapter we have taken L C C or L = C

Boundary value problems of analytic functions have been studied for a long time from different points of view. Within the past thirty years a theory of boundary value problems in the sense of distributions and generalized functions has been developed. An approach to the study of such problems is 146

developed in [39], [99], [101], [109] - [113], [135], and [138]. A distributional version of the Hilbert boundary value V

problem was first studied by Cerskii [39]; his problem can be stated as follows.

Let L be a smooth, simple closed curve lying in C and let H(L) be the space of all Holder continuous functions on L with H'(L) being its dual; find a solution of

the problem f+ = G(t) f

+ g

(3.58)

where G(t) is a given function on L and g E H'(L) is given. The unknown elements are characterized by f+ = Af+ and f

with A being an operator on H'(L). Rogozin [109] has considered the problem (3.58), among = -Af

V

others, where G(t) is a given element of the space CW(L) of infinitely differentiable functions with the topology of

uniform convergence on compact sets in L and g is a given element in the dual (CW(L))'. The unknown generalized functions f+ and f are subject to the conditions f+ = if+ and f = -If or = 0 and = 0. Here I is a certain integral operator, and c+ and p are elements of COO(L) and are boundary values of functions which are analytic inside and outside of L, respectively. Pandey and Chaudhry [101] present the solution of the

problem (3.58) when G(x) = -1 on l

,

g E D'(l), p > 1, and LP

with the boundary values f+ and f topology.

taken in the D'(IR) LP

The solution is given by

+f(z) =

2ni

=

2-1

+ P(z)

,

z E A+

and

f -(z)

- P(z), Z E A-

where P(z) is an arbitrary polynomial. 147

The multi-dimensional Hilbert boundary value problem for a tube domain TC = IR +iC = (z = x+iy: x E Qtn

where C is an open convex cone in IRn

Vladimirov in [135] and [138]. Let C

= -C+.

,

,

y E C) in Cn

was first discussed by

Let C+ be an open convex cone.

Further, let G(x) be a given CW(Mn) function

and g be a given distribution in S'(Rn).

The object is to

find functions f+(z) and f _(z) which are analytic in TC

and

respectively, and which have boundary values f+ and f respectively, in S'(R n) as y -> 0 that satisfy condition

TC

,

The key element for the construction of a solution is the Fourier transform. (3.58).

The approach to the Hilbert boundary value problem presented in this chapter originates in [84] and [86]. This approach involves the Schwartz distributions, uses the Plemelj relations obtained from the Cauchy integral of distributions as the main tool in proving the result, and preserves the direct connection with the classical theory. The intersection of the material presented in this chapter on this problem with the analysis of the cited papers and others is slight. The study of one dimensional singular integral equations in spaces of generalized functions seems to have originated with the papers [43], [44], [57], and [68] - [70].

Some general results in this area are described in [89], [90], and [105].

The method used in [43], [44], [57], and [68] - [70] is very specific and applicable to convenient classes of generalized functions. For instance, in [69] the solvability of the equation a(x) f(x) + b(x) (Uf)(x) = g(x)

(3.59)

is discussed, where U is the operator defined by Jtf

(Uf)(x) =

,r1

L

(x)

dt

.

Here L is a simple closed infinitely differentiable contour. 148

The space

of test functions on L consists of all CO(L)

functions w with norms tMax 11"II

k

°

EL CIc(t)I,

Ic(1)(t)I,...,IP(k)(t)1),

k

In (3.59) the functions a(x) and b(x) are given in 0 and g(x)

is given in the space 0' of continuous linear functionals The problem is to find f c (generalized functions) on 0 . for which the stated equality (3.59) holds. In the theory of the distributional convolution the equation R * T = V is fundamental where R and V are given distributions and T is unknown. The solution T of this convolution equation can be found if there exists an associated convolution algebra; see [105], [135], [138], and [144].

However, by introducing the distributional Plemelj

relations the equations (3.35) and (3.48) are solved by simpler techniques.

149

4 Analytic functions in Cn, cones, and kernel functions 4.1.

INTRODUCTION

In Chapters 2 and 3 we have been concerned with the representation of distributions as boundary values of analytic functions and applications in one dimension. We now pass to n dimensions in our considerations. In Chapters 5 and 6 we

consider analytic functions in tubes in Cn of which the upper We extend some of and lower half planes are special cases. the distributional boundary value results of Chapter 2 to n dimensions and also obtain boundary value results concerning the D'Lp and S' topologies which were not considered in

Chapters 2 and 3.

The purpose of this present chapter is to introduce several topics in n dimensions that will be needed in Chapters 5 and 6 and to state and prove, where appropriate, general properties of these topics which will be used in the subsequent chapters. Analytic functions of several complex variables will be defined in section 4.2.

Cones in Rn and tubes in Cn and several important properties of them will be discussed in section 4.3.

The Cauchy and Poisson kernel functions

corresponding to tubes in Cn are defined in section 4.4; the Cauchy kernel is an example of an analytic function in a tube defined by an open convex cone.

The Cauchy and Poisson kernel functions are shown to be in several test spaces of functions; subsequently generalized Cauchy and Poisson integrals of certain distributions will be formed and studied as part of the distributional boundary value analysis of Chapters 5 and 6. In section 4.5 the Hardy Hp functions in tubes are defined and several important representation and boundary value As an application of the

results are stated and proved.

Poisson integral representation of Hp functions obtained in section 4.5, we are able to prove a pointwise growth estimate 150

for Hp functions in section 4.6.

The material in sections 4.5

and 4.6 will be important for our work in sections 6.5 and 6.6 of Chapter 6 where we characterize Hp as a subspace of the analytic functions in tubes which obtain S' boundary values and prove Fourier-Laplace integral representations of the Hp functions. Section 4.7 contains some distributional boundary value results in Z' and in S' of analytic functions in tubes in Cn, and recovery of the analytic functions by FourierLaplace transforms is obtained. The results of section 4.7 will be useful throughout Chapters 5 and 6.

The present

chapter is concluded in section 4.8 with some additional comments and references concerning the topics of the chapter and extensions of them.

This section is concluded by stating n dimensional notation that will be used in the remainder of this book. (0,0,...0) will denote the origin in IR n y E

n

tnyn

,

,

is the dot product = tlyl+t2y2+...+

and , t E

defined.

0 =

For t E IRn and

.

n

,

z = x+iy E

n ,

is similarly

Let p denote an n-tuple of nonnegative integers. Pi

denotes the differential operator DR = D1 _ (-1/2ni)(6/atj),j = 1,...,n.

DP

P

D22 ...Dnn where Dj

Similarly we define Do

(ap/atp) denotes the partial differential operator without the constants (-1/2ni) being involved with a similar convention for (ap/azp). t1

For an n-tuple of integers p we define to

...tnn with a similar definition for zp

.

=

when the

components of p are nonnegative integers we also define IPI For Z E en we put pl+p2+...+pn and p! = p1!p2!...on!

=

.

IzI = j=1,.max ...n IzjI z E Cn

An equivalent definition for IzI, IzI

=

(Iz1I2+...+IznI2)1/2

,

is

.

ANALYTIC FUNCTIONS OF SEVERAL COMPLEX VARIABLES A complex valued function f(z), z E Cn, defined on an open 4.2.

subset 0 c Cn is said to be analytic (holomorphic) in 0 if 151

each point w = (wl,w2.... ,wn) E 0 has an open neighborhood N,

w E N C 0, such that the function has the power series expansion f(z) = k

C C,

1' "

k

n) kn akl...kn (z1-w1)kl...(zn-w

n= 0 which converges for all z E N. An immediate consequence of this definition is that if f(z) is analytic in a domain in Cn then f(z) is analytic in each variable separately in a domain The converse is also in C1 corresponding to each variable. true and is known as Hartog's theorem [9, p. 140]; that is, if f(z) is defined on a domain D in Cn and has the property that at each point (w1,.... wn) of D each of the functions f(wl,...,w J'-l,zJ'wJ'+1,...,wn), j

= 1,...,n, is analytic in

zi E C1 in a neighborhood of wj then f(z) is analytic in

We shall use these facts in the succeeding analysis.

z E D.

The following Cauchy type theorem will be needed and it is stated in [135, p. 198]. THEOREM 4.2.1.

If a function f(z) is analytic in a domain D,

if 0 C D is an (n+1) dimensional bounded surface with class C(l) boundary, and if an is an n dimensional piecewise smooth surface then

f

f(z) dz = 0

an

A recurring theme in the remainder of this book will be the study of analytic functions in relation to integral transforms. The following theorem gives conditions under which an integral involving the parameter z E Cn can be concluded to be an analytic function of z;. this theorem is stated in [14, pp. 295-296].

Let µ be a positive or complex measure on a measure space T, and let jµj be the corresponding total variation measure. Let 0 be an open set in Cn Assume that the function f: OxT - C1 satisfies THEOREM 4.2.2.

.

152

(i)

f(z,t) is analytic in z E 0 for almost every fixed

t E T, (ii)

f(z,t) E L1(IµI) as a function of t E T for every fixed z E 0,

(iii)

for every compact set K C 0 there exists a function gK E L1(Iwl) such that If(z,t)I < gK(t) for every (z,t) E KxT.

Then F(z) = f f(z,t) dµ(t)

,

z c O

T

is analytic in 0 and

DZF(z) = f DZf(z;t) du(t) T

for every z E 0 and every n-tuple R of nonnegative integers. 4.3. CONES IN Qtn AND TUBES IN Cn We follow the notation and definitions of Vladimirov [135, section 25] with respect to cones. DEFINITION 4.3.1. A set C C IRn is a cone (with vertex at zero) if y E C implies Xy E C for all positive real scalars X. DEFINITION 4.3.2. The intersection of the cone C with the unit sphere {y E utn: IyI = 1) is called the projection of C and is denoted pr(C). DEFINITION 4.3.3.

If C' and C are cones such that pr(C') C pr(C) then C' will be called a compact subcone of C .

DEFINITION 4.3.4.

An open convex cone C such that C does not

contain any entire straight line will be called a regular cone.

For a cone C O(C) will denote the convex hull (envelope) 1Rn+iC On of C, and TC = If C is open, TC C is a tube in Cn is called a tubular cone; if C is both open and connected, TC is called a tubular radial domain. DEFINITION 4.3.5. The set C* = (t E IRn: > 0 for all y E ,

.

C) is the dual cone of the cone C.

153

For any cone C have C

= C

the dual cone C* is closed and convex.

,

and C

= (O(C))

We

= O(C) C*

DEFINITION 4.3.6.

A cone C is called self dual if

DEFINITION 4.3.7.

The function

uC(t) =

sup yEpr(C)

= C

.

(-)

is the indicatrix of the cone C

.

Further, uC(t)
0, j=1,...,n) is a self dual cone

For each of the 2n quadrants

which we call a quadrant.

= 1 since each C

in Utn we have pC

is convex.

The forward

Fl

and backward light cones, r+ = (y E and r

IRn:

(y2+...+yn)1/2}

y1 >

= (y a ln: y1 < -(y2+...+yn)1/2), respectively, are

very important examples of self dual cones in mathematical physics.

We have p

= 1 = p +

154

.

From [135, p. 222] we have

Iti

u

t E (-r+)

,

,

(t) -

r+

-

1(1/21/2) ((t2+... +t2)1/2 _ t1), t f (-r+)

We now present two very important lemmas concerning cones and their dual cones.

These are proved in [135, pp. 222-223]. We give a separate proof of the second lemma. LEMMA 4.3.1.

Let C be an open connected cone in Rn.

O(C)

contains an entire straight line if and only if the dual cone C lies in some (n-1) dimensional plane. LEMMA 4.3.2. cone in Rn.

Let C be an open (not necessarily connected) Let y E 0(C). There exists a 5 = 6y > 0

depending on y such that

> 61YI M

,

t e C*

.

Further, if C' is an arbitrary compact subcone of O(C) there exists a 6 = 6(C') > 0 depending only on C' and not on y E C' such that (4.1) holds for all y E C' and all t e C* t E C PROOF. Since uC(t) = uO(C)(t) then > 0 for ,

all y E 0(C) and all t E C*.

,

For any y E 0(C) we have (y/IYI)

E pr(O(C)) C O(C) since O(C) is a cone, and O(C) is open since C is. Hence there exists a 6 = 8 > 0 such that the neighborhood N(y/IYI,26) = (y': IY'-(Y/IYI)I < 26) C O(C). Now let t E C* We have y' = ((y/IyI) - S(t/ItJ)) E .

N(y/Iyl,26) C O(C); thus (1/Itl) > 0, t E C* from which (4.1) follows. Now let C' be an arbitrary compact

subcone of O(C).

Define the distance from pr(C') to the

lRn\O(C), by complement of O(C) in In Rn\O(C)) d(pr(C'), = inf(Iyl-y2I: yl E pr(C'),y2 C O(C)) ,

tRn\O(C)); the which is positive here. Put 26 = d(pr(C'), preceding analysis in this proof now yields (4.1) for all y E C' and all t E C* with 6 depending only on C' and not on

155

y E C'.

The proof is complete.

Lemma 4.3.1 will be important below in the construction of the Cauchy and Poisson kernel functions corresponding to tubes and Lemma 4.3.2 yields the very important technical point (4.1) which will be used many times in our succeeding analysis. Let C be an open connected cone in Otn We denote the distance from y E C to the topological boundary of C, aC, by d(y) = inf(ly-y1j: y1 E aC). Vladimirov [136, p. 159] has in Cn

,

.

shown that inf

d(y) =

*



,

(4.2)

Y E C.

tEpr(C

Let C' by any compact subcone of C; from Lemma 4.3.2 and (4.2)

there exists a b = b(C') > 0 depending only on C' and not on y E C' such that 0 < o y

0 be arbitrary. Let g(t), t E Rn be a continuous Rn function on with support in C* which satisfies ,

Ig(t)I

< M(C',m) exp(21r(<w,t>+aI(jI)), t E utn

for all a > 0 where M(C',m) is a constant which depends on C' and on m > 0 and the growth is independent of c E (C'\(C' fl N(O,m))) (that is, the growth holds for all w E (C'\(C'

fl

N(O,m)))) where N(O,m) = (y: Iyl < m). Let y be an arbitrary but fixed point of C. Then (exp(-2,r) g(t)) E Lp for all p, 1 < p < -, as a function of t E utn. PROOF.

We prove part I.

Using Lemma 4.3.2, there exists a b > 0, y = Im(z), such that

= 6 y

II *(t) exp(2,ri)I = I *(t) exp(-2,r) C

C

< I *(t) exp(-2,rbIYIItJ) C

(4.4)

S 1; and (4.4) holds for all z = x+iy E TC and all t c IRn since I *(t) = 0, t C C (4.4) proves the desired conclusion in .

C

the case p = w.

Now let 1 < p < -. Using (4.4), [118, Theorem 32, p. 39], and integration by parts (n-1) times (or

equivalently recognizing the gamma function after the change of variable v = 2,r6plylr) we have

f IR

II *(t) exp(2,ri)Ip dt
0 depending bounded away from 0 by k > 0, say.

only on C' C C such that

Iexp(2rri) I = exp(-27r)

(4.6)

< exp(-2rr6lyl InI) < exp(-2rrbklnl ) where t E Rn and n E C*.

From (4.6) we have

II *(n) exp(2rri) I


))


))) -> 0 pointwise in Tl E

y

y E C, for any fixed w E TC. proof of (4.10) we have

Now let p E S.

I(K(z+w)-K(x+w)) p(x)I < 2 My Ip(x)I

where M

v

,

x E utn,

By the

(4.12)

is the constant of (4.10), and the right side of

(4.12) is an L1 function of x E

n

which is independent of y E C. Another application of the Lebesgue dominated convergence theorem now yields

164

lim y->0 yEC

(K(z+w) - K(x+w)) p(x) dx = 0 J

,

p C S

n

for any w C TC.

This proves the desired weak convergence of K(z+w) to K(x+w) in S' as y -i 0, y E C But S is a Montel space ([46, p. 510] or [135, p. 21].) Hence by [46, Corollary .

8.4.9, p. 510] the convergence of K(z+w) to K(x+w) in the strong topology of S' as y -> 0, y E C, follows from the weak convergence. The proof is complete.

The phrase "supp(g) C C

almost everywhere" in the following lemma means that the set of points in Rn\C* where g(n) x 0 is a set of Lebesgue measure zero. LEMMA 4.4.3.

Let h(t) E LP 5-1[h(t);n] in the LP sense.

,

1 < p < 2, and let g(n) _

Assume that (g(n) exp(2iri))

E L1 as a function of n E On for z E TC and that supp(g) C C* almost everywhere.

Then

g(i)

di1

C PROOF.

= in h(t) K(z-t) dt, z E TC

.

(4.13)

IR

Let z C TC.

The integral on the right side of (4.13)

is well defined for 1 < p < 2 because K(z-t) C D q for all q, L

(1/p) + (1/q) = 1, by Theorem 4.4.1.

Recall Lemma 4.4.1.

Let

p = 1; by Fubini's theorem we have

StIR in

h(t) K(z-t) dt =

_

exp(2iri) do dt

h(t)

J

J

C*

i e2iri I h(t)

C

= J *

e-2iri

dt do

I

g(n)

e2ri do

C

which is (4.13) for p=1.

For 1 < p < 2 g(n) is the limit in

the Lq norm, (1/p) + (1/q) = 1, of the sequence of functions 165

gk(Tl) =

h(t) e -2wl

f

dt

,

k = 1,2,3,4,...

ItI 0, y E C. (4.15) here and also the convergences (4.14) in LP, 1 < p < and in the weak-star topology of LW for p = W.

Now let p E S.

For p = 1 we have 169

I
0 for all y E C. In (4.28) ' E pr(C (Recall the first five sentences of the proof of Lemma 4.6.1.)

4.6.1.

)

,

For in (4.28) we then have 1*1 = 1 and 0


0 is the real number in (4.3) which depends only on the compact subcone C' C C

.

The proof is

complete.

we emphasize that the case p =

was not included in the

statements for Theorem 4.6.1 or Corollary 4.6.1 because the pointwise growth of the Hc(TC) functions is known by definition; the HO(TC) functions are the functions which are analytic in TC and are bounded there. The pointwise growths (4.30) and (4.33) are of independent interest as general pointwise growth estimates for the Hardy Hp functions defined on tubes in Cn

.

They are also of

interest because these estimates will be used in Chapter 6

when we relate the analytic functions in tubes which have distributional boundary values in S' to the HP(TC) functions and when we obtain Fourier-Laplace integral representations of the HP(TC) functions, 0 < p S o 4.7.

.

FOURIER-LAPLACE TRANSFORM OF DISTRIBUTIONS AND BOUNDARY VALUES

In Chapters 5 and 6 two of the main concepts to be considered will be the calculation of distributional boundary values of analytic functions in tubes and the recovery of the analytic functions by a generalized integral transform defined by the boundary value. In some cases we will consider recovery by the Cauchy or Poisson integral of the boundary value; in other cases the recovery will be by the Fourier-Laplace transform of the inverse Fourier transform of the boundary value. Of these three types of recovery, the Fourier-Laplace transform recovery is the most important for our purposes; in many cases recovery of analytic functions from the boundary value by the other integrals can be obtained from the Fourier-Laplace transform representation in a similar fashion that the Cauchy and Poisson integral representation of H2(TC) functions follow from the Fourier-Laplace representation in Theorem 4.5.1. DEFINITION 4.7.1. The Fourier-Laplace transform of a distribution V is the function of the complex variable z = 179

x+iy E en of the form e27i> 0 be a real number. For any real number m > 0 and any compact Mn+i(C'\(C' subcone C' of C put T(C';m) = O N(O,m))) where N(O,m) = (y E In: DEFINITION 4.7.2.

IyI

< m).

We shall say that a function f(z) belongs

to the class H(A;C) if f(z) is analytic in the tube In+iC = and if for every compact subcone C' of C and every

TC

m > 0 there exists a constant M(C',m) depending on C' and on m > 0 such that If(x+iy)I < M(C',m) (1+IzI)N exp(2n(A+a)Iyl),

(4.34)

z = x+iy E T(C';m),

for all a > 0 where N is a nonnegative real number which does not depend on C' or on m > 0.

We study these functions with respect to their boundary The functions H(A;C) and the growth (4.34) are

values.

motivated by associated one dimensional functions and results considered by several authors; we discuss this in section 4.8.

First the Fourier-Laplace transform of a certain type of 180

distribution will be shown to be in H(O;C); this result displays the analyticity of the Fourier-Laplace transform and is needed for subsequent analysis.

Let C be an open connected cone in Rn and let C' be an arbitrary compact subcone of C. Let m > 0 be THEOREM 4.7.1.

Let g(t), t E Itn, be a continuous function with

arbitrary.

which satisfies

support in C

Ig(t)I < M(C',m) exp(2ir( + alcwI))

,

t E Utn

,

(4.35)

for all a > 0 where M(C',m) is a constant which depends on C' and on m > 0 and (4.35) is independent of w E (C'\(C' fl

N(O,m))) (that is, (4.35) holds for all w E (C'\(C' N(O,m))), m > 0.)

fl

Let V = Dtg(t), the distributional

derivative of g(t) of order a with a being an n-tuple of nonnegative integers.

Then f(z) = is an

element of H(0;C). Distributional differentiation, which is valid here, PROOF. yields g(t) e2ri dt

f(z) = za

,

z E TC

(4.36)

.

J

C

To prove that f(z) is analytic in TC it suffices to prove that the integral in (4.36) is analytic in TC which we do by using Let K be an arbitrary compact subset of TC. For z = x+iy E K C TC there is a compact subcone C' of C and real numbers m > 0, b > 0, and d > 0 such that Theorem 4.2.2.

y = Im(z) E (C'\(C' fl N(O,m))) and

(4.37)

0 < m < b < IyI < d

Now let for all y = Im(z) such that z = x+iy E K C TC X = m/b; then 0 < X < 1. For y = Im(z), Z E K, we have .

Xy E C' since C' is a cone; and Ixyl = XIyI = (m/b)IyI Thus for ? = m/b and y = Im(z), z E K, we have > (m/b)b = m .

181

Ay c (C'\(C' fl N(O,m))).

We now choose w = Ay,

A = m/b, y = Im(z) such that z E K, in (4.35); with this choice of w (4.35) yields Ig(t)

e2,ri I

< M(C',m) exP (21r(<Xt> + aIA Yl)) Y.

(4.38)

exp (-2ir) ,

z= x+iy E K C

TC ,

t E IRn

For the chosen compact subcone C' of C we apply Lemma 4.3.2 and (4.37) and obtain a real number b = b(C') > 0 such that t E C* >> SlYlltl > 6bltl for all y = Im(z) such that z E K C TC Using this and (4.37) again, we continue (4.38) as ,

.

lg(t) e2iriI < M(C',m) exP(2iraNlYl) exP(-2n(1-A)) < M(C',m) exp(2iraXd) exp(-21r(1-X)6bltl),

(4.39)

C z= x+iy E K C T, t E C*

Here the real numbers m,b,d, and X = m/b depend only on the compact set K and are independent of z = x+iy E K; b > 0 depends only on C' which depends only on since (1-A) > 0.

K; and (4.39) holds for all a > 0.

The right side of (4.39) is thus independent of z = x+iy E K. The analysis in the concluding estimate in the proof of Lemma 4.4.1(11) shows that the right side of (4.39) when multiplied by the characteristic function of C* is an L1 function over Qtn and recall that supp(g) C C Further, the proof of Lemma 4.4.1(11) yields Mn that (g(t) exp(2,ri)) E L1 as a function of t E for each fixed z E TC Theorem 4.2.2 now yields that the ,

.

.

integral in (4.36) is an analytic function of z E TC; thus f(z) is analytic in TC The desired growth (4.34) with A = 0 remains to be proved.

Let C' be an arbitrary compact subcone of C and let m > 0 be 182

arbitrary.

Let z = x+iy E T(C';m).

Now m/2 > 0 is arbitrary

and w = y/2 E (C'\(C' C N(O,m/2))).

We apply (4.35) with m replaced by m/2 and with w chosen to be y/2 for y E (C'\(C' n

N(O,m))); this together with another application of Lemma 4.3.2 yield Ig(t)

e2ai

S M(C',m/2) exp(2n(a/2)lyl) exp(-nsm tl). (4.40)

z = x+iy E T(C';m), t E C where 6 = 6(C') > 0 depends only on C'

The estimate (4.40) together with the analysis in the concluding estimate in the proof of Lemma 4.4.1(11) show that

If* g(t) C

.

e2hi dtl

(4.41)

e-n6mltl

< M(C',m/2) exp(2a(a/2)lyl)

dt

*

C

M(C',m/2) exp(2a(a/2)lyl) 0n (n-i)! (w6m)-n S M'(C',m) exp(2n(a/2)lyl), for z = x+iy E T(C';m)

where M'(C',m) = M(C',m/2) 0n (n-1)! (a6m)-n

is a constant depending on C' and on m; here a/2 > 0 is arbitrary since a > 0 is arbitrary. (4.36) together with (4.41) yield that f(z) satisfies the growth (4.34) with A = 0.

Thus f(z) E H(0;C) as desired.

The proof is complete.

Distributions which are distributional derivatives of functions g(t) satisfying the hypotheses of Theorem 4.7.1 will Theorem 4.7.1 serves the present purpose of displaying an analytic

be encountered several times in Chapters 5 and 6.

function in the class H(0;C), and it will be used later in our work.

We now desire to obtain the distributional boundary value properties of elements in the class H(A;C).

These analytic 183

functions do not necessarily have tempered distributional boundary values but do have boundary values in the topology of The boundary value in the following result and those throughout this section are obtained as y = Im(z) --' 0, y E C' C C, for arbitrary compact subcones C' of C. THEOREM 4.7.2. Let f(z) E H(A;C), A > 0, where C is an open There exists a unique element U E Z' such connected cone. Z'.

that f(x+iy) -- U in the weak topology of Z' as y y E C; and there exists a unique element V E D' having support in SA = (t E Itn: uC(t) < A) such that U = 5[V] in Z' .

Let C' be an arbitrary compact subcone of C and let m Since f(z) satisfies (4.34) we can choose > 0 be arbitrary. an n-tuple a of nonnegative integers, which depends only on the real number N in (4.34) and hence is independent of C' C C and of m > 0, such that PROOF.

Iz-a f(z)I < M'(C',m)

(1+IZI)-n-e

exp(2,r(A+a)IyI) (4.42)

z = x+iy E T(C',m)

,

for all a > 0 where M'(C',m) is a constant which depends on C'

and on m, n is the dimension, and e > 0 is a fixed positive real number.

Put

g(t) = f z-a f(z)

e-2ri

dx

,

t E IRE,

(4.43)

n

IR

y c (C'\(C' fl N(0,m))).

Because of (4.42), g(t) is a well defined function of t E 1R for each fixed y E (C'\(C' fl N(O,m))) and in fact is a continuous function of t E Qtn.

Let C" be an arbitrary compact

From (4.42), which holds for arbitrary compact subcones C' of C and for arbitrary m > 0, we have subdomain of C.

lim XI 184

r

JC"

Iz-a f(z) e -27riI dy = 0

.

From this and an application of Theorem 4.2.1 to the function (z-a f(z) exp(-2ai)), which is analytic in TC, we have that g(t) is independent of y E C" Since C" is an arbitrary .

compact subdomain of C. g(t) is independent of y E C.

(4.43)

can be rewritten as e-2n g(t)

=

y-1[z-a

f(z);t], z = x+iy E T(C';m). (4.44)

Because of (4.42), (z-a f(z)) E L1 n L2 as a function of x E

n IR

for y E (C'\(C' C N(O,m))) arbitrary.

Thus the inverse

Fourier transform in (4.44) can be interpreted in both the L and L2 sense, and the Plancherel theory for L2 yields z-a f(z) = 9[e-2n g(t);x], z = x+iy e T(C';m),

(4.45)

in L2.

We now show that g(t) has support in SA = (t: uC(t) < A). Let t0 be an arbitrary but fixed point of (t: uC(t) > A).

From the definition of uC(t) in section 4.3 there exists a point y' E pr(C), the projection of C, such that (- - A) > c > 0

(4.46)

.

For the fixed point y' E pr(C) put C' = (y a IRn: y = ay',

X > 0); C' is a compact subcone of C.

Apply (4.42)

corresponding to this compact subcone C' = (y a R n Y = Ay', X > 0) and m > 0 arbitrary but fixed and use (4.43) and (4.46) :

to obtain Ig(t0)I

< M'(C',m) exp(2n(A+a)?Iy'I) exp(2wX m for the arbitrary but fixed m > 0; hence (4.47) holds for all X > m and for all a > 0.

constant M"(C',m), a

,

In (4.47) the

and c are independent of A

,

and c > 0

is the fixed real number chosen such that (4.46) holds. Choosing a < c and letting A --> - in (4.47) we see that g(t0) = 0.

Thus supp(g) c SA since SA is a closed set and t0

was an arbitrary point in (t: uC(t) > A). Now let + E Z and ,p C D such that +(x) = 9[.p(t);x].

The

equality (4.45) holds as an equality in Z' as well as in L2. Using (4.45) and the Fourier transform from D' to Z' we have ,

<e-21r

f(z), y(x)> =

(4.48)

z = x+iy E T(C';m),

where again C' is an arbitrary compact subcone of C and m > 0 is arbitrary. It is easy to see that (exp(-2ir) p(t)) --+v(t) p(t) in D as y--+U, y C C. Since g E D' and (exp(-21r) g(t)) E D', y c C, we have

lim

y-*0

<e-21r

yE C

g(t), 'vp(t)> = =

t

Now put V = Dag(t).

t

(Recall that Da = Dal Dal ... Dan where

DI = (-1/21ri)(8/atj), j = 1,...,n.)

supp(V) c SA = (t E

Un:

1

2

n

We have V E D' and

uC(t) < A) since supp(g) c SA

the Fourier transform from D' to Z' we have « [v], y> = =

.

Using

1)

Jai



dx

_ dx>

= «[g], xa *(x)> =

>

(4.50)

<xa 9[g], %P(x)>

where the differentiation under the integral sign is valid and we have used the fact that (xa y(x)) E Z since y(x) E Z

Now

.

(4.48) holds for each compact subcone C' of C and for arbitrary m > 0; and the n-tuple a, and hence the function g(t), is independent of both C' C C and m > 0. Thus by (4.48), (4.49), the continuity of the Fourier transform from D' to Z'

,

and (4.50) we have for y E Z

lim y->0 = y-..o yEC yEC lim

1im 0 yE C

=

xa P(x)>

= <xa 9[g], `'(x)>

(4.51)

= where we have used the fact that p E Z implies (za y(x)) E Z as a function of x = Re(z) E IRn

.

Putting U = ?[V] E Z'

,

(4.51) proves that f(x+iy) ---> U in the weak topology of Z' as

y ---. 0

,

y E C.

The proof is complete.

A technical point to note in this proof is that V has support in (t: uC(t) < A) since V is a distributional

derivative of the continuous function g(t) which has pointwise support in (t: uC(t) < A). V and g(t) will have support in

187

this same set (t: uC(t) < A) in general since (t: uC(t) < A) is a regular set in the sense of Schwartz [117, pp. 98 - 99].

If A = 0 in Theorem 4.7.2 the analytic function f(z) in this theorem can be recovered by the Fourier-Laplace transform of the constructed V E D' , with V being the inverse Fourier transform of the boundary value. COROLLARY 4.7.1.

Let f(z) E H(O;C) where C is an open

connected cone. There exists a unique element U E Z' such that f(x+iy) -- U in the weak topology of Z' as y -- 0 ,

y E C; there exists a unique element V E D' having support in (t: uC(t) < 0) = C such that U = 9[V] in Z'; and e2ni> z E TC 0 where M"(C',m) is a constant depending on C' and Since g(t) is independent of y then the growth (4.53)

on m.

is independent of y E (C'\(C' n N(O,m))).

continuous on Rn and has support in C*

;

Further g(t) is thus g(t) satisfies

the hypotheses of Lemma 4.4.1(11) with y in (4.53) being the w in the growth hypothesis of Lemma 4.4.1(11).

By the proof of

Lemma 4.4.1(11) we have (exp(-2n) g(t)) E LP for all p, 1 IRn < p < w Thus as a function of t E for each fixed y E C. ,

(exp(-2r) g(t)) E L1 O L2 here, and the Fourier transform 188

in (4.45) can now be interpreted in either the L1 or L2 sense with the equality (4.45) becoming a pointwise equality. That is, we now have f(z) = za

9[e-2w

g(t); x] (4.54) e2ni

g(t)

za

dt

,

z E T(C';m)

in IR

pointwise.

From the construction of V = Dtg(t) in the proof

of Theorem 4.7.2 and the now concluded growth (4.53) on g(t) we see that the hypotheses of Theorem 4.7.1 are satisfied; hence is well defined and is an element of H(O;C).

By the computation (4.36) for our present V = Dtg(t)

and the equality (4.54) we have f(z) =

,

z E T(C',m).

(4.55)

This equality holds pointwise for arbitrary compact subcones C' of C and for arbitrary m > 0.

Since C is open then for any

y E C there is a compact subcone C' of C and an m > 0 such that y E (C'\(C' n N(O,m))); hence any z E TC is in T(C';m) for some C' C C and some m > 0. follows from (4.55).

We conclude that (4.52)

The proof is complete.

We have concluded in Corollary 4.7.1 that V C D'

.

Since

exp(2ni) f D then the Fourier-Laplace transform of an arbitrary element in D' will not exist.

But the specific

properties of the constructed V E D' in Corollary 4.7.1 insure that the Fourier-Laplace transform in (4.52) is well defined here.

REMARK 4.7.1.

Another interesting representation of f(z) in

Corollary 4.7.1 follows from the identity (4.54); this Let y E Z and representation is of f(z) as an element of V. V E D such that y(x) = 9[.p(t);x].

For an arbitrary compact

subcone C' of C and an arbitrary m > 0 we have from (4.54), a change of order of integration, differentiation under the integral sign, distributional differentiation, and the Fourier 189

transform from D' to Z' that = In za

e2ni

g(t) J

dt y(x) dx

n

IR

=

f

g(t) f

e2,i

za y(x)

dx dt

1Rn

IRn

y(x) e27ri dx dt

(-1)lal fn g(t) Da t fn IR

f e-2a

_

(4.56)

= < [e-2n V], P(x)> for y E (C'\(C' n N(O,m))).

All of the calculations in (4.56)

are valid, and note that exp(-2ir) is a multiplier in D' Rn as a function of t E for y E n Since (4.56) holds for y .

E (C'\(C' n N(O,m))) for arbitrary C' C C and arbitrary m > 0 then y in (4.56) can be an arbitrary point of C.

We thus have

proved that f(z) in Corollary 4.7.1 can be represented as the Fourier transform 5[e-2w

f(z) =

V], z = x+iy E TC

(4.57)

in Z'.

REMARK 4.7.2.

Let us return to Theorem 4.7.1.

Using the

calculation (4.56) and the function f(z) = in Theorem 4.7.1 we have = dt y(x) dx

(4.58)

for z E T(C';m) where C' is an arbitrary compact subcone of C and m > 0 is arbitrary.

As in the proof of Theorem 4.7.2 we

have (exp(-2,r) p( t)) - (t) in D as y --> 0

,

y E C

By

.

the continuity of V E D'

lim y--->0 yE C

<e-2,r V, SO(t)> _ (4.59)

lira

y---*0 = 0 yEC

<e-2a

and for p = gyp, y E Z,

Vt,

%p

'P(t)> = , w E D

E D,

lim V y--.0 = yEC

216

(5.14)

by calculations as in section 4.7.

Z is dense in S and S is dense in S(m); thus Z is dense in S(m). We thus have that the sequence of continuous linear functionals (f(x+iy): y E C' C

C) that is bounded in norm in S(m)'converges on the set Z which is dense in S(m). By the Banach-Steinhaus theorem there is a continuous linear functional U E S(m)' C S' such that 1im y->0 _ , y E S(m)

(5.15)

yE C

By (5.14) the functional U E S(m)' restricted to Z which is dense in S(m) satisfies

E Z,

=

%p

E D.

(5.16)

The limit in (5.14) is obtained independently of how y -+ 0, y E C' C C, for any compact subcone C' of C.

Thus (5.16)

shows that the limit in (5.15) is obtained independently of how y y C C' C C, for any compact subcone C' c C. The proof is complete.

Two additional properties are immediate from the proof of Theorem 5.2.3. First, the boundary value

REMARK 5.2.1.

conclusion of Theorem 5.2.3 holds in S' Additionally, the element V C D' can be extended to be a functional on S because .

of (5.16), the fact that U E S', and Z is dense in S; thus the growth (5.9) yields the spectral function V of Vladimirov to The proof of Theorem 5.2.3 is be in S'; recall Theorem 4.7.3. a proof of Theorem 4.7.3(11).

A converse to Theorem 5.2.2 can now be obtained. = 1,...,r, be open convex cones in Rn.

j

Let C denote

arbitrary compact subcones of the Cj, j = 1,...,r. r

be analytic in

U j=1

Let Cj,

Let f(z)

Rn+iCj; and for each compact subcone C' of

r

U C. let f(z) satisfy (5.9). j=1

By Theorem 5.2.3 we obtain 217

elements Uj E S(m)

, m = R+k+n+3, j = 1,...,r, such that

lim y-.0 = , yE Cj

for j = 1,...,r.

E S(m)

(5.17)

,

Let K denote a compact (closed and bounded)

subset of Mn r

Let f(z) be analytic in

THEOREM 5.2.4.

U 1tn+iCj j=1

is an even positive integer, and satisfy (5.9).

, where r

Let the

boundary values Uj, j = 1,...,r, from (5.17) which exist in m = R+k+n+3, be equal on IRn\K, where K is a compact

S(m)

Then there exists an element U E E' with

subset of ltn.

supp(U) c K such that r

(-1)j

=

j=1 PROOF.

lim

, v E S(m)

.

(5.18)

yEC1

Put r

U =

(-1)j Uj

j=1 Then U E S(m)' C D'.

By hypothesis U1 = U2 =...= Ur on Otn\K,

and r is an even positive integer. 0 for all Thus IRn\K. E D with C This proves that U vanishes on Qtn\K and supp(U) C K.

We thus have that U E E'.

(5.18)

follows from (5.17), and the proof is complete.

In Theorem 5.2.2 the distribution U E E' was represented as a sum of boundary values of analytic functions in tubes with the analytic functions being Cauchy integrals of U We now obtain corresponding to each of the considered cones. a Poisson integral representation of U E E'; we prove that the Poisson integral of any U E E' corresponding to any regular 218

cone C has U as boundary value.

After giving this result we

discuss why this representation is different from that obtained in Theorem 5.2.2 in general though not different for half planes and tubes defined by quadrants. First we need to define the Poisson integral of U E E' Rn+iC where C is a regular corresponding to the tube TC = cone.

From Theorem 4.4.4 the Poisson kernel P(z;t) E E as a

function of t E

IRn

for z E TC.

DEFINITION 5.2.2. The Poisson integral PI(U;z), z E TC , of U E E' corresponding to the regular cone C is

PI(U;z) = , z E TC

.

(5.19)

However, in PI(U;z) is a well defined function of z e TC. general, PI(U;z) is not analytic in TC in contrast to the

Cauchy integral C(U;z); note the closing discussion at the end of this section for the reason for this. To show that U E E' is the boundary value of PI(U;z) we need several lemmas. The following lemma can be proved in a manner similar to the proof of Lemma 5.2.1. LEMMA 5.2.4. Let U E E', V C D, and C be a regular cone in

For fixed y = Im(z) E C we have = . For convenience we rephrase Lemma 4.4.4 here in a somewhat different but equivalent form through change of variable; see

Rn.

also [124, p. 105]. LEMMA 5.2.5. Let C be a regular cone in Mn and x+iy C TC.

We have K(x+iy) K(x+iy)

0, x+iy E TC

(5.20)

K(21y)

In

K(x+iy) K(x+iy) dx = 1, y E C; K(21y)

(5.21)

219

lim

K(x+iy) K(x+iy)

if a > 0, y->0 Ix >a

yEC

K(2iy)

dx = 0.

(5.22)

Let p E D and C be a regular cone in Itn.

LEMMA 5.2.6.

have -, V(t) in E as y = Im(z)-'0 PROOF.

,

We

y E C.

A change of variable yields K(x+iy) K(x+iy)

= In p(x+t)

dx

(5.23)

K(21y)

Let p be any n-tuple of nonnegative integers.

Using (5.21)

and (5.23) we have DQ - Dpw(t) _

(y(x+t) - ,P(t)) K(x+iy) K(x+iy) dx

Pn

(5.24)

K(21y)

where fi(t) = Daf(t).

From (5.24), (5.20), and

Let 6 > 0.

(5.21)

IDt -

Iy(x+t) - y(t)I K(x+iy) K(x+iy) dx K(2iy) Ix a


a

K(x+iy) K(x+iy) dx K(21y)

.

Let t vary over an arbitrary compact set in Utn

Since y = E D, y is uniformly continuous on the compact set; hence we may choose 6 such that sup Iy(x+t) - y(t)I is uniformly .

IxIo

dx = 0

K(21y)

Thus by (5.25),

Dep(t) uniformly on

compact sets in Rn as y = Im(z) --' 0

,

y E C

,

for arbitrary

p; the proof is complete.

We now prove that an element U E E' is the boundary value of its Poisson integral. THEOREM 5.2.5. Rn

Let U E E', p E D, and C be a regular cone in

For y = Im(z) E C

.

lim_ y--70 = . yEC PROOF.

The proof is immediate by combining Lemmas 5.2.4 and

5.2.6 and the continuity of U E E'.

The convergences in Lemmas 5.2.2, 5.2.3, and 5.2.6 and

Theorems 5.2.2 and 5.2.5 are all obtained as y - 0 arbitrarily in the respective cones; y does not need to be restricted to arbitrary compact subcones in these results as

y-) U. REMARK 5.2.2. half plane.

Suppose that TC in Theorem 5.2.5 is the upper

Suppose that the cones in Theorem 5.2.2 are (y: y

> 0) and (y: y < 0); so that the dual cones are (t: t > 0) and (t: t S 0), respectively.

For y = Im(z) > 0 and U E E' we

have

PI(U;z)

0) in R1

,

C(U;z) is not

analytic for z = x+iy, y > 0.

However, PI(U;z) in (5.26) is a harmonic function, and the Poisson integral corresponding to the tube defined by any quadrant is n-harmonic. 5.3.

ANALYTIC REPRESENTATION OF VECTOR VALUED DISTRIBUTIONS OF COMPACT SUPPORT

The results of section 5.2 are extended to vector valued distributions of compact support in this section. This is the only section in this book in which we consider vector valued

Thus we freely reference works of other authors in this section rather than add the considerable distributions.

amount of definitions and results needed for vector valued distributions for just this one section. Throughout this section VS will denote a locally convex separable quasi-complete topological vector space, where quasi-complete is in the sense of Schwartz [120, p. 198]. L(A,VS) will denote the set of all continuous linear transformations of the given function space A into VS.

We

refer to [119] or [132] for the needed definitions concerning 222

vector valued distributions.

For the present we assume further that VS is a space of type (DF) [55, p. 63] and the topology of VS is defined by the sequence of seminorms (Nb), b E B. Let U E L(D,VS) and have compact support; by Tillmann [132, Theorem 1], U E L(E,VS) C L(D,VS).

Combining [119, Corollary 2, p. 85] with [119, Proposition 24, p. 86] and the note at the bottom of

page 87 in [119] we have U =

DQfp(t)

(5.27)

IPI<m

where the fp are continuous functions having values in VS and having support in an arbitrary neighborhood of the support of U.

Recalling that pr(C) denotes the projection of the cone C,

the following is a generalization of Theorems 5.2.1 and 5.2.2. THEOREM 5.3.1.

Let U E L(D,VS) and have compact support. Let VS be a space of type (DF). Let Cj, j = 1,...,r, be regular cones which satisfy the property (5.7). analytic functions fj(z) in

Then there exist

1Rn+iCj,

having values

j

in VS which satisfy -k -1

M(p,b,Cj) IIm(z)I

Nb(fj(z))
O Nb( A>0

r

) = 0, b E B

,

(5.29)

j=1

223

where y E pr(Cj), j = 1,...,r.

As noted above [132, Theorem 1] implies that U E Rn+iCj L(E,VS). For z E j = 1,...,r, we put PROOF.

,

f j (z) = , j

= 1, ...,r.

Ci

Because of (5.27), a proof analogous to that of Theorem 5.2.1 yields that fj(z) is analytic in Rn+iCj and satisfies (5.28). The proof of Lemma 5.2.1 can be adapted to the present case to yield

= >

(5.30)

Cj

for p E D and y = Im(z) E Cj, j = 1,...,r.

By (5.30), Lemma

5.2.2, and the continuity of U E L(E,VS)

lim X->0 Nb( -

(5.31)

A>0

- 0 and has values in E.

Using (5.32) and exactly

the same construction as in the proof of Theorem 5.2.3 we obtain a function A(-k-1)f1(p;x,y), y E pr(C'), which is analytic in the half plane A = Im(p) > 0, has values in VS1 and satisfies N(A(-k-1)f1(p;x.Y))

(5.33)


1, where M(k+1)(C') is a constant depending on C'; and f(x+py) = f1(p;x,y) =

=

a

k+1

(A(-k-1)f1(p;x,Y)),

(5.34)

a = Re(p)

aak+1

Choose m = R+k+n+3.

From (5.33),

(: y E pr(C'), 0 < X < u, u > 1)

is a bounded subset of VS for each fixed p E S(X)a).

By the

uniform boundedness theorem (A(-k-')f 1(p;x,y): y E pr(C'),

0 < X < u, u > 1) is a bounded set in L(S(m), VS), m = R+k+n+3.

By (5.34) the same is true of (f(x+py) _

fl(p;x,y): y E pr(C'), 0 < X < u, u > 1). 226

Using this fact and

an estimate as in (5.13) we have (: y E pr(C'), 0 < A S u, u Z 1) is a bounded subset of VS for qp in a bounded set of S(m).

Applying the uniform boundedness theorem again we obtain that (f(x+iXy): y E pr(C'), 0 < X S u, u Z 1) is a bounded set with respect to the norm in L(S(m),VS), M = R+k+n+3.

If K is an arbitrary compact (closed and bounded) subset of C, then K C C' C C for some compact subcone C' of C; and from (5.32)

N(f(z)) S M'(C',K) (1+Ixj)R , z = x+iy E TK We note that [119, Proposition 22, p. 76] can be stated and proved equally well for IRn+ir rather than r+iotn , where r is

Thus by [119, Proposition 22, p. 76] there exists an element v e (S'(C))(VS) such that f(z) = 9[exp(-2,r) Vt](x), y C C. From the Fourier

an open convex set in Otn.

transform on L(D,VS)

= <e 2n Vt,

(5.35)

W(t)>

where W C D, y = .p E Z, and z = x+iy E IRn+iC. (exp(-2Tr) p(t)) -

f(t)

in D as y

Now

y C C' C C.

Since V E L(D,E) then lim y->0 yE C

N(<e-2n

Vt,

*(t)> - ) = 0

(5.36)

Combining (5.35) and (5.36) we have lim y--;O- N( 0, j = 1,...,n. Let Cj, j = 1,...,r, be

Let U E Oa and tp E D.

THEOREM 5.4.1.

regular cones which satisfy the property (5.7). There exist Rn+iCj, j = 1,.... r, for functions fj(z) which are analytic in which r

lim

j =l

yEC j

y-+0

.

(5.46)

The analytic functions are the Cauchy integrals

PROOF.

di>, z E Rn+iCj , j = 1, ... , r,

fj (z) = 0, j = 1,...n, as a function of t E IRn for each fixed

z E TC

.

This allows us to form the Poisson integral of

U E 0' as a PI(U;z) = , z E TC

.

(5.47)

For U E 0' and f E D we have

a

= , y = Im(z) E C, (5.48) and

231

lim y->O = V(t)

(5.49)

yE C

in 0a which when combined yield the following Poisson integral representation similar to that for E' in Theorem 5.2.5. THEOREM 5.4.2.

Let U E Of

,

a1

lim y---Z =

1,...,n.

> 0,

We have

E D.

yEC The discussion in the paragraph of section 5.2 containing equation (5.26) of the connection in the representations obtained by the Cauchy integral and the Poisson integral holds for our present Theorems 5.4.1 and 5.4.2 exactly as in the case for E'. Results similar to Theorems 5.4.1 and 5.4.2 for arbitrary a also hold; in this case a distributional derivative of U E 0'

a

Further discussion concerning the boundary value representation of elements in Oa will be given in the is represented.

commentary section 5.6. 5.5.

ANALYTIC REPRESENTATION OF DISTRIBUTIONS IN D'p L

The distributions in D'p were not considered in Chapter 2. L

Thus the analysis of representing these distributions as boundary values of functions in arbitrary tube domains in en is presented in full detail in this section.

The D'p L

distributions are represented using both the Cauchy and Poisson integrals of these distributions.

Additionally the Cauchy integral of certain elements in D'p is related to the L

Fourier-Laplace transform of corresponding S' distributions which will yield Cauchy integral representations of certain analytic functions in tubes.

Before proceeding to these results we first present some analysis concerning the Fourier transform of convolutions 232

First recall that the

which will be used in this section.

Fourier transform of an LP function, 1 < p < 2, exists and is It is well known that if f a function in Lq, 1/p + 1/q = 1. and g are in L2 then 9-1[f(x)

(5.50)

g(x); t] = (f*g)(t)

where * denotes the usual function convolution.

f and g are in LP

1 < p < 2, and if g is in Lq

,

1/p + 1/q = 1, then (5.50) holds.

Similarly if ,

Furthermore, if f E L2 and

g E L1 then 9[(f*g)(t); x] = f(x) g(x) in the sense of L2

.

(5.51)

It is obvious that under the above

conditions both f*g and fg in (5.50) and (5.51) are elements

of S'

.

1 < p < 2, and Let f and g be elements of LP 1/p + 1/q = 1. Then 9[f*g] _ let g be an element of Lq LEMMA 5.5.1.

,

,

fg in S'. PROOF.

Let p E S.

By (5.50), = >

"I<m IQI

1

z = x+iy E Rn+iC, and IyI < 1

where m is the order of U E D' LP

.

By the characterization theorem for D'p

PROOF.

L

U=

DR fR (t)

,

fR E Lp

1 k > 0 for all z = x+iy E K C TC

.

For z e K C

TC (5.59) yields

fn ICTl)

e2,ri dnI

Tip


0 depending only on C' such that (4.1) holds; using (4.1) we have for z = x+iy E Utn+iC' 239

in IIC*(TI)

TIP e2TriIp

dTl

fn

S

IR

IC*(TI) ITIIPIPI e-27rp dTl

IR

f

I *(,a) Ig1PIRI exp(-2,rp6IYI ITII)

nn < IZn

drl

C

S0

u PIPI+n-1 exp(-2,rp6lylu) du

(2TrpbIYI)-PIPI-n f' vPIPI+n-1 a-v

= 0n

0

= Un r(PIPI+n) (2,p,jyj

(5.63)

dv

)-PIPI-n

where 0n is the surface area of the unit sphere in IRn and the change of variable v = 2Trpblylu was used to obtain the gamma function r.

(We have used Schwartz [118, Theorem 32, p. 39] again here as we did in (4.5).) Combining (5.57), (5.62), and (5.63)

IC(U;z)I


1

,

z = x+iy E IRn+iC' and IyI

,

< 1,

1 < p < 2, which is (5.56)

L

for 1 < p < 2. The growth (5.56) for p = 1 remains to be proved. Again let C' be any compact subcone of C. Let z = x+iy E IRn+iC' and let 13 be any n-tuple in (5.57). Apply (4.1) and the analysis of (5.63) with p = 1 to obtain

in

IC*(T1) n9

e2ai dlil

an r(IRI+n) (2,,61yl ) -113

for z = x+iy E Dtn+iC' and 6 = b(CI) > 0.

1-n

Using this estimate

in (5.57) for p = 1, the same estimate (5.56) is obtained with The proof is the same constant M(1,n,C') as in (5.65). complete.

As is evident from the proof of (5.56) the separation of the two cases for lyl

1 and lyl

< 1 was done for

convenience; we could have separated the two cases corresponding to lyl > e and lyl < e for any e > 0 and obtained the same fundmental growth for C(U;z).

The point is

that for y E C' C C bounded away from the origin 0

,

C(U;z) is

bounded; while C(U;z) can get arbitrarily large as y -' 0, y E

C' C C

.

To represent U E D'p

,

1 < p < 2, as the boundary value of

analytic functions in tubes, several lemmas will be needed. 1 < p < 2, and C be a regular cone LEMMA 5.5.6. Let U E D' Lp ,

in IRA.

Let W E S.

For fixed y = Im(z) E C,

=

.

(5.66) 241

By a proof as in Theorem 4.4.1 we have

PROOF.

*

exp(2iri<x+iy,n>) do E B n D q

,

1/p + 1/q = 1,

J

L

C

as a function of x e

IRn

for y C C.

1/p + 1/q = 1, we have D

C D'

Lq

;

For 1 < p < 2, and

Lq

C D' .

By the

L

theorem of Schwartz [117,Theoreme XXVI, p. 203](Theorem 1.7.1) exp(2,ri<x+iy,n>) dn) E D'w

(U * f C

for U E D' LP

(5.67)

L

,

By a change of variable

1 < p < 2.

= <J

*

exp(2,ri<x+iy,n>) dn, p(x+t)>

C

for p C S and = ) dn, tp>.

(5.68)

*

C

By (5.67) and (5.68) the right side of (5.66) is well defined.

Using the characterization theorem of Schwartz for U E D'p L

1 < p < 2, and change of order of integration we have

= (-1)IQI

IRI<m

IRI<m

=

in IR

DRK(z-t) pp(x) dx dt

Mn

IR

(-1)IRI

242

f

i f(t)

,P (X) i

In

f(t) DRK(z-t) dt dx

for y E C, and (5.66) is obtained.

The Cauchy integral C(U;z) of U E D'p

,

1 < p < 2, is not

L

necessarily an element of D'p

.

Thus we can not take V in

.

But by Theorem 5.5.1, C(U;z)

L

Lemma 5.5.6 to be in D q or B

is continuous and bounded as a function of x = Re(z) C Rn for In y = Im(z) E C; thus C(U;z) E S' as a function of x E For .

this reason we can let p E S in Lemma 5.5.6. Let C be a regular cone in Mn and p E S. LEMMA 5.5.7.

We

have

lim_

y-.O 0 = yEC PROOF.

.

C

The proof is obtained by combining Lemmas 5.5.6 and

5.5.7 and using the continuity of U E D'p

,

1 < p
is analytic in

= y E C, for w E S.

This proves (5.76) and

the proof is complete.

We note that y

U, y c C, has the interpretation y

y E C' C C, for all compact subcones C' of C in Theorem 5.5.3, Corollary 5.5.1, and Theorems 5.5.4 and 5.5.5. Recall the functions H(A;C) defined in section 4.7. As a corollary to Theorem 5.5.3 elements in H(O;C) which have D'p L

boundary values in the S' topology can be recovered by the Cauchy integral of the boundary value. COROLLARY 5.5.1.

Let C be a regular cone.

converge in the S' topology to U E D'p

,

1

Let f(z) E H(O;C) < p < 2, as

L

y = Im(z) -* U, y E C.

Then f(z) = C(U;z), z E TC

as

,

elements of S' and U = 5[V], V E S', where V has the form as in (5.74) and supp(V) C C By Corollary 4.7.2 there exists V E S' with supp(V) c U = ?[V], and

PROOF. C

,

f(z) = , z E TC

both in S' and pointwise.

f(z) = C(U;z), z E TC , now follows

by (5.75).

The form of V as in (5.74) follows from the proof of Theorem 5.5.3.

Corollary 5.5.1 prompts us to ask for necessary and sufficient conditions that an element U E D'p be the S' L

(Remember that boundary value of a function f(z) E H(O;C). the weak and strong topologies of S' are equivalent since S is

a Montel space.)

THEOREM 5.5.4.

Let C be a regular cone.

converge in the S' topology to U E D'p

,

Let f(z) E H(O;C) 1 S p S 2, as

L

y = Im(z)

y E C.

Then C(U;z) = 0, z E

1n+iCi ,

where C1

is any regular cone such that c* fl C1 is a set of Lebesgue 249

measure zero.

From Corollary 5.5.1 we obtain V E S' with supp(V) C

PROOF.

C

,

U = 5[V], and V has the representation (5.74).

Let

I *(t) and if * (t) denote the characteristic function and coo C1,E C1

function corresponding to C1

C(U;z) = ,

,

(5.79)

and the right side of (5.79) is 0 since supp(V) C C

THEOREM 5.5.5.

Let U E D' LP

,

< p < 2, and C be a regular

1

Let Cj, j = 1,...,r, denote regular cones such that

cone.

U \ ( Ur C. U C*) , C* fl C. j=1 j

z E Rn+iCl

C1,E

C1

in S'

By the proof of (5.75) we have

.

and C. fl Ck

k,

= 1,...,r, k = 1,...,r, are sets of Lebesgue measure zero

and such that C(U;z) = 0, z E Un+iCj

,

j

= 1,...,r.

Then U is

the S' boundary value of a function f(z) E H(0;C) as y = Im(z)

-4 0

,

PROOF.

y E C. Let I *(t) and I *(t), j = 1,...,r, denote the C

C. J

respective characteristic functions of the dual cones.

V = 5-1[U] has the form (5.74); and by the hypotheses on the dual cones we can decompose V as

r

I *(t) V

V = I *(t) V + C

j=1

CJ

For p E S

r



= + C

j=1

Cj

By hypothesis and a calculation as in (5.79) 250

.

(5.80)

0 = C(U;z) = ,

z E IRn+iCj ,

j

(5.81)

= 1, ... , r .

The proof of (5.78) yields

lim Y--;O- O

_ [I ye(t) V] C.

Cj ,e

Cj

in S', j = 1,...,r.

(5.82)

(5.81) and (5.82) imply 5[I *(t) V] = 0, C. J

j

= 1,...,r, and from (5.80) U = ?[I

(t) V].

But U = 9[V]

C

also, so that V = (I *(t) V) in S' C

f ( z )

=



= ,

( t )

z E TC

we have f(z) E H(0;C) and f(x+iy) --+ 5[V] = U in S' as y --- 0,

y E C, by Theorem 4.7.4.

(In applying Theorem 4.7.4 with A = 0, inequality (4.70) implies inequality (4.34) for z = x+iy E T(C';m), C' C C, m > 0.) The proof is complete. REMARK 5.5.1. In the case that C is either (0,o') or (--,0) in one dimension or one of the 2n quadrants in utn, the Lebesgue

measure zero assumptions of Theorems 5.5.4 and 5.5.5 are valid for these cones and are exactly the properties of the cones that make these two theorems valid for these special cases. For example, if C is any fixed quadrant in Rn then we can take

the other 2n-1 quadrants in Rn to be the r other regular cones in Theorem 5.5.5. The dual cones of the 2n quadrants have the assumed properties of the dual cones in Theorem 5.5.5. The theorems in this section to this point have involved We now consider the Lp the Cauchy integral of elements in D'.

251

Poisson integral of these elements.

Let C be a regular cone

and recall the Poisson kernel function P(z;t) defined in section 4.4 for z E TC and t E Utn. E B fl D q for all q, 1 < q < L

,

By Theorem 4.4.4, P(z;t)

as a function of t E

IRn

for z

E TC DEFINITION

5.5.2.

The Poisson integral PI(U;z) of U E D L'

p

is defined by

1 < p

PI(U;z) =

,

z E TC

(5.83)

As for the Poisson integrals of elements in E' and Of

the

,

Poisson integral of U E D'p is a well defined function of z E L

TC but is not an analytic function of z E TC in general.

It has been known for many years that the Poisson integral of a function g(t), t E IRn corresponding to the tube TC has ,

g(t) as boundary value as y = Im(z) --> 0

y E C, in several topologies [66, Proposition 3, p. 280] depending on the properties of the function g(t). For U E D'Lp we 1 < p < m ,

,

,

now show that the Poisson integral PI(U;z), Z E TC

,

has U as

boundary value distributionally as y y E C. (Here, by y - 0, y E C, we mean that y -+ 0 independently within C.)

To do this several lemmas will be needed.

The following lemma

was obtained by Koranyi [66, Proposition 3(c), p. 280] who stated it without proof. A sketch of the proof will be given here; for the method used will be important in obtaining the boundary value result for the Poisson integral of U E D'p L

LEMMA 5.5.9.

Let C be a regular cone and g E LP

We have lim y-1.0

yEC

252

in J

g(t) P(x+iy;t) dt = g(t)

,

1 < p
0, y E C, can now be proved. THEOREM 5.5.6. Let C be a regular cone and U E D L'

p

1< p< m. For p E D 1 lim y->0 =

(5.93)

yE C

For any p E D 1 the proof of Lemma 5.5.10 yields that

PROOF.

L

exists for y E C. U E D'p

,

The continuity of and Lemma 5.5.11 combine to prove

1 < p

L

lim

y->0 =

(5.94)

yEC

is well defined for p E D

and

since D L1

q, 1 < q < m

,

and D 1 c B

.

L

1-

D

Lq

for all

(5.93) is now obtained by

L

combining (5.85) and (5.94).

If p = m in Theorem 5.5.6, then (5.93) proves that PI(U;z) - U in exactly the weak topology of D', as y -REMARK 5.5.2.

L

0, y E C, since (5.93) holds for each p E D 1 whose dual space L

is D'w L

.

In the paragraph succeeding the proof of Lemma 5.5.6

we noted the reason for working with p E S in the results concerning the Cauchy integral; another reason is that the symmetry of S under the Fourier and inverse Fourier transforms For our Poisson was needed in the proof of Theorem 5.5.2. integral results, it suffices to take p E D 1 essentially L

because of (5.88) which makes the right side, and hence the 257

left side, of (5.85) well defined. Oa

,

As in the cases of E' and

in general the Cauchy integral of U E D'p has a boundary L

value corresponding to each regular cone C which depends on But as shown in Theorem 5.5.6

the cone; recall Lemma 5.5.8.

the Poisson integral of U E D'p has U as boundary value L

independently of the cone C for which PI(U;z), z E TC defined.

This property of PI(U;z), U E D' Lp

,

is

is like the

classical Poisson integral of a function g(t) in that the Poisson integral of g(t) attains g(t) as boundary value in a topology depending on the properties of g(t) and does so independently of the cone C with respect to which the Poisson integral is defined. (See Koranyi [66, Proposition 3, p. 280] as noted before.) 5.6.

COMMENTS ON CHAPTER 5

The famous paper [67] of G. Kothe is the original work on the representation of distributions as boundary values of analytic functions. Many of the original papers of representation of

distributions as boundary values of analytic functions in general tube domains are referenced in Vladimirov [135]. The proof of Theorem 5.2.3 is an adaptation of the proof of the result of Vladimirov [135, p. 235] which we have stated in Theorem 4.7.3(11).

The result [119, Proposition 22, p. 76] of

Schwartz, which was used in the proof of Theorem 5.3.3, is a generalization of Theorem 4.7.3(I). For R = 0 the boundedness condition (5.9) (see also (5.8)) is analogous to the slowly increasing functions of Kothe and Tillmann for half planes and tubes defined by quadrants.

However, when we take R = 0 in (5.9) our converse Theorem 5.2.4 for functions analytic in general tube domains is not as

strong as the E' converse theorem of Tillmann [129, Theorem 10].

In Theorem 5.2.4 the domain of analyticity may not be of

the form Cn\K where K is a compact subset of IRE.

Our

extension of the results of section 5.2 to the vector valued

case in section 5.3 is like that of Tillmann [132] for the quadrant case. 258

Further results concerning the analytic representation of 01 in n dimensions can be found in the papers of Carmichael [18] - [21] including growth considerations on the Cauchy integrals involved. The original work on the representation of D' was done by LP Tillmann [130] and Luszczki and Zielezny [73] for the scalar valued case and by Bengel [4] for the vector valued case.

Many of the results of section 5.5 are reminiscent of Hp space theory and related analysis concerning Cauchy and Poisson integrals of LP functions. We have already noted in section 5.5 that the result of Theorem 5.5.6 is like that for the Poisson integral of an LP function in that in both cases the Poisson integral of the defining distribution or function attains that element as boundary value in the relevant Similarly Corollary 5.5.1 and Theorems 5.5.4 and 5.5.5 remind us of Hp space results in their structure. The topology.

Hp functions are replaced by those of H(0;C), and the set of admissible boundary values has been enlarged from LP functions to D'

distributions.

The distributional setting is obtained

LP

without altering the essential structure of the classical setting; that is, the relations between the analytic functions, the corresponding Fourier-Laplace, Cauchy, and Poisson integrals, and the boundary values are retained in the distributional setting.

Of course the topology employed in

obtaining our distributional results is weaker than that of pointwise or norm convergence.

259

6 The Cauchy integral of tempered distributions and applications in n dimensions 6.1.

INTRODUCTION

The theme of this chapter is the calculation of distributional

boundary values of analytic functions in tubes in Cn in the topology of S', the tempered distributions, and the recovery of the analytic functions by a Cauchy integral of the boundary

value and by the Fourier-Laplace transform of the inverse Fourier transform of the boundary value.

H.G. Tillmann [131] was the first to study distributional boundary values in S' as previously noted in section 4.7. He obtained a growth, recall (4.67), which characterizes those

analytic functions in (C1-f1n, the union of the 2n tubes 1Rn+iC11 for the quadrants Cµ for which the boundary value ,

mapping lim

(-1)1111

f(x+ieyu)

attains a limit in S'; here yµ = ((-1)l, (-1)4 2'...,(_1)un)

where µ = (µ1'µ2"

is the n-tuple whose components are 0

or 1 which defines the quadrant Cµ

.

Tillmann's analysis

suggests ways to recover his analytic functions by an integral or transform involving the boundary value. Meise ([76], [77]) extended the S' boundary value analysis of Tillmann to tempered vector valued distributions and in this more general

setting obtained the recovery of the vector valued analytic functions in terms of a Fourier-Laplace transform. In sections 6.2 and 6.3 of this chapter we show that the recovery

of Tillmann's analytic functions by a Cauchy integral of the boundary value can also be accomplished as is suggested by the original analysis of Tillmann in [131].

In section 6.2 the

Cauchy integral of S' elements is con:.tructed and properties 260

of the integral are obtained.

The recovery of the analytic

functions by the Cauchy integral of the boundary value is obtained in section 6.3.

By a different construction than that used here, V. S. Vladimirov has defined a Cauchy integral of S' distributions corresponding to functions which are analytic in arbitrary tubes defined by regular cones.

We present Vladimirov's

Cauchy integral in section 6.4 and use it to recover analytic functions in tubes TC which attain S' boundary values on the distinguished boundary of the tube.

The growth which characterizes the analytic functions in arbitrary tubes TC that have S' boundary values has the pointwise growth for HP(TC) functions obtained in section 4.6 as a special case.

In fact we proved in Theorem 4.5.2 that any element f(z) E HP(TC), 1 < p < w where C is a regular cone, has an S' boundary value without using growth ,

considerations; and the S' boundary value is an LP function. It is natural to ask if any analytic function in TC which satisfies the growth that characterizes the existence of S'

boundary values can attain an LP function as S' boundary value other than an Hp function.

The answer is no as we show in

section 6.5.

An important tool in the analysis of section 6.5 is the Fourier-Laplace integral of functions and distributions. This integral has been used to represent H2 functions beginning with the analysis of Paley and wiener [100, p. 8].

In section

6.6 we represent Hp functions in tubes as Fourier-Laplace The particular tube integrals of functions and distributions. setting and the use of a function or of a distribution that is

not a function in the representation depends on the value of p, 0 < p < W

.

As in the previous chapters, a commentary section concludes this chapter; in section 6.7 analysis related to that of Chapter 6 is discussed.

261

6.2.

THE CAUCHY INTEGRAL OF TEMPERED DISTRIBUTIONS: THE CASE CORRESPONDING TO THE QUADRANTS IN n

n t E The Cauchy kernel function K(z-t), z E TC corresponding to any regular cone C in IRn is not an element of the test space S. However, given U E S' a smoothing ,

,

function can be obtained such that U applied to the product of this smoothing function and the Cauchy kernel is well defined and in fact is an analytic function.

The Cauchy integral

(generalized Cauchy integral) of U E S' can then be constructed to be a certain equivalence class of analytic functions which are equivalent to the one obtained by applying

U to the product of the smoothing function and the Cauchy kernel. This Cauchy integral of U E S' is constructed in this section and is used in the next section 6.3 to represent the analytic functions which obtain S' boundary values. The Cauchy integral presented here is motivated by analysis of Sebastiao e Silva [122, sections 4 and 5] and Tillmann [131].

The construction of the Cauchy integral in this section will be obtained corresponding to the quadrants CA in IRn.

Recall the notation (C1_C1)n j

=

(z E Cn: Im(zj)

0,

= 1,...,n) stated in Theorem 4.5.5 and in section 4.7.

We begin our construction of the Cauchy integral by defining psuedo-polynomials. DEFINITION 6.2.1.

A pseudo-polynomial P(z) is any function of the complex variable z E (C1_R1)n which is a finite linear combination of functions of the form zj

where the s are nonnegative integers and the gs'j(z[j]) are analytic functions of the variable z[j]

(zip ...,zj_i, =

zj+i,...,zn) E

(C1_C1)n-1

,

j = 1,...,n.

Thus a pseudo-polynomial is any element of the form

262

r.

n

P(z) =

j

zj

z E (C1-Utln

j=1 s=0

where the gs,j(z[j]) are analytic functions of z[j] E (11_I1)n-1,

s = 0,1,...,rj and j = 1,...,n, and the as,j are

complex constants for s = 0,1,...,rj and j = 1,...,n.

If the

dimension n = 1, a pseudo-polynomial is just a polynomial in C1.

We shall show that elements in S' yield analytic

functions and pseudo-polynomials in (C1-Ut1)n when applied to

certain products involving the Cauchy kernel function.

Then U can be written in the form

Now let U E S'. aR

n II

II

8tR ll llj=1

(,+t2 7j] h(t),, t e Utn

for some n-tuples R = (/31,82,...,pn) and ry =

nonnegative integers and some function h(t) E L2.

(_Y1'_Y2...... n) of

(Recall

Choose a polynomial p(t) without zeros in which

section 1.4.)

any component is real such that

rn II

` j=1

71 (l+t2) jl h(t) J J

E L1 n L2

P(t)

:

(6.2)

for example p(t) can be chosen as +2

n

p(t) =

II

j=1

(1+t2

)

j

J

Of course there are infinitely many choices of p(t) here for which (6.2) holds. With such a choice of p(t) it follows that

263

I

`

II

(1+t2 )~

j=1

z E

a0

j ) h(t)

(C1_R1)n

ata

p(t)

j

n if

E L1 fl L2 ,

1

(6.3)

(tz

for any n-tuple a of nonnegative integers.

,

THEOREM 6.2.1.

Let U E S'

.

With the representation of U E

S' as in (6.1) choose a polynomial p(t) without zeros in which any component is real such that (6.2) holds. We have that 1

R(z) =

(t

j

J

n [¼PH + T(z) (t j-zj

a

1

(o(z))

p(z) 0 and Nj > 0, j = 1,...,n, are real constants and the kj > 1, j = 1,...,n, are integers.

(Recall (4.67).)

In a manner which will be made precise below, we shall show that these analytic functions obtain distributional boundary values in the strong topology of S' and can be recovered by the Cauchy integral of the boundary value with this Cauchy integral being that which we defined in section 6.2.

Conversely we shall show that for any U C S' and any f(z) e C(U;z), f(z) is analytic in (C1_l1)n satisfies (6.31), and obtains U as its strong S' boundary value. (In both ,

directions a representation of the analytic functions in terms of a Fourier-Laplace transform will be obtained; this will be 280

discussed in section 6.7 below.)

As previously noted in section 4.7 and in the introductory section to the present chapter, Tillmann [131] obtained the first results on representing tempered distributions as boundary values of analytic functions. The growth estimate (6.31) is essentially Tillmann's growth condition which characterizes those analytic functions in (C1_IR1)n that obtain S' boundary values, and the boundary value mapping considered by Tillmann is lim E

(-1)1µl

f(x+iey

4O+µ

which is a sum of boundary values from each of the 2n tubes µJ

utn+iCA defined by the quadrants C j

= (y E

Utn

(-1)

yj > 0,

= 1,...,n) where µ = (µ1,µ2,...,µn) is any of the 2n

n-tuples whose components are 0 or 1 and yµ = ((-1)µl, (-1)µ2, This is the S' boundary value

...,(-1) n) E Cµ for each µ.

mapping which we consider here.

The analysis of this section

should be viewed as a natural extension of the work of Tillmann in [131]; our work here builds directly upon that of The new result here is the recovery of the analytic Tillmann. functions from the boundary values by the Cauchy integral that we defined and studied in section 6.2. In addition some of the technical details of our work here are different from those of Tillmann; in particular we make use of some interesting calculations involving the Hardy H2 spaces We shall corresponding to tubes defined by the quadrants. emphasize and describe the results of Tillmann as we use them. Before proceeding to the main results of this section some preliminary theorems will be obtained. Let f(z) be analytic in THEOREM 6.3.1.

If(z) j< M

n

(1+Iz I) i J=1 U

N

(C1_O1)n

and satisfy

z E (c1-Qtl)n 281

where M > 0 and Nj > 0, j = 1,...,n, are real constants. There exist 2n polynomials qA(z) without zeros in IRn+iCU and 2n functions FU (z) E H2(IR +iCU) such that for each of the 2n

n-tuples g we have f(z) = gU(z)FA(z), z E Rn+iCU. PROOF.

Let e > 0 be fixed.

For f(z) satisfying the assumed

growth and for each p choose a polynomial % (z) without zeros

such that

in Dtn+iC

n If(z)/gU()I < Mu

(1+Iz.I)-1-e

R

for some constant MA depending on U. (f(z)/qA(z)) E H2(IRn+iC

z E T

CU ,

z E T CU

We then have FU(z) _

from which f(z) = gA(z)FU(z),

The proof is complete.

follows.

DEFINITION 6.3.1.

,

j=1

Define 2n sets of functions A

corresponding to the 2n tubes TCU

=

1n+iCU

A

as follows:

Au =

(gA(z)FU(z): qU(z) is a polynomial without zeros in 1n+iCU

and

FU (z) E H2(1tn+iCU)). If f(z) is analytic in

(C1-C1)n

and satisfies the growth of Theorem 6.3.1 then this result yields that f(z) E AU for each of the 2n n-tuples U.

We have the following representation theorem for elements in AU in terms of the Cauchy integral in (6.14).

Let f(z) be analytic in

THEOREM 6.3.2.

IRn+iCU

f(z) E AU, z E

,

(C1-l1)n

such that

for each of the 2n n-tuples U.

There

exist 2n functions h1(t) E L2 and 2n polynomials q1(t) such that

[f(z)] =

E (CR Cf (-1)g11(t)hA(t);zz l, 1-l)n 1111

u 282

J

.

(6.32)

PROOF. or 1.

Let p be any of the 2n n-tuples whose components are 0 Rn+iC Since f(z) E AA then f(z) = g11 (z)F9(z), z E for

some polynomial qA(z) without zeros in Mn+iCU and some F(z) E H2(Cn+iCU).

Put

(-1)IuI(f(z)/q(z)), z E TC (z) =

(6.33)

U

O

z E Cn+iCU,

U' - u

where µ' denotes any of the 2n-1 n-tuples of zeros and ones other than µ.

Since (f(z)/qU(z)) = FU(z) E H2(Cn+iCU),

Theorem 4.5.5 yields a function hu(t) E L2 such that

%0

(C1-IR 1)n

(z) = I(h(t);z), z E

(6.34)

where n

(2ni)-n

I(hI(t);z) =

hU(t) J Cn

(C1-C1)n

R

(tj-zj)-1 dt,

(6.35)

j=1

is the classical Cauchy integral of the L2 function h(t). Since each h(t) E L2 then h(t) E S' and z E

,

(qU(t) hu(t)) E S'.

Applying Theorem 6.2.5 and (6.28) we have

C(qu(t)hu(t);z) = qU(z) C(hU(t);z) =

= qU(z) [I(hU(t);z)], z E

(C1-IR1)n ,

which when combined with (6.34) yields

283

C(qu(t)hu(t); z) = qu(z) [%*,(z)], z E

for each of the n-tuples µ.

(C1-IR 1)n

(6.36)

,

Because of the definitions of the

2n functions pA(z) in (6.33) we have

f(z) = C (_1)1µI qu(z) µ(z), z E

(-1) IWI

Now

qu(t) hµ(t)] E S'

.

(C1-lR1)n

(6.37)

.

From the representation

J

(6.37), the equality (6.36), and Theorem 6.2.4 the desired representation (6.32) follows.

Using Theorems 6.3.1 and 6.3.2 we extend the Cauchy integral representation to analytic functions which satisfy the growth (6.31). Let f(z) be analytic in (C1-IR1)n and satisfy

THEOREM 6.3.3. (6.31).

There exist 2n functions h(t) E L2 and 2n

polynomials qu(t) such that

[f(z)] =

8

K+1

Oz

C( (-1)IPI qµ(t) hµ(t):zl, K+1

µ

(6.38)

J

z E

(C1_Rl)n

where K = (kl,k2,...,kn) is the n-tuple of integers kj, j

= 1,...,n, in the growth (6.31) and 1 is the n-tuple

(1,1,...,1). PROOF.

For each of the 2n n-tuples u = (ul,u2,...,11n) and

each of the corresponding 2n tubes T P = IRn+iCP put

fzI(m+l)(f(z))

ji,n

=

A.

I(mn(f(zl...,zn-1'f)) df, W,

(6.39)

i(-1)

C

m = 0,1,...,k 284

n,

z E T

C

with the convention that I(O)(f(z)) = f(z), z E T µ p,n f(z) is analytic in z e

IRn+iCµ

.

Since

then f(z) is analytic in the µ

variable zn in the half plane {zn:(-1) n Im(zn) > 0) for fixed µi

(zip....Zn-1) E ((z1,....Zn-1) j

:

Im(Z1) > 0,

(-1)

Thus I(i)(f(z)) is analytic as

= 1,...,n-1) (section 4.2).

p,n

µ

a function of zn in Czn:(-1) n Im(zn) > 0) and the path of integration in (6.39) can be taken over the straight line i(-1)un

with zn since the integral is independent

connecting

A similar argument yields that Iumnl)(f(z)) is

of path.

µ

analytic in zn E (Zn:(-1) n Im(zn) > 0), m = O,l,...,kn Using the growth (6.31) and integrating in (6.39) over the I1

straight line connecting i(-1) n with zn we obtain (kn+1)

(f(z))I

Ilµ.n

n1 < M

ff

(6.40)


0 and rn > 0.

Further, we now have

(k +1) I

(f(z)) is analytic as a function of z E

n

In+iCµ

satisfies (6.40), and kn+1

k +1 (Iµ,n

a

8z

(k +1) n

n

C

(f(z))) = f(z), z E T

n

we now iterate this process by putting

285

(µk n+1n)

Iµmn11(I

(f(z))) _

zn-1 i(-1)

(kn+1)

(m) I

un-1

µ,n-1

(I

(f(z ,...,z

µ,n

1

n-2'

g,z n))) df

C

m = 0,1,...,kn-1' z E T

(k +1)

(k +1)

(...(Iµ,n

umll)(IU.2 zi

(k+i)

(k2+1)

(m)

Ill Iu.l(IU,2 S i(-1)

(6.41)

(f(z)))...)) _

(f(f,z2,...,zn)))...)) df

(...(Iµ,n

C

m = 0,1,...,k1, z E T

Beginning with the growth (6.40) on the analytic function (kn+l)

Iµ,n

n (f(z)), z E IR +iC

and the first equality in (6.41) we

iterate the process to obtain (6.40) from (6.31) and (6.39) and obtain (k1+1) Ilµ'l

(k2+1)

(IU.2

(kn+1)

(f(z)))...))I
0 and rj > 0, j = 1,...,n.

Further

the function in the absolute value on the left of (6.42) is C analytic in T u and satisfies

286

(k1+1)

aK+1

µ,1

(k2+1)

(kn+1)

(f(z)))...))) _

(...(IU,n

µ,2

azK+1 C

= f(z), z E T

where K = (ki k2,...,kn) and 1 = (1,1,...,1).

(6.43)

The

calculations in this paragraph hold with respect to each of the 2n tubes TCµ, and for each tube TCµ we have constructed the function given by the last equality of (6.41) with m = k1 C

which is analytic in T µ and satisfies (6.42) and (6.43). (C1_C1)n

We now construct a function F(z) defined for z E by putting (k1+1) F(z)

= Iµ.1

(k2+1)

(Iµ.2

(kn+1)

(...(Iµ,n

(6.44)

(f(z)))...)), C

z E T

for each of the 2n tubes fn+iCµ (C1-f1)n

.

Then F(z) is analytic in

and satisfies the growth (6.42) in (C1_C1)n

.

(Without loss of generality we can assume that the constants Qj and rj, j = 1,...,n, in (6.42) are the same for each of the C

2n tubes T j

we can replace the finite number of Qj and rj,

= 1,...,n, for the finite number of tubes Cn+iCµ by suitable

(C1_f1)n.) Thus for each of the majorizing constants for z E IRn+iCµ 2n tubes Mn+iCµ Theorem 6.3.1 yields F(z) E Aµ, z E

for each of the 2n sets A

defined in Definition 6.3.1.

Hence

by Theorem 6.3.2 there exist 2n functions hµ(t) E L2 and 2n polynomials qu(t) such that

[F(z)] = CIA (-1)Iµ

q (t) h(t);zz E (C1-1R1)n

,

(6.45)

'µ

287

where we note that

qµ(t) hµ(t)] E S'.

But by

J

(6.43) and (6.44) we have a K+1

(C1-IR1n

(F(z)), z E

f(z) =

(6.46)

azK+1

Combining (6.45) and (6.46) we have (6.38) and the proof is complete. We now obtain that the analytic functions in

(C1-l1)n

which

satisfy the growth (6.31) have distributional boundary values

in S' based on the work of Tillmann; and using our Cauchy integral defined in (6.14), we recover these analytic functions by the Cauchy integral of the boundary value. Recall the notation yµ = ((-1)µl, (-1)µ2,...,(-1)µn) E C each of the 2n quadrants Cµ

for

.

(C1-IR1)n Let f(z) be analytic in and satisfy There exists an element U E S' such that

THEOREM 6.3.4. (6.31).

[f(z)] = C(U;z), z E (Cl-I1)n

(6.47)

lim

(6.48)

with

e--+0+ 2

-1)141 f(x+ieyµ) = U

in the strong (and weak) topology of S'. PROOF.

Put

aK+1 Ut =

qµ(t) hµ(t)1

I

at

K+1

Il

µ

J

where the n-tuple K = (k1,...,kn), the 2n functions hµ(t) E

288

L 2,

and the 2n polynomials qu(t) are as obtained in Theorem

Then U E S' and (6.47) follows from (6.38) and

6.3.3.

Corollary 6.2.1.

We can rewrite U in the form

Ut = a

K+1

n

h(t)

R

atK+1

j=1

for some n-tuple 'r =

2"'" 1n) of nonnegative integers

and some function h(t) E L2.

Since f(z) E C(U;z), by applying

Corollary 6.2.1, Theorem 6.2.5, the fact (6.28), and the notation (6.35) for the classical Cauchy integral in succession we have

f z (

)

E

a

n

K+1

azK+1

z E

(C 1 _f 1 n )

f (z)

=

,

IT

n

azK+1

j=1

IT

(C1_IR1)n,

1+

I

h t );z)]

,

(

6

.49)

(

6

.

so that

K+1

a

(

j=1

(

l +z ?

)

I

(

h t );z) (

+ P (z),

5 0)

j

for some pseudo-polynomial P(z).

The partial derivative term in (6.50) is now exactly in the form which Tillmann showed has as strong (and weak) S' boundary value the z E

element U E S' in the sense that (6.48) holds for this partial derivative; and the pseudo-polynomials form the kernel of the

boundary value mapping lim L

in (6.48).

f(x+ie y

The details of these results of Tillmann are

289

contained in [131, Satz 1.1 and Satz 1.3]; thus by these results and (6.50) we have

lim

f(x+iey

=

)

µ

8 (7

K+1

n Q

K+1

(1+x?) 3

h(x)

= U

j =1

in the strong and weak topology of S'.

(6.48) is obtained and

the proof is complete.

It is interesting to note that the element U E S' constructed in Theorem 6.3.4 is uniquely determined by any element F(z) E [f(z)]; and for any such element F(z), (6.48) (With respect to holds in the strong and weak S' topology. the strong and weak topology of S' , we recall that S is a Montel space ([46, p. 510] or [135, p. 21]) as we have noted

before; thus weak convergence in S' and strong convergence in S' are equivalent [46, Corollary 8.4.9, p. 510].)

We have (C1_IR1)n

shown in Theorem 6.3.4 that any analytic function in which satisfies (6.31) has an S' boundary value in the sense

that (6.48) holds, and the analytic function can be recovered from this boundary value by the Cauchy integral defined and The boundary value mapping in (6.48)

studied in section 6.2.

is the mapping studied by Tillmann in [131]; our analysis here

builds directly upon that of Tillmann and is a natural extension of it.

In section 6.7 of this chapter we also show that the analytic function in Theorem 6.3.4 has S' boundary values from each tube Rn+iCV and that these 2n boundary values sum to U in S'.

Further the function f(z) can be recovered by a

Fourier-Laplace transform defined by the inverse Fourier transforms in S' of these boundary values from each tube Ut n+iC

µ

The following is a converse to Theorem 6.3.4 and also builds upon the work of Tillmann. THEOREM 6.3.5. Let U E S' and let f(z) E C(U;z). Then f(z) is analytic in (C1_LR1)n 290

,

satisfies (6.31), and (6.48) holds

in the strong (and weak) topology of S'.

U E S' can be written in the form (6.1).

PROOF.

Using this

form of U E S' and combining Corollary 6.2.1, Theorem 6.2.5, the fact (6.28), and the notation (6.35) in succession as in

the proof of Theorem 6.3.4 we obtain that f(z) E C(U;z) satisfies

[I(h(t);z)]J z E (C1-IR 1)n

9

E

f(z)

l l j= 1

p RR

for n-tuples /3 and 7 of nonnegative integers and some function h(t) E L2.

f(z) =

Thus n

8R

II

a

j=1

7

(l+z?) 3

j

I(h(t);z)

+ P(z), z E

for some pseudo-polynomial P(z).

Now that we have this form for f(z), the facts that f(z) is analytic in (C1-IR1)n. satisfies (6.31) in (C1_IR1)n, and (6.48) holds in the strong (and weak) topology of S' again follow by the results of Tillmann [131, Satz 1.1 and Satz 1.3].

In fact the

analyticity of f(z) is obvious from its form, and the growth The proof is complete. condition follows easily also. Let us return to our discussion of the Dirac delta function 6 and the Heaviside function H(t) in one dimension at the end of section 6.2.

We noted there that

C(6;z) = C(H'(t);z) =

TZ_

C(H(t);z), z E

(C1_11)1

Let f(z) E C(H(t);z); then g(z) = (df(z)/dz) E C(S;z) and by Theorem 6.3.5 lim

(g(x+ie) - g(x-ie)) = 0 = H'(t)

e--40+ in the strong and weak topology of S'.

This fact suggests the 291

following general result in n dimensions which is a corollary of our calculations concerning the Cauchy integral of tempered distributions. COROLLARY 6.3.1.

Let p be any n-tuple of nonnegative

Let U E S' and V = ap(U). atp

integers.

boundary value of g(z) =

ap

Let f(z) E C(U;z).

The

(f(z)) in the sense of (6.48) is

azp V; that is lim

e40+

V

L

in the strong and weak topology of S'. V E S' since it is the distributional derivative of U PROOF. E S'.

ap

f(z) E C(U;z) implies g(z) E

(C(U;z)).

By Corollary

azp p = C(V;z); now the desired boundary

6.2.1, g(z) C C 12070 (U);z

value result follows from Theorem 6.3.5.

The proof is

complete.

Thus the boundary value of any derivative of an element in the Cauchy integral C(U;z), U E S'

,

is the corresponding

distributional derivative of U.

We close this section with some examples. EXAMPLE 6.3.1.

Recall (6.29).

The function

n

f(z) = (-1/27i)n

R

j=1

z-1

,

z E

(C1-fl

1 )n

J

is in the equivalence class C(b;z) as noted in (6.29). f(z) is an example of an analytic function in

(C1-C1)n

This to

which our theory in this section applies; that is, it is (C1_tR1)n, analytic in satisfies the growth (6.31) as is easily seen, and

292

lim F_

f(x+ieyµ) = 6

10

in the weak and strong topology of S'.

This f(z) defines the Cauchy integral C(6;z) of the Dirac delta function. EXAMPLE 6.3.2. We now give an example in one dimension of an (C1_1R1)1

analytic function in

section are not applicable.

to which our results in this This is somewhat surprising for

the function that we consider because it has a pointwise boundary value on the real axis of the complex plane that is

an element of S. However, the Cauchy integral of this S' element does not contain the analytic function, and this analytic function cannot belong to the Cauchy integral of any element in S'. The function which we consider is Z t 0

L(z) = 1

where z = x+iy E C1.

,

z = 0

,

L(z) is an entire analytic function.

Now put

h(x) _

h(x) is the pointwise limit of L(z) as y =

for x E 1R1.

Im(z) - 0+

,

and h(x) E L2 c S.

It would be natural to

conjecture that the Cauchy integral C(h;z) equaled [L(z)]; but this is false as we shall see by considering growth properties and also considering boundary value properties.

First note that L(z) defined in this example does not satisfy the growth (6.31) and in fact has exponential growth.

Since the growth (6.31) characterizes those analytic functions in (C1-Ut1)1 which belong to the Cauchy integral of some U E S', according to Theorems 6.3.4 and 6.3.5, then L(z) can not belong to the Cauchy integral of any element U E S. We also show that L(z) and g(z) E C(h;z) have fundamentally 293

different pointwise boundary value properties.

As noted

above, h(x) is the pointwise limit of L(z) as y = Im(z) -> 0+.

To see what pointwise boundary value g(z) E C(h;z) has as Im(z) -o 0+ it suffices to consider the particular function G(z) defined by

2iri J_ t-z dt, z c (C1-1R1)1

G(z)

the Cauchy integral of h E L2 (recall (6.28)), since g(z) and G(z) differ by a polynomial if they differ at all.

Taking

real and imaginary parts we have

t-z

is

+ i

a

(x-t)2+y2

(x-t)2+y2

z = x+iy '

n

Hence for z E (C1-D21)1 we have m

2 G(z) =

-

n

y2

h(t)

(x- t) +y

2

dt +

PCO

+ i

h(t)

x 2

(x-t) +y

2

dt = I1+iI2.

I1 is the Poisson integral of the continuous, bounded, L2

function h(t), and we have the classical fact

lim Y--+O+

I1 = h (x)

pointwise for all x E 121.

lim

y-'0+

1

2

Further

= fi(x)

is the Hilbert transform of h E L2 ([133, Chapter 5] or [124, pp. 185 - 186]); and fi(x) exists for almost every x E 294

(-co,to)

and is in L2 [133, pp. 122 - 125].

E(x) cannot be identically

zero since h and h satisfy reciprocal formulas [133, Theorem These facts combine to show that G(z), which defines C(h;z), has a pointwise boundary value as 91, pp. 122 - 123].

y -4 0+ that is entirely different from the similar pointwise limit of L(z), which is h(x).

Again we see that L(z) is

fundamentally different from elements of C(h;z).

The analysis of this section is not applicable to L(z). However, L(z) is an example to which the Paley-Wiener-Schwartz

theorem [117, p. 272] is applicable in one dimension; in fact L(z) equals one-half the Fourier-Laplace transform of the characteristic function of the interval [-1,1]. 6.4.

THE CAUCHY INTEGRAL OF TEMPERED DISTRIBUTIONS: THE CASE

CORRESPONDING TO ARBITRARY REGULAR CONES IN Rn The domain of definition for the Cauchy integral C(U;z) of sections 6.2 and 6.3 is union of the 2n tubes Cµ.

(C1_[R1)n

Rn+iC,

which should be viewed as the

corresponding to the quadrants

In this section we prove a result which yields a Cauchy

integral of S' distributions corresponding to tubes TC defined

by arbitary regular cones C in Rn of which the quadrants C are special cases.

Analytic functions in tubes TC which

y E C, are considered obtain S' boundary values as y - 0 and these analytic functions are shown to be recoverable by a ,

Cauchy integral involving the boundary value. For C being a regular cone and C* being its dual cone, the Cauchy integral will be constructed corresponding to tubes TC for which C* has an admissible differential operator associated with it. We now define such an operator. DEFINITION 6.4.1. A homogeneous differential operator with constant coefficients 0((-1/2ni)at), a

= (at 1

at ), n

'

at

2

and the corresponding polynomial O(t) are said to be

admissible for the cone C* if for any V E S' with supp(V) c C* there exists an integer m Z 0 such that for every integer m > 0

mo there exists a unique continuous function g(t) of 295

polynomial growth with support in C

such that V =

0 m((-1/2iri)8t)(g(t))

As an example of an admissible differential operator note the following.

For the cone C being regular, C is closed, convex, and contains interior points (section 4.3). We say that the vectors ek a C*, lekI = 1 for k = 1,2,...,n, form an C*

admissible set of vectors for the cone k = 1,...,n, form a basis for IRn

.

if the ek

Vladimirov has proved the

following theorem in [140]. THEOREM 6.4.1.

Let C be a regular cone in 6tn

Let V E S'

with supp(V) C C* + N(O,R) where N(O,R) = (t E IRn:

Let m

0

denote the order of V.

Iti

< R).

Every admissible set of vectors

ek, k = 1,...,n, for the cone C* and every integer m > mo correspond to a unique continuous function g(t) such that n

V =

II

k=1

a

m+2

a

<ek

et >

'

(g(t))

w here supp(g) C C* + N(O;R) and Ig(t)l < Km (l+1tj)3m+1

t E

,

n

.

This theorem shows that admissible differential operators corresponding to dual cones C* of regular cones C do exist. For a regular cone C assume C* has an admissible set of C* vectors ek, k = 1,...,n. For V E S' with supp(V) C ,

,

Theorem 6.4.1 states that

0((-1/2iri)at) = (-2iri)n

n R

<ek

,

(-1/2iri)8t>

k=1

is an admissible differential operator for the cone C*

.

For

example, if C = Cu we know that C* = Cu has an admissible set 296

of vectors; hence the dual cone of any quadrant has an admissible differential operator associated with it.

For our work here we need to define a space of distributions which form a subspace of S'.

S2 will denote the set of all Lebesgue fn measurable functions g(t), t E such that DEFINITION 6.4.2.

,

(g(t) (l+1t12)-m) E L2 for some m > 0.

The importance of the space S2 is that it is exactly the image of the space of distributions D'2 under the S' Fourier L

transform as we now note. THEOREM 6.4.2.

The S' Fourier and inverse Fourier transforms

map D'2 one to one and onto S

and conversely.

L PROOF.

U E D'2 implies U has the representation given in L

Theorem 1.6.1.

(Recall the first sentence of the proof of Theorem 5.5.1, for example.) The result now follows by using the S' Fourier or inverse Fourier transform, distributional differentiation, and the fact that both the Fourier and inverse Fourier transform are one to one mappings of L2 onto L2 The converse is similar. .

Let C be a regular cone and let C' denote an arbitrary compact subcone of C. If(x+iy)I

Consider the growth

< M(C') (l+IzI)N (1+Iyl

),

z = x+iy E TC

,

(6.51)

where the constants N > 0 and k > 0 are independent of C', and the constant M(C') depends on C' C C.

If f(z) is analytic in

TC and satisfies (6.51), there exists a unique element V C S' with support in C

such that

lim y-->0 f(x+iy) = 9;[V]

(6.52)

yE C

and

297

f(z) _

,

z E TC

(6.53)

by Theorem 4.7.4. Using the facts developed to this point in this section we

can now obtain a Cauchy integral corresponding to elements in S' and corresponding to a regular cone. For C being a regular

cone let 0((-1/2ni)) and 0(t) be an admissible differential operator and an admissible polynomial, respectively, for the dual cone C in the following result. THEOREM 6.4.3. If for some integer m > 0 f(z) = 0m(z) < m1

U, K(z-t)>, z E TC

(6.54)

O (t)

where U E S' such that Uo = ((1/Om(t)) U) is the unique solution of the equation Om(t) Uo = U

with U

0

(6.55)

E D'2 and supp(5-1[U0 ]) c_ C* L

,

in TC and satisfies the growth (6.51).

then f(z) is analytic Conversely, if f(z) is

analytic in TC and satisfies the growth (6.51) then there exists an integer m > 0 such that for every integer m > m 0 0

(6.54) holds in S' where U E S' is the strong S' boundary value of f(x+iy) as y --i U, y E C, and Uo = ((1/Om(t)) U) is

the unique solution of the equation (6.55) with Uo E D2 and L

supp(9-1[U0]) C C* PROOF.

.

Recall from Theorem 4.4.1 that K(z-t) is an element of

D 2 as a function of t E 0n for fixed z E TC

.

By hypothesis

L

in the sufficiency, Uo = ((1/Om(t)) U) E D'2; thus by Theorem L

5.5.1 f(z) is holomorphic in TC and satisfies the growth (6.51). 298

we now prove the converse.

As previously noted, for f(z)

being analytic in TC and satisfying (6.51) there is a unique element V E S' with supp(V) c_ C such that (6.52) and (6.53) For the admissible differential operator 0((-1/2ai)at)

hold.

which we are assuming exists here, there exists an integer m > 0 such that for every corresponding to the cone C

,

0

integer m > m

0

there exists a unique continuous function g(t)

of polynomial growth with support in C

such that (6.56)

V = 0m((-1/2,ri)at) (g(t)).

Taking the S' Fourier transform of both sides of equation (6.56) we obtain the equation (Om(t) ?[g]) = 9[V] = U in S';

thus (6.57)

g[g] = (1/Om(t)) U in S'.

Since g(t) is continuous and has polynomial growth, it

easily follows that g E S2

.

Thus Uo = ?[g] E D'2 by Theorem L2

6.4.2 and 5-1[U0] = g has support in C

.

From (6.53) and

(6.56),

f(z) =

e2iri

(6.58)

dt

C

=

Om(z) 5[I *(t)

e-2n g(t);x],

C

TC, where I *(t) denotes the characteristic function of C and by the proof Recall that g E S so that 5[g] E D'2 C*. z E

;

L

299

of Lemma 4.4.1, (IC*(t) exp(-2n)) E L2 C DL2.

Thus using

Lemma 5.5.5 (recall the proof of Theorem 5.5.3) and (6.57) we continue (6.58) as f(z) = Om(z) (9[g] * 9[I *(t) e-2n]) C

(6.59)

Om(z) (((1/Om(t)) U) * g[I *(t) e-2n]) C for z E TC, and this equality now holds in S'

.

As in the

proof of Theorem 5.5.3, (6.59) is continued to obtain U, K(z-t)>, z E TC

f(z) = Om(z) < OM(t) (t)

as an equality in S' which is (6.54).

The proof is complete.

In this tube domain setting the crucial assumption in obtaining the Cauchy integral representation in (6.54) is that the dual cone C* of C has an admissible differential operator.

In this case we are able to multiply the S' element U by the reciprocal of a polynomial which makes the product an element of D'2 and hence makes the Cauchy integral defined. Note that L

the Cauchy integral representation (6.54) of the given analytic function in terms of its boundary value in the converse of Theorem 6.4.3 is obtained as an equality in S' and not as a pointwise equality. The Cauchy integral in (6.54) is

not unique because m can be any integer m > mo in the converse.

But for fixed m > m

unique corresponding to M.

0

,

U = 9[V] _ (Om(t) 9[91) is

This is the same situation as in

the Cauchy integral constructed and studied in sections 6.2 and 6.3; as we noted there, the choice of p(t) in (6.2) is not unique so that the form

300

P(z) (2



which defines C(U;z) in (6.14) is not unique for a given U E S'.

However, C(U;z) given in (6.14) is unique; and this was

the principal reason for considering equivalence classes Even though the Cauchy integral in (6.54) is not

there.

unique, (6.54) still gives the desirable recovery of a given f(z) as a Cauchy integral in terms of its boundary value; and the Cauchy integral representation in (6.54) is unique for fixed m. 6.5.

ANALYTIC FUNCTIONS WHICH HAVE S' BOUNDARY VALUES

AND WHICH ARE Hp FUNCTIONS The distributional boundary value calculation is of importance in quantum field theory where the vacuum expectation values are distributional boundary values in a relevant distribution topology of holomorphic functions in tubes in Cn. In the various field theories different distribution topologies are used but central among these is the S' topology.

We

reference, for example, Streater and Wightman [126], Reed and Simon [107], and Simon [123] for S' boundary value results associated with field theories and for associated references. In [41] Constantinescu constructs local fields which are a category of fields larger than the strictly localizable ones and which contain the tempered fields; he proves that the vacuum expectation values in local fields are distributional boundary values in the topology of (Si), of holomorphic functions in a tube domain in en defined by the forward light cone in O. Here (S1)' is the dual space of the space S1 of type S; see Gel'fand and Shilov [54] for a general discussion of the spaces of type S and their duals.

Raina [106] built upon the analysis of Constantinescu in one dimension by attaining a condition on the (Si), boundary value of an analytic function in the upper half plane such that the analytic function could be concluded to be in the Hardy space Hp corresponding to the half plane; the condition 301

is that the (S1)' boundary value be an element of Lp. The Hardy Hp spaces have been used in particle physics and

especially in form factor bounds; see Raina [106] and the references given there for a discussion of these applications of Hp spaces to mathematical physics.

In this section we are going to consider the problem studied by Raina in [106], that of obtaining conditions on the distributional boundary value of an analytic function under which the function can be concluded to be in Hp; but we are going to be concerned with arbitrary dimension and with the topology of the tempered distributions S' rather than the topology of the dual spaces of the spaces of type S. Recall the pointwise growth estimates obtained in section 4.6 for the For a regular cone C, f(z) E HP(TC), < p < -, satisfies the growth estimate (4.30) for z E TC and

Hp functions. 1

satisfies (4.33) for z E TC

for any compact subcone C' C C.

Further, f(z) E HP(IRn+iCµ), 0 < p < any quadrant Cu.

,

satisfies (4.21) for

Upon comparing these growth estimates with

the estimate (6.31) of Tillmann for quadrants that we studied in section 6.3 and the estimate (4.70) with A = 0 that was considered in section 4.7 (recall also (6.51)) we see that the

pointwise growth of Hp functions is a special case of the growth which yields S' boundary values for analytic functions. More explicitly, any element f(z) E HP(IRn+iCU), 0 < p < ,

has a distributional boundary value in the strong topology of S' for each quadrant Cu because (4.21) is a special case of the growth (6.31) with which Tillmann [131] characterized the existence of S' boundary values of analytic functions in tubes corresponding to quadrants. For an arbitrary regular cone C, f(z) E HP(TC), 1 < p < w

,

attains a strong S' boundary value

because the pointwise boundedness for p = w and the growth (4.33) for 1 < p < - are special cases of (4.70) (also (4.69)) which yields S' boundary values as noted in Theorem 4.7.4 (Theorem 4.7.3). Because of the relation of the pointwise growth of Hp functions with the growth which characterizes the 302

existence of S' boundary values and because of the central importance of the S' topology in quantum field theory, the analysis that is presented in this section will be concerned with the existence of boundary values of analytic functions in the strong topology of S'. In section 6.7 results similar to those obtained in this section for other distribution topologies, including that of the dual spaces of spaces of type S, will be discussed.

The growth with which we will be concerned in this section Let C be a regular cone and let C' be an arbitrary compact subcone of C. We consider the growth

will be stated now.

If(x+iy)I < M(C') (l+IzI)N

11-k ,

z = x+iy E TC'

,

(6.60)

where M(C') is a constant which depends on C' and where N Z and k >

0

0 are real constants which are independent of C' and

depend only on the cone C and of course on the function.

We

emphasize that an analytic function in TC which satisfies

(6.60) has a unique strong S' boundary value as y 3 0, y E C, which is attained independently of how y -+ 0, y E C, by All of the S' boundary values of analytic

Theorem 4.7.3(11).

functions in this section are obtained uniquely and y c C' C C, independently of how y -+ 0, y E C (i.e. y for any compact subcone C' of C.)

Instead of lyI-k in (6.60) we can have (1+lyl-k); recall (4.70) with A = 0 and also (6.51). All of the results of this section will hold with (1+lyl-k) in place of 1y1-k in (6.60). We have put jyl-k in (6.60) to emphasize the generalization of the pointwise growth of Hp functions obtained in (4.33), the growth of Tillmann (4.67), and the general tube growth (4.69) of Theorem 4.7.3(11). To motivate the main question of this section let us recall Theorem 4.5.2.

For C being a regular cone in fin, let f(z) E

Theorem 4.5.2 states that there exists a HP(TC), 1 < p < y E C, function h(x) E LP such that f(x+iy) -+ h(x) as y .

in the weak and strong topologies of S' and 303

f(z) = in h(t) P(z;t) dt, z E TC

(6.61)

.

I

The boundary value of f(z) here is obtained uniquely and independently of how y -i 0, y E C. As noted before, f(z) E and satisfies the growth (4.33) if HP(TC) is bounded if p = in either case f(z) E HP(TC), 1 < p < -, satisfies 1 < p < w ;

(6.60).

Thus by Theorem 4.7.3(11) or Theorem 4.7.4, f(z)

attains a 0, y E C; must be a HP(TC), 1

unique S' boundary value independently of how y -> by the proof of Theorem 4.5.2 this boundary value function h(x) E L. In summary, any function f(z) E satisfies the growth (6.60), attains a < p < ,

function h(x) E LP as strong S' boundary value uniquely and independently of how y - 0, y E C, and is recoverable by the Poisson integral of the boundary value as in (6.61). We now ask if there are any functions f(z) which are holomorphic in TC, which satisfy (6.60), and which attain an LP function, 1 < p < w as S' boundary value other than the HP(TC) functions. The answer is no as we show in the following theorem. ,

THEOREM 6.5.1.

Let C be a regular cone.

in TC and satisfy (6.60).

Let f(z) be analytic

Let the unique strong S' boundary

value of f(z), which exists, be h(x) E LP, 1 < p < . Then f(z) E HP(TC), 1 < p
0

(6.62)

with µ = (µ1,...,µn) being the n-tuple whose entries are 0 or 1 that defines the quadrant C dimension.

, N Z 0 is fixed, and n is the

We have XE(x+iy) - XE(x) in the weak and strong

topologies of S' as y

0, y E C. Further, if there exist elements U C S' and h(x) E LP, 1 S p Ssuch that (X (x) U) = h(x) in S' then U = h(x)/X, (x) in S'. PROOF.

Since a limit result is desired as y -i 0, y e C, we

can assume without loss of generality that IyI S M, y e C, for µj

a fixed M > 0.

Since ((-1)

yj) > 0, j = 1,...,n, for y e C

, we have

C Cµ

IXE(x+iy) - XE(x)I S

S

n U

IXE(x+iY)I + IXE(x)l n

E2x2)(N+n+2)/2 +

j=1

II

(1+e2x2 (N+n+2)/2

j=1

The product of the right side of this inequality and an IRn

which is independent element p E S is an L1 function of x E Using this fact and the fact of y C C C C such that jyj S M. 11

that XE(x+iy) -> XE(x) pointwise as y -a 0, y E C, the Lebesgue dominated convergence theorem yields

lim yCC

In

XE (x+iy) p(x) dx =

In

Xa (x) p(x) dx, P e S 305

which proves that XE(x+iy) -> X(x) in the weak topology of S'

as y - 0, y E C. As we have noted in the proof of Lemma 4.4.2, the strong convergence in S' now follows from the weak convergence.

To obtain the second result in this lemma first note that (1/X6(x)) is a multiplier in S and hence in S'; and h E LP, 1

< p < -

implies h E S'.

,

By hypothesis (XE(x) U) = h(x) in

S'; thus for any p E S =

which proves U = h(x)/XE(x) in S' as desired.

Let the cone C be as in Theorem 6.5.2 and the function XE(z) be defined by (6.62). We have LEMMA 6.5.2.

I1/X(z)I

1

z E TC

,

> 0

9j

If ((-1)

yj) > bj > 0, j = 1,...,n, y E C C Cu, then

IXE(x+iy)I >

n U

(Kj(e,bj))

N+n+2

(1+IZj

N+n+2 )

j=1

for some constants Kj(e,bj) depending on e > 0 and on bj j

= 1,..

,

n. I1j

PROOF. 1,...,n.

z E TC

C C C

,

implies ((-1)

yj) > 0, j

=

For each j = 1,...,n,

I1-i6(-1)u3zj

306

,

(1+6(-1)Ajyj )2 + e2 x] ) 1/2

>

(1+62x] ) 1/2

>

1

from which I1/Xe(z)I < 1, z C TC

,

e > 0, follows.

To prove the other inequality note that for z E TC, C C CA

,

((l+Iz11)/Iz1I) -+ 1 as IzjI

CO

= 1,...n.

j

,

Applying the definition of limit with e = 1/2 for each j = 1,...,n we obtain a number MI > 0 such that

1 + I zj I < (3/2) I zj I

if

I zj I

> Mj > 0

.

Thus

e

if

IzjI > (2e/3)(1+lz11) if IzjI > Mj > 0.

then for ((-1)

I zj I < Mj , z C TC , C C Cu ,

µ

yj) > 5i > 0

we have

(1+1zj1)/Iz1I < (1+Mj)/6 j

,

hence e

Iz11

>

(e 6 j/(1+Mj))(1+1zj1)

It follows that 6

1z1I

> Kj(e,61)(1+1Z11), ((-i)

µ

7

yj) > sj > 0, j = 1,...,n,

where K1(e,Sj) = min(2e/3, e61/(1+M1)) for z = (zl,...,zn) E P TC. j

Thus for yj = Im(zj) satisfying ((-1)

j yj) Z sj > 0,

= 1,...,n, we have

307

I1 - if(-1)Aj

zjI = ((1+e(-1)Pjyj)2 + e2 xj)1/2 > EIzjI

> Kj(E.bj)(1+IzjI)

for each j = 1,...,n.

The second inequality in the conclusion

of this lemma follows from this and from the definition of The proof is complete.

XE(z).

LEMMA 6.5.3.

Let the cone C be as in Theorem 6.5.2.

be analytic in TC and satisfy (6.60). gE(z) = f(z)/XE(z), z E TC

,

Let f(z)

Put

e > 0

where XE(z) is defined in (6.62) with N being the N in the growth (6.60) of f(z).

For each e > 0 there exists a

continuous function GE(t) with support in C* such that

ge(z) = In Ge(t)

PROOF.

e2ai

dt, z E TC

By Lemma 6.5.2, I1/XE(z)I S 1, z E TC , e > 0.

Thus

gE(z) satisfies (6.60) since f(z) does, and ge(z) is analytic in TC.

By Theorem 4.7.3(11) or Theorem 4.7.4, for each e > 0 there is a unique UE E S' such that gE(x+iy) -> UE as y - 0,

y E C, in the strong topology of S'; and Ue is obtained independently of how y -> 0

Now let C' be an arbitrary compact subcone of C C Cµ and let b > 0 be arbitrary.

,

y E C.

From the second inequality obtained in Lemma 6.5.2

together with (6.60) we have the existence of a constant M'(C',b,e) such that

Ige(z)I

0) which is a compact subcone of C.

Now choose

6 > 0 arbitrary but fixed; for this 6 we use the growth (6.64), which holds for arbitrary 6 > 0, applied to the compact subcone C' = (y c C: y = ]iyo, X > 0) of C and for z = x+iy E IR

n+i(C'\(C' fl N(O,b))) to obtain

309

Gto)I < exp(2,r)

IgE(x+iy)l dx

(6.66)

J

In

n < M'(C',b,E) exp(2a)

J IR

t

n j=1

(1+Ixj)- n-2 dx

< M"(C',b,E) exp(2,rX)

< M"(C',6,E) e-2-ir'n where M"(C',b,E) is a constant which is independent of X, n is

a fixed real number such that n C (0,1), and this inequality holds for all X > 6. GE(t0) is independent of X; thus letting A -> - in (6.66) we see that GE(t0) = 0.

Since to was an

arbitrary point in C* = In\C* and C* is a closed set we thus have supp(GE) C

as desired.

C

From (6.64), gE(x+iy) C L1 n L2 as a function of x C Qtn for

each fixed y E C.

> 6 for some b > 0, and y E C' for some compact subcone of C since C is open.)

(For any fixed y C C,

Jyj

Thus (6.65) can be rewritten as

e-2,r GE(t)

=

9-1[gE(x+iy);t],

Y E C

where 9-1 can be interpreted as either the L1 or L2 inverse Fourier transform. By the Plancherel theory, (exp(-2,r) GE(t)) E L2, y C C, and

g6(x+iy) =

?[e-2a

GE(t);x], z = x+iy C TC

,

(6.67)

with this Fourier transform being in the L2 sense. For any compact subcone C' C C and any 6 > 0 the growth (6.64) and analysis as in (6.66) yields

310

IG(t)I
0, EE(t) = 1 for t on an a neighborhood of C* and f6(t) = 0 for t E Rn but not on a (Recall the proof of Corollary 4.7.2 for a similar previously constructed function EE (t).) Then Rn for each (EE (t) exp(2ai)) E S as a function of t E 2e neighborhood of C*.

z E TC.

Using (6.70) and (6.71), we obtain

Ge(t)

e2ni

dt =

J *

C (6.72)

=

313

e2ni

HE (t)

= J

dt

C

for z E TC

.

g(z) =

From (6.63), (6.72), and Lemma 4.4.3 we have e2ni

HE (t)

dt

(6.73)

J

C

= J

(h(t)/XE(t)) K(z-t) dt, z E TC

In

Now let w be an arbitrary but fixed point of TC. (K(z+w) g6(z)), z E TC, is analytic in TC.

The function

From the proof of

Lemma 6.5.3 g6(z) satisfies exactly the growth (6.60) of f(z). Thus (K(z+w) gE(z)) satisfies this same growth multiplied by the constant My

,

v = Im(w), of (4.10) in Lemma 4.4.2 which is

independent of z E TC.

By the boundary value result of Lemma 4.4.2, K(x+iy+w) -> K(x+w) in S' as y - 0, y e C, for each w E TC; by this convergence, the property ga(x+iy) -> UE as y E C, in S' contained in the proof of Lemma 6.5.3, and Lemma 6.5.4 we have

y

lim y-0 K(x+iy+w) gE(x+iy) = K(x+w) UE

(6.74)

yE C

= K(x+w) (h(x)/XE(x))

in the strong topology of S'; and (K(x+w) (h(x)/XE(x))) E LP, 1

< p < 2, since h E LP, 1 S p S 2, here and both K(x+w) and

1/XE(x) are bounded for x c Itn.

We thus have the same type of

hypothesis for (K(z+w) gE(z)), z E TC in obtaining (6.73).

as we did with gE(z)

By (6.74) and the same proof as in

obtaining (6.73) we have

314

,

K(z+w) gE(z) =

(6.75)

(K(t+w) h(t)/XE(t)) K(z-t) dt, z E TC

J

.

fn

For z = x+iy E TC we now choose w = -x+iy E TC

.

With this

choice of w, (K(t+w) K(z-t)) = IK(z-t) I2 and K(z+w) = K(2iy); and (6.75) becomes gE(z) =

(h(t)/XE(t)) P(z;t) dt, z E TC

.

(6.76)

fn

For the considered cases 1 < p < 2, (6.69) is now obtained by combining (6.63), (6.73), and (6.76).

We now obtain (6.69) in the remaining cases that 2 < p < . For any e > 0 it is obvious that (1/XE(x)) E Lq for all q, 1

By hypothesis in the present case, h(x) E LP Thus (h(x)/XE(x)) E L1 fl LP 2 < p < . If

< q


0, y E C, from Theorem 4.5.2, we apply Corollary 4.7.2 to obtain an element V

E S' with support in C

such that h = 5[V] in S' and such that

(6.91) holds for z E TC .

This completes the proof of Theorem

6.6.1.

Case IV was included in Theorem 6.6.1 because the conclusion of the existence of an element V E D'2 is somewhat L

more precise than can be obtained in case V for the general 325

case of an arbitrary regular cone since D'2 C S' L

By combining results contained in sections 6.5 and 6.6 we have two corollaries of Theorems 6.5.1 and 6.6.1. COROLLARY 6.6.1.

Let the cone C and the analytic function

f(z), z E TC, satisfy the hypotheses of Theorem 6.5.1 for 1 < Then (6.61) holds and there exists a function g(t) E (1/p) + (1/q) = 1, with supp(g) c C* almost everywhere

p < 2.

Lq

,

such that (6.92) holds.

If p = 2 we further have

f(z) = In h(t) K(z-t) dt, z E TC

(6.95)

where h(x) E L2 is the unique strong S' boundary value of f(z) in the hypothesis of Theorem 6.5.1. PROOF.

By Theorem 6.5.1 f(z) E HP(TC), 1 < p < 2, and (6.61) holds. The existence of g(t) E Lq (1/p) + (1/q) = 1, with supp(g) C C almost everywhere such that (6.92) holds now ,

follows by Theorem 6.6.1, part III.

If p = 2 there exists a

function H(t) E L2 such that f(z) = In H(t) K(z-t) dt = In H(t) P(z;t) dt, z E TC

with H(t) = 9[g(i); t] E L2 by Theorem 4.5.1.

(6.96)

But (6.61)

holds.

As at the end of the proof of Theorem 6.5.1 we have

Ilh(x)

H(x)IIL2 < IIh(x) - f(x+iy)llL2


h(x) as y - 0, y E C, in Lp if 1 < p < in the weak-star topology of L" if p = w, and in the strong topology of S' for 1 < p S W PROOF. f(z) is analytic in TC, satisfies the growth (4.70) with A = 0, and has 9[V] as its S' boundary value as y y E C, by Theorem 4.7.4. The growth (4.70) with A = 0 is such that the proof of Theorem 6.5.1 is applicable as we have noted in section 6.5, and ?[V] = h E Lp, 1 < p < -, by hypothesis here. Thus by Theorem 6.5.1 f(z) E HP(TC), 1 < p < and (6.61) holds. We have already obtained the conclusion that 5[V] = h is the S' boundary value of f(z). The facts that f(x+iy) --> h(x) as y -. 0, y E C, in LP if 1 < p < - and in the weak-star topology of Lm if p = - follow from the representation (6.61) by the proof of Theorem 4.5.2. COMMENTS ON CHAPTER 6 The construction of the Cauchy integral of tempered distributions in section 6.2 was motivated by one dimensional 6.7.

analysis of J. Sebastiao e Silva [122, sections 4 and 5] and by the analysis of H. G. Tillmann [131]. The analytic functions in Theorem 6.3.4 and 6.3.5 can also be shown to be recoverable by a Fourier-Laplace transform.

Consider the function

g(z) =

K+i

8

n II

Oz

K+1

IT

(1+zj)

j

I(h(t);z)

,

z E

(C1-Otl)n

,

(6.97)

j=1

which is part of the representation of f(z) in (6.50).

g(z)

(C1-IR 1)n

and satisfies the growth (6.31). Thus is analytic in g(z) has a strong and weak S' boundary value Uµ E S' from each of the 2n tubes Dtn+iC

lim

[131, Satz 1.1 and Satz 1.2]; hence

(_1)IµI g(x+iey

u

(-1)1111

U

u 327

Also

in the strong and weak topology of S'. lim

C (_1)IµI P(x+i6yu) = 0

e--,0+ µ

in S' for any pseudo-polynomial P(z) because the pseudo-polynomials form the kernel of the boundary value mapping [131, Satz 1.3]. We conclude that the S' boundary

value U in the conclusion (6.48) of Theorem 6.3.4 satisfies U =

(-1)µl Uµ

in S' where each Uµ is the strong S' boundary value of g(z) Un+iCµ. Because g(z) satisfies from each of the 2n tubes (6.31) it also satisfies the growth

n lg(z)l