Richard D Carmichael Wake Forest University
Dragisa Mitrovic University of Zagreb
Distributions and analytic functions
m Longman
N NW
W Scientific &
- Technical
Copublished in the United States with John Wiley & Sons, Inc, New York
Longman Scientific & Technical, Longman Group UK Limited, Longman House, Burnt Mill, Harlow Essex CM20 2JE, England and Associated Companies throughout the world. Copublished in the United States with John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158
© Longman Group UK Limited 1989 All rights reserved; no part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without either the prior written permission of the Publishers or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, 33-34 Alfred Place, London, WC1E 7DP. First published 1989
AMS Subject Classification: (Main) 46F20, 46F12, 46F10 (Subsidiary) 32A10, 32A07, 32A40 ISSN 0269-3674
British Library Cataloguing in Publication Data Carmichael, Richard D. Distributions and analytic functions. 1. Calculus. Bounded analytic functions 1. Title
H. Mitrovic, Dragia
515'.223
ISBN 0-582-01856-0 Library of Congress Cataloging-in-Publication Data Carmichael, Richard D. Distributions and analytic functions/Richard D. Carmichael, Dragisa Mitrovic. p. cm.- (Pitman research notes in mathematics series, 0269-3674 206) Includes bibliographical references and index. ISBN 0-470-21398-5
1. Distributions, Theory of (Functional analysis) 2. Analytic functions. 1. Mitrovic, Dragi§a, 1922- . II. Title. III. Series. QA324.C37
515,7'82-dcl9
1989
88-34332
CIP
Printed and bound in Great Britain by Biddies Ltd, Guildford and King's Lynn
Contents
Preface
Dedication Chapter 1
1
1.1
Spaces and properties of distributions Introduction and preliminaries
1
1.2
The spaces D and D'
5
1.3
The spaces E and E'
15
1.4
The spaces S and S'
18
1.5
The spaces 0a and Oa
23
1.6
The spaces DLp and D'jjp
27
1.7
29
1.8
Convolution of distributions The Fourier transform
1.9
The spaces Z and Z'
39
Chapter 2
Distributional boundary values of analytic functions in one dimension
35
42
2.2
Introduction Distributional analytic continuation
46
2.3
Analytic representation of distributions
54
2.1
42
in E' ([R) 2.4
Analytic representation of distributions
72
in O' a(IR ) 2.5
Distributional Plemelj relations and boundary value theorems
83
2.6
Representation of half plane analytic and meromorphic functions
89
2.7
Equivalence of convergence in DI(M) and 0a- (IR)
103
2.8
Comments on Chapter 2
106
Chapter 3 3.1
3.2
Applications of distributional boundary values
Introduction Applications to boundary value problems
111
111
116
3.3
Applications to singular convolution equations
135
3.4
Comments on Chapter 3
146
Chapter 4
Analytic functions in Cn kernel functions
,
cones, and
150 150
4.3
Introduction Analytic functions of several complex variables Cones in Rn and tubes in en
4.4
Cauchy and Poisson kernel functions
157
4.5
Hp functions in tubes
167
4.6
Growth of Hp functions in tubes
174
4.7
Fourier-Laplace transform of distributions and boundary values
179
4.8
Comments on Chapter 4
201
4.1
4.2
Chapter 5
151
153
Distributional boundary values of analytic functions in n dimensions
205
Introduction Analytic representation of distributions in E': the scalar valued case
205
5.3
Analytic representation of vector valued distributions of compact support
222
5.4
Analytic representation of distributions in 0'
230
5.5
Analytic representation of distributions in D'p
232
5.1 5.2
206
LP 5.6
Comments on Chapter 5
Chapter 6
The Cauchy integral of tempered distributions and applications in n dimensions
258
260
6.1
Introduction
260
6.2
The Cauchy integral of tempered distributions:
262
the case corresponding to the quadrants in Rn 6.3
Cauchy integral representation of the analytic functions which have S' boundary values
280
6.4
The Cauchy integral of tempered distributions: the case corresponding to arbitrary regular
295
cones in Rn
6.5
6.6
6.7
Analytic functions which have S' boundary values and which are Hp functions Fourier-Laplace integral representation of Hp functions Comments on Chapter 6
301
321
327
References
333
Index
345
Preface
Analysis concerning the representation of distributions in the sense of L. Schwartz as boundary values of analytic functions in one and several variables is presented in this book. The analysis is based on the research of the authors, and the basic material presented here associated with the topic of study is not contained in any other book. Previous research books concerned with the topic under consideration here have been written by Bremermann [11], Beltrami and Wohlers [2], Roos [114], and Vladimirov ([135], [136].) The present book is a companion work to these; much of the analysis in this book has been developed since the publication of these companion works.
Research in the area
considered here finds applications in quantum field theory, partial differential equations, and convolution equations in addition to other areas; and research in this area continues with developments now occuring as well in the representation of ultradistributions in the sense of Beurling and Roumieu as boundary values of analytic functions. Basic for the study of this book are a knowledge of the rudiments of the distributions of L. Schwartz and of basic complex variable in one dimension.
For the convenience of the
reader, a review of the test spaces and distributions to be used in this book is given in Chapter 1 along with a brief discussion of the convolution and Fourier transform of distributions. The concept of analytic function of several complex variables is defined at the beginning of Chapter 4, and some basic facts of these functions have been collected No previous working experience with several complex there. variables is assumed on the part of the reader or is needed to study this book.
A review by the reader of the basic facts
concerning the Fourier transform for LP functions, 1 < p <
2,
would be helpful for the reading of some parts of Chapters 4-6.
Distributional boundary values of analytic functions in one dimension in the topologies of D'(IR) and O'(IR) will be studied Analyticity and growth properties and boundary value results of the Cauchy integral of elements in E'(IR) and in Chapter 2.
O'(6t) will be obtained; conversely, analytic and meromorphic
functions in half planes with appropriate conditions are studied with respect to distributional boundary values in the In some cases recovery of the D'(IR) and 0;(It) topologies. analytic or meromorphic functions in terms of the Cauchy integral of the boundary value is obtained.
Of particular
interest throughout the development of Chapters 2 and 3 is the construction and application of the distributional Plemelj relations concerning the distributional boundary values involving both E'(IR) and Oa(IR) distributions.
These relations
are the natural extension to distributions of the boundary value relations of half plane analytic functions introduced by the Yugoslav mathematician J. Plemelj in the early twentieth century.
The distributional Plemelj relations are used
systematically throughout Chapters 2 and 3 and in particular in generalizations of results of Bremermann and Beltrami and Wohlers. The generalization of the Plemelj relations to
distributions was given independently at approximately the same time (1966-1967) by Mitrovic and by Beltrami and Wohlers but by using different techniques.
Several new results are published and proved for the first time in Chapter 2 including the whole of section 2.7.
Revised proofs from those given in the original papers of some of the other results are given in Chapter 2.
Chapter 3 contains applications of the distributional boundary value results of Chapter 2.
In particular
generalizations of boundary value problems of Plemelj, Hilbert, Riemann-Hilbert, and Dirichlet to the setting of
Further, singular convolution equations in the distributional setting are stated distributions are stated and solved.
and solved using the distributional boundary value results and the Plemelj relations. The analysis in Chapters 4-6 is obtained in n dimensions.
After defining the concept of analytic function of several complex variables in Chapter 4 we then proceed to introduce
topics in the remainder of the chapter that are needed in 1n and tubes in Cn are defined and Cones in Chapters 5 and 6. studied. The Cauchy and Poisson kernel functions corresponding to tubes in Cn are shown to be in relevant test spaces.
Hp functions in tubes are recalled, and a pointwise
growth estimate for these functions is obtained.
The
Fourier-Laplace transform of distributions is related to analytic functions which satisfy various growth conditions; in some cases the analytic functions can be recovered by the Fourier-Laplace transform of the inverse Fourier transform of the boundary value.
Some of the results in Chapter 5 extend corresponding results in Chapter 2 to n dimensions. The Cauchy and Poisson integrals of distributions in E' = E'(Dn) and Oa = 0a(fn) ,
,
DLp = D'
(,n) are studied in Chapter 5 and are shown to have
distributional boundary value properties.
The Cauchy
integrals are analytic functions in tubes in Cn and satisfy Certain analytic functions are related to distributional boundary values in these spaces. Except for growth conditions.
section 5.3, the entire book concerns scalar valued distributions in the sense as originally defined by L.
Schwartz; in section 5.3 we present some boundary value results of vector valued distributions of compact support. H. G. Tillmann was an original investigator of the representation of distributions as boundary values of analytic functions.
In Chapter 6 we define a Cauchy integral of
tempered distributions which we use to recover the analytic functions in tubes defined by quadrants that Tillmann showed obtained S' = S'(IRn) boundary values; this analysis builds
upon that of Tillmann.
In addition, a Cauchy integral of S'
distributions that was defined by Vladimirov corresponding to arbitrary tubes in Cn is studied. The pointwise growth of Hp functions in tubes, which is obtained in Chapter 4, is a special case of the growth that characterizes boundary values in S'; necessary and sufficient conditions are given in Chapter 6 for an analytic function in a tube which has an S' Fourier-Laplace integral representations of Hp functions in tubes, 0 < p are boundary value to be an Hp function.
given at the conclusion of Chapter 6.
Examples of the topics introduced are given throughout the book.
Each chapter concludes with a section containing comments. Frequently other conclusions that can be made on the topics of the chapter are discussed, and the relations between the results presented with those of similar investigations are indicated.
Always we have attempted to give a historical
perspective of the analysis presented in this book with the development of the subject matter in general although we do not claim completeness in this effort.
The authors take great pleasure in acknowledging the contributions of three individuals in the realization of this We thank Mrs. Teresa Munnell of the Department of
book.
Mathematics and Computer Science of Wake Forest University for her expert work on the word processor at all stages in the development of the manuscript. We also thank the reviewers of the manuscript for their thoughtful and helpful suggestions.
Winston-Salem and Zagreb July 25, 1988
R.D. Carmichael and D. Mitrovic
To Jane and Mary Jane R.D.C.
To the memory of my mother Anka D.M.
1 Spaces and properties of distributions
1.1.
INTRODUCTION AND PRELIMINARIES
For the convenience of the reader, in this chapter we review the definitions, constructions, and properties of the test functions and distributions that will be considered in this book.
In addition we recall the definitions and properties of
the convolution and Fourier transform of distributions, both of which are operations on distributions that will play an important role in the analysis presented here.
We shall state important representation theorems concerning the structure of
the distributions and give references for their proofs. The analysis of Chapters 2 - 6 is built upon the theory of distributions of L.
Schwartz [117].
For a complete
development of the distributions and the topological vector space preliminaries upon which distributions are based we refer to Horvath [60] and Treves [134].
We shall use some topological vector space terminology here; we refer to [60] and [134] and to the very readable book [108] by Robertson and Robertson for definitions. The development of the theory of distributions is related to Heaviside's operational calculus, Dirac's formalism of the 6 function, contributions by J. Hadamard, M. Riesz, and S. Bochner, and S. L. Sobolev's generalized solutions of partial differential equations. A systematic and unified exposition of the distribution theory based upon topological vector spaces in functional analysis was given by L. Schwartz in 1945 - 1950.
Since the appearance of Schwartz's theory, many
mathematicians have contributed in various ways to the theory of distributions.
A very nice history of the evolution of the
mathematics leading to the theory of distributions together with an extensive bibliography has been given by Synowiec [128].
we now give a brief discussion concerning some of the ways that the theory of distributions has overcome some 1
difficulties that arise in classical analysis.
A very
familiar tool of applied mathematics is the so-called Dirac delta function 6(x) which is usually defined by
10
,
x # 0
,
x = 0
,
6(x) _
with 6(x) dx = 1 1J
fR1
and if T is a continuous function on
IR1
then
6(x) ,p(x) dx = p(0) fJ
1R1
No function exists in classical analysis which has the properties ascribed to this 6 function. The theory of distributions gives a rigorous mathematical foundation for the purely formal calculus of Dirac; the 6 function is interpreted as a continuous linear functional. Dirac's remarkable equality 6(x) = H'(x) is interpreted to mean that 6(x) is the distributional (generalized) derivative of the Heaviside function
1, x >
0
H(x) = 0
,
x < 0
.
One property of distributions which is essentially different from the situation pertaining to locally integrable functions is that distributions are infinitely differentiable.
Every locally integrable function has a distributional
derivative since it can be identified with a certain distribution. In contrast to classical analysis, a convergent sequence of distributions can always be differentiated and the resulting sequence converges to the derivative of the limit. 2
The theory of distributions treats differential equations in a qualitatively new way by introducing distributional solutions instead of the continuous function solutions of classical analysis.
This yields the proofs of the very
general theorems on the existence of solutions of partial differential equations.
In particular a linear differential
equation whose right side is a discontinuous function can not be considered in the classical sense.
Note that the equation
xy' = 1 has no classical solution on all of IR1; its
distributional solution is given by y(x) = logIxj + c1 H(x) + c2
, where H(x) is the Heaviside function and c1 and c2 are
arbitrary real constants.
The classical Fourier transform of a polynomial or the Heaviside function does not exist. In distribution theory the class of functions which are Fourier transformable is greatly enlarged and includes the polynomials and H(x). As a final contrasting idea between distributions and classical analysis we recall that the density of a mass that is distributed continuously along an interval in R1 is a continuous function. However, if the total mass is concentrated at a finite set of isolated points of R1 then the corresponding density function does not exist in classical analysis. The density of a point unit mass located at the origin is equal to the Dirac b distribution.
we complete this section by giving some important Let TVS definitions and notation for the work in this book. be an arbitrary topological vector space. A linear functional U on TVS is a mapping DEFINITION 1.1.1. from TVS into the complex numbers
C1
such that
= c1 + c2 for all W1 and W2 in TVS and all cl and c2 in O1; here denotes the complex number obtained by
operating with U on p E TVS.
The technical definition of continuity of a functional U on TVS is given, for example, in [108, p. 8]. DEFINITION 1.1.2.
A function f(x) defined on In is locally 3
integrable if it is measurable and
JK
If(x)I dx <
fin;
for all compact sets K C
Lloc(ln) will denote the set of
all equivalence classes of locally integrable functions on IR
Lloc(,n)
is a large class of functions; the piecewise
continuous functions on 1n are in Lloc(In) as are the integrable functions and also the measurable and locally bounded functions.
The support of a function f(x) defined on In is the closure in IRn of the set (x E 1n: f(x) x 0) and will be denoted by supp(f). DEFINITION 1.1.3.
The support of a function f(x), supp(f), is the smallest closed subset of In outside of which f(x) is identically zero. If supp(f) is a bounded set then supp(f) is compact, and we say that f has compact support.
The following containments
hold:
supp(f1f2) c supp(fl) fl supp(f2)
and
supp(fl+f2) c supp(fl) U supp(f2)
.
Let a = (al,a2,.... an) be an n-tuple of nonnegative integers, n = 1,2,3,...
For notation, the order of a is jal = a1 + a2 + ... + an and a! = al!a2!...an!. We define the .
differential operator Da = Dt
,
t = (t1,t2,...,tn) E IRn
putting
a
a
a
Dap(t) = Dtv(t) = D11 D22...Dnn ap(t) 4
,
by
where -1
_
Dj
2ai
a
= 1,...,n
j
atj
Here the inclusion of the term (-1/2zri) is simply for
convenience of notation in relation to analysis concerning the Fourier transform given in this book, and the inclusion of the subscript t in Dtp(t) emphasises that the differentiation is with respect to the variable t E t11 t22...tnn for t E IRn
Rn .
We also put to =
Throughout 0 will denote the
.
origin (0,0,...,0) in IRn
1.2. THE SPACES D AND D' DEFINITION 1.2.1.
D = D(IRn), n = 1,2,3,..., will denote the
vector space of all complex valued infinitely differentiable functions on Rn which have compact support, and the elements of D will be called test functions.
An example of a function in D is given by a2
wa(x) =
1
0 ,
Ixl < a
x12- a 2 1x1
>a
,
for any fixed real number a > 0 where x E 1x1
2 = (x1 + x2 +...+ x2)
fn
and as usual
1/2 .
Other test functions can be
constructed from wa(x) by the method of regularization. We now topologize the vector space D.
Let K be a fixed
compact subset of IRn and denote by DK = DK(IRn) the subspace of
D consisting of all functions in D which have their support in K.
We consider the locally convex topology on DK defined by
the sequence of norms
5
jalpm
IDatp(t)I
Sup
, m = 0,1,2,...
The sequence (,p,) of functions w, E DK converges in DK to the
function p E DK as X -i X0 if and only if the sequence Pa
ttpx
(t)) converges uniformly to Dap(t) on K for all n-tuples
(Here X varies over an indexing a of nonnegative integers. set which may be a discrete or continuous set.) We recall that DK is a Frechet space.
Let us choose an increasing sequence (Kj)jC1 of compact IRn
subsets of
whose union is Rn and consider the locally
convex topological vector spaces DK
,
j
= 1,2,...
We have
.
J CO
D = JU1
,
and the topology of DKJ is
DKJ C DKj+1 ,
DKJ
identical to the one induced on it from DK
Since the
J+1 conditions are satisfied for applying the inductive limit construction, we define on D = D(Itn) the inductive limit topology of the spaces DK
,
j
= 1,2,...
.
This topology is
J
independent of the choice of the (K1), and the topology of DK J
is identical to the one induced on it by that of D.
We
LRn
further have that for any compact set K C the topology of DK is identical to the one induced on it by the topology of D. ,
It is interesting to note that the space D is not metrizable or normable. REMARK 1.2.1. From the topology defined on D, we have the following criterion for convergence in the space D: a sequence (,p,) of functions in D converges in the topology of D to a
function tp e D as X --, X0 if and only if there is a compact subset K C n such that supp (,px) c K for each X
,
supp (gyp)
c K,
and for every n-tuple a of nonnegative integers the sequence converges to
6
uniformly on K as A -i Xo
.
It is easy to prove that the operation of differentiation Also multiplication of elements
is continuous from D into D.
in D by a function g E Cw(IRn), the set of all infinitely
differentiable functions on Mn is continuous from D into D. DEFINITION 1.2.2. The dual space D' = D'(IRn) of D equipped ,
with the previously stated topology is the space of distributions; D' is the set of all continuous linear functionals on D equipped with this previously stated topology.
Let T E D' and p E D.
The value of T at c will be denoted
The vector space structure of D' is defined in the and usual way as follows: for T, T1, and T2 in D' w E D .
,
,
c E C1 we put
= c and
= + .
Among the topologies on D' that are compatible with the vector space structure, the most important are the weak topology (convergence on finite subsets of D) and the strong topology (convergence on bounded subsets of D.) The weak dual topology is defined by the family of seminorms (p
p E D)
where
pw(T) = Jj, T E D'
.
This implies the following criterion of convergence for sequences of distributions: a sequence {T.} of elements of D' converges weakly to zero in D' as X -> X0 if and only if for every W E D the sequence of complex numbers {} converges to zero as X
X0
.
7
A subset B of D is bounded in D if and only if the supports of all functions p E B lie in some fixed compact subset K C Otn
and for every a there is a number Ma such that
SUP JDap(t)j tUK
< Ma
E B
The strong topology is introduced on D' by means of the seminorms pB(T) = SUB Il
Thus in the strong topology of D' the sequence (T.) of elements in D' converges as B varies over all bounded sets in D.
to zero as X --> X0 if and only if
lim X)X O sEB = 0 qp
That is, a sequence (T.) of elements in D' converges strongly to zero in D' as X -. X0 if the sequence of complex numbers
converges to zero uniformly on every bounded subset
of D as X
X
0
From the above two definitions it is clear that strong convergence in D' implies weak convergence in D' .
Conversely, if TX -' T weakly in D' as X -, X0 for a sequence (TX) in D' then T E D' and T. --+ T strongly in D' as A
A0
the fact that weak convergence implies strong convergence in D' follows from the fact that D is a Montel space [60, p. 241 and p. 314] and the result [46, Corollary 8.4.9, p. 510]. (In Chapters 2 and 3 we restrict ourselves to the weak topology of D'.)
The following theorem gives equivalent conditions for a linear functional on D to be a distribution. 8
Let T be a linear functional on D.
THEOREM 1.2.1.
Then the
following assertions are equivalent: (i)
T is a distribution;
(ii)
T is sequentially continuous on D; that is, for every
sequence (pA) which converges to zero in D as A --+ X0 then
converges to zero as A --b Ao ; (iii) for every compact subset K C Rn the restriction of T on DK is continuous on DK ;
(iv)
for every compact subset K C Rn there exist a
positive real number M and a nonnegative integer m depending only on K such that Il
< M Is al
s sup p IDaw(t)I, w e DK
One of the most important examples of a distribution is the Dirac b functional defined by w E D. The functional T from D to O1 defined by
EXAMPLE 1.2.1.
00
=
j=0
p(1)
is a distribution.
The functional T from D to O1 defined by = Iw(0)I is not a distribution. The simplest distributions, but among the most useful, are those generated by locally integrable functions. integrable function f on IRn
,
For every fixed locally
the functional Tf from D to O1
defined by
=
fn
f( t) p(t) dt
,
pED
is a distribution. REMARK 1.2.2.
A distribution T is called a regular
distribution if there exists a locally integrable function such that the equality (1.1) holds. All other distributions 9
are said to be singular.
For example the Dirac 6 distribtuion
is a singular distribution.
The function 1/x, x E IR1
,
is not
locally integrable on IR1; hence it does not define a regular
But let us set
distribution.
f
p> = vp
°xx
dx = lim
Wxx
a D.
dx,
IX >£ The limit on the right side is called the Cauchy principal value of the integral
and vp
x
is a distribution.
Two distributions T and U are said to be
DEFINITION 1.2.3.
equal if
for all qp E D
.
The basic identification between regular distributions and locally integrable functions is contained in the following theorem.
THEOREM 1.2.2.
Let f and g be two locally integrable The regular distributions Tf and Tg are
functions on stn.
equal in D' if and only if f(x) = g(x) almost everywhere on n IR
Now denote by D'(Otn;r) the space of all regular
distributions generated by locally integrable functions. Consider the map A:
D' (IRn,r)
Lloc(IRn)
given by A(f) = Tf
.
We recall that an element of Lloc(IRn) is
the class formed by all functions that are equal almost everywhere to a given locally integrable function on IRn is easy to show that the map A is linear.
10
.
It
Since A(f) = A(g)
implies Tf = T9 and this implies f(x) = g(x) almost everywhere, the map A is an injection from Lloc(IR') to D'(IR n;r).
By definition A(Lloc(IR n))
map A is a bijection.
=
D'(IR n;r) so that the
The map A is thus an algebraic
isomorphism from Lloc(Rn) onto D'(IRn;r).
Since these spaces
are isomorphic, we can identify an equivalence class of locally integrable functions with the distribution generated by one of its representatives. In particular two continuous functions on
IRn
are identical.
which generate the same regular distribution Hence we can identify a continuous function f
with the regular distribution Tf and write for . Since D'(Mn;r) C D'(IRn) the notion of distribution generalizes that of continuous function. Therefore a continuous function In can be interpreted in two different ways, first as an f on In ordinary function f: -, C1 and secondly as a distribution
f: p -+ In f( x) p(x) dx from D to
O1 .
Similarly a constant M has three meanings,
first as a complex number, secondly as a constant function, and thirdly as a constant distribution
1
In
M p (x) dx
A distribution T E D' equals zero on an In if = 0 for every p E D with support open set 0 C DEFINITION 1.2.4. in 0
.
For example the Dirac 6 distribution equals zero on 0 = Rn\{0}.
The regular distribution which corresponds to the Heaviside function H(t) equals zero on the set 0 = {t a 1R1: - < t < 0). This is a particular case of the result that if a locally integrable function equals zero on an open subset 0 of IR
n then the corresponding regular distribution equals zero 11
on 0 also.
Two distributions T and U in D' are equal fn that is if T - U equals zero on R on an open subset 0 C if = for every p E D with support in 0 are For example the distributions T and T + 6, T E D' Rn that does not include the equal on every open subset of DEFINITION 1.2.5.
,
,
origin.
The support of T E D' is the complement in DEFINITION 1.2.6. n of the largest open subset of IR n where T equals zero and is IR denoted by supp(T).
Equivalently, a point belongs to supp(T) if an only if there is no open neighborhood of the point on which T equals zero.
EXAMPLE 1.2.2.
We have supp(8) _ (0) and supp(H(t)) _
(t E f1: 0 < t < w).
Often one uses the following result: if p E D and T E D' are such that supp(T) O supp(p) = 0 then = 0. If T and U belong to D' we have supp(T+U) c supp(T) U supp(U). We now discuss and define several important operations on distributions, the first being multiplication by an infinitely differentiable function.
To motivate a general definition
first let us consider the product of a regular distribution with a function in Cw(IRn).
Let f E Lloc(,n) and g E Cw(ln).
For all p E D we have the equality = fn (g (x) f(x)) w(x) dx
=
f f(x) (g(x) w(x)) dx =
,
g(x) p(x)>.
Recalling that gp E D we are led to the following definition. The product of a distribution T E D' with a DEFINITION 1.2.7. function g E Ct(IRn) is the functional gT defined by = , T E D. 12
It is easy to prove that gT as given in Definition 1.2.7 is a distribution for T E D' and g E C*(Rn).
The linearity is
obvious and = -+ 0 for V. -+ 0 in D as X --> X0 . Note, for example, that if g E CO'(IRn) then _ = (g(0) p(0)) = g(0) , E D, which yields tip
g5 = (g(0) 0) in D'
Note also that supp(gT) c supp(g) fl
.
supp(T).
In general it is not possible to define multiplicaton of two arbitrary distributions; it may not be possible to do so in D' even if the two distributions are regular.
However,
this disadvantage of distribution theory does not exclude the possibility of defining a product of distributions in certain Progress on multiplication of distributions during the last three decades appears to have significant subspaces of D'.
application in quantum field theory.
Differentiation of distributions is a basic operation which has significant applications in pure and applied mathematics. The definition of distributional differentiation is motivated by considering the situation for regular distributions. Let f Then f and all of its partial
be a function in C1(Dtn).
derivatives ef(t)/atj
= 1,2,...,n, define the following
j
,
regular distributions:
= in f( t) tip(t) dt,
<
'P> =
of J
af(t) I
tip
ED
4p(t) dt
,
E D
j
J
in
Integrating by parts in the last integral we obtain
Of <
p> _ -
)>
a
pED
.
(1.2)
J
13
The equality (1.2) suggests the definition of distributional differentiation.
The partial derivative aT/atj
DEFINITION 1.2.8.
j
= 1,...,n,
of T E D' is the functional given by < aT
atj
.p> _ -
ap(t)
atj
The functional aT/atj
,
j
>, j = 1,...,n
= 1,...,n
,
w E D
.
(1.3)
is a distribution.
,
Linearity is obvious; continuity follows from the fact that a'px(t)/atj converges to zero in D as A -+ A 0 for each j
= 1,...,n when wX --> 0 in D as A -> X0
.
By iteration we
have
= (-1) lal , f E D
(1.4)
,
for any n-tuple a of nonnegative integers, and DaT E D' for T E D'
.
For example,
= - _ -
J 0
%p '(x) dx = %p(0)
for p E D and H(t) being the Heaviside function; thus H' = b in D'.
Let f be a function of
Let x0 be a fixed point in 1t1.
class C1 on IR 1\(x0) with a discontinuity of the first kind at
the point x0
Suppose also that the classical derivative f'
.
of the function f is locally integrable on R1
.
Then the
distributional derivative Df of the function f is given by
Df = f' + bx in D' where b
X
0
is defined by
- f (x0-e) ),
[e--+O+ ( f (x0+e ) ,
w> = p(x0), w E D
0
We see that in contrast to classical differentiation of 14
functions, every distribution has derivatives of all order which are also distributions.
In particular every locally
integrable function has distributional derivatives of all order; these derivatives, in general, are not regular distributions. In the case of a continuous function possessing continuous derivatives, the distributional derivatives coincide with the classical derivatives. Moreover, in contrast to classical differential calculus, we have the following result: if (T.) is a sequence of distributions which converges to T in D' as A -+ X0 and a is any n-tuple of nonnegative integers then DaTx X -4 X0
.
This follows directly from (1.4)
.
DaT in D' as
We conclude
that distributional differentiation is a continuous linear operator from D' to D' .
1.3. THE SPACES E and E'
There are a number of important subspaces of the vector space of distributions D' such as the space DK = DK(IRn) which
consists of the continuous linear functions on DK functions that was introduced in section 1.2. section deals with another subspace of D'
,
,
a space of
The present
the distributions
with compact support.
We denote by E = E(on) the vector space of all infinitely differentiable complex valued functions on IRn In order to topologize E let (Kj) 1 be an increasing DEFINITION 1.3.1.
,
sequence of compact sets in Rn whose union is In j
.
For each
= 1,2,... and each m = 0,1,2,... define the seminorm pm,j by
pm,JOP
DaP (t)I
ialPm
tEK.
E E
J
The family of seminorms (pm,j) defines a locally convex topology on E.
This topology does not change if we replace
the sequence (Kj)
1
by another increasing sequence of compact
subsets of IRn whose union is Rn
.
The topology of E
,
often 15
called the natural topology, is the topology of uniform convergence on compact subsets of Un for sequences of functions in E and for the corresponding sequences of derivatives of all order. we make this explicit in the following criterion for convergence in E. A sequence (gyp.) of functions V. E E converges
REMARK 1.3.1.
to a function ,p in E as A -> X0 if and only if for each
n-tuple a of nonnegative integers the sequence (Dt4px(t))
converges to Dtp(t) uniformly on every compact subset of
Rn
as
Since the family of seminorms (pm'j ) is countable then E is metrizable.
Additionally E is complete.
Thus the topological
vector space E is a Frechet space. > -' 0 as
A linear functional T on E is continuous if
---> 0 in E as X -> No
-4 X0 when q,
.
The set of all
continuous linear functionals on E is denoted by E'
= E'(IRn).
We have the following characterization of elements in E': a linear functional T on E belongs to E' if and only if there is a constant M > 0
,
an integer m > 0, and a compact subset K of
IRn such that
M IaIPm
for all 'p
EE
tsup uK
IDa'p(t)
.
Every element T of E' is an element of D'; that is, T is a distribution. To see this first note that if T E E' then Further, if
is well defined for all 'p E D since D C E.
the sequence {gyp.} converges to zero in D as X -- X0 then
YX -* 0 in E also; hence -' 0 as A -* X0 and T E D' We conclude that E' C D'
.
Let f be a locally integrable function on support.
16
IRn
The linear functional Tf from E into
with compact C1
defined by
= fn f(t) p(t) dt
p e E
,
is a regular distribution in E'
.
One of the most important distributions in E' is the Dirac We recall that supp(b) = (0).
6 distribution.
As in the case of D' we shall describe the weak and strong topologies on E'. A sequence (TX) of distributions T. E E' is
said to converge weakly to zero in E' as X -1 X0 if for every p E E the numerical sequence () converges to zero. sequence (T.), TX E E'
,
A
converges strongly to zero in E' as
X -' X0 if the numerical sequence () converges to zero uniformly on bounded sets of E
We recall that a subset B of
.
IRn
E is bounded if for every compact set K in
and every a
there is a constant M depending on K and a such that IDap(t)I < M for all t E K and p E B.
The space D is dense in E; that is, for each p E E there is a sequence of functions in D which converges to p in E. Indeed, let (Kj)1
be an increasing sequence of compact
1
subsets of Dn such that their union is In
,
and let (pj}1W1 be
a sequence of functions in D such that yj(t) = 1 on a neighborhood of Kj j
= 1,2,...
.
,
j = 1,2,...
We have that p
.
For p E E
E D for each j.
show that pj -> V in E as j - w
,
put Vj = wyj It is easy to
.
This result implies the following useful fact: if T and U are in E' and if = for all V E D then _ Another useful fact concerns a for all v E E. modification of test functions outside the support of a distribution.
If T E D' and g(t) E Co(LRn) such that g(t) = 1
on a neighborhood of the support of T then _ for all V E D.
This result also holds for T E E if T E E'
Additionally we have that E' is dense in D' and the The elements of
canonical injection E' -- D' is continuous.
17
,
E' are distributions with compact support. We shall state two results of L. Schwartz [117] that give additional structure for E' distributions, and we shall use these results in the succeeding material of this book. Every distribution T E E' with [117, p. 91] THEOREM 1.3.1. Rn Rn in infinitely can be represented in compact support K C
many ways as a sum of a finite number of distributional derivatives of continuous functions which have their support contained in an arbitrary neighborhood of K. Every distribution T E E' whose [117, p. 100] THEOREM 1.3.2. support is the origin can be represented in a unique way as a finite linear combination of distributional derivatives of the Dirac 6 distribution. 1.4. THE SPACES S AND S'
In order to define a distributional Fourier transform, L. Schwartz introduced the space of functions of rapid decrease S = S(IR n)
and the corresponding dual space of tempered distributions S' = S'M n). DEFINITION 1.4.1. S = S(Rn) is the vector space of all COO(Un) functions 'p such that
ItI
Ito Da,p (t)I
= 0
for all n-tuples a and p of nonnegative integers; equivalently
S is the vector space of all CW(Rn) functions p such that for
each a and p there is a constant Map for which sup
It'3
Da,P(t)I
<_ Ma,p
tCIR
An example of a function in S is exp(-IxI2).
For n-tuples a and p of nonnegative integers, define the seminorms Pa'p(W) = Sup
Ito
Da.p(t)I
,
(p E S
tEIRn
The space S equipped with the topology generated by the 18
countable family of seminorms (p a,
is metrizable and
complete; hence S is a Frechet space. REMARK 1.4.1. From the topology of S we have that a sequence (yp1) of functions in S converges to a function W E S in S as
) X0 if and only if for all a and p
A
limy
supra
O tEIR
Ito Dt(wx(t) - w(t))I = 0
We have D C S and convergence in D implies convergence in
S.
If Ox -i p in D as X --1 AO for the sequence (c1) in D and
p E D then all supports supp(p1) and supp(p) are contained in a fixed compact subset of Rn
p in S as X -> XO
wX
Hence (1.5) will hold, and
.
.
The product of the function p E S with an arbitrary infinitely differentiable function may not be a function of S. For example (exp(Ix12) exp(-1x12)) = 1 which is not in S.
A
set of CO(Mn) functions whose elements are multipliers of S is the space OM = OM(IRn) of Schwartz [117, p. 243]. DEFINITION 1.4.2.
OM = 0M(IRn) is the set of all functions W E
CW(Mn) such that for every a there is a polynomial Pa(t) for which
I Da'p (t) I
< Pa (t)
,
tE
OM is called the space of
CCO
IR n
;
(IRn) functions which are slowly
increasing at infinity.
We collect several important facts concerning S in the following result. THEOREM 1.4.1. (i)
The operation of multiplication by a function in OM
is continuous from S into S; the operation of differentiation is continuous from S (ii) 19
into S;
(iii) D C S C E with continuous injections; (iv)
D is dense in S and S is dense in E;
(v)
S C L1 with continuous injection.
The dual S'
S'(Pn) of S is the set of all continuous The elements of S' are called
=
linear functionals on S.
It is known that a linear functional
tempered distributions.
T on S is continuous if and only if there is a constant M > 0 and a nonnegative integer m such that II
sup
< M
IaI<m
Ito Daw(t)I, w C S
tEIRn
I1I<m
From this result it follows that every tempered distribution is of finite order.
(The order of an element T C S' is
explicitly defined in the alternative definition for S and S' which is given in Remark 1.4.2 below.) EXAMPLE 1.4.1. Every continuous function f(t) from IRn to C1 which is slowly increasing at infinity, that is which is bounded by a polynomial, generates a regular tempered distribution Tf
.
As in the case of D' and E' we shall consider the weak and strong topologies on S' A sequence (T.) of distributions in .
S' converges weakly to zero in S' as X --+ X0 if
0
X
To introduce the strong topology on S' we must define the boundedness of a set in S. A set B C S is said to be bounded in S if for each pair of n-tuples a and p of nonnegative integers the products (to Dap(t)) are uniformly bounded on IRn as p varies over B. A set B' C S' is bounded if 'up II < MB
for all T C B' and all bounded sets B C S. 20
Now we have the
following condition for strong convergence in S': a sequence (T.) of distributions in S' converges strongly to zero in S'
asl' -->A 0
if
lim
N-'N0
sup pEB
, p> = 0
for all bounded sets B C S
.
Obviously strong convergence in S' implies weak The converse is also true; the fact that weak
convergence.
convergence implies strong convergence in S' follows from the fact that S is a Montel space [135, p. 21] and the result [46, Corollary 8.4.9, p. 510]. Let T E S'
Since D C S and convergence in D implies
.
convergence in S then T is continuous on D.
Hence T E D' and
S' C D' with proper containment since there are distributions in D' that are not in S'.
Further, convergence of sequences
in S' implies convergence in D'; for if Tx -p 0 weakly in S'
as X
--i 0 for all rp E D and hence T. -p 0
A0 then
weakly in D' as X --b N0
From Theorem 1.4.1(iii) we have E' C S' C D' with continuous injections.
Let T E S' and let g E OM _
ES
,
.
The product gT given by
is an element of S'
,
and the map T --> gT is
continuous in S'.
The differentiation of a tempered distribution T is given by the relation
= (-1) Ia I
Dap>, 4p E S
,
for every n-tuple a of nonnegative integers. Following V. S. Vladimirov [135, pp. 20-22] we REMARK 1.4.2. give an alternative definition for S and its topology.
Consider the countably many norms
21
sup
=
II,P II
(1 + ItI)m IDaw(t)I
, m = 0,1,2,...,
t E IRn
m
IaI<m
for tip
A sequence {'h} converges to zero in S if and only
E S.
-i 0 for all m as X
if IIpxII
AO
.
Denote by S(m) the
m
completion of S with respect to the mth norm. The spaces S(m) are separable Banach spaces, and D is dense in S(m) A .
function p belongs to S(m) if and only if ,p has continuous
derivatives up to and including order m and (Itlm as Iti -i - for all a such that IaI < m. S(0) D S(1) D S(2) D
-i 0
Now we have
...
and
s=
fl
s (m)
m>0
This construction of S as the intersection of the S(m) spaces also yields an alternative way to construct S' The dual spaces S(m)' of S(m) form an increasing chain .
S(0), C S(1)I C S(2)' c
...
and we have U S'
_
S(m)
m>0
Thus if U E S' then U E S(m)' for some m > 0
the smallest M for which U E S(M)' will be called the order of U. For each U E S' we introduce the decreasing sequence of norms
22
;
-m
sup
=
11M
II'
m = M, M + 1,...
=1
where M is the order of U
We shall use this alternative construction of S and S' in terms of S(m) and S(m)' in Chapter .
5.
We conclude this section by stating the following characterization theorem of S' which has been given by L. Schwartz.
THEOREM 1.4.2.
[117, p. 239] A distribution U E D' is an element of S' if and only if U has the form U = Dot((1+It12)k/2 g(t))
for some n-tuple p of nonnegative integers and some real number k > 0 where g(t) is a continuous bounded function on It easily follows from this characterization theorem that if U E S' then there exists a function h(t) E L2 and n-tuples p and -r
of nonnegative integers such that n
U = DR((
II
(1+t2 )ry3) h(t))
j=1 1.5.
THE SPACES 0a AND 0' a
In order to represent as many distributions as possible with the Cauchy integral, H. Bremermann [11] introduced the test spaces 0a = Oa(Rn) and distribution spaces 0' = 0I(Rn) give two definitions of spaces of this type in Otn
,
.
We
one for a
being a real number and another for a being a n-tuple of real numbers.
DEFINITION 1.5.1.
For a being a given real number, we say
that a function N E 0a = Oa(,n) if p is infinitely 23
differentiable and_ if for each n-tuple p of nonnegative
integers there exists a constant MP such that
Mp (1+ItI)a
t e n .
,
Convergence in the vector space 0a is defined as follows:
Xo if
converges to p in 0a as X
a sequence (i)
each pX E Oa
(ii)
for each p the sequence (Dpw,) converges uniformly
;
on every compact subset of
IRn
to Dpw as X -+ X0 ;
and (iii)
for each p there exists a constant MP , which is
independent of X
ID13
,
such that
< MP (1+ItI)a
(t)I
,
t E In
(1.6)
.
Note that the space 0a is complete; that is, the limit function w belongs to 0a
.
Also observe that (w.) converges
in 0a if it converges in E and (1.6) holds.
The vector space 0a is the space of all continuous linear functionals on 0a with continuity having the usual meaning
that -> if P -p w in 0a for U E 0' We have D C 0a for every a E R1
;
and if a sequence
converges to 'p in D then the sequence also converges to 'p in Oa
.
Accordingly every continuous linear functional in 0a is
also a distribution; that is, 0a C D' Let U E 0a
.
We define the derivative A by
= (-1)I1I , 'p E 0 a
24
.
The right side in (1.7) is well defined since qp E 0a implies DRv E 0a.
For U E 0a the functional DOU is linear; it is also
continuous since
X - X0
0 when V. - 0 in 0a as
Therefore DRU E 0' if U E 0' a
If al < a2 then 0a
C 0a
a
Also we have
c 0'
and 0' 2
1
a
2
a
1
D C 0a c E and E' C 0' C D' with continuous imbedding.
D is
a
dense in Oa for all a E 6t1
.
As in the case of D' we define the convergence in 0' as
a
weak convergence: the sequence (U.) of distributions in Oa
converges to U if ---+
for every p E 0a as X --' X0.
The space Oa is closed under convergence.
Following Bremermann [11, p. 53] we give the following definition of asymptotic bound of a distribution. A distribution U E D' is said to have the asymptotic bound g(t) Z 0, and we write U = 0(g(t)), if there
DEFINITION 1.5.2.
exist constants R and M such that for all qp E D with support in (t E IRn:
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
<_
M
tE[R
n IDap(t)
IPI<<m
for all w E Oa
U
(1+It.1)-ail
j=1
.
THEOREM 1.5.3.
Let m be a fixed positive real number and let p be an n-tuple of nonnegative integers. Let U = 13fP(t) where for each p fp(t) is a Lebesgue IPI < m ,
,
Ipkm measurable function which satisfies
n If p(t)I
e > 0
26
,
-a . -1-E
< Rp ]nl (1+It1I)
for all t c IRn with R0 being a constant depending on
Then U E 0'
p.
a
In Chapters 2-6 we shall always state explicitly whether a is a fixed real number or an n-tuple of real numbers in Oa and 0'. a
THE SPACES DLp AND DLp
1.6.
The definitions and results of this section are taken from L. Schwartz [117, pp. 199 - 203]. DEFINITION 1.6.1. DLp = DLp(,n)
1 < p <
,
,
is the space of
all infinitely differentiable functions 'p for which Dp'p(t) E
LP for each n-tuple p of nonnegative integers.
B = D
_
L
(,n) is the space of all infinitely differentiable
D L
functions which are bounded on Rn B is the subspace of B consisting of all functions which vanish at infinity together .
with each of their derivatives.
The topology of D p is given in terms of the norms L
11`-11
(f IDpw(t)IP `Jn
m.P
A sequence of functions the topology of DLp
pED
,
< p <
1
dt]1/p
.
I3
< m , m = 0,1,2,...
converges to a function w in ,
as X -> X0 if each V. E DLp
and
L
0
lim for every p
.
A sequence of functions (gyp.) converges to a function w in
if each w, E B
,
p E B
,
and (1.9) holds for p = -
.
27
.
We have that D is dense in D Lp
not in B = D L IRn
in
If W E D
.
w
,
1
,
and in B but
,
then p is bounded
Lp
and converges to zero at infinity with the same being
true for all derivatives of W
1
1
,
We have D C DLp C DLq c B for
.
.
DEFINITION 1.6.2.
DLp = D'
(IRn),
is the space of
1 < p
continuous linear functionals on D
,
1/p + 1/q = 1; D'
Lq
_
L
D'1(IRn) is the space of continuous linear functionals on L
The space DLp is a subspace of D' since D C D q
,
.
Indeed, let U E DLp
is defined for all W E D
.
Clearly U is
L
Since convergence in D implies convergence in D q then converges to zero when (,p,) converges to zero linear on D. L
in D as X -, X0
Thus U E D'
.
.
Furthermore, knowledge of
for p E D uniquely determines
for T E D
q because L
D is dense in D
.
Similar reasoning yields D'
Lq
D C B
C D' since
L1
.
The following theorem gives the structure of D'p L
THEOREM 1.6.1. D'p
1 < p <
co
[117, p. 201] A distribution U belongs to if and only if U is a finite sum of ,
L
distributional derivatives of functions in LP that is U E D' is in D 1 < p if and only if there is an ;
integer m > 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
.
P(X,Y)> = In f( x) p(x,y) dx
,
P(x,Y)> = 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)
g(Y)
,
w(x,y)> dx =
,
9(y)
,
w(x,Y)>>.
In the same manner we obtain =
f
,
P(x,y)>>
.
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 =
,
, I(x,y)>>
,
,
p(x,y)>>.
In particular if p(x,y) = y(x)x(y) where y E D(Mn) and X E D(IRm) we have =
, y>
9(y)
'
X>
.
The preceding comments motivate the following definition obtained by L. Schwartz [117, pp. 106 - 109]. DEFINITION 1.7.1. Let T E D'(IRn) and U E D'(IRm). distribution, denoted by T®U
,
The unique
in D'(IRnxIRm) which satisfies
= for all y E D(IRn) and X E D(IRm) is the direct product (tensor
product) of the distributions T and U
.
In general the direct product T®U is calculated as = > = > ,
,
for all p E D(RnxRm).
The support of T®U is the cartesian product of supp(T) and supp(U).
30
If Tx - T in D'(Dn) and U, -> U in D'(0km) as X -> 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
=
,
qp (x+y) > ,
'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 =
,
,
O(x+y)>>
=
,
,
.p (x+y) >> ,
(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 is a C-(R n)
The second result is derived
function with compact support. from the limiting equality
Aim
32
0
= lim
0
,
f (y)
,
W(x+y)>> =
where f(y) is a function in D which is equal to 1 on a neighborhood of K. In the following we describe three simple but REMARK 1.7.1.
important cases for the convolution to exist Put D+(IR1) = {T E D'(R1): supp(T) C [0,-)) (i) .
.
If T
and U belong to D+(IR1) then T*U defined by Definition 1.7.2
exists, and T*U E D+
.
We know that the convolution T*U is a function if both T and U are functions; this is also true if one is a distribution and one is a Cm(Rn) function. In fact let U be (ii)
an arbitrary distribution in D' and let g(t) E Co(Dn).
We
assume that at least one of U or g(t) has bounded support. For every W E D we have =
(x)>
,
with
Cx) = fn g(y) v(x+y) dy = fn g(t-x) (t) dt = <,P(t), g(t-x)>.
S ince
the direct product is commutative we may write
=
,
<w(t), g(t-x>> = <w(t),
Now put h(t) =
,
g(t-x)>.
,
g(t-x)>>
.
Since the function h(t) is a
Co(Rn) function we have =
.
We conclude that if U E D'
,
g e
Cco(IRn),
and U*g exists then
(U*g)(t) =
33
in D' and (U*g)(t) is a CC(Rn) function.
Following L.
Schwartz this convolution (U*g)(x) is called the regularization of U. (iii)
Note that the constant 1 cannot be the unit element
in the convolution operation because in general T*1 X T; for every T with bounded support the convolution T*1 = . But for an arbitrary T E D' we have =
and T*S = b*T = T
,
<5y
,
p(x+y)>> =
,
p(x)>
,
p E D
,
Thus the Dirac 6 distribution is the unit
.
element for the convolution operation when considered as a multiplication.
The convolution T*g of a tempered distribution T and a test function g E S plays an important role in applications.
when
T is in S' and g is in S their convolution T*g is defined as a functional on D by =
,
<w(y), g(y-x)>>, W E D
It is known that T*g is a distribution. function of slow growth.
.
In addition it is a
If gx -* g in S as X - X0 then
(T*gX) - T*g in S' as X -> 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
(T)
, V> = =
,
Y(x+T)>
and for every p E D we have (T*U) (T) = T * U(T) = T(T) * U
(v) support: supp(T*U) c supp(T) + supp(U)
.
Using convolution Schwartz has proved a fundamental result in the theory of distributions, namely that the space D considered as a subspace of D' is dense in D'; thus each distribution is a weak limit in D' of a sequence of test functions.
We end this review of convolution by stating the following theorem due to L. Schwartz [117, p. 203] which we shall use in this book. THEOREM 1.7.1. If T E
(i)
D'r with r >
1
DLp ,
and g E DLq then the product gT belongs to
1/r < 1/p + 1/q; and under these conditions
L
x D the bilinear map (g,T) -+ gT from D' into D' is Lr Lq LP hypocontinuous. (ii)
E D'r
,
If T E DLp and U E DLq
,
1/p + 1/q - 1
>
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 <
ap(t)
We also know that 1p(x)j -* 0 as lxi -> 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 <
2
P11Lq
we have the Parseval inequality
,
-
IIT11Lp ,
1/p + 1/q = 1
with equality if p = 2.
However, not every element y E Lq is
the Fourier transform of some p E LP 1/p + 1/q = 1 36
.
(1.14)
,
,
1 < p < 2
,
All of these facts concerning the Fourier and
inverse Fourier transforms of LP and Lq functions hold independently of which sign is chosen for the exponent in the exponential term of the Fourier or inverse Fourier transform. In this book we collectively refer to the facts presented in this paragraph as the Plancherel theory. These facts can be found for one dimension in [65, pp. 139 - 146]. They hold equally well in n dimensions and the explicit n dimensional ,
analysis can be found in [8, pp. 111 - 121] for the case p = 2.
Starting from the definition of the L1 Fourier transform for p E S and using properties of S together with the inversion formulas
gp-1[w(x);t]]
=
9-1[9[w(x);t]] = P(x)
Schwartz [117, Chapter 7] has proved that the Fourier transform is a topological isomorphism of S onto S.
The Fourier transform of distributions in S' is motivated by considering the situation for regular distributions. Let T E S' be generated by the integrable function f E L1 = = in f( t) p(t) dt
The Fourier transform of f
,
e2ni
f(x) = 5;[f(t);x] = in f(t)
in
f(x) P(x) dx = f
p(x) $
IRn
f(t) in
ip(x)
dt
f(t)
e2ni
dt dx
IR
e2ni
= J
v E S.
,
dx dt =
IRn
37
for V E S
.
The change of order of integration is justified
by Fubini's theorem since IP(x) f(t)
e2niI = IP(x) f(t)l
This and the product of E L1 for f E L1 and p E S calculation suggests the following definition. The Fourier transform of V is Let V E S' DEFINITION 1.8.3. .
.
the element U = ?[V] = V such that =
,
p E S
(1.15)
.
The right side of (1.15) is well defined since p E S for p E S.
From this definition we have U = V E S' for V E S'
and the Fourier transform is a topological isomorphism of S' onto S'
.
Now we define the inverse Fourier transform of a distribution in S' DEFINITION 1.8.4.
.
Let U E S'
The inverse Fourier transform
.
of U is the element V = 9-1 [U] such that
=
,
p E S
(1.16)
.
The stated results for the Fourier transform of S functions
and S' distributions hold also for the inverse Fourier transform of elements in S and S'; the inverse Fourier transforms of S functions and S' distributions are topological respectively. isomorphisms of S onto S and S' onto S' The Fourier transform of the convolution of certain elements in S' will be of interest to us in this book. Here ,
we recall the basic result of the distributional Fourier transform of convolutions as presented by L. Schwartz [117, Chapter 7], the result being that this Fourier transform converts certain convolutions into multiplication. First, recall the space of functions OM = 0M(,n) (Definition 1.4.2
38
and [117, p. 243]) and the space of distributions 0, = O (ln) [117, p. 244].
OM is the space of infinitely differentiable
functions of slow growth; an element p E OM if and only if is infinitely differentiable and every derivative of p is bounded by a polynomial whose degree depends on the order of the derivative. 1.4.1(i)).
OM is the space of multipliers of S (Theorem
OC is the space of distributions of rapid
decrease, and this space is characterized in [117, Theoreme IX, p. 244]. If U E OC and V E S' Schwartz [117, Theoreme ,
XI, pp. 247 - 248] has shown that the convolution U*V is a distribution in S'.
Thus U*V has a Fourier transform in S';
and according to the following basic result of Schwartz [117, Theoreme XV, p. 268], this Fourier transform converts the convolution into multiplication.
THEOREM 1.8.1. If U E O and V E S' then U E OM , V E S'
?[U*V] = U V
,
and
(1.17)
.
On the right side of (1.17) we have that U V is the product of the element V E S' by the multiplier U E OM of S; hence this product U V is well defined as an element of S' All of these facts concerning OM and the Fourier transform 0c .
,
,
property given in Theorem 1.8.1 are summarized in Vladimirov [135, pp. 21 and 25]. 1.9.
THE SPACES Z AND Z'
The equation (1.15) which defines the Fourier transform in S' is not well defined for all V E D' with p E D since p may not be a function in D.
For this reason Schwartz introduced his
spaces S and S' with which to define the Fourier transform for
After the appearance of Schwartz's books on distributions a theory of the Fourier transform on the whole
distributions.
of D' was obtained by Ehrenpreis ([47], [48]) and Gel'fand and Shilov ([53, Chapter 2], [54, Chapter 3]) by introducing the spaces Z = Z(lRn) and Z' = Z'(ln) which we now define. 39
Z = Z(ln) is the space of all infinitely differentiable functions 4, which can be extended to be entire analytic functions in On such that there exists (al,a2,...,an)
DEFINITION 1.9.1.
with aj > 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.
=
40
Let V E D'
,
.
v E D, P =
E Z
,
(1.18)
where vo(t) = p(-t).
The equality (1.18) is, in fact, in the same form as the equality (1.15).
We easily obtain that the Fourier transform so defined is and this Fourier transform is a topological
an element of Z'
,
isomorphism of D' onto Z'
.
Further the inverse Fourier transform is a topological isomorphism of Z onto D from which we can define an inverse
Fourier transform from Z' to D' by the identity V
,
P E Z
,
w(t) _
-1[p(x);t] E D
(1.19)
with y(x) = y(-x) , where U E Z' and V = 5-1[U] is the inverse Fourier transform of U. This inverse Fourier transform is a topological isomorphism of Z' onto D'
.
41
2 Distributional boundary values of analytic functions in one dimension INTRODUCTION The theory of the Cauchy integral (the Cauchy representation) 2.1.
of distributions developed in this chapter is motivated by the classical theory of the Cauchy integral ([52], [97].) The major themes are boundary values of the Cauchy integral of distributions and recovery of analytic functions by the Cauchy The main theorems yield distributional relations of the Plemelj type and applications of these relations in boundary value problems. In particular, the analytic representation of distributions in E'(IR) and in Oa(U) will be obtained using the Plemelj relations. The integral of the boundary value.
material in this chapter is based on the papers [79] - [95] and is influenced and stimulated by the works of Beltrami and Wohlers [2], Bremermann [11], and Roos [114]. In Chapters 2 and 3 of this book the open upper half plane and the open lower half plane will be denoted by A _ {z E C: Im(z) > 0} and A = {z C C: Im(z) < 0), respectively; also we put A = A+ U A so that A = C\R Of course by C and C in .
Chapters 2 and 3 we mean one dimensional complex space C1 and one dimensional real space R
,
respectively.
Let a function g from a nonempty set E1 c C to C be given.
If f, which maps E to C, is analytic in the domain E, E1 c E and the values of f and g coincide on E1
,
then f is said to
be an analytic continuation of g from E1 into E.
If E1 has a
limit point in E, we have from the uniqueness theorem of complex variables that f is unique.
In section 2.2 we shall use the technique of analytic
continuation across a boundary which is derived from the classical theorem of Painleve [96, p. 46] and is described as follows. 42
Consider the bounded domains E+ and E
contained in
the half planes A+ and A
respectively, such that E+ and E
,
have an open interval Cl C l as their common boundary.
Let us
assume that the function f+(z) is analytic in E+ and is Similarly the function f (z) is analytic in Furthermore, let us assume that and is continuous on 0
continuous on Cl. E
.
the given functions f+(z) and f -(z) take the same boundary value on D
.
Then there exists a unique function f(z) which
is analytic in E = E+ U 0 U E in E+ and with f -(z) in E
.
and which coincides with f+(z)
The function f(z) is the
analytic continuation of the functions f+(z) and f -(z) into the domain E.
The main result of section 2.2 is the
formulation and proof of a distributional version of the Painleve theorem.
The results in section 2.3 are concerned with the analytic representation of distributions from the viewpoint of the Plemelj formulas which will be established in the sense of distributions with compact support. In particular, if the given distribution is the Dirac 6 distribution then the Plemelj relations imply immediately well known formulas of quantum mechanics. The condition T C E'(IR) is sufficient, but not necessary, for the Cauchy integral of T C(T;z)
to exist.
2ni
tlz >
For instance consider Co
C(T;z)
2ni
J m
f(t)
dt
where f is a continuous function with f(t) = O(Itja) for some then C(T;z) exists although T = f(t) is a < 0 as I tI -+ ;
not an element of E'(R).
Note also that C(T;z) does not exist
for all T E D'(IR) since the function 1/(t-z) is not a test In view of these function in D(D) as a function of t e R. remarks the dual space D'(IR) is too large for the study of the
Cauchy integral C(T;z) of a distribution T, and the dual space 43
E'(IR) is too small.
In order to extend the class of
distributions which are representable by the Cauchy integral, Bremermann [11] has introduced the distribution spaces O'(IR) which are intermediate spaces between E'(l) and D'(IR).
Section 2.4 is devoted to the analytic representation of distributions in the spaces O'(IR). Plemelj formulas are used to obtain this representation, and distributional Plemelj formulas are stated for the spaces Oa(l). As will be seen,
the obtained results extend the analytic representations of distributions found in [11].
In section 2.5 necessary and sufficient conditions that a distribution in O'(IR) be the boundary value in the DI(R) or O'(IR) topology of a half plane analytic function are given.
Additionally an interesting property of the distributional boundary values of the derivatives of the Cauchy integral of distributions is established. In the theory of distributional behavior of analytic the functions the following two topics are central: representation of distributions in terms of boundary values of analytic functions and the representation of analytic functions in terms of distributions.
While the first topic
will be considered in sections 2.2 - 2.5 of this chapter, section 2.6 will be devoted to the second topic. The initial result in section 2.6 (Theorem 2.6.1) is surprising because it shows how a half plane analytic function
can be generated by the real or imaginary part of its distributional boundary value. This is useful since, among other things, a solution of a distributional Dirichlet boundary value problem for the half plane (section 3.2) is obtained, and the proof of the distributional version of a result due to Harnack [97] is facilitated.
Hence the real
valued distributions are required.
The second main result of section 2.6 is concerned with the
representation of a given function by means of its boundary We assume that the considered
values on IR from A+ and A 44
.
function is analytic in C except for a finite number of poles and for a closed set K C It; the representation of
in A+ U A
the analytic function is then obtained in terms of its assumed
boundary values on C from A+ and A To every distribution T with compact support in C there .
corresponds the Cauchy integral C(T;z)
=
2iri
,
t-z >
which is analytic in C\supp(T) and vanishes at infinity It is natural to ask (Definition 2.3.1 and Theorem 2.3.2.) whether the converse is true. Namely, given a function f(z) which is analytic in C except on a compact subset K in Ot and does there exist a distribution T which vanishes as Izi -> ,
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.
<
x
Ixl
> 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
pn (x) > -- 0 as n -> -
We must show that
Let r > 0. Then we can
.
write
h+(x+ie) pn(x) dxI
f
<
lh+(x+ie) Vn(x)
lxl
I
(2.4)
<
dx +
f
lh+(x+ie) 'pn(x)l dx
lxl>r
Letting 6 be an arbitrarily small positive real number, we may choose r > R so that lh+(x+ie) pn(x)I dx < A MO
f
lxl-1+a
dx < 6
(2.5)
J
lxl>r
lxl>r
for all n = 1,2,3,...
The closed interval [-r,r] now being fixed, we have from the convergence of {,pn} to zero in Oa(IR) .
and the Lebesgue dominated convergence theorem that
48
li
f
h+(x+ie) pn(x) dx = 0
(2.6)
IxI
The bound (2.5) and the limit (2.6) together show that the estimate (2.4) can be made arbitrarily small for large enough n.
Consequently the linear functional h+(x+ie) defined by
(2.3) is a distribution in O'(O), a < 0 Now let a < 0 and let R > 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
IxI
h+(x+ie) p(x) dx
J
IxI>r 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 <
for all e > 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
But D(IR) is dense in 0a(IR).
= U on D(IR).
Hence h+ = U on OC(IR)
From part (i) we know that As is noted at the beginning of section
We now consider part (iii). h+ E 0'(IR), a < 0.
2.4, 1/(t-z) E 0a(M) for all a > -1 as a function of t c IR for
Im(z) A 0; thus we must choose a in the interval -1 < a < 0. For such a the inclusion 0; (R) c 0'1(M) holds, and the distribution h+ is an element of 0'1(M). integral of h+ is well defined.
Thus the Cauchy
To prove (iii) we shall first
evaluate lim E-40+
1
2rri
lim
1
t-z
>
E->0+
1
27ri
1' °
J
h
t+i6 -z
dt .
(Observe that the integral here exists for each e > 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
2iri
t-z
0
.
z E A-
,
This identity combined with the result of part (ii) yields the representation (2.2) since 1/(t-z) E Oa(D), a > -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
(x-ie) p(x) dx
ei0+
for all V E D(IR).
In the case of (iii) the corresponding
identity is
-
1
2ni
1
t-z >
zE
0
_
h-(z)
,
A+
(2.7)
z E A-
Let hf(z) be an analytic function in A} Then respectively, with h±(z) = 0(1/IzJ) as Izi - -
THEOREM 2.2.1.
.
hf (xf ie) converges to hf in D' (R) as a -> 0+ if and only if ht(xfie) converges to h} in O'(R), a < 0 PROOF.
,
as a - 0+
.
The sufficiency follows at once from Lemma 2.2.1 and
Remark 2.2.1.
The necessity holds since convergence in Oa(D)
implies convergence in D'(IR).
A distributional version of the Painleve theorem adapted for our purposes can be formulated and proved as follows. 51
Let f+(z) and f -(z) be analytic functions in
THEOREM 2.2.2. A+ and A
as
respectively, with the property ff(z) = 0(1/IzI)
,
Suppose that f+ (x+ie)
-->
IzI
f (x-ie) -- f
f+ E D' (61) and
as a --> 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
t-z >
1
2ni
t-z >
By Lemma 2.2.1 and Remark 2.2.1 we have f(z) =
Jf(z)
,
z E A+
f (z)
,
zEA
To complete the proof we must show that f(z) is analytic in some neighborhood of ft Let X(t) be a Cm(61) function which is equal to 1 on supp(f+-f that is on the complement of U in 61 and is equal to 0 on an arbitrary closed interval 0' C 0 We may write .
)
,
,
.
f(z)
=
2ni
2n1
=
i
t(z) >
.
As z tends to a point x0 E 0' then by [11, p. 56] the function X(t)/(t-z) converges to A(t)/(t-x0) in 0_1(61). is continuous on 0_1(61) we have
52
Because f+-f
lim f(z) = z->x0
1
2ni
X(t) >
t-x0
This shows that f(z) is continuous on 0' The function f(z) is analytic in A = A+ U A and continuous on A+ U 0' U A .
.
Since 0' is an arbitrary closed interval in 0 the fact that f(z) is analytic in A+ U 0 U A follows. The proof is ,
complete.
Theorem 2.2.2 implies the distributional uniqueness of analytic functions and uniqueness of their distributional In fact we have the distributional version
boundary values.
of the following classical result: if an analytic function vanishes on a path which is part of the boundary of its domain of analyticity then the function vanishes at all points of this domain.
Let f+(z) be analytic in A+ and satisfy
COROLLARY 2.2.1.
f+(z) = o(1/IzI) as Izi -> 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
that f (z) = 0, z E A
0 for all f E
D(IR); and, in particular, = 0 for all p E D(M) with support in 0 . Thus = for all p E D(It) with support in a
f(z)
=
By Theorem 2.2.2 we thus have that
.
+
1
27ri
<
1
ft' t-z >
is analytic in A+ U 0 U A f(z) =
f+ (z)
,
z E A+
{rcz)
,
zEA
1 - 2,ri
-
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 <6,T> = 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)
=
2ni
t 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)
=
2ni
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
21
t-(x+ie) > p(x) dx
n
lim n-
2ni
k=1
1
t-(xk+ie)
> `p(xk)
(xk
xk-1)'
Because T is a linear functional on E(R) we can write
= lim
t
n -1
2ai
P(x
xk-
)
k
(t-ie. )
(x
k - xk
)>.
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-ie)>
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
e-+0+
t' a-+0+
For each e > 0 consider the regular distribution C(T;x-ie) defined by
= J
C(T;x-ie) 4p (x) dx
.
_ao
By a slight modification of the previous argument, we obtain for a fixed e > 0 through the Cauchy integral
pp(t+ie) =
that
66
"0 (x)
1
2iri
cc
x-(t+ie)
dx
.
=
and
lim
=
+(t)>
where w+ (t) is the limit corresponding to V(t+ie) as a -+ 0+ given by (2.12).
By applying the Plemelj formulas derived from (2.12), (2.13), and (2.14) in the paragraph succeeding Theorem 2.3.6 and noting the above calculations in the proof of the present theorem we have lim e-*0+
_
(2.20)
= =
and
lim e-),0+
=
(2.21)
= = - = -2
for all c E D(l).
The proof is complete.
We are going to formulate Theorem 2.3.7 in a more condensed form now. We showed above that the functions C(T;x+ie) and Since C(T;x-ie) converge in D'(IR) as a ---> 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
x
P> __
1
Vi
Here vp denotes the Cauchy principal value and p C E(IR).
Thus
we can state Theorem 2.3.7 in the following equivalent form. THEOREM 2.3.8. Let T C E'(IR). The limits lim F--+O+
C(T;x+ie) = T+ E D'(IR)
and
lim C(T;x-ie) = T e-*0+
E D'(IR)
exist in D'(IR) with T+
-T =T
(2.22)
and
T++T
=
-1
ni
(T * vP
(2.23)
x
in D' (IR). Two relations which are equivalent to (2.22) and (2.23) and which will be used often are T
and 68
=
2
T-
1
27ri
(T* v P
1
x
(2.24)
T
=
-
2
T -
(T * vp
2ni
x
)
(2.25)
.
Equations (2.24) and (2.25) are obtained from equations (2.22) and (2.23) and conversely by addition and subtraction. we shall call the relations (2.22) and (2.23) or (2.24) and (2.25) the distributional relations of Plemelj (distributional Plemelj relations.)
Consider the Dirac b distribution defined on The Cauchy integral of 8 is given by
EXAMPLE 2.3.3. E(IR)
.
C(S;z) =
z E C\(O)
2niz
b+ = lim
-1
e-0+
=
2ni(x+ie)
-1
2ri(x+iO)
and
s- = lim
-1
e-O+
=
2ni(x-ie)
exist in D'(l).
Since 6 * vp
-1
2ni(x-iO)
x
= vp
x
,
a direct
application of the formulas (2.24) and (2.25) corresponding to the Cauchy integral C(S;z) in Theorem 2.3.8 leads to the well known formulas of quantum mechanics
1
x+iO
1
= -iwS + vp
x
and 1
xli0
=
ins + vp
x
These formulas are known as the formulas of Sochozki-Plemelj. A direct proof is given in [137, pp. 89 - 90]. Recall that 69
are called the Heisenberg distributions
6+ = 6+ and 6- = -0 (delta "functions").
Let T E E'(IR) and consider again the Cauchy
EXAMPLE 2.3.4. integral C(T;z)
tlz >
2ri
Since the limits C(T;xtie) - Ti exist in D'(IR) as a - 0+ and +,W> = on n = IR\supp(T), then by Theorem 2.2.2 the
c T ;z) = J C(T+;z)
,
z E A+
-C(T;z)
,
z E A
,
(
Denote by Ea(IR) the subspace of all functions V E E(IR) with
the property
p = 0,1,2,...,
I'G(P)(t)I
Itla
for a given a > 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
tlz >
(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
-
P(x
)
xk-(tkie)
2ai
(xk-xk-1)
k=1
where as before (a = x0,x11x2,...,xn = b) is a partition of Let us now consider the functions
[a,b) = supp(p).
n
1
%n(t-ie) =
4p(xk)
xk-(t-ie)
2ai
(xk-xk-1), n = 1,2,...,
k=1
defined for t c IR with e > 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
=
t'
- ;(t-iE)>
.
Since %p E D(R), each derivative of p is bounded on supp(p) in R.
Hence all derivatives (P)(t-iE) with respect to
t E IR are Holder continuous on supp(,p) and on every compact
subset of l
According to the classical theorem of Plemelj, Theorem 2.3.6 and (2.16), the derivatives with respect to .
t E IR ;p(P)(t-iE)
=
2
'p(x)
I
,J
00
(x-(t-iE))P
dx =
+1
1
27f i
F.
v(P)(x) x- (t-iE )
dx
converge uniformly to (P)(t) on every compact subset of l as
6 4 0+
On the other hand, by the theory of the Cauchy integral we have that ;(P)(t-iE) = 0(1/ItI) as Iti --i m .
.
(As
can be seen in Gakhov [52, Chapter 1], this is a consequence of the fact that the function p(x) and its derivatives have common compact support.) Moreover the inequalities
I(1+ItI)
,p(P)(t-i6)I
< AP
which are valid for large Itt and all e > 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
<-.L-
t
1= 1
t-z
2Tri
,t (t-z ) 1
dt
-CO
which can be calculated by means of the Cauchy integral 79
theorem to yield
C(vp
t ;z) _
2z
zE
-1
z E A-
2z
A+ .
Restricting a to -1 < a < 0, Theorems
(See also [11, p. 64].)
2.4.4 and 2.4.2 yield elements T+ and T
lim e-+O+
in OI(IR) such that
= T+
1
2 (x+ie )
and
lim e-'O+
-1
T
2(x-i&)
in both the 0-(M) and D'(IR) topologies.
T+
T
Further we have
1
= vp
x
and
T+ + T
-1
=
,ri
(vp
in both Oa(R) and DI(R) with T+ and a
with T
1
* vp
x
.
1
)
x Recall Example 2.3.3.
Comparing 6+
we see that T+ = -ina+ and T
= ina
Hence
a = a+ - a
=
-1
(T+ + T-) =
1
n
(vp
x
* vP
x
)
in both D' (R) and 0;(R), -1 < a < 0. While Theorem 2.4.1 gives the Plemelj representation of distributions in O'(IR), a > -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
1
(t-z)
k+1 >
is analytic in C\supp(T) with lim e->0+
f
(F(x+ie) - F(x-ie)) p(x) dx
=
and
lim e->0+
_
(F(x+ie) + F(x-ie)) ip(x) dx = -2
for all ' E D (M) where (t) is as in Theorem 2.4.3. PROOF.
Consider the Cauchy kernel 1/(t-z) for z E C\supp(T). First let a be an arbitrary negative real number. We can find a value of k = 0,1,2,... such that 1/(t-z) k+1 = o(ItIa) as Iti -p w This implies that .
dp 1
dtp
=
(t-z)k+1
as Iti - - for p = 0,1,2,...
.
Thus 1/(t-z)
k+1 E 0a(JR),
a < 0, as a function of t E IR for the chosen value of k. a > 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
=
(-1)-k-1 c(k)(t_ie)>
=
(-1)-k-1
and
where the derivatives here are with respect to t E IR for fixed
As a -4 0+ the terms on the right of these two
e > 0.
equalities converge to
(_1)-k-1 ;(k)(t)>
- _(t)>
=
_
and
(-1)-k-1
respectively.
P+k)(t)>
+(t)>
The conclusions now follow by Theorem 2.3.7.
Finally we give a Plemelj representation of distributions in Oa(IR) for arbitrary a using division by a polynomial. THEOREM 2.4.6.
Let T E O'(IR), a E IR, and let P(t) be a
polynomial without real roots and of degree k.
Let a > -k-1
and
f(z)
2ni
P(t)1(t-z) > z E O\sUpp(T)
We have
lim e-),0+
82
J, [[;](x+1e)
-
I
P
(x-ie)J P(x) .p (x) dx =
and
F (fp, (x+ie) +
e-'0+
P(x) v(x) dx = -2
I
for all p E D (R) where ap (t) is as in Theorem 2.4.3. PROOF.
The order relation 1/P(t) = 0(1/ItIk) as Itl - w implies that 1/P(t) E O_k(R). Since the Cauchy kernel is an element of 0_1(M) we have 1/(P(t)(t-z)) E O_k_1(R) and hence belongs to 0a(IR), a > -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(t)E) >
< P I (x-ie), P(x)> =
'pPtie)
and
P( t)
>
These two identities together with an application of the Plemelj formulas (2.12) and (2.13) lead to the conclusions of the theorem. 2.5.
DISTRIBUTIONAL PLEMELJ RELATIONS AND BOUNDARY VALUE THEOREMS
Our purpose in this section is to find conditions under which a distribution is the boundary value of a half plane analytic function.
Assume that for a distribution T E 0'(IR), -1 < a < 0, we have
lim f+ (x+ie) e-+0+
=
f+ = T 83
in D'(IR) where f+(z) is an analytic function in A+ with
In order to characterize the distribution T we use its Cauchy integral f+(z)
= 0(1/IzI) as Izi -+
C(T;z)
tlz >
21
in A+.
z E C\supp(T).
,
Recall that C(T;z) is a locally analytic function with the boundary of the domain of analyticity consisting of supp(T) on By the assumption that is stated in the first sentence in this paragraph and Lemma 2.2.1(ii) we have the real axis.
1
C(T;z) =
2,ri
lim E-40+
+
1
2,ri
lim
1
J0,
>
- e-40+
f+(t+ie)
t-z
1
2,ri
1
>
dt
According to part (iii) of Lemma 2.2.1 we have C(T;z) =
f+(z) 0
,
z E A+
,
z E A-
(2.33) .
Let T+ be the boundary value of C(T;x+ie) as a -. 0+ from Theorem 2.4.2. We then have that T+ = f+ = T in DI(R) In view of the Plemelj relation .
T+=
1
2
T -
2,1
(T * vp
x
(2.34)
which can be obtained from Theorem 2.4.2 similarly as (2.24)
and (2.25) were obtained from (2.22) and (2.23), we thus have
T + Ti
(T * vp
)
=0
(2.35)
x Conversely, let T be the boundary value of C(T;x-ie) as a --a 0+ from Theorem 2.4.2 and assume that T in D'(IR)
.
satisfies the relation (2.35). 84
This hypothesis and the second
Plemelj relation
T-
_-
1
2
1
T
1
(T * vP
),
x
(2.36)
which is obtained from Theorem 2.4.2 similarly as was (2.34), implies that T = 0 on R Now from T = T+ - T we obtain .
T=T =f
in D' (1R)
Correspondingly assume that lim f (x-ie) = f e->0+
= T
in D'(IR) where f -(z) is analytic in A
as Izi - -
.
C(T;z) =
In this case we prove as above that 0
-f-(z)
,
z E A+
,
z E A-
,
(2.37)
This leads to the equality T = -f view of (2.36) we get the formula
T-
in
with f -(Z) = 0(1/IzI)
(T * vp
1
)
= -T in D'(IR)
=0
(2.34) this means that T+ = 0 on JR
Thus in
(2.38)
Conversely, let T satisfy the relation (2.38). f
.
,
Because of
and T = T+ - T
= - T
=
follows.
The preceding analysis in this section proves the following two theorems. THEOREM 2.5.1.
A necessary and sufficient condition that a
distribution T E 0a(M)
,
-1 < a < 0, be the boundary value in
the D'(IR) topology of a function fi(z) which is analytic in Ai
with f±(z) = 0(1/lzl) as Izi -.
is that the formula (2.35)
or (2.38), respectively, hold.
A necessary and sufficient condition that a -1 < a < 0, be the boundary value in distribution T E O'(IR) THEOREM 2.5.2.
,
the D'(IR) topology of a function f}(z) which is analytic in A} 85
with ff(z) = 0(1/IzI) as jzi - m is that the formula (2.33) or (2.37), respectively, hold. As a complement to Theorems 2.5.1 and 2.5.2, the uniqueness
theorem for distributional boundary values of half plane analytic functions is now obtained. THEOREM 2.5.3.
If a distribution T E 0'(l), -1 < a < 0, is
the boundary value in D'(R) from A+ or A
of a function which
is analytic in A+ or A with the order relation 0(1/Izl) as Izi -> 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
Let T E E' (IR) and
.
t-z >,
z E C\supp(T)
We know that C(n)(T'z)
-
2ai
(t-z)n+l > =
2iri
tlz >
for n = 0,1,2,... where T(n) is the nth distributional derivative of T. Let lim
C(n)(T;x+ie)
e->O+
=
(T(n))+
and
(T(n))-
li0+ C(n)(T;x-ie) =
in D'(IR)
Since T(n) E E'(IR) for every n = 0,1,2,... we can
.
apply the general Plemelj relations T+ =
1
T-
1 27r i
{T*
v
p
xJ
(2.39)
and 87
T
2
2ni
T
IT * vp 1
(2.40) J
in D'(IR) to C(n)(T;z) and obtain (T(n))+
T(n)
2
(n)
2ni
* vp
IT
1
(2.41) J
and T(n)
2
IT(n) * vp
21
.
1
(2.42)
J
Differentiating the Plemelj relations (2.39) and (2.40) we obtain (n)
(T+) (n)
=
2
T(n)
2ni
IT
* vp
x
(2.43) J
and
(T-)
(n)
_ -
2
T(n)
2ni
IT(n) * vp
.
1
(2.44)
J
(Recall that a convolution may be differentiated by
differentiating either one of the distributions in it.) Comparing (2.41) with (2.43) and (2.42) with (2.44), we obtain for each n = 0,1,2,... that in D'(I) (T(n))+
(T+) (n)
(2.45)
(T(n))- = (T-) (n)
(2.46)
=
and
Thus the distributional boundary values of the derivatives of the Cauchy integral of distributions in E'(IR) coincide with the derivatives of its distributional boundary value.
88
REPRESENTATION OF HALF PLANE ANALYTIC AND MEROMORPHIC
2.6.
FUNCTIONS A distribution T E DI(M) is called real valued if is a real number for all real valued functions p E D(IR).
Hence
= for all 1p E D(l) for a real valued distribution T.
This equality also holds for T E 01 (R) and p E 0a(M) since
D(l) is dense in Oa(M). a sequence
(That is, for each p E Oa(R) there is
n) of elements in D(l) which converges to c in
THEOREM 2.6.1.
Let T E O'(l), -1 < a < 0, be the D'(l)
boundary value from A+ of f+(z) which is analytic in A+ and is
0(1/Izl) as IzI - W f+(z)
.
We have
n1
tiz
(2.47)
>
tlz >
for z E A+
For the main step in proving this theorem we use
PROOF.
Theorem 2.5.2.
21
Thus we may write
tiz > = 0, z E A
or
2i-i
> = 0, z E A+
1
t-z
Decomposing the distribution T into its real and imaginary
parts, the latter equality becomes
2ni
1
t-z
> =
2n
1_
>, z E A+
t-z
89
Since Re(T) and Im(T) are real valued distributions, taking conjugates we have
1
tai
1
_
-i
J> =
27r
t-z
l
1
J
t-z
and finally
tai
>
2a
t-z >, z E A+
.
(2.48)
But by Theorem 2.5.2, f+(z) equals the Cauchy integral of T; that is +
f (z) =
2ai
for z e A+
1
1
2ai
(2.49)
t-z >
t-z >
Comparing (2.48) with (2.49) the desired
.
conclusion follows. The similar representation for f -(z) = 0(1/Izl) as JzJ -. - and which is analytic in A
follows correspondingly.
Because of Lemma 2.2.1, Theorem 2.6.1 implies the following theorem. Let T E O'(R), -1 < a < 0, be the Oa(R) THEOREM 2.6.2. boundary value of a function f+(z) which is analytic in A+ and We have that (2.47) holds. which is 0(1/IzJ) as Izi --> THEOREM 2.6.3. Let T be a real valued distribution in 01(R) .
for -1 < a < 0 and let C(T;z)
=
2ai
t1z >, z E C\supp(T).
If C(T;z) = 0 for all z E A+ or for all z E A
90
then T = 0.
PROOF.
Let us consider the case C(T;z) = 0 in A
.
By this
assumption and the analysis used to prove Theorem 2.5.1, the distribution T is the DI(R) boundary value of f+(z) from A+ that is
lim e-0+
f+(x+ie) p(x) dx
=
_ , p E D(IR)
We note that if = 0, U E O'(tR), for all p E D(l) then = 0 for all V E Oa(IR) since D(R)
is dense in Oa(l).
Here T is a real valued distribution in O'(R), -1 < a < 0 which implies = 0 for all w E D(IR); hence
t-z
> = 0, z E A+
Now by Theorem 2.6.1 we obtain f+(z) =
Vi
tlz > = 0 , z E A+
Hence = 0 for all p E D(IR).
This finishes the proof for
the case of A-.
The proof for A+ is similar. Observe that Theorem 2.6.3 can be easily proved by the
analytic continuation method; in this case this theorem contributes to section 2.2. Also, if in Theorem 2.6.3 the condition C(T;z) = 0 in A+ or in A is replaced by Re(C(T;z)) = 0 in A+ or in A then the same assertion follows by the first Plemelj relation and properties of analytic functions.
Theorem 2.6.3 implies the following facts about real valued distributions. If the Cauchy integrals of two real valued coincide on A+ or A then distributions in O'(IR), -1 < a < 0 ,
If T(t) is a real valued these distributions coincide on R continuous function with order 0(1/Itla), a > 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)
+
27i
=
A k,p
1
t-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
= .
Thus
= 0 for all such gyp; this proves supp(T) C K.
Now
if supp(T) C K then there exists an open interval (a,b) C K\supp(T) on which T is zero in D'((a,b)).
By the analytic
continuation principle, f(z) would be analytic on (It\K) U 92
(a,b) contrary to the hypothesis; we conclude in the case that K is a closed proper subset of ER that supp(T) = K. same reasoning supp(T) = ER in the case K = R.
the boundary values f+ and f Also (f+-f
all a < 0.
)
By the
By Lemma 2.2.1
are distributions in 01(E) for
E 0'(E) for all a < 0.
Since the
Cauchy kernel belongs to 0'(R) for all a > -1, we must take -1 < a < 0 in order to form the Cauchy integral of (f+-f ).
Thus we consider (f+-f take (f+-f
)
)
Also we can
E 01(R), -1 < a < 0.
E O'1(It) since 0'(E) C O'1(Qt) for a > -1.
Therefore the function
F(z) =
1
2,ri
+
,z> 1
(2.51)
is defined and analytic in the domain C\K.
By a standard
argument with the assistance of Lemma 2.2.1 we have F(z) _ 0(1/Izl) as jzj --> -
.
In addition by the Plemelj relations
(Theorem 2.4.2,) F(xfie) converges in DI(M) to a distributional boundary value Fi
,
respectively, as a - 0+.
(In fact the convergence of F(xtie) to F± in 01(E), -1 < a < 0, as e -i 0+ holds here by Theorem 2.4.4.)
Now let us put
Obviously the function H(z) is defined
H(z) = f(z) - F(z).
and analytic at least in C\It except at the poles of f(z). Further, H(z) = 0(1/jzI) as IzI --> - and
- H-, V> = -
for all %p E D(IR).
- F-, p>
According to the first Plemelj
distributional relation in Theorem 2.4.2, from (2.51) we have
,
p> =
,
p> for all f E D(IR). Hence =
for all pp a D(IR).
By the analytic continuation theorem
(Consequences 2.2.1(ii)) the function H(z) is analytic everywhere in C except at the poles ak, k = 1,2,...n.
By
virtue of the generalized Liouville theorem, H(z) is a rational function in C which vanishes at infinity. 93
Consequently the partial fraction expansion of H(z) is possible and gives the coefficients Ak,p in (2.50) as can additionally be seen in [88].
The representation (2.50) is
proved.
REMARK 2.6.1.
If we replace the convergence in the D'(O)
topology with that of O'(R), -1 < a < 0, in Theorem 2.6.4 then we get an equivalent result by Theorem 2.2.1.
Theorem 2.6.4 implies immediately the following corollaries. COROLLARY 2.6.1. Let f(z) be sectionally analytic in C except for a finite number of simple poles ak, k = 1,2,...,n, which are located in A+ U A a closed set K C C lim
.
and with a boundary on M consisting of Let f(z) = 0(1/IzI) as jzj Let .
f(x+iE) = f+
e-40+
and lim
f(x-ie) = f-
&-+0+
in D'(IR)
.
For z f K and different from the ak, k = 1,2,...n,
we have
1
f(z) =
2ni
+
1
-
n
t-z > +
Res[f(z);ak] z-ak
k=1
where Res[f(z);ak] is the residue of f(z) at ak, k = 1,...,n. COROLLARY 2.6.2.
Let f(z) be sectionally analytic in C and
with a boundary on R consisting of a closed set K CR f(z) = 0(1/IzI) as Izi - Let .
lim f(x+ie) = f+ e40+
94
.
Let
and
lim
in DI(It)
f(x-ie) = f
For z ff K we have
.
1 + f(z) = 2ni
1
'
t-z
>
As an illustration of Theorem 2.6.4 and Corollary 2.6.1 we give the following two examples. EXAMPLE 2.6.1.
Let 2
f(z) =
Here we have n = 1, a1 = i, a1 = 2, K = CO), and
A1,
p=1
=
A1,1
(z-i)p
A1,2
+
z-i
(z-i)2
where
z-i
(z-i)2
2
2
= -2i
z(z-i)
and _
A1,1
lim
d
z-'i dz
[(z -i) 2
2
z(z-i) 2
=2
The distributional boundary values are f+ _
2
(x-i)2 (x+iO) and 95
f
2
=
(x-i)2 (x-iO)
with
f+
- f-
-2 (x-i)
2r16
2
Thus
f(z) = -2
EXAMPLE 2.6.2.
1 2
2i
2
> +
z-i
(t-z)
Let
z E A+ with z X i
1
f(z) _
-1
2z(z-i) {_2z(z-i)
'
z E A
-
The distributional boundary values are 1
f+
2(x-i)(x+i0)
and
f
=
-1
2(x-i)(x-iO)
We have Res[f(z); i] = -i/2 f(z) =
1
1
t '
.
Consequently 1 (t-i) (t-z) > -
i 2(z-i)
We now are able to give necessary and sufficient conditions for an analytic function to be represented by the Cauchy integral of a distribution in E'(D) THEOREM 2.6.5. Let T E E'(IR) with support being the compact .
96
(i)
If f(z) = C(T;z) then f(z) is analytic in C\K;
(ii)
f(z) = 0(1/Izas Izi ->
set K C R.
;
and
the distributional boundary values
(iii)
lim e40+
f(x+ie) = f+
and
lim e-*0+
f(x-ie) = f
exist in D'(IR)
.
Conversely, if f(z) satisfies (i), (ii), and (iii) then f(z) is the Cauchy integral of some T E E'(D) with supp(T) = K.
Let T E E'(R) with support being the compact set
PROOF.
K C R. 2.3.2.
For the conclusions (i) and (ii) we refer to Theorem The conclusion (iii) follows from the proof of Theorem
2.3.7.
Conversely, let f(z) satisfy the conditions (i),(ii), and Then f(z) is sectionally analytic in C with boundary consisting of the compact set K. From the proof of Theorem (iii).
2.6.4 we know that the distributions ff , and ff belong Put T = f+ - f
to 0'(R), -1 < a < 0.
2.6.4 shows that supp(T) = K.
The proof of Theorem
The space E'(R) is a proper
subset of O'(R) for all a E R; and T E 01(R), as a distribution in DI(R) with compact support, can be extended Thus T E E'(C) with uniquely to a distribution in E'(l). supp(T) = K. C(T;z)
2ai
Now we can put
t1
>
21
By Corollary 2.6.2 we get C(T;z) = f(z)
.
t1z ,
>
(2.52)
The proof is
complete.
The following theorem shows that the convergence in D'(IR) 97
in condition (iii) of Theorem 2.6.5 can be replaced by convergence in O'(IR), -1 < a < 0, with the same result being obtained.
Let T E E'(O) with support being the compact
THEOREM 2.6.6. set K C R
.
If f(z) = C(T;z) then
(i)
f(z) is analytic in C\K;
(ii)
f(z) = 0(1/Izl) as Izi -
;
and
(iii) the distributional boundary values
lim f (x+ie) = f+ e-40+ and
lim
f(x-ie) =
exist in 01(R) a
,
fe-40+
-1 < a < 0
Conversely, if f(z) satisfies (i), (ii), and (iii) then f(z) is the Cauchy integral of some T E E' (M) with supp(T) = K.
The conclusions (i) and (ii) in the sufficiency are obtained as in Theorem 2.6.5. Since T E E'(IR) implies T E O'(IR), (iii) is obtained from Theorem 2.4.3. The proof of PROOF.
the converse is similar to that of the converse in Theorem 2.6.5.
Observe that Theorems 2.6.5 and 2.6.6 are equivalent. By Lemma 2.2.1, condition (iii) of Theorem 2.6.5 implies condition (iii) of Theorem 2.6.6; the converse is always true because of the topologies.
Further, these two theorems show
that there exists a one to one correspondence between the distributions in E'(IR) and the functions f(z) with the properties and 2.6.6.
and (iii) of each of the Theorems 2.6.5 A meromorphic variant of Theorem 2.6.5 is given in
[95].
We now obtain a Cauchy integral representation of an analytic function in a strip. 98
THEOREM 2.6.7.
Let f(z) be an analytic function in the strip r = (z E C: yl < Im(z) < y2) with f(z) = 0(1/Izll+') for some X > 0 as IzI -> - in r lim 6-0-0+
Let
.
f(x+i(yl+e)) = fl
and
lim e-*0+
f(x+i(y2-e)) = f2
in the DI(R) topology.
For yl < Im(z) < y2 we have
-
1'y2
f ( z )
= 2ni
2ni
+ i l - z > -
>
(2.53)
where the Cauchy integral of f1 is analytic in the upper half plane Im(z) > yl and the Cauchy integral of f2 is analytic in the lower half plane Im(z) < y2
Let z E r be arbitrary but fixed.
PROOF. a
.
and
b
Choose real numbers
such that yl < a < Im(z) < b < y2
tends uniformly to zero as IzI -> - in r
,
Since f(z)
.
an application of
the Cauchy integral formula [127, Lemma 1, p. 293] leads to the decomposition f(z) = f+ (z) + f -(z), z E r where ,
+
f (z) =
1
2ni
03+ia
f J-w+ia
f(l) c-z
dS
and _
f(z)
-1 =
2ni
+ib +ib
f (c
dC
C-z
we have that f+(z) is analytic in the upper half plane Im(z) > a and f (z) is analytic in the lower half plane Im(z) < b. 99
The preceding analysis in this proof holds for any given z e F and yields that both f+(z) and f -(z) are analytic on r
In
.
order to investigate the behavior of these functions as IzI -
consider the equality
,
z f+ z (
1
)
=
2iri
--
1 2iri
-+ia
J
-+ia
J
-oo+ia
z f (c) S-z
d
(2 54) .
+ia
S f(S) dr - 1 2rri [-z
f(r) dr
J -oo+ia
The integral of the Cauchy type in the last equality in (2.54)
vanishes as Izi -> - in the upper half plane Im(z) > yl while the other integral in the last equality of (2.54) converges From this and (2.54) we From a similar conclude that f+(z) = 0(1/IzI) as Izi -> integral representation for (z f (z)) we also infer that since f(c) = 0(1/1511+)), X > 0.
f -(z) = 0(1/Izl) as Izi -. w in the lower half plane Im(z) < y2 .
we must now verify that the functions f+(z) and f -(z) converge in the D'(IR) topology to boundary values on
Im(z) = yl and Im(z) = y2, respectively, from the interior of Let z = x+i(a+e), e > 0, be a point in the half plane Im(z) > a Recalling f+(z) above we have r
.
.
f+(x+i(a+e))
=
2,ri l
1
-(x+ie)
>
and
f+(x+i(yl+e))
=
lim
f+(x+i(a+e)) 1
lim a-yl
1
2ai
The analyticity of f(z) in r 100
,
1
t-(x+ie)
>
.
the fact f(z) = 0(1/Izl) in r
as IzI -i -
,
and the assumed convergence of f(t+i(yl+e)) to
fl in D'(IR) as e - 0+ together with Lemma 2.2.1 yield that fl E O'(IR), -1 < a < 0
.
Additionally by Lemma 2.2.1 we have
Now by the distributional Plemelj formulas (Theorem 2.4.4) we obtain f+ =
in D' (IR)
elim
-0+
f+(x+i(yl+e))
=
fl
2
21ri
(fl * VP
x
)
.
Now let z = x+i(b-E) be a point in the half plane Im(z) < y2 Starting from .
f (x+i(b-e)) =
2rr1
t-(Xlie) >
and proceeding with similar analysis as before we have
f (x+i(> y2-e))
-1
=
21ri
1
t-(x-ie)
and
f
l-imO+
=
f (x+i(y2-e)) =
2
f2 +
21ri
(f2 * VP
x
)
in the D'(IR) topology with f2 E 0'(IR), -1 < a < 0.
We have now proved that the function f+(z) (f (z)) is analytic in the half plane Im(z) > y1 (Im(z) < y2), has the and converges in the order property 0(1/Izl) as IzI -> w topology to f+ on Im(z) = y1 (to f on Im(z) = y2). D'(IR) ,
By
Lemma 2.2.1 we have
101
12ai
f+(z), Im(z) > Y1
1
j
t+iyl-z >
'
0
,
Im ( z )
10
'
Im(z) > y2
'
f (z), Im(z) < y2
.
<
Y1
and
-1
-
1
2ai
t
t+iy2-z
Now we shall show that
1tai
1 t+iy2-z > = 0, Im(z) < y2
(2.55)
For such z the function f+(c)/(c-z) is analytic as a function of c inside the closed path which consists of the segment [-r+iy2, r+iy2] and the semicircle Lr of radius r lying in Im(z) > y2
.
According to the Cauchy integral theorem we may
write 1
r
2ni
f-r
f+(t+iy2)
1 t+iy2-z dt + 2ai
cz dC = 0
f
f+
J L
r
The integral along Lr tends to zero as r - holds for Im(z) < y2
.
.
Thus (2.55)
By the same analysis just used we also
have 1
tai
t+i1
yl
z
> = 0
,
Im(z) > y1
.
(2.56)
Combining the representations of the Cauchy integrals of ft and ft above with (2.56) and (2.55) we have
102
f+(z) =
21 i
t+iy1-z >' Im(z) > y1
and
f -(Z)
=
2,ri
t+iy2-z >' Im(z) < y2
From the decomposition f(z) = f+(z) + f -(z) we see that
f1 = ft + f (t+iy1) is the boundary value of f(z) on Im(z) = yl in the D'(IR) topology, and f2 = ft + f+(t+iy2) is the boundary value of f(z) on Im(z) = y2 in D'(IR)
Consequently
f+(z)
1
=
21i
+iyl-z
=
2wi
+iy2-z >' Im(z) < y2
>
Im(z) >
yl
'
and
f -(Z)
Since f(z) = f+(z) + f -(z), the desired result (2.53) is obtained. 2.7.
EQUIVALENCE OF CONVERGENCE IN D'(fl) AND O'(IR)
In some of the previous sections we have touched upon the subject of the present section several times.
Now we shall
treat the equivalence of convergence in DI(R) and O'(R) in more detail.
Consider again the distributions
b+ =
lim e->O+
-1
2,r i (x+ie )
(2.57)
and
103
b-
=
lim
-1 2r i (x-iE )
E-40+
(2.58)
with the limits being taken in the weak D'(R) topology. In Example 2.3.3 we showed that these distributions satisfy the Plemelj relations b+
=
S-
2
1
vp
1
2ir i
and
-
6-
1
2
1
2ir i
(2.59)
x
vp
i
(2.60)
x
In developing the theory of the distribution spaces 01(R)
Bremermann [11, Chapter 7] has proved the formulas (2.59) and (2.60) by taking the limits (2.57) and (2.58) in the 01(It) topology for a < 0
.
The formulas (2.59) and (2.60) in Oa'(IR)
are immediate consequences of Theorem 2.4.3 for -1 < a < 0 however, we would like to extend the original formulas (2.59) ;
and (2.60) which were obtained using the D'(IR) topology to 0a1(R) for all a < 0 by using the knowledge of the convergences in (2.57) and (2.58) in D'(IR)
At this point recall that b E O'(R) for all a E C and vp(1/x) E O'(IR) for all a < 0. .
In
order to obtain the stated desired extension let us consider the function C(6;z) =
2ri
t-z >
-
27riz
z # 0
which is involved in (2.57) and (2.58).
C(6;z) is analytic in C\(0), satisfies the order property 0(1/Izl) as Izi -+ and ,
has the boundary values 6+ and b planes A+ and A respectively. ,
in D'(IR) from the half
Hence, according to part (i)
of Lemma 2.2.1, C(b;x+ie) and C(b;x-iE) converge in D'(I2) to b+ E O1(Qt) and b
104
E Oa(O), respectively, for all a < 0 as
a --> 0+
.
So the formulas (2.59) and (2.60) hold in D'(ll)
with distributions 6+, 6-, 6 and vp(1/x) in O'(IR), a < 0.
By
part (ii) of Lemma 2.2.1, C(b;x+ia) and C(b;x-ia) converge in O'(IR) to b+ and 6 for all a < 0. Thus the formulas (2.59) and (2.60) obtained from the DI(R) topology imply the formulas (2.59) and (2.60) in the 01(M) topology, a < 0.
Since weak
convergence in O.(IR) always implies weak convergence in D'(IR),
the properties obtained in this paragraph yield the following result.
THEOREM 2.7.1.
The formulas (2.59) and (2.60) obtained from
(2.57) and (2.58) with convergence in D'(IR) are equivalent to
the formulas (2.59) and (2.60) obtained from (2.57) and (2.58) with convergence in O'(ll), a < 0.
A more general version of Theorem 2.7.1 for certain values of a is obtained now. THEOREM 2.7.2.
If T E 01(R), -1
0(1/lzl) as Izj -
F--40+
< 0, and C(T;z) _
then the relations T -
C(T;x+ia) = T+ =
1
2n i
(T * vp 2
(2.61)
2
and
lim
C(T;x-ia) = T
= -
2
T
(T * vp ---) X
2ni
(2.62)
in the topology of D'(IR) are equivalent to the relations (2.61) and (2.62) in the topology of O,(IR). PROOF. A
The function C(T;z) is analytic in the domains A+ and
(section 2.4), satisfies the order property 0(1/lzl) as
Izi -> - in A+ and A
by assumption, and by Theorem 2.4.1 has
By part (i) of Lemma the boundary values T+ and T in D'(IR) 2.2.1 and Remark 2.2.1 we have that T+ E 0;(R) and T- C 0;(M) .
for -1 < a < 0 here. in O'(IR), -1 < a < 0.
Note that T * vp(1/x) is a distribution Now by part (ii) of Lemma 2.2.1, 105
C(T;x+ie) and C(T;x-ie) converge in 0;(U) to T+ and T respectively, for -1 < a < 0 here.
,
Thus the relations (2.61)
and (2.62) stated in the D'(IR) topology imply the relations
(2.61) and (2.62) in the 0.(l) topology for -1 < a < 0
.
Conversely, by Theorem 2.4.3 the limits (2.61) and (2.62) exist in O'(l), -1 < a < 0.
But weak convergence in O'(IR)
implies weak convergence in D'(IR) hold also in D'(IR) 2.8.
.
Thus (2.61) and (2.62)
.
COMMENTS ON CHAPTER 2
The definition of the Cauchy integral of a distribution is an extension to distributions of the definition of the usual Cauchy integral for functions, and the two Cauchy integrals coincide if the distribution is defined by a function.
The
generalized Cauchy integral is systematically used in Kothe's paper [67].
It should be mentioned that the indicative term
"a generalized integral depending on a parameter" for a function I(z) = is due to Schwartz [117, pp. 104 - 105].
Here K(t,z) is a suitable function defined on l for
certain values of the parameter z E C
.
A well known problem in classical analysis is to find a representation of a given function f(x) defined on R by means of a function ?(z) which is locally analytic in C
A classical solution to this problem is due to Carleman [13]. The same problem relative to a simple closed smooth curve L instead of IR , was first solved by Plemelj [104]; see also [114, pp. 233 - 234]. Namely, let f(t) be a continuous function on L and denote by z' and z" two points on the inside and outside of L, respectively. If ?(z) is the Cauchy integral of f(t) then, defining ?+(t) - ?_(t) as the limit of T(z') - ?(z") when z' and z" approach t E L
we have ?+(t) - ?_(t) = f(t).
,
.
Iz'-tl = Iz"-tl,
We recognize the notion of the
analytic representation of a distribution as the distributional version of the problem cited above.
The theory of the representation of distributions as boundary values of
analytic functions has been discussed by many authors and was 106
initiated by Kothe [67] and Tillmann [129].
Theorem 2.2.2 on distributional analytic continuation is used for solutions in applications in Chapters 2 and 3. can be extended to functions of n complex variables as 1Rn and put y2 = follows. Let y = (y1,y2.... ,yn) E ...-yn G
The domains G+ = {y: y2 > 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
(2ni)
j=1
t - z j
of a distribution T E OI(Rn), 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
<
1
< M 1x2 - x1I°
(3.4)
If the function f is Holder continuous on
.
[a,b] it is continuous on [a,b]. The function f is said to be Holder continuous at the point at infinity if it has a definite limit value f(-) as $x$ -->
which satisfies the
order relation f(x) - f(OD)
= 0(1/Ixla)
for some a > 0 as Ixi - If(x) - f(o)I <
,
M
that is if (3.5)
Ixla
for sufficiently large Ixi
.
The number M in (3.4) or (3.5)
is called the Holder constant; v and a are called the Holder indices.
In the algebra of Holder continuous functions only
the values of v or a are important, u in the case of Holder continuity on [a,b] C C and a in the case of Holder continuity at w Where it is necessary to specify a Holder condition, we shall write H(u) or H(a), respectively. If two functions f1(x) and f2(x) which map IR to C satisfy
the conditions H(a1) and H(a2) at the point at infinity, respectively, it is easy to show that the functions f1(x) + f2(x), f1(x)f2(x)
,
and fl(x)/f2(x) satisfy the condition H(a)
at infinity with a = min(al,a2); we assume that f2(x) x 0 on IR in the case of the quotient. 113
Let h(x) be a complex valued function on IR
.
Denote by
(h(x))O. the total change, the variation, of h(x) as x traverses the real line D in the positive direction; that is, (h(x))mm = h(oo) - h(-m) =
lim
h(x) -
h(x)
Now let f(x) be a complex valued function defined and continuous on f with f(x) x 0 on IR including the points at
An integer A , which is positive, negative, or zero, defined by infinity (f(co) # 0 and f(-w) # 0.)
2ai
=
(arg(f(x)))
2n
will be called the index of f(x) and will be denoted by Ind(f(x)).
From the definition we derive the following rules:
Ind(fl(x) + f2(x)) = Ind(fl(x)) + Ind(f2(x))
,
Ind((f(x))n) = n Ind(f(x)), and
Ind(f1(x)/f2(x)) = Ind(fl(x)) - Ind(f2(x))
.
Also if f(x) is the boundary value of a function which is analytic everywhere in A+ except for a finite number of poles then according to the principle of the argument we have Ind(f(x)) = N - P where N denotes the number of zeros of the function and P denotes the number of poles in A+ For example, we have .
Ind((x-i)/(x+i)) = 1 because the function (x-i)/(x+i) is the boundary value on IR of the function (z-i)/(z+i) from A In .
particular, if A = Ind(f(x)) = 0 then log(f(x)) is single valued and continuous on M. If A = Ind(f(x)) is a positive or negative integer then the function 114
11
log(fo(x)) = log(( x+i
f(x)J
,
is single valued since Ind(f0(x)) _ -X Ind(
X-i
+ Ind(f(x)) = -1 + X = 0
Frequently we shall use implicitly the following result. If a function f(x) from l to C satisfies the Holder condition
H(a) at the point at infinity then log(f(x)) satisfies the same condition. (Here we assume that f(x) # 0 on D and Ind(f(x)) = 0.)
We recall from Chapter 2 that a distribution T is real if is a real number for every real valued p C D(R). Any distribution T can be represented in the form T = T1+iT2
,
with T1 and T2 being two real distributions, such that
+ i for every real valued c E D(IR).
A
distribution T = T1-iT2 is said to be the complex conjugate of the distribution T = T1+iT2 ; we write TI = Re(T) and T2 = Im(T).
EXAMPLE 3.1.1. defined by
=
Let us consider the distribution b+ E D'(IR)
lim
-1
e>0+ 2iri
m
'O(X) x+ie
-oo
dx
,
p E D(R)
We can write
lim e- O+
P> =
-1 2,ri roo
lim dx + e--).O+
x 4p(x) X2+e2
for all real valued T E D(IR)
.
1 2-w
(
J
a p(X) oo
dx
X2+E2
From here we get immediately 115
that Re(S+) = 6/2 since se(x) = e/1r(x2+e2) is a delta sequence that tends to 6 in D'(IR) as e -> 0+
.
We recall that throughout Chapters 2 and 3 the symbol f(p), p = 0,1,2,..., denotes the pth derivative of the function f with respect to its variable unless explicitly stated otherwise. As usual f(0) = f. 3.2.
APPLICATIONS TO BOUNDARY VALUE PROBLEMS
First we shall consider a few problems of the Plemelj type. PROBLEM 3.2.1. Let T E E'(IR) be given. Find a function which
is sectionally analytic in the complex plane C cut along supp(T), vanishes at infinity, and has boundary values T+ and respectively, T on IR in the DI(R) topology from A+ and A ,
which satisfy the boundary condition (3.6)
Comparing the boundary condition (3.6) with the SOLUTION. first Plemelj relation (2.22) in Theorem 2.3.8 we conclude that the solution of the problem under consideration is the Cauchy integral of T ,
C(T;z)
2ni
t1z >, z E C\supp(T)
.
(3.7)
The function C(T;z) satisfies the required conditions as we recall from Theorems 2.3.2 and 2.3.8. We now show that this solution is unique. Let us assume that another solution F(z) of the above stated problem exists which is distinct from Put H(z) = C(T;z) - F(z). The function H(z) is C(T;z). sectionally analytic at least in C\supp(T), vanishes at infinity, and has boundary values H
+ =
lim 6-40+
e-
H(x+ie)
lim C(T;x+ie) e-40+
116
em0+ F(x+ie) = T+ - F+
and
H-
lim
e)0+ lim
H (x-ie ) C (T; x-ie ) - e->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 .
.
=
for all pp E D(IR).
, p> -
= - = 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
t-z
> + 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)
2ni
> +
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.
We have
'P> = lim J f+(x+ie) v(x) dx _OG
and put
< f+' T> =
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
= <2 Re(f+) , p> = <2U, p> 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
<2Ut,
t-z
z E C\supp(U)
Thus the solution of the present Problem 3.2.6 is the function
f+(z)
ai
=
tlz >, z E A+
For example, the solution of Problem 3.2.6 with U = 6 in From Example (3.10) is the function f+(z) = -1/,riz, z E A+ 3.1.1 we have .
f (x+i&) = 26
Hence Re(26+) = 6 and Re(6+) = 6/2. In the remainder of this section we shall consider the
in D'(Ot)
.
distributional Hilbert and Riemann-Hilbert boundary value problems for half planes.
To be precise in some of our
calculations we need the following two simple results [89]. (i) Let f(z) be a sectionally analytic function in the domain C\R except perhaps at a finite number of complex points with imaginary part different from zero.
Let the pth derivative of f(xfi&) converge pointwise to the function dpff(x)/dxp p = 0,1,2,..., everywhere on Qt as E -4 0+. ,
Let
h(z) be a sectionally analytic function in the domain C\Qt and
suppose that h(xfi&) converges to hi in the D'(R) topology as 6 --4 0+.
lim
and 122
Then (f(x+ie) h(x+i6)) = f+(x) h+
(3.13)
lim
(f(x-ie) h(x-ie)) = f (x) h -
F--40+
in D' (IR) (ii)
(3.14)
.
Secondly, if the functions h+(x) and h -(x) are
bounded on JR together with all their derivatives and h(xfie)
converges to h± in O'(IR) and (3.14) hold.
,
-w < a <
then the limits (3.13)
,
The multiplication of a distribution in
O'(IR) by an infinitely differentiable function is justified under the conditions stated on h+(x) and h -(x).
In fact if
h(x) is a complex valued Cw(IR) function with the property Ih(p)(x)I < Ap
,
p = 0,1,2,...,
x E IR
,
where the Ap are positive constants, p = 0,1,2,..., then h(x) is a multiplier in Oa(IR) and hence in O (J).
Recall that any
CW(IR) function is a multiplier in D(R) and E(M). We now state and solve a Hilbert boundary value problem for distributions. PROBLEM 3.2.7.
(Hilbert)
Let G(x) be a given complex valued
CW(IR) function which does not vanish on K and which satisfies
a Holder condition at infinity. Assume that every derivative G(p)(x) satisfies a Holder condition at infinity depending on p = 1,2,...
.
Let U E E'(IR)
be a given distribution.
Find a
function F(z) which is sectionally analytic in the domain C\IR,
is of order 0(1/IzI) as Iz1F
,
and has boundary values F+ and
in the D'(IR) topology from A+ and A
,
respectively, which
satisfy the boundary condition F+ = G(x) F
in D' (IR)
+ U
(3.15)
.
We approach the general solution of the nonhomogeneous problem (3.15) in steps. First the general SOLUTION.
solution of the homogeneous Hilbert problem 123
=
,
p E D(R),
(3.16)
At the beginning it is convenient to introduce
will be given.
the sectionally analytic function
X(z) = exp(T(z)), z E A+ X(z) _ 1.
X (z) =. z+i
exp(r (z)), z E A-
J
where
I- (Z)
=
1 f 2iri J_
f ( tt +-ii,
log11
with r+(z) = r(z), z E A+
,
G(t)I
t-z dt
(3.17)
and F -(z) = T(z), z E A
(The
notation r(z) here is generally adopted in Russian literature.) Here X will take the value X = Ind(G(t)), and for X = Ind(G(t)) the integrand in (3.17) is a single valued From the hypothesis, the function G(x) function of t E M .
and all of its derivatives are Holder continuous on compact subsets of R.
In addition, the hypothesis on the derivatives
of G(x) implies that every derivative of the integrand in (3.17) satisfies a
Holder condition at infinity.
In general
the integral in (3.17) is understood to be a Cauchy principal value integral at the point at infinity. Note that the Plemelj formulas remain valid for this integral. The function X(z) is a particular solution of the problem (3.16) as we proceed to indicate; that is
<X+, p> = , * 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
<
,
p>_<
X+ (x)
F
-
X (x)
,
p>
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
<
.p(x)> =
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 =
for all w E D(IR) with H(z) = 0(1/Izl) as Izi - reasoning similar to the case A = 0 and A < 0 that H(z) = 0 In summary we can say that homogeneous problem (3.16) is
.
p>
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)
2,ri
By the first Plemelj relation of Theorem 2.3.8 we get U
=
T+
-T-
X+ (x) where lim
C(T; xiie) = Ti
Therefore the boundary condition (3.21) can be written in the form in D'(!R)
.
- T
- T
=
(3.22)
X (x)
in D'(ll)
.
The left and right sides of (3.22) are boundary
values in D'(IR) of the function H(z) =
X z)
as Im(z) - 0t
,
- C(T;z)
respectively.
Here H(z) is sectionally
analytic in the domain C\It except at the point z = -i, where it has a pole of order X
,
by the hypothesis on F(z).
and vanishes as 1/Izi for IzI In addition, by (3.22) we have the
boundary condition = for all w E D(ll).
Applying
the principle of analytic continuation, Theorem 2.2.2, and then the generalized theorem of Liouville, we get
F(z)
- C(T;z) _
(z+i)
with P(z) being an arbitrary polynomial of degree m < X-1; hence we obtain
128
ta1
F(z) = X(z)
i
l
z e A = A+ U A
,
1
+t
<
t1 z > +
,
X (t)
P(z)
(3.23)
J,
(z+i)
as the solution to the problem (3.15) when
The solution F(z) is general since it
A = Ind(G(x)) > 0.
contains the general solution (3.20) of the homogeneous problem (3.16) as a part of the sum on the right side of (3.23).
we shall now show that the solution of the problem (3.15) is given by Ut < X+(t)
X(z)
F(z) =
2iri
1
t-z
>
z
E A,
(3.24)
in the case X = Ind(G(x)) = 0 and has the form (3.24) in the
case X = Ind(G(x)) < 0 if and only if certain conditions concerning U/X+(t) are satisfied. If X = 0, the function F(z) given in (3.24) is analytic in A = A+ U A and satisfies the conditions of a solution to this problem including (3.15). For the problem (3.15) to be solvable for X < 0 the
function F(z) in (3.24) must be sectionally analytic in A = A+ U A
This will be accomplished if the function
.
C
1
U :
(z+i)-
z
X+(t)
is analytic at the point z = -i. Let us develop the Cauchy integral of U/X+(t) in the Taylor series about z = -i. The
radius of convergence is equal to the shortest distance from Since the coefficients of the series are z = -i to supp(U). defined by the derivatives C(n)
U
X+(t)
z '
=
Ut
n! 2iri
<
X+(t)
1
(t-Z) n+1
>
evaluated at z = -i, we have that F(z) is analytic at z = -i 129
if and only if
<
+t 1 X (t)
> = 0
,
k = 1,2,...,-X
(3.25)
.
(t+i)
We thus conclude that the general solution to Problem 3.2.7 for X > 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
<
t
(a(t)+ib(t))X
where K is a real constant.
+
tl z > + 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
<
X(<
Ut
)
[;i
t-z
(a(t)-ib(t))X+(t)
X(Z)
t
1
Ti
1
< (a(t)+ib(t))X+(t)
t-z
> + K
= F(z)
Consequently we have proved that the function F+(z) defined by F(z) in (3.33) for z e A+ is the solution of for z E A
.
the Riemann-Hilbert problem. In particular, for a(x) = 1 and b(x) = 0 on It, we obtain a
solution to the Dirichlet problem, Problem 3.2.6, from (3.33). In this case i
X(z) =
-i
,
z e A+
zEA
Hence, with F(m) = 0 F +(z) _
Ti
,
tlz >, z e A+
APPLICATIONS TO SINGULAR CONVOLUTION EQUATIONS First we shall concentrate on the following problem. 3.3.
PROBLEM 3.3.1.
Let a(x) and b(x) be given complex valued
CW(IR) functions which satisfy, together with all of their
a Holder condition at Assume that the infinity that depends on p = 0,1,2,... Let U functions a(x)+b(x) and a(x)-b(x) do not vanish on IR derivatives a(p)(x) and b(p)(x)
,
.
.
be a given distribution in E'(D)
.
Find a solution T to the
equation 135
a(x) T +
b(x)
IT * vp
(3.35)
U
x
where the equality holds in D'(O) By hypothesis the three distributions involved in SOLUTION. .
(3.35) act on the space D(IR).
We shall seek the unknown -1 < a < 0, using
distribution T in O'(O) for an a
,
properties of the Cauchy integral of T for such a we begin by formulating an alternative problem whose .
solution will aid in the solution of the present one.
To
begin, assume that a solution T of (3.35) exists and introduce the sectionally analytic function F(z) in C\lt defined by the
Cauchy integral
F(z)
2iri
t1z >, z C A
(3.36)
With the aid of the distributional Plemelj relations given in Theorem 2.4.2 we can immediately reduce (3.35) to the Hilbert boundary problem (a(x)-b(x)) F+ - (a(x)+b(x)) F
= U
(3.37)
This shows that F(z) is a solution of (3.37) which vanishes at infinity. in D'(IR)
.
Conversely assume that the sectionally analytic function F(z) which vanishes at infinity is a solution of the problem (3.37).
Now we desire to show that T satisfies (3.35).
For
this purpose let T be defined by
T = F+ - F-
(3-38)
Solving the Plemelj boundary value problem (3.38), the function F(z) may be written in the form (3.36); and hence the relation in D'(IR)
ri 136
.
(T * vp
x
)
=
F+ + F
(3.39)
Starting from (3.37) and using (3.38) and (3.39) we obtain (3.35). This proves that T is a solution of (3.35),
holds.
and recall that F(z) is a unique solution to the problem (3.37).
In summary, we have that T is a solution of (3.35) if and only if the function F(z) given in (3.36) is a solution to the problem (3.37) with the supplementary condition F(-) = 0. This fact is the key element in the solution of this Problem 3.3.1.
Thus the problem F
a(x)+b(x) a(x)-b(x)
+ =
in D'(fl)
F
+
U
(3.40)
a(x)-b(x
which is equivalent to (3.37), will be solved now.
,
Let
a(x)+b(x) a(x)-b(x)
K(x) =
(3.41)
and suppose that X = Ind(K(x)) > 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
<
F
F
<
X+(x)
for all V E D(R).
(3.42)
,.p>
U
X+(x) (a(x)-b(x))
X (x)
Using the solution of the general Hilbert
problem (section 3.2, Problem 3.2.7) we obtain the following solution of the problem (3.42) for z e A :
F(z) =
(3.43)
= X(Z)
U
1 21ri
<
t-z > +
X+(t)(a(t)-b(t))
where X(z) = X+(z) for z e A+
,
P(z)
1
(z+i)X
X(z) = X (z) for z e A
P(z) is an arbitrary polynomial of degree m S X-1
.
,
and
Also, if
X = Ind(K(x)) S 0, then
F(Z)
=
X(z) tai
<
Ut
1
X+(t)(a(t)-b(t))
t-z
>, z e A
For X < 0 we require Ut <
1
+
X (t)(a(t)-b(t))
> = 0
,
k = 1,2,...,-X
(3.44)
(t+i)
Now we can compute the unknown distribution T of (3.35) by means of the relation (3.38). Applying the distributional Plemelj relations in Theorem 2.3.8 for the function F(z) we obtain for X > 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) _
2 rri
1
t
which vanishes at infinity. A+ - A
= b(x) A
A + + A
=
t-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
<
(t)
(t)+bbt)) Bt
> = 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
t-z
> + P(z)
,
z E A+
and
f -(z)
tlz > - 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) <
We have C* = (t E utn: uC(t) S 0).
then uC(t) = uO(C)(t) [135, p. 219].
uO(C)(t); and if t E C
IRn\C*
For a cone C put C* =
PC
The number
.
tEC* uO(C)(t)/uC(t)
characterizes the nonconvexity of the cone C. Vladimirov [135, p. 220] has proved that a cone C is convex if and only if pC = 1, and if a cone is open and consists of a finite number of components then pC < +-.
Some examples of cones and of the above notions are easily constructed. If C = (0,w) (= (-w,0)) then If uC(t) _ -t (= t), and pC = 1 (= 1). C = [0,w) EXAMPLE 4.3.1.
C = IR n then C* = (0), uC(t) _ ItI, and pC = 1. (µ1,...,µn)
Let µ =
be any of the 2n n-tuples whose entries are 0 or F1
1; Cµ = (y E Utn: in
C
IRn
(-1) J yj > 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) <(Y/IYI) - act/ItI),t> > 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
<_
d(y) K jyj, y E C' C C
(4.3)
.
We now make a
Let C be an open connected cone in Itn.
convention concerning the notation y -* 0
y E C. Frequently in the remainder of this book y - 0 y E C, will mean y -> 0, y E C' C C, for every compact subcone C' of C. We shall distinguish between this convergence of y --> 0 y E C, as opposed to y 0 arbitrarily within C where appropriate; ,
,
,
in most relevant material the analysis clearly shows which interpretation of y --. 0, y E C, is meant for the problem being considered.
Let U be a distribution or generalized function and let f(z) be a function of z = x+iy E TC By f(x+iy) -> U in the weak topology of the distribution space as y --3 0, y E C, we .
mean --3 as y -> 0, y E C, for each fixed element p in the corresponding test function space.
By
f(x+iy) - U in the strong topology of the distribution space
as
yE C, we mean
as y->0,
y E C, where the convergence is uniform for N on arbitrary 156
bounded sets in the corresponding test function space. U is then called the weak or strong, respectively, distributional boundary value of f(z); this boundary value is defined on the distinguished boundary of the tube TC , {z = x+iy: x E stn y = 0), which is not necessarily the topological boundary of TC. 4.4.
CAUCHY AND POISSON KERNEL FUNCTIONS
Let C be a regular cone in R
n ;
that is, C is an open convex
cone such that C does not contain any entire straight line. DEFINITION 4.4.1.
The Cauchy kernel K(z-t), z E TC = IR n+iC,
t E Utn, corresponding to the tube TC is
K(z-t) =
J
exp(2iri) di1
,
zE
t E 1tn
TC ,
C
If C = Cu is any of the 2n quadrants in Rn then K(z-t) takes the classical form
K(z-t) = Kµ(z-t) _
since C
µ
= C
u
n lul n R (t.-z.z e 6tn+iCµ, t E IRn (2rri) j=1 _
(
1)
in this case.
DEFINITION 4.4.2.
The Poisson kernel corresponding to the
tube TC defined by the regular cone C is
P(z,t)
__
K z-t K(2iy)
K(z-t) K(z-t) K(2iy)
2
z = x+iy E T
C
t E
n IR
.
The Poisson kernel reduces to the classical form in the case C = Cµ is a quadrant; in this case
P(z;t) = Pµ (z;t) _
n (-1)141 n
n R
y 72
2
,
j=1 (tj-xj) +yj
z = x+iy E ftn+iCµ
t E Utn
157
We could have taken the cone C above to be open and connected but not necessarily convex, but then we would have defined K(z-t) and P(z;t) for z E TO(C) and would have proceeded to obtain the properties concerning them for z E TO(C) Thus we have assumed that C is convex at the .
beginning here without any loss of generality. Recall from Lemma 4.3.1 that C will lie in some (n-1) dimensional plane
if C contains an entire straight line; in this case the *
Lebesgue measure of C would be zero. Hence K(z-t) would equal zero in this case and P(z;t) would be undefined. To avoid this triviality concerning K(z-t) and to make P(z;t) well defined we must have that C does not contain any entire For these reasons we are considering a regular cone throughout this section unless explicitly stated otherwise.
straight line.
In Chapters 5 and 6 we will form Cauchy and Poisson integral transforms of certain generalized functions and compute boundary values of these transforms.
For these generalized integral transforms to be well defined the Cauchy and Poisson kernel functions must be in appropriate test spaces.
We now derive properties of K(z-t) and P(z;t)
including their containment in several test spaces of functions.
To prove the properties of the Cauchy kernel K(z-t) we need the following lemma and its proof which also will be applicable later in this chapter and in Chapter 6. This lemma is stated in two parts; the proof of each part uses a similar technique. We prove the lemma for C being an arbitrary open connected cone. LEMMA 4.4.1. I.
Let C be an open connected cone in Rn If z E TC = 11n+iC is arbitrary but fixed and I *(t) C
denotes the characteristic function of C then (I *(t) exp(2ni)) E LP for all p 1 5 p C ,
as a
function of t c IRE. II.
158
Let C' be an arbitrary compact subcone of C.
Let
m > 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 <
n
exp(-2,rbplylltl) dt
J
C
(rn-1 CO
Un JO
exp(-2vbPlylr) dr
(4.5)
= fln (n-1) ! (27r6plyl)-n where 0n is the surface area of the unit sphere in Rn. proves the desired conclusion for 1 < p < -.
(4.5)
The proof of 159
part I is complete.
To prove part II let y be an arbitrary but fixed point of There exists a compact subcone C' of C and a fixed real
C.
number m > 0 such that y e (C'\(C' n N(O,m))) since C is open.
Choose w in the assumed growth on g(t) as w = Ay where X > 0 such that 1 > X > (m/IyI) > 0. (For y E (C'\(C' n N(O,m))), Thus 1 > (m/IyI) > 0, and m/IyI is fixed since both IyI > M. m > 0 and y are fixed. We can then choose a real number X > 0 such that 1 > A > (m/IyI) > 0. Since C' is a cone and y E C' then Ay C C'.
Thus w = Ay a (C'\(C' C N(O,m))) since Ay E C'
and IXyl = AIyI > m.
By assumption the growth on g(t) holds
for this particular choice of w.)
By Lemma 4.3.2, given the
compact subcone C' of 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* Using the above choice of w = Ay, the assumed growth on g(t), and (4.1) corresponding to C' we have .
Ie
-2ir g(t)I < M(C',m) exp(2,raXIYI) exp(2,r(1-A)(-)) < M(C',m) exp(2,rcAlyl) exp(-2ir(1-A)6IyIItI)
which holds for all t E C* and the arbitrary but fixed y E C. Now let 1 < p < w and recall that supp(g) C C* Using the .
integration by parts (or gamma function) technique in (4.5) we have r Ie-2w
In
g(t)IP at = f Ie-2n g(t)IP at C*
<
exp(-2ir(1-A)6pIyIItI) dt
(M(C',m))p exp(21raXplyl) J
C =
*
0
(M(C',m))p fln exp(2yaXplyl) Sr1 exp(-2n(1-A)6plylr) dr 0
=
(M(C',m))p f2n exp(2iraXplyl) (n-i)! (2n(1-X)6plyl)-n
The right side here is finite and (exp(-2n) g(t)) E LP as 160
desired.
As noted above, the open cone C will be a regular cone in our analysis concerning the Cauchy and Poisson kernel functions K(z-t) and P(z;t). THEOREM 4.4.1. K(z-t) is an analytic function of z E TC for Rn. K(z-t) E each fixed t E For 1 < p < 2 and fixed z E TC ,
B fl D q for all q,
(1/p) + (1/q) = 1, as a function of t E Otn.
L
Let I *(n) denote the characteristic function of C*.
PROOF.
C
By the proof of Lemma 4.4.1, (I *(rl) exp(2rri)) E L1 as C a function of 1 E Utn for fixed z E TC and t E Rn.
Now let K
be an arbitrary compact subset of TC. For z E K C TC, y = Im(z) E C' for some fixed compact subcone C' C C and y is Thus for z = x+iy E K C TC we have by Lemma 4.3.2 that there is a b = b(CI) > 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
<
I *(n) exp(-2,rbklr,I)
(4.7)
C
C
for z = x+iy E K C TC, t E utn, and rl E stn; and the right side of (4.7) is an L1 function of ri C IRn for z C K and t E Rn by the proof of Lemma 4.4.1 (i.e. the calculation (4.5).) Since (I *(n) exp (2rri)) is analytic in z E TC for each fixed C
Rn and R E
IRn
we now apply Theorem 4.2.2 and obtain that K(z-t) is an analytic function of z E TC for each fixed t E
t E IRn. Now let z E TC be arbitrary but fixed and let p be an By the analysis of arbitrary n-tuple of nonnegative integers. 161
(4.5) we have
fn
IC* (n )
e27ri
Tip
<
fl
fIn
dry
rIRI+n-1
n
= Dn (Ipl+n-1):
exp (-2rrs I y l l n l) do
I nR l
exp(-27r6lylr) dr (27rsly1 )-I13
(4.8)
I-n
which shows that
(4.9)
DQK(z-t) = f n0 exp(2,ri) dry C
where the integral converges absolutely and uniformly with IRn. Thus K(z-t) E CO as a function of t E IRn respect to t E for fixed z E TC since 0 was any n-tuple of nonnegative (1/p) + (1/q) integers. We now show that DQK(z-t) E Lq ,
Arguing as in (4.8) we have that (I *(n) n C 1 < p < 2, as a function of n E IRn exp(2rri)) E L1 n LP for fixed z E TC. From (4.9) DRK(z-t) = 5-1[I *(n) TI 13 exp(2,ri); t] = 1, 1 < p < 2.
,
C
with this inverse Fourier transform being interpreted either in the L1 sense or in the LP sense, 1 < p < 2, as a limit in the mean.
By the Parseval inequality
IIDRK(z-t)IILq <
and DRK(z-t) E Lq for fixed z E TC.
,
III C
*(n)
nR
e2TriIILp
(1/p) + (1/q) = 1, as a function of t E IRn
We have thus proved that K(z-t) E D q for L
all q, (1/p) + (1/q) = 1, 1 < p <
we have the containments D Lq
162
But by [117, pp.199-200]
2.
C B C D m for all q, 1 < q < w. L
We thus conclude that K(z-t) E B fl D q for all q, (1/p) + (1/q) = 1, 1 < p <
The proof is complete.
2.
Let a be an n-tuple of real numbers such that We have K(z-t) E 0a as a function of t E aj > 0, j=1,...,n. THEOREM 4.4.2.
IR
for each fixed z E TC. Any derivative DPK(z-t) exists and is bounded as a
PROOF.
function of t E Rn for fixed z E TC by the analysis in the The proof is Thus K(z-t) E O_.
proof of Theorem 4.4.1.
0
C 0a for any n-tuple a of real
completed by noting that 0 0
numbers such that aj > 0, j = 1,...,n.
The following theorem will not be used here in the boundary value results but is stated for completeness.
It is proved by
an integration by parts technique. Let a be any n-tuple of real numbers.
THEOREM 4.4.3.
There
exists an n-tuple (3 of nonnegative integers such that DQK(z-t) E 0
a
as a function of t E ptn for each fixed z E TC.
Of course Theorem 4.4.3 reduces to Theorem 4.4.2 in the 0, j=1,...,n, for the n-tuple a; in this
case that each aj case (3
= 0 in Theorem 4.4.3.
An obvious but important point to note is that K(z-t) E E IRn
as a function of t E
for fixed z E TC by the proof of
Theorem 4.4.1.
Two technical lemmas that will be used later concerning the Cauchy kernel will be obtained now. LEMMA 4.4.2.
Let w = u+iv E TC be fixed. exp(2,ri) dR
K(z+w) =
,
Then
z E TC
J
C
is analytic in TC and IK(z+w)l < My <
,
z E TC
,
(4.10)
163
where My is a constant which depends only on v = Im(w).
Further, K(x+iy+w) -- K(x+w) in the weak and strong topology of S' as y = Im(z) y c C, for each w E TC PROOF. The proof that K(z+w) is analytic in z e TC is the same as in the proof of Theorem 4.4.1. For y E C and rl E C*, > 0; by Lemma 4.3.2 there is a 6 = 6v > 0 such that
> 6lvi lk
i
for v E C and rl E C*; from these facts and
calculations as in (4.5) or (4.8) the growth (4.10) follows. We now prove the desired convergence in S'. For w E TC and z E TC
II *(TI) (exp(2,ri) - exp(2,ri<x+w,il>))
<
C
(
I
(n)
(e-2,r e-2tt