Clifford Analysis

F Brackx, R Delanghe & F Sommen State University of Ghent, Belgium

Clifford analysis

Pitman Advanced Publishing Program BOSTON·LONDON · MELBOURNE

PITMAN BOOKS LIMITED 128 Long Acre, London WC2E 9AN PITMAN PUBLISHING INC 1020 Plain Street, Marshfield, Massachusetts 02050 Associated Companies Pitman Publishing Pty Ltd, Melbourne Pitman Publishing New Zealand Ltd, Wellington Copp Clark Pitman, Toronto First published 1982

© F Brackx, R Delanghe & F Sommen 1982 AMS Subject Classifications: (main) 30 A97, 46 F15, 42 A68 (subsidiary) 46 E20, 31 B65, 32 A30 Library of Congress Cataloging in Publication Data Brackx, F. Clifford analysis. (Research notes in mathematics; 76) Bibliography: p. Includes index. 1. Holomorphic functions. 2. Clifford algebras. 3. Distributions, Theory of (Functional analysis) 4. Harmonic functions. I. Delanghe, Richard. II. Sommen, F. III. Title. IV. Series. QA331.B77 1982 515.9'8 82-14993 ISBN 0-273-08535-2 British Library Cataloguing in Publication Data Brackx, F. Clifford analysi~>.-(Research notes in mathematics; 76) 1. Functions I. Title. II. Delanghe, R. III. Sommen, F. IV. Series 511.3'3 QA331 ISBN 0-273-08535-2 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 and/or otherwise without the prior written permission of the publishers. This book may not be lent, resold, hired out or otherwise disposed of by way of trade in any form of binding or cover other than that in which it is publish<:d, without the prior consent of the publishers. This book is sold subject to the Standard Conditions of Sale of Net Books and may not be resold in the UK below the net price. ISBN 0 273 08535 2 Reproduced and printed by photolithography in Great Britain by Biddies Ltd, Guildford

Contents

List of symbols Preface Chapter

Clifford algebras and modules 1. 2. 3. 4. 5. 6. 7.

Clifford numbers Modules over Clifford algebras : general properties The spaces C0 (K;A) and C 0 (~;A); A-measures The space V(~;A); A-distributions The space S(ffir;A); tempered A-distributions The space E(~;A); A-distributions with compact support Hilbert A-modules Notes to Chapter 1

Chapter 2 Monogenic functions : general theory 8. 9. 10. 11. 12. 13. 14. 15.

A decomposition of the Laplace and Laplace-Beltrami operator Cauchy's integral formula and its applications Morera's Theorem Taylor series Laurent series and pointwise singularities Adjoint monogenic functions A product of monogenic functions Elementary monogenic functions Notes to Chapter 2

Chapter 3 Spaces of monogenic functions 16. The space ~(r)(~;A) 17. The space M(r)(~;A) 18. Runge type theorems 19. The equation Df = g

13 23 25 32 34 35 40 44 44 51 60 65 84 102 108 115 137 142 142 143 150 160

20. 21. 22. 23. 24.

Primitives of monogenic functions Mittag-Leffler theorems The dual of M(r)(~;A) The bidual of M(r)(Q;A) Hilbert modules with reproducing kernel Notes to Chapter 3

Chapter 4 Boundary values of monogenic functions and Fourier analysis on the sphere 25. 26. 27. 28. 29.

Distributional extension properties of harmonic functions Distributional boundary values and canonical extensions of monogenic functions Representation of distributions by monogenic functions Analytic functions and functionals on the unit sphere L2-functions on the unit sphere- the Szeg~ and Bergmann kernels Notes to Chapter 4

Chapter 5 Transform 30. 31. 32. 33.

References Author index

165 167 169 181 184 193

199 199 203 210 215 224 238

in Euclidean space

244

Hypercomplex Fourier transforms The generalized Laplace transform in S(l)(Rm;A) A generalized Fourier-Borel transform Connections with complex analysis- a generalized Radon transform Notes to Chapter 5

245 253 265

a~alysis

284 297 302 304

List of symbols

A denotes the real Clifford algebra C(V n,o ) (see 1.10) Ap is the subspace of p-vectors of A Ac

denotes the complex Clifford algebra over Vn,o (see 33.1) a(~;A) is the module of A-valued analytic functions in Q (see 8.4) a(Sm;A) is the module of A-valued analytic functions on the unit sphere (see 28.4) S is a multi-index (S 0 , ••• ,Sm) with Sj non-negative integers, with !BI -+

s

=

60 + •.• + Bm and S!

=

S0 ! ... sm!

is a multi-index (6 1, ••• ,Sm)

B(a,R)

denotes the ball in Rm+ 1 with center a and radius R

Bm(a,R)

denotes the ball in Rm with center a and radius R

B*(O,R)

see 33.2

BVf = a+f -a-f is the boundary value off E M(r)(~;A) (see 26.5) B(x,t) module C(Vn , s)

is the Bergmann kernel, i.e. the reproducing kernel of the Hilbert ML 2 (r)(~;A)

(see 24.8)

is the real Clifford algebra constructed over the quadratic vector

space Vn, s (see 1.3) CP is the subspace of p-vectors of C(Vn,s) (see 1.4) C0 (K;A)

is the module of A-valued continuous functions on the compact set K

(see 3.1) C 0 (~;A) is the module of A-valued continuous functions in~ (see 3.1)

dx, dxi, do, dS : see 9.1 aB is a shorter notation for ax~ X

0

-~

()~

is a shorter notation for

(J

X

B 1 x1

Cl

Bm xm

Clr' Cl ul : see 8.7 a+f,a-f are the boundary values of f E M(r)(SNi;A) for x0

0+ and xo

+

0-

V(D;A) is the module of A-valued Coo-functions with compact support in

~

(see

+

respectively (see 25.5) D is the generalized Cauchy-Riemann operator (see 8.3)

B

is the conjugate of D (see 8.3)

OJ, Di : see 19.1

4. 1)

stands for the gradient operator (Clx , ••• ,a ) o xm 6m+ 1 is the Laplace operator in Rm+ 1 6;+ 1 is the Laplace-Beltrami operator in Rm+ 1 eh

h

is a basic element of the Clifford algebra A (see 1.3)

1• • • r

eA

is the conjugate of eA (see 1.10)

E(x)

is the fundamental solution of the operator D (see 8.8)

E(t,x) and E(t,x)

see 15.4 see 15.5

Exp(R 1 ,R 2 )

see 32.11.1

Exp{R1oR 2

see 32. 12. 1

E(D;A)

)

is the module of A-valued Coo-functions in D (see 6.1)

'\.,

f

denotes the adjoint of the function f (see 13.12)

F denotes the Fourier transform '\.,

'\.,

r, r* and r, r* are the spherical Cauchy-Riemann operators and their adjoints (see 8.7) H(u,y) : see 15.8.3 Harm(D;A) is the module of A-valued harmonic functions in D

Hk(Sm;R) is the space of all real-valued spherical harmonics of order k (see 11.1.1) Kk,m+ 1 ,y(x) : see 11.4.2 (Kj)jEN denotes the special compact exhaustion [\] 0

of~

(see 18.10)

denotes the e0 -component of the Clifford number\ (see 1.11)

[\]A denotes the eA-component of\ E A (see 1.11) (\,~) 0

denotes the inner product on the Clifford algebra A (see 1.11)

l\1 0 denotes the Clifford norm of\ E A (see 1.11) LI 0 c(~;A)

is the module of A-valued locally integrable functions in ~ (see

4.5) Lk,m+ 1(x) : see 11.1.2 L denotes the Laplace transform +

~+

A(R), A(R) : see 33.2 M(m,k)

is the number of coefficients in a homogeneous polynomial of degree

k in m variables M(r)(~;A)

is the right module of left monogenic functions

~

M(r)(~;A)

is the submodule of

functions

in~

M(r)(~;A)

in~

(see 16.1)

consisting of the left monogenic

which are regular at infinity with respect toE (see 17.1)

M(r)(Sm;A) : see 28.5 M(r)(Rl,R 2 ) , M(r)(R 1 ,R 2 ) ML 2 (r)(~;A)

:

see 32.2, 32.3, 32.5

is the submodule of

which are square integrable in ML 2(r)(aBR;A)

see 24.13

M(r)V*(~;A)

see 27.9

M(r)S*(~;A)

see 27.9

M(r)E*(~;A)

see 27.9

M(r)(~;A)

of the left monogenic functions

~

O(~*;Ac) is the module of Ac-valued holomorphic functions in ~* (see 33.3)

n open subset in Rm+ 1

n open subset in Rm Pk(x) is an inner spherical monogenic of order k (see 11.2.1) Pkf denotes the left inner spherical monogenic of order k appearing in the Laurent series expansion off (see 11.3) P(u,z) : see 33.7 k P(r) is the module of left inner spherical monogenics of order k (see 11.2.2) P(T) is the generalized Radon transform ofT E o )(n*;Ac) (see 33.10)

0

ITk(k E l) : see 29.3 IT + , IT - : see 29.5

. . "'"* IT(R), IT(K), "'ITj(Rj), IT * (R) : see 33.2 Qk

is an outer spherical monogenic of order k (see 12.1.5)

Qkf denotes the left outer spherical monogenic of order k appearing in the Laurent series expansion off (see 12.1) k Q(r)

is the module of left outer spherical monogenics of order k (see

12.1.5) Rm+1 Rm+1 Rm+1 Rm+1 : see 20.2 +'-'=''/:

R(r)(S) is a right module of left meromorphic functions with poles in S (see 18.6)

R(r)(L) : see 18.11 sf denotes the spherical transform of the function f (see 13.9) Sm

is the (m+1)-dimensional unit sphere

Si

is a reflection operator (see 30.7)

Sk(x) denotes a spherical harmonic of order k (see 11.1.1) S(u,t)

denotes the Szeg6-kernel, i.e. the reproducing kernel of the Hilbert

module ML 2 (r)(aBR;A) (see 24.17, 29.6.4) S(R 1 ,R 2 ), S(R 1 ,R2 )

:

see 32.4, 32.6

S(Rr;A)

is the module of rapidly decreasing A-valued C -functions in Rr (see C'•

5. 1 ) S(st;A) :

a(T)

denotes the generalized Fourier-Borel transform ofT (see 32.16)

1(A,~) 1

eA

T

see 5.5

denotes the trace of an H*-algebra (see 1. 12, 1.14)

: see 1• 14 eo denotes the Cauchy-transform ofT E ,

1

E0 )(Rm;A)

(see 27.6) or of

T E a( 1 )(Sm;A) (see 28.6) vn,s(O v1

~

s

~

n) is a quadratic vector space (see 1.2)

k 1 (x), (1 1 , .•• ,1 k) E {1, •.• ,m} , are the basic polynomials in the

1 ... k

Taylor series expansion (see 11.2.2) W11 .. .lk(x), (l 1 , ... ,lk) E {1, ••. ,m}k, form a basis for Q~r) (see 12.1.6) X(fJg

* X(l)

denotes the algebraic dual of the left A-module X denotes the dual module of the left A-module X, i.e. the right module

of bounded left A-linear functionals on X Z(r)(m;A) : see 30.12 (f,g) = JH f(x)g(x)

d~ is the inner product on L 2 ,(r)(H;A;~) (see 7.14)

lim J fgdS is the inner product on ML 2(r)(aBR;A) (see 24.13) r(R aBr 1 (f,g) m J m f(w)g(w)dS is the inner product on a(r)(Sm;A) (see 28.10) S wm+1 S w and on L2(r)(sm;A) (see 29.2) (f,g)R

is the area of the (m+1)-dimensional unit sphere Sm ·

e denotes the Cauchy-Kowalewski product (see

14.11)

Preface

Tne aim of this book is twofold: on the one hand to present a function theory in higher dimension generalizing the classical theory of holomorphic functions of one complex variahle and refining the theory of harmonic functions; on the other hand to show how this function theory may be applied in such fields as the theory of distributions, transform analysis in Euclidean space and harmonic analysis on the unit sphere. The origin of these notes lies in research started some ten years ago and which, at the very beginning, was essentially focused on the function theory itself. A systematic use of functional analysis techniques has made this research grow to its present form. Nowadays it is oriented towards applications in micro-local analysis and electromaqnetism and in the study of generalized Cauchy-Riemann operators on manifolds. The reader is assumed to have some background in real analysis, distribution theory and functional analysis. Nevertheless, in order to make these notes as self-contained as possible, we have gathered all prerequisites needed in the first chapter. In the first two sections the basic material on modules over a Clifford algebra may be found. Combined with classical results from functional analysis, they yield the necessary information concerning the function and distribution modules used throughout the book. Chapter 2 is devoted to the function theoretic results: besides the fundamental theorems such as Cauchy's Theorem and Formula, Taylor and Laurent series expansion. Liouville's and Morera's Theorem, etc., a product between monogenic functions is defined using a monooe~ir. version of the Cauchy-Kowalewski Theorem. This product enables us to define some special kernels which are revealed to be of b~~ic imoortance in transform analysis. In Chapter J, Runge's Theorem is proved leading up to the solution of the non-homogeneous Cauchy-Riemann equations in respectively the E-, V'- and S'context, and to Mittag-Leff&er·s Jheorem. Moreover duality theory is worked out, while the classical H2 -and HL 2 -spaces with their respective Szega and Bergmann kernels are also generalized.

Chapter 4 deals with the problem of representing several types of distributions as boundary values of m~nogenic functions. Furthermore, analytic functions on the unit sphere are studied, yielding a decomposition of spherical harmonics into so-called spherical monogenics, which in its turn gives rise to a ret1nement of harmonic analysis on the sphere. Finally the Bergmann and SzegB kernels are calculated explicitly in the case of the unit ball and sphere. Chapter b is devoted to transform analysis in Euclidean space. In particular the Fourier, Laplace, Fourier-Borel and Radon transforms are discussed within the monogenic setting. The latter transform enables us to obtain an optimal version of the Cauchy-Kowalewski extension theorem, providing a link between Clifford analysis and complex analysis. ~acn chapter ends with comments in which we tried not only to point out similar investigations, but also to put the problems dealt with in a broader perspective. More particularly the impact of techniques from partial differential equations has been stressed and the problems treated have been compared with their analogues in the theory of several complex variables. Of course we do not claim any completeness, either for the comments themselves or their listed bibliographies. It is hoped that the diversity of the problems tackled in this book illustrates the possibilities offered by the function theory considered, so that it might be a source of inspiration to both mathematicians and theoretical physicists. The first author owes very much to the late F. Wuytack, who suggested the subject at the end of the sixties and to the late H. Behnke and A. Dinghas whose encouragements were very stimulating. He also wishes to express his thanks to H.G. Garnir and his collaborators, who initiated him in the theory of localLY convex spaces and distributions_. Thanks are also due to H.G. Tillmann, R. Meise, D. Vogt and J. Schmets for the interesting discussions we had on the theory of the representation of distribut1ons oy noaumorphic functions. The third author wishes to thank M. Morimoto who invited him to come and work at his Seminar, and the Department of Mathematics of Sophia University for its hospital1ty during his stay. The second author wishes to express his thanks to F. Kuliasko for his continuous interest and encouragements. We are also very indebted to the National Fund for Scientific Research,

Belgium (Nationaal Fonds voor Wetenschappelijk Onderzoek): it offered the third author a grant as a research assistant during the last four years, and its continuous financial support moreover enables us to have fruitful contacts with foreign research groups. We also thanK our home institution, the State University of Ghent (Rijksuniversiteit-Gent), for the material support and working conditions we enjoy. Furthermore we want to thank R.P. Gilbert for suggesting we write the book and K. Habetha for encouraging this project. It is also a pleasure to thank Christiane Lamproye and Caroline Verstraeten for typing the first draft and Terri Moss for typing the final draft of the manuscript, Roland Maerivoet for his practical assistance and F. Kuliasko for his constant support. Finally, our thanks go to Pitman Books for the realization of this monograph.

Gent, February 1982

R. Delanghe, F.F. Brackx, F. Sommen

1 Clifford algebras and modules

In this first introductory chapter the reader is acquainted with the basic tools handled throughout this book. Section 1 deals with the real, associative, but non-commutative algebra of Clifford numbers. Next to the general universal Cliffo~d algebra C(Vn,s) over a real quadratic vector space vn,s' special attention is paid to the specific Clifford algebra A= ClVn,o> used in the sequel of this book. Section 2 describes the general properties of modules over the Clifford algebra A and establishes some relationships between the A-linear and reallinear functinnals on such modules. In particular A-modules provided with a proper system of seminorms and their duals are dealt with. The remainder of the chapter is devoted to the study of some specific function A-modules and their respective duals, as there are the module of the continuous functions on a compact and on an open subset of Rr and A-measures (§3), the moaule of C -testtunctions with compact support and A-distributions (§4), the module of testtunctions of slow growth and tempered A-distributions (§5) and the module of C -functions and A-distributions with compact support (§6). The chapter closes with a section on Hilbert A-modules, including the example of the Hilbert A-module of square integrable functions. 00

00

1.

Clifford numbers

1.1 In this first section a brief introduction is given to the notion of Clifford algebra, presenting all the necessary material which will be needed in the course of reading this book. The way the universal Clifford algebra is defined here is rather constructive, leading to a geometrical interpretation of the Clifford numbers, as contrasted with other possible approaches which are mostly formal algebra. We treat of the inner and outer product of vectors, and of the even subalgebra, including the examples of the DiracPauli-, quaternion- and complex algebras. Special attention is paid to the specific Clifford algebra A which is used throughout the whole of the book. Provided with an inner product and the corresponding norm it turns out to be a finite dimensional (reaU H*-algebra.

A.

T~e

universal Clifford algebra : generalities ~

1.2 Let vn,s(O basis

s

~

n) be ann-dimensional (n

~

1) real linear space with

and provided with a bilinear form (v lw),

v,w E Vn,s

such that (e.le.)

I j

0

J

1

(ei lei)

1 ' ••• 's

(eilei)

s

-1

+

1, ••. ,n.

n

If v

I

viei E vn,s then the associated quadratic form reads

i=1 s (vlv)

I

2 Vi

i =1 1.3

n

- I

2 Vi.

i=S+1

Consider the 2n-dimensional real linear space C(V n, s) with basis {eA = eh

< ••• < h

h

1• • • r

where N stands for the set {1, ••• ,n}and where Now a product on C(V n, s) is defined by

e~

r

;;; n}

is written as e 0 •

( 1. 1)

where S stands for the set {1, ••• ,s}, n(A) p(A,B)

L

#A,

p(A,j), p(A,j) = # {i E A : i

>

j}

jEB and the sets A,B and A~B are ordered in the prescribed way. It follows at once from the multiplication rule (1.1) that e0 is the identity element and that in particular 2

1 ' ••• 's -1

s + 1, ••• ,n

-e .e., J

while for 1

~

h1

-F j

1

< •••

<

hr

~

n,

and for any vectors v, w E vn,s vw

+

wv = 2(vlw).

It is an easy matter to check that in this way C(V n, s> is turned into a linear, associative, but non-commutative algebra over m. It is called the universal Clifford algebra over vn,s· 1.4 The linear subspace of the universal Clifford algebra C(Vn s) spanned by the(~) products eA, #A= p, is denoted by CP. The elements of'cp are called p-vectors. In particular we have that C0 is the 1-dimensional subspace of a-vectors or real scalars with basis {e 0 }, c 1 is the n-dimensional subspace of 1-vectors or the vectors of vn,s with basis {e 1,e 2 , ••• ,en}; Cn is the 1dimensional subspace of n-vectors or pseudo-scalars with basis {e 1• • • n }• From the geometrical point of view each p-vector can be interpreted as a piece of an oriented p-dimensional subspace of the underlying space Vn,s with a certain magnitude. This interpretation is the more clear if we take a closer look at the product of two 1-vectors, which can be split up into a symmetric and an anti-symmetric part: vw = ~(vw

wv)

+

+

~(vw - wv)

or vw = (vlw)

+

v

A

w.

The 'inner product' (vlw) coincides with the bilinear form on Vn,s ; it is symmetric and represents a scalar; it vanishes whenever the two vectors are orthogonal with respect to the given bilinear form on Vn, s" Thus the 'inner product' is related to the projection of one vector on the other. 3

The 'outer product' v

A

w = 21(vw - wv)

is anti-symmetric and is a 2-vector; it vanishes whenever the vectors are collinear. It can be interpreted as the oriented surface area of the parallelogram constructed on the two vectors. Notice that for the basic vectors

and

1.5 The 'outer product' of a 1-vector and a 2-vector is by definition the 3-vector given by 1 UA(VAW) = ~(U(VAW)

+

(VAW)U).

The 'outer product' of a 1-vector u and a p-vector v(p) is the {p+1)-vector:

and, more generally, the 'outer product' of a p-vector and a q-vector is the (p+q)-vector

The 'outer product' is associative and moreover

for any p-vector v(p) and any q-vector w(q). The 'outer product' of a number of 1-vectors is zero if and only if those 1-vectors are linearly dependent. 1.6 The 'inner product' of a 1-vector and a p-vector is given by

while for a p- and a q-vector we have 4

(V 1A•.• AVplw 1A••• AWQ) = (V 1A••• AVp_ 1 (VpiW 1A••• AWQ)). so notice that the 'inner product' of a p-anda q-vector is a lp-ql -vector. Moreover

Notice also that

which means that the product in the Clifford algebra of a 1-vector and a pvector is the sum of a (p-1)-vector and a (p+1)-vector. 1.7 The universal Clifford algebra C(Vn,s) can be split up into two 2n- 1dimensional subspaces containing respectively the even and odd multi-vectors:

p even and

c

I p odd

Moreover C+ is a subalgebra of C(Vn,s), called the even subalgebra, which is isomorphic to the universal Clifford algebra C(Vn- 1,s') for some value of 0 ::;; s' :;; n-1. This concept of the even subalgebra of a Clifford algebra can beautifully be illustrated by the following examples. Consider the Dirac-algebra which is the 24-dimensional universal Clifford algebra over v4 , 1• If {y 1,y 2 ,y 3 ,y4} is a basis of v4 , 1 then the product in C(v 4 , 1) is governed by the rules yiyj + yjyi

= 0,

i Fj

y~ = 1 2 y2

2

2

= y3 = y4 = - 1•

A basis for the 23-dimensional even subalgebra of C(V 4 , 1) is given by

5

or, with a new but straightforward notation, {1,o 1 ,o 2 ,o 3 ,o 1o2 ,o3o1,o 2o3 ,o 1o2o3 }. Clearly this even subalgebra of C(V 4 , 1) is isomorphic to the 23-dimensional Pauli-algebra, which is the universal Clifford algebra over v3 , 3 and where the multiplication obeys f: j '

A basis for the 22-dimensional even subalgebra of C(v 3 , 3 ) is given by

or, with a new but straightforward notation, {1,i,j,ij}. Clearly this even subalgebra of the Pauli-algebra is isomorphic to the 22dimensional algebra of real quaternions ~.which is the universal Clifford algebra over v2 , 0 and where the multiplication rules are {

ij

+

ji

=

0

.2 =J.2 = -1. 1

A basis for the 2-dimensional even subalgebra of C(V 2 , 0 ) is given by {1,ij}. Obviously this even subalgebra is isomorphic to the algebra of complex number-s [,which is the universal Clifford algebra over v1 , 0• 1.8 The above decomposition into the even and odd subspace for the complex numbers coincides with the decomposition into the real and purely imaginary numbers. This inspires an involution on C(V n, s) which can be looked upon as the analogue of the complex conjugate and is 'invariant' in this sense that ij is independent of any particular basis in C(Vn,s). This main involution, also called inversion, is given by 6

for any p-vector v(p) E Cp.

In particular we have for any basic vector:

or

where n(A)

= # A.

1.9 Next we define the main anti-automorphism in the Clifford algebra; it is called PeVePsion or hePmitian conjugation and given by (eh

h )

1""" r

t

= eh

h r··· 1

or

B.

The Clifford algebra A

1.10 In the sequel of this book the Clifford algebra under consideration will be the universal Clifford-algebra C(V n, 0 ), which will be denoted by A for short. Its subspace CP will be denoted by AP. Recall that in A the multiplication is given by #

j

= 1, .•. ,n. We shall constantly use the following involution, which is a combination of the inversion and the reversion introduced above. It is defined for a basic vector by e

h1 ••• hr

(eh

h )*t = (-1)r(r+1)/2 eh h 1""" r 1""" r

or

7

and for any Clifford number A

I

eAAA by

A \ =

I

eAAA.

A Notice that for any p-vector v(p) E A p v(p) = v(p)

whenever

p = 0,3(mod 4)

while v(p) = -v(p) whenever

p

1,2(mod 4).

Notice also that A,~

1.11

Let A=

I

E A.

eAAA be a Clifford number.

The coefficient AA of the eA-

A

component will also be denoted by

In particular the coefficient A0 of the e0 -component will be denoted by

it is called the scalar part of the Clifford number A. is defined by putting for any A,~ E A

Notice that

The corresponding norm on A reads

8

An inner product on A

IAI0 =

I( A• A) 0

= 2n 12 /[A XJ

(L

= 2n/2

0

A~)1/2.

A

In this way A is a real Hilbert space and at the same time it is a Banach algebra with

It should be noticed that for all A E PN

and

Furthermore for the special Clifford numbers n ..... A = A0 e 0 + L Aiei E A0 ~ A1 i =1 holds - ..... ..... n .......... A A = A A = L A~1 i=O and

while n

1!1 02 = 2n L A~1 i=O

and r-1 = 2n

r-

for

X f. 0.

lXI~ 9

1.12 In A : A is an inner A,ll,V E

[1] Ambrose gave the following definition of a (complex) H*-algebra a (complex) Banach-algebra, the norm I ·I* of which is derived from )* on A, with an involution A ~X such that for all product ( ' A:

We know from [2] that a proper H*-algebra is a trace algebra where the trace -r is defined by

and satisfies A,ll E A. This trace is not singular; indeed, define the real linear functional -rA:A by

Then, if <-rA,ll>

= 0 for all

l1

~R

E A,

= Re IAI*2 = 0 whence A = 0. Moreover if A is supposed to be finite-dimensional, then -r is a two-sided trace on A which represents the real linear functionals on A, that is, if T is a real linear functional on A, then there exists a unique A E A such that

1.13 Proposition The Clifford algebra A is a finite-dimensional real H*algebra. Proof In view of the considerations made in 1.11 it suffices to prove that for all A,ll,V E A:

10

First we have 2n[>..~vJ 0

(A.~,v)o

2n [A.v~] 0

(A.,vii.)o. Next n 2n[~A.v] 0 = 2 [~v>..] 0

(~.Xv) 0

(vii.,X) 0

(vii.,A.)o

= (A.,vii.)o. 1.14

0

In accordance with 1.12 the function< given by

n

2

2n

[~A.]o

= 2n [A.~]o

L (- 1)n(A)(n(A)+1)/2

"A~A

A

is a non-singular trace on A satisfying >..,~

EA

antl representing the real linear functionals on A, i.e. for any fixed A. E A the function '>..:A ~R defined by

is a real linear functional on A, and conversely, for any real linear functional T on A there exists a unique >.. E A such that

In particular for >..

<'e

'~

A

= eA we obtain that

> -_ 2n( - 1',n(A)(n(A)+1)/2 ~A· 11

In the special case where A = ~ we have n

2 JJo· Moreover

1.15 In a classical way a matrix representation of the Clifford algebra A may be obtained as follows. Order the basic elements eA, A E PN, of A in a certain way: n

{e(K) : K = 1,2, ••• ,2} and associate to each A E A the real (2n x 2n)-matrix 8(A), the entries of which are given by n

8(A)K,L = [Ae(K)](L)' K,L = 1, ••• ,2 , where [JJ](L) denotes the coefficient of the e(L)-component of JJ. This representation is an isomorphism between A and ~2n x2n. Notice that e(e 0 ) is the identity matrix I and that

If v(p) denotes a p-vector then the associated matrix e(v(p)) is symmetric when p = 0,3(mod 4), while the matrix e(v(p)) is skew-symmetric if p = 1,2 (mod 4). In this way it turns out that whatever p may be (0 :S p ~ n), e(v(p))

= (e(v(p)))T

T 2n X 2n where a denotes the transpose of a E ~ • Hence for any A E A

12

z. Modules over Clifford algebras: general properties z. 1 The aim of this section is to give some general information

concerning a class of topological modules over Clifford algebras and their duals. As will be seen, nice relationships between real linear and A-linear functionals on such modules may be established. They turn out to be of basic importance when function and distribution modules will be introduced in the following sections. In the sequel definitions and properties will be stated for left A-modules and their duals, the passage to the case of right A-modules being straightforward. In order to avoid any confusion, we use systematically the subscripts '(1)' or '(r)' to indicate that the module under consideration is a left or a right A-module. In the case of a bi-A-module, no subscript is used, although when it is e.g. considered as a left A-module, the subscript '(1)' is then added. 2.2 Let X(l) be a unitary left A-module, i.e. X(l)'+ is an abelian group and a law (A,f) + Af from Ax X into X(l) is defined such that for all A,~ E A andf,gEX(l)

(i)

(A+

~)f

Af +

~f

(A~)f

A(~f)

(iii)

A(f+g)

Af + Ag

( i v)

e0 f = f

(ii)

.

Notice that X(l) becomes a real vector space if m is identified with me 0

A0

c

=

A.

Moreover when speaking of a submodule E( l) of the unitary left A-module X(l) we mean that E(l) is a non empty subset of X(l) which becomes a unitary left A-module too when restricting the module operations of X(l) to E(l)• l) and Y( l) are unitary left A-modules then a function T:X( l) +Y( l) is said to be a left A-linear operator if for all f,g EX( l) and A E A

2.3

If X(

T(H+g)

H(f)

+

T(g).

The set of all left A-linear operators from X( l) into Y( l) is denoted by 13

Notice that, by defining for each T E L(X(l)'y(l)) and~ E A, (T~)(f) = T(f)~. f EX( l)'L(X( l)'Y( l)) is turned into a unitary right A-module. In the special situation where Y(l) coincides with A, the laiter being considered as a unitary left A-module over itself, L(X(l)'A) is called the algebraic dual of X(l) and it is denoted by X(~1g· Its elements are called left A-linear functionals on X(l) and for any T E x(~1g and f E X(l) we denote by the value ofT at f. By previous considerations X(~1g is a unitary right A-module. Finally observe that if X is a unitary bi-A-module then for each T E x(~1g and ~ E A we can introduce, next toT~ already defined, the left A-linear functional ~T by putting L(X(l)'Y(l)).

<~T,f>

= . f EX.

Similarly, under the same assumption upon X, we have for each S E x*alg and ( r) ~ E A, the right A-linear functionals ~Sand S~ given by <~S,f>

= ~<S,f>, f E X,

<S~,f>

=

and S,~f>,

f E X.

2.4 Let X(l) be a unitary left A-module. Then a family P of functions p : X(l) ~R is said to be a proper system of semi-norms on X(l) if the following conditions are fulfilled: [P 1J

There exists a constant C0

~

1 such that for all p E P, A E A and

f,g E x(l) (i)

(ii)

p(f+g) s p(f) + p(g) p(Af) s C0 IA1 0 p(f), and p(Af)

IAip(f) if A € R.

[P 2J For any finite number p1 ,p 2 , ••. ,pk E P, there exist p such that for all f E X(l) sup j=1, .•• ,k

14

p.;(f) s C (f). ..J

p

€

P and C > 0

[P 3]

= 0 for all p E P then f = 0.

If p(f)

Notice that when considering X(l) as a real vector space, (X(l)'P) turns out to be a (real) locally convex Hausdorff space. Obviously a lot of definitions and properties from the general theory of locally convex spaces may be carried over without any restriction to the class of topological modules over a Clifford algebra which was just introduced. We shall restrict ourselves here to mention those notions which will be currently used throughout the book. For more details concerning the classical situation the reader is referred to e.g. [4]. 2.5 Let P and Q be two proper systems of seminorms on X(l)" Then Pis said to be weaker than Q (or Q is stronger than P), notation P < Q, if for each pEP there exist q E Q and C > 0 such that for all f E x( 1) p(f )

~

Cq ( f) .

Moreover P and Q are called equivalent, notation P ~ Q, if P

<

Q and Q < P.

2.6 If X(l) and Y(l) are provided with the proper system of seminorms P and Q respectively then T E L(X(l)'Y(l)) is said to be bounded if for each q E Q there exist p E P and C > 0 such that for all f E X(l) q(Tf)

~

Cp(f).

The set of bounded left A-linear operators from L(X(l)'Y(l); it clearly constitutes a submodule In the special case where Y = A an element T there exist pEP and C > 0 such that for all f Il 0

~

X(l) into Y(l) is denoted by of L(X(l)'y(l)). E X(~Jg is called bounded if E X(l)

Cp(f).

The submodule of X{~~g which is obtained in this way is denoted by x( 1) and it is ca 11 ed the dual of X( 1). Furthermore if (X(l)'P) is looked upon as~ real locally convex space then its real dual, i.e. the set of bounded (real) linear functionals on x(l)' is denoted by X(l)" 2.7 Now one may define proper systems of seminorms on L(X( l)'Y(l)) and X{ l) by considering special families F of bounded subsets of (X(l)'P). 15

Let B c X(l) be bounded, i.e. for each pEP there exists Cp sup p(f) fEB

~

0 such that

Cp;

then, for each q E Q, Pq,B p B(T) q,

>

L(X(l)'Y( l)) ~R, given by

= sup q(Tf), fEB

defines a seminorm on L(X(l)'Y( 1)). Now, if F is a family of bounded subsets of X( l) satisfying (i)

(ii)

u B = x(l) BEF if B1, B2 E F, then there exist C > 0 and BE F such that B1 u B2 c CB,

then PQ,F

= {Pq,B : q E Q, BE F} determines a proper system of seminorms on

L(X( l)'y( 1)).

In the special situation where Y = A we put for each B E F and T E pB(T)

x(1 )

= sup Il 0 • fEB

In this case we pay special attention to the families Fs and Fb consisting of either all finite or all bounded subsets of X. If X(l) is endowed with the corresponding proper system of seminorms we denote it respectively by X(l)s and XCl)b and we say it carries the weak or strong topology respectively. Of course, if X(l) is considered as a (real) locally convex space, then X(l)s and X(l)b stand for the classical dual spaces. 2.8 In what follows we shall establish some basic relationships between Alinear and real linear functionals on an A-module.

Proposition Let X(l) be a unitary left A-module and letT be a left A-linear functional on X. Then for each A E PN, T T is a real linear functional on eA X( l) satisfying (Te T,f) = (Te T, eAf) , A o Moreover for all f E X(l)' 16

f E X(l)"

= 2-n

I A

eA << T,f> eA

conversely, letT be a real linear functional on X(l)" Then for each A E PN there exists a unique left A-linear functional TA on X(l) such that 'eTA= T. This functional TA is given by A

I

= 2-n

eB .

B

Finally T = 0 if and only if T T = 0 for some A E PN. eA Proof (i) If T E x*alg then clearly 'e Tis a real linear functional on X(l) for (1 ) A which <<e T,f> = <<e , > A A

= <'e/• eAf>

,

f E x(l )"

Next for any f E x( 1 ) we have <<e T,f> = <<e , > A A = 2n(-l)n(A)(n(A)+1)/2A and hence = I eA[JA A

= 2-n

I

eA(-1)n(A)(n(A)+1)/2 A A 17

2-n

I

eA (Te T,f>

A

2-n

I

A

eA
A

(ii) Conversely letT be a real linear functional on X(l )" A E PN. Then the functional tf,A : A~ ffi determined by

Fix f E X(l) and

,.\ E A is real linear on A. Hence there ought to exist a unique wf,A E A such that for all,.\ E A (see 1.14)

so that, putting ,.\

= eA,

~

Now define TA : X(l)

we obtain

A by f E X( 1 ) ;

= wf,A'

then one may easily check that TA E X*alg ( 1 ) ' while Te TA = T. A

It still remains to prove that = 2-n

I

eB ,

f E x(l)"

B

Call the expression at the right hand side vf,A; then in view of the above considerations it suffices to show that for all f E X(l)'

vf,A = lJf,A' or equivalently
vf,A

,,.\)


wf,A

,>.>,

,.\ E A.

Now for any A. E A, (T

vf ,A

,>..> = T(A.vf A) •

L eB

[A.

Jo

B

I >-c c



:. (T

llf,A

,A).

(iii) If T = 0 then obviously 'e T = 0 for each A E PN. Conversely suppose that for some A E PN, 'e T = 0. AThe unique left A-linear functional TA on X(l) such that, TA =,AT is apparently T. Hence by (ii) eA eA = = 2-n

I

eB = 0, v f E x(l )'

B

which means that T = 0.

c

2.9 Proposition Let X(l) be a unitary left A-module provided with a proper system of seminorms P. Then T E x*alg is bounded if and only if , T is (1) eA bounded for some A E PN. Proof

If T is bounded then for any A E PN and any f E X(l )' l<'e T,f>i A

n

2

I. AI

2n/2 iAio

< 2n/2 ilo and so 'e T is bounded. A

19

Conversely suppose that for some A E PN, , T is bounded, i.e. there exist e C > 0 and pEP such that for all f E X(l)' A [<<e T,f> [ : ; C p(f). A

By Proposition 2.9 we have 2 [[ 0

- 2 = 2-2n ,L (<<eAT,eAeBf>) B

: ; c2.2-2n

I

(p(eAeBf))2

B

: ; 2- 2n c 2 c~!eAI ~(p(f)) 2 Lie 8 ! ~ B

Hence, there exist c* Il 0

::;;

* )' whence T E X(l 2.10

>

0 and p E P such that for all f E X(l )'

c* p(f) c

Now we state a Hahn-Banach type theorem.

Theorem Let X(l) be a unitary left A-module provided with a proper system of seminorms P, let Y(l) be a submodule of X(l) and letT be a left A-linear functional on Y(l) such that for some C > 0 and p E P I[ 0 ::;;c p(g) for all g E Y(l)' Then there exists a left A-linear functional T* on X(l) such that (i) (ii)

T*IY(l)

= T;

for some c* > 0, Ilo::;; c*p(f),

f E X(l)'

Proof As for any g E Y(l) l<<e T,g>l : ; [e 0 10 i[ 0

::;;

C[e 0 [ 0 p(g),

0

there ought to exist, in view of the classical Hahn-Banach theorem for real 20

locally convex spaces, a bounded real linear functional Ton X(l) such that TIY(l) = Te T and Il ~ Cle 0 10 p(f) for all f E x(l )" 0

By means of Proposition 2.9 and its proof, there exists T* E x(~~g such that Te T* =Ton X and Il 0 ~ C*p(f) for all f E X(l) with C* C0 Cie 0 I0 • 0

As T IY( 1) 1

eo

(T-T*)

Te 0 T we have again by virtue of Proposition 2.9 that on Y(l )'

o so

that T = T*IY(l)'

c

2.11 Corollary Let X(l) be a unitary left A-module provided with a proper system of seminorms P and let Y(l) be a submodule of X(l)' Then Y(l) is dense in X(l) if and only if for each T E x( 1) such that TIY(l) = 0 we have T = 0 on X( 1). 2.12 As will be seen in the next sections most of the A-modules of test functions which will be used currently, fit into the following framework. Suppose that X is a unitary bi-A-module and that Y is a real vector space such that, considered as rea 1 vector spaces, X = IT Y. AEPN Furthermore assume that each f E X may be written as f = I fAeA with fA E Y A

for all A E PN, the subsets A of N being ordered in a certain way. Finally suppose that X(l) andY are provided respectively with the systems of seminorms P and p* such that the locally convex topology on X induced by P is equivalent to the ::>roduct topology defined on X = IT Y. AEPN Then the following proposition characterizes completely the dual x( 1 ) of X( 1).

2.13 Proposition LetT be a bounded left A-linear functional on X(l)' Then for each BE PN there exists a unique bounded real linear functional r 8 on Y such that for any~= L ~AeA E X(l)' A

=

L

eAeB.

A,B Conversely if (T8 )BEPN is a sequence of bounded real linear functionals on Y then T : X(l) ~A defined by

21

= I

eAeB' ~=I ~AeA E X(l)'

A,B A sa bounded left A-linear functional on X(l)' roof

Given

any~=

I

eA~A E

X we have, using Proposition 2.8,

A

= A

=I eA A

2-n

~here

I eAeB <•e T,~A> A,B B

I eAeB A,B we have put for each B E PN, = 2-n <,_ T, -I-> , -I- E· Y. eB

We know from Proposition 2.9 that 'e T is a bounded real linear functional on X(l) and hence on IT Y and finally ~n Y itself. So r 8 is a bounded real AEPN linear functional on Y; its uniqueness follows from its construction. The converse is easily checked by computation. c 2.14 We close this section by considering inductive limits of topological A-modules. Let ((xi 1},Pj))jeN be a sequence of unitary left A-modules xt 1 ) each of which is provided with a proper system of seminorms Pj such that (i)

(ii)

for all j E IN,

xt

l) is a submodule of

xt;~;

the canonical injection Ij : X~)+ xt;~ is bounded.

Then on X(l) = _u xtl)' which is clearly a unitary left A-module, a filter.1eN ing set of seminorms Pind is determined in the following way. Let n = {p 1 ,p 2 , ••• ,pJ., ••• ) be a sequence of seminorms belonging to n P. jeN J and let y = (c 1 ,c 2 , ••• ,cj'''') be a sequence of positive real numbers. Furthe~ more let an arbitrary decomposition off E X(l) be written as f = ~ f., (J)

22

J

j

fj E X(l)" p

n,y

Then we put (f) =

f

I

inf = \f. L J

cJ.pJ.(fJ.).

(j)

( j)

One may easily check that P1- nd = {P

n,y

: n E 11

jEl-l

P., J

y

E 11 lR+ } jEl-l

is a filtering set of seminorms on X(l)" The topological left A-module (X(l)' ~ind) obtained in this way is called the inductive limit of the sequence ((X~l)'Pj))jSN and it is also denoted by x(l) = limjSNind xil)" * j Of course one has that T E X(l) if and only if Tj = TJX(l) E xj*(l) for each j E IN.

3.

The spaces C0 (K;A) and

c0 (n;A);

A-measures

3.1 Let K be a compact subset of lRr (r ~ 1). Then C0 (K;A) stands for the unitary bi-A-module of A-valued continuous functions on K. Of course C0 (K;A) =

11 C0 (K;lR) AEPN

and each f E C0 (K;A) is endowed with the norm llfll =sup Jf(x)J 0 , f E c(l)(K;A). xEK Clearly this norm is equivalent to the one defined in a classical way on C0 (K;A) = 11 C0 (K;lR). AEPN 3.2 Now ·;et r2 c lRr be open and let (~B)BEPN be a sequence of real measures inn (see e.g. [5]). Then for any semi-interval I contained in n,~(l) is defined by ~(I)

L

~B(I)eB

BEPN and the

function~

thus obtained is called an A-valueJ measure inn. 23

Notice that the support of supp

~

= u

~.

notation supp

~.

is given by

~8 •

BEPN If N(n;A) stands for the set of A-valued measures in n, then clearly M(n;A) is a unitary bi-A-module. An A-valued function f = L fAeA is said to A

be ~-integrable inn if for all A E PN and BE PN,fA E L 1 (n;~ 8 ) where L 1 (n;~ 8 ) denotes the classical space of R-valued ~ 8 -integrable functions. The set of all A-valued ~-integrable functions inn is denoted by L 1 (n;A;~). It clearly constitutes a unitary bi-A-module. In the particular case where ~ 0 is the Lebesgue measure in n and ~B is the zero measure in n for each B F ~. we write L1 (n;A). For any f E L 1 (n;A;~) we put

I f(x)d~ = L n

eAeB

A,B

I fA(x)d~B Q

and

I d~f(x) = L Q

B,A

e8eA

I fA(x)d~ 8 • n

Using a classical result and Proposi~ion 2.9 the dual C0 ( 1 )(K;A) of c( 1)(K;A) may be described in terms of c0 (K;R).

3.3

Theorem: (Riesz representation theorem) LetT E C 0 ~ 1 )(K;A). Then there exists a unique A-valued measure ~ in Rr with support contained in K such that for all f E c( 1 )(K;A)

= IK f(x)d~.

3.4 Now let C0 (n;A) denote the unitary bi-A-module of A-valued continuous functions in n. Obviously

c0 (n;A) = rr

C0 (n;R)

AEPN

and each f E C0 (n;A) can be written as f

c( 1)(n;A) K~ 24

= L fAeA, fA E C0 (n;R). The module A

is equipped with the following proper system of seminorms P. n be compact and put

Let

pK(f) = sup If (X ) I0. xEK

f E

c0 ( Q; A) •

Then P = {pK : K c n compact} induces the topology of uniform compact convergence on c( 1)(n;A). If (Ks : s = 1,2, ••• ) is the compact exhaustion of n determined by Ks = {x E Q: lxl :;; sand d(x,ll{~) ~ then Pis equivalent toP*= {pK

s

-}l,

s

1,2, ••• ,

: s = 1,2, ••• }.

Moreover P is also equivalent to the system of seminorms defined in a classical way on C0 (n;A) = rr C0 (n;JR). LetT E Ca*(l )(Q;A); then there exist AEPN s ~ 1 and C > 0 such that Il 0 < CpKs(f), f E

c( 1)(n;A).

As c( 1 )(n;A) is a submodule of c( 1)(Ks;A), by the Hahn-Banach theorem, T admits an extension T* to c( 1)(Ks;A) and hence, in view of Theorem 3.3, an A-valued measure ~ in Rr can be found with support contained in Ks such that for all f E c( 1)(Ks;A) =

Jnf(x)d~.

which implies that for all f E = 4.

c( 1)(n;A)

Jnf(x)d~.

The space V(n;A); A-distributions

4.1 Let K and n be respectively compact and open subsets of Rr. Then V(K;A) and V(n;A) denote respectively the unitary bi-A-modules of A-valued infinitely differentiable functions in Rr with support in K and of A-valued infinitely differentiable functions in Q having aompaat support contained in n. Obviously V(K;A) V(n;A) and for

each~

IT V(K;JR), AE.PN n V(n;JR) AEPN E V(K;A) (resp.

~

E V(n;A)),

~

L 4AeA with

~A

E V(K;R)

A

25

( res p.



A E V (rl ;lR) ) •

Notice also that if

<1>

=I AeA E V(Q;A), then its A

support is given by supp¢ =

u AEPN

supp A

4.2 Taken as a left A-module V(l )(K;A) may be provided with the proper system of seminorms PK = {pk : k E ~} with

aE]N r ,

r a1 = I aj, aa =_a__ a1 ax 1 j=1

Ia!

(J,

" r " ar axr

Clearly this system is equivalent to the one defined in a classical way on V(K;A) = II V(K;JR).· Next V(l)(rl;A) may be equipped with an inductive limit AEPN topology by putting

where (Ks)sElN is a sequence of compact subsets of ( i) (ii)

r2

such that

-

~s = Ks

KS

c

KS+1 00

( i i i ) II =

U

s=1

Ks •

4.3 Obviously this locally convex topology is equivalent to the one defined in a classical way on V(S"l;A) = II V(st;JR). Hence the dual V(l )(rl;A) of AEPN V(l )(rl;A) is completely characterized by Proposition 2.13. It consists of the so-called left A-distributions in rl. 4.4 If a= (a 1, ••• ,ar) is any multi-index in ~r then for any T E V(l )(rl;A), aaT is defined by ('daT,¢> = (-1)jaj, obviously 'daT E V{l)(ll;A). 26



E V(l)(rl;A);

Notice also that, as DU1;A) is a unitary bi-A-module, for any T E v( 1 ) (Si;A) and ~ E A, ~T and T~ may be defined. For each ~ E ~r. ~n(~T) and aa(T~) are then given by

(-1)[a[ (-1)[a[ and

= <(aaT)~.¢> = ~

¢

E V(l )(Si;A).

Now call an expression of the form P(D)

[a[$k complex differential operator with constant coefficients and put for any

TE

D( l) (Si;A) P(D)T = ( L [a[-
and

L

T(P(D)) = T (

~aao.) = L

[a[<;k i.e. for all

<(

l: [a[$k

¢

[a[$~

E V( l)(Si;A) ~

a

Clo.)T ,¢> =

l:


T),¢>

[a!$k

l: [a[;;k

l: [a[;;k

(-1)[a[<~ T Cla¢> a ' (-1) [a[ a ' 27

while
L

L

lal~:k

L

, ) ,cp> a

(-1)1al

lal~k

Then P(D)T and T(P(D)}both belong to v( 1)(n;A). Obviously if T E V{r)(Q;A) then analogous definitions may be given for T(P(D)) and P(D)T. 4.5 Standard examples of A-distributions in Q are of course elements from M(n;A) and L~ 0 c(Q;A), where L~ 0 c(n;A) is the unitary bi-A-module of A-valued locally integrable functions in Q with respect to the Lebesgue measure in Q, i.e.

To each f E L~ 0 c(Q;A), a left and right A-distribution, still denoted by f, may be associated by putting

= Incp(x)f(x)dx =

L

A,B

eAeB

I

cpA(x)fB(x)dx, cp

E

v(l )(Q;A),

Q

and

= Inf(x}cp(x)dx

= L eBeA I fB(x}cpA(x)dx, cp A,B

Analogously to iated whereby

and 28

each~

E v(r)(Q;A).

Q

E ~(n;A), a left and right A-distribution may be assoc-

<~.~> = J~ d~ ~(x),

~ E V(r)(~;A).

In this way L1(~)(~;A) and M(r)(~;A) (resp. L~(~)(~;A) and M(l)(~;A)) are embedded into V(l)(~;A) (resp. V(r)(~;A)). Finally remark that for each T E V( 1 )(~;A) its support is given by supp T = u supp T T BEPN eB where for each BE PN, supp Te T stands for the support of the real distri8 bution T T. eB 4.6 Next we introduce the notions of tensor product and convolution of Adistributions. To fix the ideas, let f E L~ 0 c(Rr;A), g E L1°c(Rr' ;A). Then their tensor products f ® g and g ® f defined by f ® g(x,y)

= f(x)g(y)

g ® f(x,y)

= g(y)f(x)

and

belong to L~ 0 c(Rr+r' ;A). Hence they may be interpreted as left or right Adistributions. Notice that in general f ® g F g ®f. A straightforward compution shows that, when f and g are considered as left A-distributions, then e.g. f ® g, being considered as a left A-distribution too, satisfies for all r+r' ~ E V(l)(R ;A) ( 4. 1)

On the other hand, if all of f,g and f ® g are interpreted as right A-distrir+r' butions, then for all ~ E V(r)(R ;A)

= >.

(4.2)

The above f~rmulae (4.1) and (4.2) suggest t~at for S E V( 1 )(Rr;A) and T E V(l)(:ll{ ;A); we defineS® T E VCl)(Rr+r ;A) by <S ® T,~>

= > 29

while for S E v(r) (l!{;A) and T E V(r)(l!{' ;A), <SeT,¢> = <Sx,>, and this for all ¢belonging to respectively V(l)(lRr+r' ;A) and V(r)(lRr+r' ;A). The distributionS~ T is called the tensor product of SandT. Notice again that in general S 0 T 1 T 0 S. Now let S,T E v( 1 )(lRr;A), at least one of them having compact support. Then S*T E V(l)(lRr;A) is defined by <S*T,¢>

= <Sx

0 Ty,¢(x+y)>>

= >, and this for all ¢ E V(l )(lRr;A). Analogously, let S,T E V(r)(lRr;A), at least one of them having compact support; then S*T E V(r)(lRr;A) is given by <S*T,¢>

<Sx 0 Ty,¢(x+y)>

= <Sx, >, and this for all ¢ E V(r)(lRr;A). In both cases S*T is called the convolution of SandT. Notice that the convolution of A-distributions is not commutative and that, if e.g. S,T,U E v( 1 )(lRr;A), at least two of them having compact support, then the associativ~ law holds, while the distributive law remains valid in v*(l) . ,+,* As to the a-distribution, defined as usual by = ¢(0), ¢ E V(lRr;A) it may be considered as well as a left or as a right A-distribution. Moreover one may easily check that for all T E v( 1)(lRr;A) or T E V(r)(lRr;A) O* T = T*o = T. Furthermore, i f S,T E v( 1 )(lRr;A) are such that S*T is defined and P(D) is a hypercomplex differential operator with constant coefficients, then P(D)(S*T) = (P(D)S)*T.

* (lR r ;A) Analogously, for S,T E V(r) (S*T)(P(D)) = S*(T(P(D))).

30

Finally, for f =I fAeA and g A

I g8e 8 belonging to L1(:n{;A), define their· B

convolutions f*g and g*f by f*g(z)

J f(x)g(z-x)dx lR r

and g*f(z)

J r

g(x)f(z-x)dx

lR

g8 (x)fA(z-x)dx. I eBeAJ lRr B,A Then f*g, g*f E L 1 (1Rr;A) and in general f*g I g*f. A straightforward computation yields that if f and g are considered as left A-distributions, then their convolutions f*g and g*f, considered too as left A-distributions, coincide with the ones in distributional sense. Analogous results of course still hold when dealing with the case V(r)(lRr;A). 4.7 Taking account of Proposition 2.13 and the theory of [-valued distributions, the following fundamental theorems may be formulated. For the classical situation the reader is referred to [16]. 4.7.1 Theorem ("Principe du recollement par morceaux") Let {Sii:i E I) be a family of open subsets in lRr and let (Ti:i E I) be a family of distributions such that for each i E I, TiE v( 1 ){Sii;A) (resp. V(r)(Sii;A)). Furthermore assume that, if Si. n D. 1 ¢, then T. lSi. n Si. = T.ISi. n Si .• Then there exists 1

J

11

J

J1

J

a unique distribution T E V(l){Si;A) (resp. V(r){Si;A)) where~= i~I Sii' such that Tls-2.1 = T.1 for all i E I. 4.7.2 Theorem Let Si and V be open subsets of lRr such that V is compact and contained in r.. furthermore letT E v( 1 ){s-I;A). Then there exist a E ~rand an A-valued function f continuous in a neighbourhood of V such that T = 3af in V, i.e.

31

= (-1)iai J

aa~(x)f(x)dx, ~ E V(l)(V;A).

Rr 4.7.3 Theorem (1) Let n eRr be open and let V c n be open such that V c n is compact. Furthermore let (Ta)aE]0, 1] be an s-bounded family in v( 1 )(n;A) such that Ta s-converges to T0 if a~ 0+. Then there exist B E ~r and a family of A-valued functions (fa,fo :a E ]0,1]), all of them being continuous in a neighbourhood Wof V, such that fa converges uniformly to f 0 on Vand Ta = a8f a , To = a8f o in V. (2) If moreover (Ta>aE]0, 1] depends s-continuously upon a, then fa may be chosen in such a way that the function f given by f(a,x) = f a (x), f(O,x) = f 0 (x) is continuous in [0,1] x W. 5.

The space S(Rr,A); tempered A-distributions

5.1 Call S(Rr;A) the unitary bi-A-module of A-valued Papidly decPeasing . . Rr • Of course f unct-z-ons 1n S(Rr ;A) = IT

S(Rr ;R)

AEPN

and each~ E S(Rr;A) can be written as~

I AEPN

5.2 Considered as a left A-module S(l)(Rr;A) may be provided with the proper system of seminorms P = {pk : k E ~}where for each~ E S(l)(Rr;A)

k E

~.

a E

~

r

•

Its dual SCl)(Rr;A) consists of the so-called tempePed left A-distPibutions or left A-distPibutions of slow gPowth in Rr. 5.3 Using classical results (see [16]) and again Proposition 2.13 the following fundamental theorem may be proved. First recall that a continuous A-valued function g is said to be of slow gPowth inRr if there exist k E~ and C > 0 such that inRr, lg(x)l 0 :;; C(1+lxl 2 )k. Obviously, if g is continuous and of slow growth in Rr its associated A-distribution is tempered. 32

* r ;A). Then there exist a E ~ r and a continuous 5.4 Theorem LetT E S(l)(m A-valued function g of slow growth in mr such that T = aag in mr. 5.5 Now let n cmr be open. Then we define s(n;A) to be the set of functions ¢ E s(mr;A) such that supp ¢ c nand ¢together with all its derivatives aa¢ vanish on an, i.e.

Clearly s(n;A) is a submodule of S(mr;A) and hence s(l)(n;A) may be endowed with the topology induced by S(l)(mr;A). Its dual is denoted by S(l)(Q;A) and it is called the space of left A-distributions of slow growth inn. By the Hahn-Banach Theorem, each T E s*(l)(n;A) admits an extension " ' . * r ;A), TinE s(l)(n;A). * T E* s(l)(m r ;A) and obv1ously for every T E s(l)(m Notice t~at if n is bounded then S(l)(n;A) c V(l)(n;A) and that in this case T E S(l)(n;A) implies that T E V( 1 )(n;A). Moreover if n is bounded and an= an, then clearly s(l)(n;A) = V(l)(n;A), both algebraically and topologically, so that in this case s( 1 )(n;A) = v( 1 )(n;A). Furthermore an A-valued continuous function g is said to be of slow growth in n if it can be extended to a function h which is continuous and of slow growth in mr. Finally an A-valued function g is said to be of weak slow growth inn if there exist a constant C > 0 and k,k' E ~such that inn

I g ( x) I 0

::;:

c(1

+

I x 12 )

k

1

(

1 + ~a-n})

kI

•

Of course if g is of slow growth in n then it is of weak slow growth in n. Theorem Let Q cmr be open. (i) If T E s( 1)(n;A) then there exist a E ~rand g, continuous and of weak slow growth in n such that T = aag in n. (ii) I~ n is convex and f is continuous and of weak slow growth inn, then f E s(l)(r.;A). 5.6

* * r ;A) so that, (i) If T E s(l)(n;A) then T admits an extension "'T E S(l)(m in view of Theorem 5.4, a E ~r and an A-valued continuous function g, which '\, is of slow growth in mr, may be found such that T aag. Consequently '\, Tin = T = aagln. (ii) Without loss of generality we may assum~ that 0 En. Consider

~

33

the following diffeomorphism 8 between the open halfsphere Sr = {X E ffir+ 1 : r+1 + r (x 0 + 1 )2 + x1 + .•• + x~ = 1 and x0 > -1} and ffi= = {x E ffir+1 : x0 = 0} = ffi , determined by the projection from S~ onto ffir with centre (-1,0, .•. ,0). Then clearly the space S(~;A) is transformed into the space S(~' ;A) = V(~;A) where -1 r = {x E ffi r+1 : (x + 1) 2 + x2 +•.. +x 2 =1}. where~· = 8 ~. ~· being taken inS 1 o r Now define for each ¢EV(l)(~;A), ¢'(x') = ¢(0(x' )) and associate to f the distribution f' E D( 1 )(~' ;A) given by = . Then one may show that f' can be extended to an element of s( 1 )(~';A) and hence, using density arguments, that f can be extended to an element of s(n(~;A).

6.

o

The space

E(~;A);

A-distributions with compact support

6.1 Let~ be an open subset of ffir; then E(~;A) is the unitary bi-A-module of all A-valued C -functions in ~. Obviously 00

E(~;A)

IT AEPN

and for each ¢ E

E(~;JR)

E(~;A),

¢=

I

¢AeA with ¢A E

E(~;JR).

A

6.2 Considered as a left A-module, E(l )(~;A) may be provided with the proper system of seminorms P = {Ps,k s, k E W} where for each¢ E E(l )(~;A) sup

sup

icxi~k

XEKS

(Ks)seN being a compact exhaustion of~ and a E Wr. Its dual r( 1 )(~;A) consists of the so-called left A-dist~ibutions with compact

6.3

and

34

suppo~t

in

~.

Of course we have the inclusions

1.

Hilbert A-modules

To fix the ideas let H(r) be a unitary right A-module. Then a function (,) : H(r) x H(r) +A is said to be an inner product on H(r) if for all f,g,h E H(r) and A E A, 1.1

(i)

(f,g +h)

(f,g) + (f,h)

(ii)

(f,gA)

(f,g)A

(iii)

(f,g)

(g,f)

o

~

( i v) ( v)

<1e ,(fA,fA)>

0 and <1 ~ IAI~

eo

,(f,f)> = 0 if and only if f = 0

<1e ,(f,f)>.

0

0

Notice that from the definition itself it immediately follows that (O,f) = (f,O) = 0 and that (fA,g) = ~(f,g). MOI~eover, as for each A E A,= eo <1e , ~>. <1e ,(,)> is an inner product on H(r) the latter being considered 0

0

as real vector space.

Consequently, putting for each f E H(r)

llfll 2 = <1e ,(f,f)> 0

we have that for all f,g E H(r) l<1e ,(f,g)>l ~ llfllllgll and llf +gil~ llfll + llgll· 0

Hence 11·11 is a proper norm on H(r) turning it into a normed right A-module. 7.2 Definition Let H(r) be a unitary right A-module provided with an inner product (,). Then it is called a right Hilbert A-module if it is compiete for the norm topology derived from the inner product. 7.3 In a classical way the elements f,g E H(r) are called orthogo·wl if (f,g) = 0 while a subset {fi:i E I} of H(r) is said to be orthonormal if (fi'fj) = 0, i I j, and llfill 2 = 1, i E I. 7.4 Proposition (Bessel's inequality) Let (fi) be a finite or countable orthonormal subset of H(r)" Then for all f E H(r)

35

Proof M E lN

As for each i E lN, llfi(fi,f)ll 2

;;;

I

(fi,f)l~

we obtain that for any

M

o

l:

:sllf-

fi(fi,f)JI 2

i =1 M

= llfll 2

M

-l:

llfii 2

-L

l(fi'f)l~·

l:

l
i=1

i =1 M ;S

l:

l(fi,f)l~-

M

llfi(fi,f)ll 2

i=1

D

i =1

7.5

Corollary (Cauchy-Schwarz inequality). I (f,g)

;;;

10

For all f,g E H(r)

llfll II gil·

7.6 The following theorem generalizes the classical result concerning the dual of a Hilbert space. Theorem (Riesz representation theorem) Let H(r) be a right Hilbert A-module and let T E H(~1g· Then T is bounded if and only if there exists a (unique) element g E H(r) such that T(f) = (g,f) for all f E H(r)" Proof For each g E H(r)' the right A-linear functional Tg defined by Tg(f) (g,f), f E H(r)' is bounded. Conversely let T E HCr)" If H(r) is considered as a real Hilbert space provided with the inner product Teo(,), then Te 0T E H(r) whence there exists a unique element g E H(r) such that for all f E H(r) (Te ,T(f)) = (Te ,(f,g)) = (Te ,(g,f)). 0

0

0

Consider the right A-linear functional Tg associated with g; then Tg is bounded and Te T = Te T so that by means of Proposition 2.8, T = Tg. o 0

0

g

7.7 Let (~1) E c H(r) and call E~ = {f E H(r) : (f,g) = 0 for all gEE}. Then E~ is a closed submodule of H(r) and if moreover 0 E E, then E n E~ = {0}.

36

7.8 Proposition Let E(r) I H(r) be~a closed submodule of H(r)" exists f E H(r)'{O} such that f E E (r)"

Then there

~

Take g E H(r)'E(r)' call g0 the unique element in E(r) for which c5 = llg 0 -gll = inf llh-gll and put f = g0 -g. Then f 1 0 hEE(r) and for all A E Rand hE E(r)' g0 + Ah E E(r)" Hence, using a classical result, we find that = 0. eo Now call for hE H(r) fixed, ~ = (f,h). Then for all A E A we obtain that (T ,A) = T(~A) ~

=

T((f,h)A)

= T((f,hA)e 0 ) (Te ,(f,hA)) 0

0

and so

~

= 0 or f E E~ (r)"

c

7.9 Let E c H(r) and denote by spA(E) the right A-linear envelope of E. Then E is said to be total if c(spA(E)) = H(r) where c denotes the closure operator in H(r)" 7.10

From Proposition 7.8 we get

Corollary A subset E of H(r) is total if and only if E~ 7.11

= {0}.

Again using Proposition 7.8 we immediately get

Propositi on (i)

E(r)

(ii)

E(r)

Let E(r) and F(r) be closed submodules of H(r)" E(r) +

Then

.Ll

F(r) is a closed submodule if (f,g)

0 for all

f E E(r) and g E F(r)" 7.12

Theorem

Let E(r) be a closed submodule of H(r)"

Then H(r)

= E(r)~E~ (r)"

7.13 If E(r) is a closed submodule of H(r)' then the foregoing theorem tells us that any f E H(r) admits a unique decomposition f = g + h with g E E(r) ~

and h E E(x).

37

Calling PE

(f) = g, f E H( )' we obtain that PE is a projection operator (r) r (r) in H(r)' i.e. PE(r) is a bounded right A-linear operator in H(r) with (PE(r)f,f') = (f,PE(r)f') for all f,f' E H(r)' and P[(r)

PE(r)

Conversely

if P is a projection operator in H(r) then p

E(r) = {f E H(r) : Pf = f} is a closed submodule of H(r) with (E~r))~ = (1-P)H(r)" For any T E L(H(r)' H(r)) we put IITII = sup IIHII· II f II~ 1 Notice that if P is a projection operator in H(r) then IIPfll:;; Jlfll for all f E H(r)" Moreover if the projection operators P and Q are orthogonal, notation P ~ Q, which means that PQf = 0 for all f E H(r)' then P + Q is again a projection operator. This implies that, when (Pj)jSN is a sequence of projection operators in H(r) which are mutually orthogonal, then for each f E

H(r)

7.14 We close this section by extending the standard Hilbert space of square integrable functions. Let (X,M,~) be a measure space and let H E M. Denote by L 1 (H;A;~) and L 2 (H;A;~) the sets of functions f:H ~A which are respectively integrable or square integrable with respect to~. i.e. iff= I fAeA, then for each AEPN, fA E

L 1 (H;~). respectively fA E L (H;~). Then on~ may easily check that

L 1 (H;A;~)

and Lz(H;A;~) are unitary bi-A-modules and, by the definition itself, considered as real vector spaces, we have: IT

L1 (H;p)

AEPN and

Again for any f =

I A

38

fAeA E

L 1 (H;A;~)

we put (see also 3.2)

Furthermore, for any f,g E L 2 (H;A;~), f E L 2 (H;A;~) while fg E L 1 (H;A;~), where f(x) = fTX) and (f,g)(x) = f(x)g{x), x E H. Considered as a right Amodule, define for all f,g E L 2 ,(r)(H;A;~) (f,g) = IH

f(x)g(x)d~.

Then it is clear that ( , ) satisfies the conditions (f,g +h)

(f,g) + (f,h)

(f,gA)

(f,g)A

( f ,g)

(g,T)

for all f,g E L 2 (r)(H;A;~) and A EA. Furthermore for any real linear functional Ton A = o

f(x)g(x)d~> = IH d~.

=1

eo

= <1e , o

we find that

I f(x)f(x)d~> = I H

H

<1e

o

,f(x)f(x)>d~

=I H if(x)izo d~. Hence for all f E L2

'

(r)(H;A;~).

only if f = 0 ~ a.e. in H. Finally, as !fA!~ ~ !fA!~ (fA,fA)>

~ !AI~

<1e ,(f,f)>. 0

<1

iAI~

eo

,(f,f)>

~

0 and <1

eo

,(f,f)> = 0 if and

for all A E A and f E L 2 ,(r)(H;A;~),<1e 0 '

So all conditions for

L 2 (r)(H;A;~)

to be a

unitary right 1nner product A-module are satisfied. Since L 2 (H;A;~) IT L 2 (H;~) we have that L2 ( )(H;A;~) is complete; in other words AEPN • r L 2 ,(r)(H;A;~) is a right Hilbert A-module.

39

Notes to Chapter 1 Ever since the nineteenth century, mathematicians have developed different algebras adapted to meet the need of physicists. Perhaps the best known among them is the algebra of real quaternions invented by Hamilton in 1843 and presented in a final form in [Ham]. As is well known, the quaternionic approach to handling vectors in Euclidean space was left once Gibbs and Heaviside introduced vector calculus {see [GW] and [Hea]). However, observe that using the Clifford algebra C(V 3 , 0 ), the product of two vectors a = ajej and j=1 3 b = L bjej in m3 , equals ab = (alb)+ a A b, where (alb) is the scalar j =1 product of a and b, and where the 'outer product' a A b behaves in the same way as the vector product a x b. Hamilton also used quaternions explicitly in his definition of the v-operator: v = i + j a~ + k a~· which, acting on a vector v = v1i + v2j + v3k, where the functions vi(x,y,z) are real valued, yields the equation Vv = - div v + curl v. This way of separating the scalar and vector part of Vv was first pointed out by Maxwell, who thus introduced the divergence and curl of a vector function. Maxwell also remarked that v2 = - ~ 3 (see [Max]). Here already we see how algebras, in casu the algebra of real quaternions, can be used to take 'square roots' of certain differential operators. In fact the same idea is followed in geometry when passing from quadratic forms (i.e. distances) to their 'square roots' (i.e. spinors). Hence it is not surprising at all that algebras entered the game when Dirac tried to linearize the Klein-Gordon equation. In fact it appeared that the algebra needed was a special type of Clifford algebra. Clifford algebra was constructed by Clifford in 1878 (see [C1]) and, as cited above, it has been intensively studied since the thirties in connection with the theory of spinors. Among the literature treating this subject we refer to the works [Ri] and [Hes 1] of respectively M. Riesz and Hestenes. Note that in his treatise M. Riesz wrote out Maxwell's equations in a remarkably simple form using Clifford algebras. Solutions to Maxwell's equations using regular functions of a biquaternion variable are obtained by Imaeda in [Im], while Ryan in [Ry 2] extends these ideas to the n-dimensional wave operator. A complexified version of the function theory developed in Chapter 2 of this book plays an important role in this approach to problems in

I

a:

40

electromagnetism. The cited authors thus obtained generalizations of equations which were established by Fueter in [F1] and recently used for analyzing four dimensional a-models and Yang Mills fields (see e.g. [GT], [JST], [Ka]). In this context also Sou~ek's work [Sou] should be mentioned. It thus becomes clear that quaternions, especially complex quaternions, and their natural generalization, namely (complex) Clifford algebras, enjoy a revival in tackling problems in mathematical physics (see also [Ed]). For a pure algebraic approach to Clifford algebras, we refer to the works [Che], [Cr] and [Po] of Chevalley, Crumeyrolle and Porteous respectively. It should be noted that from the algebraic and geometric point of view, Clifford algebras are employed to construct twofold coverings for the orthogonal groups and for the rotation groups in particular. An important link between deformation theory for linear differential equations, quantum fields with critical length and the theory of the Clifford group was made by Sato, Miwa and Jimbo in [SMJ]. The Clifford algebra A which is introduced in Section 1.10 and which is systematically used throughout the book, is the universal Clifford algebra C(V n, 0 ). It gives rise to a linearization of the Laplacian in (m+1)-dimensional Euclidean space (see Chapter 2) and the Laplace-Beltrami operator on the unit sphere Sm (see Chapter 4). It should also be noted that the same algebra plays a basic role in the paper [ABS] on Ktheory by Atiyah-Bott-Shapiro. The fact that A becomes a trace algebra in the sense of Bonsall-Goldie is revealed to be of a fundamental significance, since it allows us to pass freely from A-linear functionals to real linear functionals and vice versa. By means of this relationship we can take over without much difficulty basic principles from the general theory of locally convex spaces and its appllcations to function and distribution spaces. Heavy use of these concepts is made in Chapters 3, 4 and 5. As references for the Sections 2-7 we have taken the books [GDS], [Sch] and [Hal]. In connection with Section 7 we should also mention the paper [Saw] by Saworotnow. Bibliography [ABS] M.F. Atiyah, R. Bott and R. Schapiro, Clifford modules, Topology 3 (1964) 3-38. [Che] C. Chevalley, The algebraic theory of spinors (Columbia University Press, New York, 1954). 41

[C1] W.K. Clifford, Collected Mathematical Papers (1882; Chelsea (reprint), 1968). [Cr] A. Crumeyrolle, Alg~bres de Clifford et spineurs (Universit~ Paul Sabatier, Toulouse, 1974). [Ed] J.D. Edmonds, Jr., Quaternion quantum theory: new physics or number mysticism, Amer. J. Physics 42 (1974) 220-223. [F1] R. Fueter, Analytische Funktionen einer Quaternionen-variablen, Comment. Math. Helv. 4 (1932) 9-20. [GDS] H.G. Garnir, M. De Wilde and J. Schmets, Analyse Fonctionnelle, Vol I, II, III (Birkh11.user Verlag, Basel, 1968, 1972, 1973). [GT] F. G~rsey, H.C. Tze, Complex and quaternionic analyticity in chiral and gauge theories I, Annals of Physics, 128 (1980) 29-130. [GW] J.W. Gibbs and E.B. Wilson, Vector Analysis (1901) (Dover (reprint), 1960). [Hal] P.R. Halmos, Introduction to Hilbert space and the theory of spectral multiplicity, 2nd Ed. (Chelsea, New York, 1957). [Ham] Sir William R. Hamilton, Elements of Quaternions, 2 Vols, (1866, 2nd Ed. 1899-1901; Chelsea (reprint) 1969). [Hea] 0. Heaviside, Electromagnetic Theory, Vol 1 (Dover (reprint), 1950). [Hes1] D. Hestenes, Space-Time Algebra (Gordon and Breach, New York, 1966). [Im] K. Imaeda, Solutions to Maxwell's equations by means of regular functions of a biquaternion variable, to appear. [JST] M.A. Jafarizadeh, M. Snyder and H.C. Tze, Quaternionic multi-s 4 instantons in general covariant SU(2) Yang Mills and HP(n) a-models, Nuclear Physics Rulletin 176 (1980) 221. [Ka] Yu. N. Kafiev, Four dimensional a-models and geometry of Yang Mills fields, Nuclear Physics Bulletin 178 (1981) 177. [Max] J.C. Maxwell, The Scientific Papers, 2 Vols (Dover (reprint), 1965). [Po] I.R. Porteous, Topological Geometry (Van Nostrand, London, 1967). [Ri] M. Riesz, Clifford numbers and spinors (Chapters I-IV), Lecture Series no 38 (Institute for Physical Science and Technology, t4aryland, 1958). [Ry2] J. Ryan, Complexified Clifford Analysis, to appear. [SMJ] M. Sato, T. Miwa and M. Jimbo, Holonomic quantum fields, I; II; III, Publ. RIMS, Kyoto Univ. 14 (1977) 223-267; 15 (1979) 201-278, 15 (1979) 577-629.

42

(Saw] P.P. Saworotnow, A generalized Hilbert space, Duke Math. J., 35 (1968) 191-197. [Sch] L. Schwartz, Th~orie des distributions (Hermann, Paris, 1966). [Sou] V. Sou~ek, Regulfni funkce quaternionove prom~nn~ (Thesis, Charles University Prague, 1980).

43

2 Monogenic functions: general theory

In this chapter a theory of monogenic functions in Euclidean space ~m+ 1 is developed. Monogenicity is introduced in Section 8 by means of a generalized Cauchy-Riemann operator which linearizes the Laplace operator ~m+ 1 • Introducing spherical coordinates the spherical Cauchy-Riemann operators are defined which yield a decomposition of the Laplace-Beltrami operator ~;+ 1 • Central theme in Section 9 is Cauchy's Integral Formula; as direct applications the Mean Value Theorem and Maximum Modulus Theorem should be mentioned. Section 10 is devoted to Morera's Theorem. The tryptich Cauchy-Morera-Weierstrass is then achieved in Section 11 by the local expansion of a monogenic function into its Taylor series. The building blocks in this Taylor expansion are special monogenic polynomials which are referred to as inner spherical monogenics on the analogy of the spherical harmonics. They replace the positive powers of z in the classical situation. In Section 12 the local behaviour of a monogenic function in an annular domain is investigated. This leads to a Laurent series expansion where the negative powers of z in the classical case are replaced by the so-called outer spherical monogenics. In this context singularities and residue theory are discussed, as well as entire and meromorphic functions. In Section 13 the adjoint of a monogenic function is defined by means of a spherical transformation which turns monogenic functions in the interior of a ball into monogenic functions in the exterior of a ball and vice versa. In Section 14 a product between monogenic functions is defined. This is done by means of a Cauchy-Kowalewski type theorem on the extension of analytic functions in Rm to monogenic functions in Rm+ 1• This chapter concludes with a section on elementary monogenic functions. Special attention is paid to the exponential function E(t,x) and to H(u,y), which generalize respectively e-itz, t E R, z E [,and eu/~,(u,~) E [ x [ 0 •

f

8. 8.1 44

A decomposition of the Laplace and Laplace-Beltrami operator First we set up the framework in which a function theory will be developed·

on the one hand we have the Euclidean space Rm+ 1 with m ~ 1; points in Rm+ 1 are denoted alternatively by x = (x 0 ,x 1, ••• ,xm) = (x 0 ,x)

x

where clearly = (x 1, ••• ,xm) lays in the hyperplane x0 = 0 which is identified with Rm. Frequent use will be made of the notation R~+ 1 which means Rm+1 = Rm+1,{0}. 0

On the other hand we have the Clifford algebra A, already introduced in Section 1, the vector-subspace A1 of which has dimension n. It is assumed that m ~ n. For the points x and in Rm+ 1 and Rm respectively we put m m m

x

X=

'r L

eiXi' X=

'r L

-

ei

X;o -+X=

'r L

e;X;• i =0 i =0 i =1 By the way notice that the Euclidean norm and the Clifford norm of x differ by a constant: lxlo = 2n/21xl. 8.2 Let n be an open subset of Rm+ 1• Then functions f defined in values in A are considered. They are of the form f(x) =

n and with

L eAfA(x) A

where the functions fA are real-valued. Whenever a property such as continuity, differentiability, etc. is ascribed to fit is clear that in fact all the components fA, A E P{1, ••• ,n}, possess the cited property (see also Section 2). The conjugate of the function f is the function f given by

8.3

Let D denote the differential operator m

D=

L i=O 45

its action on functions from the left and from the right being governed by the rules

I i ,A

eAe.a fA. 1 x. 1

The conjugate operator is given by m

D=

I

eiax .• 1

i=O

Definition A function f E C 1 (~;A) is said to be left (Pight) monogenic in Q if and only if Of = 0 (fD = 0) in ~. Introducing the components of f the left monogenicity condition reads

I

. A

eieA ax. fA = 0

(8.1)

1

1'

which is equivalent to a linear system of 2n homogeneous partial differential equations of the first order with constant coefficients. To fix the ideas let us examine how this system looks like in a few special cases. If n = m = 1 then A =Re 0 E9 Re 1 "" 0:, f = f 0 e0 + f 1 e 1 and D = e a + e 1 ax ; so (8.1) reduces to the system 0 xo d f - dX 1f1 { xo 0 d f1 + d f x x1 o 0

0 0

which is nothing else but the classical Cauchy-Riemann system for the holomorphic functions of one complex variable. Notice however that for a complete analogy one has to think of x 1 as the real part and of x0 as the imaginary part of the variable and to identify e 1 with -i; the operator D then takes the form

This approach will become more clear when introducing exponential functions (§15) and studying boundary values of distributions (§§25-27). -+ m -+ If f(x) = I eifi(x) then the system takes the form i=l

46

{ div f curl f

=0 =0

and so describes, in the particular case where m = 2, an irrotational fluid flow without sources nor sinks. It will be proved in 8.7 that the system of differential equations associated with the operator D is elliptic and even strongly elliptic in the sense of [3]. 8.4 The set of the left (right) monogenic functions inQ is denoted by

Clearly when provided with the classical laws of addition and multiplication with Clifford scalars it is a right (left) A-module. As DD = DD = ~m+ 1 e 0 where ~+ 1 denotes the Laplacian in Rm+ 1 , any monogenic function in Q is harmonic in Q and hence infinitely differentiable; even more, it is an Avalued analytic function in Q, i.e. each of its components is real-analytic in n. Denoting by a(Q;A) the bi-A-module of A-valued analytic functions in n, we thus have that

8.5 Proposition The linear system of partial differential equations associated with the operator Dis strongly elliptic. Proof Let 8 denote the left regular representation of A introduced in Section 1, which associates with each A E A a 2n x 2" matrix 8(A) over R. As 8(D)T = O(D) we have (det 8(0)) 2

det 8(D).det 8(D)T det 8(D).det 8(0) det (8(D).O(D))

47

det


(fm+1 )2n.

Each entry in the matrix o(D) being of order one, it is obtained that

which implies the strong ellipticity of the system o(D).

o

8.6 It follows from the considerations made in 8.4 that if f is left monogenic in~ then all components fA, A E PN are harmonic functions in ~. We wonder if, given a real-valued harmonic function u:~ ~ ~. there exists a left monogenic function f in ~ such that [f] 0 = u, where [f] 0 denotes the scalar part f 0 of the function f. It can be proved in a straightforward manner that Proposition u:~ ~~is

If ~ is open and star-shaped with respect to the origin and harmonic in~ then the function

f(x) = u(x)e 0 + is left monogenic in 8.7

~

J:

tm- 1 Du(tx) x dt-

[J:

tm- 1 Du(tx) x ctt] 0

and its scalar part is precisely the function u.

Introducing spherical coordinates x0

r cose 1

x1

r sin8 1 cose2

xm_ 1

r sine 1 sine 2

sinem_ 1 cosem

r sin81

sinem+ 1 sinem

xm

sine 2

where 0 < r < +m, 0 < e 1 , ••• ,em_ 1 < TI, 0 <em< 2TI, for a point x E ~~+ 1 can write x = rw, x-

rw-

where m w=

I i=O

48

m eiwi' w

I i=O

eiwi

we

with wi

X. 1

X. 1

=TxT= r-·

i

= 0,1, •.• ,m.

Notice that w represents a point on the unit sphere sm. For the operators D and Dthe following spherical form, valid in R~+ 1 , is obtained: D = war where

1r aw, D= ~a r

+

+

1r aw

m aw

I i =1

1

aw

a .

2 • as.1 • s.1 • 1~1 asi

for completeness let us subjoin that aw l2 = s1n · 2 81 I asi

· 2 si_ • ••• s1n 1

This allows us to introduce the following differential operators acting on the unit sphere sm. Definition The operators r = ~a w and r* = wa

(!.)

are called spherical Cauchy-Riemann operators. '\,

The operators

'\,

r = aww and r * = aw~ and their respective adjoints. Notice that these operators are intended to act from the left; for acting from the right we have the formulae '\,

'\,

fr

rf, fr*

'\,

fr

'\,

rf, fr*

= r*f '\,

= r*f

where f is sufficiently differentiable on Sm. Also notice that the generalized Cauchy-Riemann operator D and its conjugate D may now be written as D = w(ar

+

1

rr) = (ar

+

1'\,

rr*)w 49

and

Those spherical Cauchy-Riemann operators can be used for instance to decompose the Laplace-Beltrami operator ~~+ 1 appearing in the spherical expression for the Laplacian:

Indeed we have consecutively


dnd in an analogous way

~m+ 1 = Cl 2r 2 + .!(r* + ?*)a r + 1...(?*r* - r*) r rc and hence •-u

'\,

r + r

['*

+ I'*

=

m!1

'\,

~~+1

( f-1)

((m-1)1-r)r

(8.2)

'\,

~~+1

(r*-l)r*

= ( (m-1

)1-r*)r*.

In illustration we give the explicit form for the spherical Cauchy Riemann operators in the particular case n = m = 1 of the holomorphic functions of one complex variable:

r = -i "e, r* r = 1+i a8 , r*

'\,

'\,

where we have replaced e 1 by -i. 8.8 Of the highest importance in the sequel is the so-called Cauchy kernel E defined in R~+ 1 by 50

E(x)

=-

wm+1

X

lx lm+1

'

X

F0

where um+1 stands for the area of the unit sphere Sm. This function E enjoys the following properties.

E E M(R~+ 1 ;A) since E E a(R~+ 1 ;A) and satisfies

(i)

DE = ED = 0 in R~+ 1 while lim E(x)

=

0

X-><x>

(ii) E EL! 0 c(Rm+ 1);A) so with E a left and a right A-distribution, again denoted by E, may be associated. It can be proved that, in distributional sense,

DE = ED = o as well for E E V(r) (Rm+ 1;A) as for E E V(l) (Rm+ 1;A). In view of these two properties the function E is obviously a left and a right fundamental solution of the operator D. 9.

Cauchy's integral formula and its applications

9.1 By Mwe denote an (m+1)-dimensional differentiable and oriented manifold contained in some open subset n of Rm+1 • By means of the m-forms

an A-valued m-form is introduced by putting m

do

= L i=O 51

If dS stands for the 'classical' surface element and m n

I

=

e.n.

1 1

i=O where ni is the i-th component of the outward pointing normal, then the Avalued surface element do can be written as do = ndS. Furthermore the volume-element

is used. 9.2

If f,g E C1 (Q;A) then for any (m+1)-chain con Mc Q,

Proposition

fac Proof

f dog=

fc

(fD.g + f.Dg)dx

Using Stokes'sTheorem for real-valued functions we have successively

J3C

f

do g =

r

L

J 3C

(-1)i eAeieB f 8g8dxi

A, i, B

L

(-1)i eAeieB

Jc

(-i)i 3xi (f 8g8 )dx

A, i ,B

fc fc

A,i,B (fD.g + f.Dg)dx.

9.3 Corollary If f is right monogenic in then for any (m+1)-chain con Mc Q f dog = 0.

D

Q

and g is left monogenic in

I

3C

9.4

Remark

fc 52

Putting f =

Dg dx =

fac

dog

in the above formulae yields

Q

for any g E

Iac

C 1 (~;A),

da g

for any g E Theorem.

while

=0

M(r)(~;A).

This last statement is nothing else but Cauchy's

9.5 Theorem LetS c ~be an (m+1)-dimensional compact differentiable and oriented manifold-with-boundary; then if f E C 1 (~;A)

I~s

0

f

E(y-x)doy f(y) - Js E(y-x).Df(y)dy

a

f f(x) for x E S

=1 0 l

for x E

~s.

Proof First let x E ~S. It then follows from Proposition 9.2, which also may be applied to S, that the left hand side of the stated formula reduces to

Is E(y-x)

D .f(y) dy

y

which apparently equals zero. Now take x E S to be fixed and choose R > 0 such that B(x,R) c Then again by Proposition 9.2 and taking into account that E(y-x) is right monogenic in y E S,B, we get

S.

J

Cl(S,B)

E(y-x)do f(y) y

=

I

S'B

E(y-x).Df(y) dy.

( 9. 1 )

Observe that E(y-x).Df(y) is integrable on S since Of is continuous on S and

where R' > 0 is such that S c B(x,R'). of (9.1) tends to

fs

So if R ~ 0+ then the right hand side

E(y-x).Df(y)dy.

As to the left hand side of (9.1) it can be put into the form

Jas

E(y-x)do

Y

f(y) -

I

aB(x,R)

E(y-x)do f(y) Y

the second term of which can also be written as

53

I

e.e.eA (y.-x.)fA(y)dy. 1 ClB(x,R) J J J 1

j,i,A

Rm~ 1

L

I

eJ.eieA

B(x,R)

j, i ,A

(6 .. fA(y) + (y.-x.)Cl fA(y))dy lJ J J Y;

I

I

1 (m+1) f(y)dy + - 1Rm+l B(x,R) wm+1

1 G-x)Df(y)dy. Rm+ 1 B(x,R)

As for sufficiently small R f(y) = f(x) + e(R)

with

lim e(R) = 0 R+O+

and sup IDf(y) 1 0 yEB(x,R)

$

C

it is obtained that

I

lim R+O+

ClB(x,R)

E(y-x)do f(y) y

1-(m+1) vm+1 f(x) wm+1

= -

f(x)

where Vm+ 1 = ~+1 1 is the volume of the (m+1)-dimensional unit ball. taking limits for R + 0+ in (9.1) yields

Ias

E(y-x)do f(y) - f(x) Y

= I E(y-x).Df(y)dy. s

9.6 Corollary (Cauchy's Integral Formula) left monogenic in n then

Hence

c

Let S c n be as in 9.5.

Iff is

0

Ias

E(y-x)do f(y) Y

=~ L

f(x) for x E S

o

for x E sz-....s.

9.7 Proposition (Mean Value Theorem) then f(a)

=

I

B(a,R) Rm+ 1v m+1 for each R > 0 such that B(a,R) 54

f(u) du c

n.

Iff is left monogenic inn and a En

-

proof Take R > 0 such that B(a,R) on the ball B(a,R) gives

c

n. Applying Cauchy's Integral Formula

I

f(a)

CJB(a,R)

or f(a) As (u-a)Du

= m+1 one gets by Proposition 9.2

I

m+1

f(a)

B(a,R)

f(u) du.

c

9.8 Proposition (Cauchy's Integral Formula outside a ball). Let f be left monogenic in Rm+l,B(O,R) with lim f(x) =A· Then for each xSRm+l,B(O,R) f(x) = A -

J

aB(O,R')

E(y-x)do f(y) y

where R' is suitably chosen such that R < R' < lxl. Proof Take x E Rm+l,B(O,R) arbitrarily; choose R' such that R < R' < lx I and R" such that B(O,R') c B(x,R"). Applying Cauchy's Integral Formula on 0 B(x,R"),B(O,R') one gets f(x)

I

=

aB(x,R")

- as(6,R') r

E(y-x)do f(y). Y

(9.2)

But introducing spherical coordinates the first integral becomes

J and so

m

f(x+R"w)dS

s

w

lim J E(y-x)doy f(y) =A. R"-++oo aB(x ,_R") Hence taking 1imits for R" ->-+ oo in (9.2) it is obtained that f(x) =A -

r

.

l E(y-x)do f(y). aB(b,R') y

c

55

9.9 Theorem (Maximum Modulus Theorem). Let f be a left monogenic in the open and connected set n. If there exists a point a E n such that lf(x) 10

~

lf(a) 10 for all x

E

n

then f must be a constant function in rt. Proof

Put lf(a)l 0 =A and consider the subset nA rt>, = {x E n: lf(x)

of~

given by

= A}.

10

Of course nA I~ since a E nA. So let y E ~QA; this implies that lf(y)I 0 0 such that lf(u)l 0 0 such that B(z,R) c n. By the Mean Value Theorem

I

B(z,R)

f(u)du

from which it follows that n

A2 = lf(z)l2 =_,,..,..,2;r-_ o R2m+Zv2 m+1

and still, using H6lder's inequality,

L(

J

I

B(z,R)

J

du) (

A B(z,R)

fA(u)du)

B(z,R)

lf(u)l~du.

Hence 0

$

J ( If ( u)

1

- Rm+1v m+1

B(z,R)

1

2 - A2 )du

0

:£

0

which yields that lf(u)l 0 = A for all u E B(z,R); this means that B(z,R) c nA and hence that nA is relatively open in n. As n is supposed to be connected 56

it follows that n = nA or if(x)l 0 Now if A = 0 then clearly f(x) then for all x E n

A for all x E n. 0 for all x E n.

So assume that A > 0;

and by differentiating twice

I

(ax.fA(x)) 2 +

A

Summing up over i

I

I

0, 1 , .•• ,m.

0,

1

= 0,1, •.• ,m yields

(ax.fA(x)) 2 + 1

i ,A

fA(x).a~~fA(x)

A

1

I

fA(x).~m+ 1 fA(x)

0

A

or

I

(ax.fA(x)) 2

. A 1•

= 0 for all x En.

1

This results finally into

a fA(a) = 0 inn for all

0,1 , ••• ,m and all

Xi

A E P{1, .•. ,n}, which means that f is constant inn.

c

9.10 Corollary Let n be a bounded open set in ~m+ 1 and suppose that f is continuous in n and left monogenic in n. Then sup lf(x)l 0 = sup lf(x)l 0 • xEn xEan

n

Proof As is compact and lf(.)i 0 is a continuous function on point a E n such that sup lf(x)l 0

=

n there

is a

lf(a)l 0 •

XEQ

If a E an then of course sup lf(x)l 0 = lf(a)l 0 xEan and the assertion is true. So assume that a E n. ponents

Decompose n into its com-

57

all of them being bounded, open and connected. components, say~ .• As

Then a belongs to one of the

J

if(x)

10

~

lf(a) 1 0 for all x

E

nj

it follows from Proposition 9.9 that f is a constant function in nj. sup lf(x)l 0

=

lf(a)l 0

=

sup

~

lf(x)l 0

XE()[2J.

XE n"

which yields the desired result.

sup

lf(x)l 0

~sup

XECl\-:

Hence

lf(x)l 0

XE\2

o

9.11 Theorem (Weierstrass) Let (fj)jeN be a sequence of left monogenic functions in n. If for each compact set K c g and each E > 0 there exists a natural number N(E,K) such that sup lf.(x)- f.(x)l xEK

1

J

0

<

~

E whenever i,j

N(E,K)

then there exists a function f in n such that (i) (ii)

f is left monogenic in~; the sequence (a 6fj)j8N converges uniformly on the compact subsets of~ to a6f, for any multi-index BE ~m+ 1 .

Proof Take an arbitrary compact subset K c ~. Let (Ki)iSN be a sequence of regular compact sets filling up g (see also 4.2); then there exists an i 0 E ~ such that K c K1. • By Cauchy's Integral Formula, the function f. -f. being 0

0

1

left monogenic in~. we have for all x E K and for any multi-index

Hence V( aK. ) • 10

sup yEaKi 0

sup

lfi(y)- fj(y)l 0

yEClKi 0

58

J

s E ~m+l,

if i,j

~

N(E,Ki 0 ), where d

= d(K,aKi 0 ).

For the components of the functions fj, j EN, we thus get sup ia 6f. A(x)-a 8f. A(x)l ~ C'.E 1' xEK J'

if i,j

~ N(c,K 1.

).

o

This means that for each A E P{1, ••• ,n} the sequence (f. A) "ElN is a Cauchy J' J sequence in the space E(n;R) provided with the topology of compact uniform convergence. As this space is sequentially complete it follows that there exist functions fA' A E P{1, ••• ,n}, such that the sequences (a 8fj,A)j8N converge uniformly on the compact subsets of n to a8fA, for any multi-index B E lNm+ 1. Put f = I eAtA; then it is clear that (a 8fj)j8N will converge uniformly A

on the compact subsets of n to a8f, for any (3 E lNm+ 1• In particular we will have that (Dfj)jElN converges uniformly on the compact subsets of n to Of. As all Dfj' j E lN, vanish inn, it follows that f is left monogenic in n. c 9.12.

Remarks

(i) The above theorem in fact means that the module M(r)(n;A) provided with the topology of uniform convergence on compacta is sequentially complete (see a1so § 16) . (ii) The above theorem holds a fortiori if the given sequence of left monogenic functions converges uniformly on the compact subsets of n to a function f. (iii) In particular it follows that if a series of left monogenic functions converges uniformly on the compact subsets of n to a function f, then this sum f is left monogenic in Q and any derived series converges uniformly on the compact subsets of n to the corresponding derivative of f; in other words under the given assumptions differentiation and summation may be switched.

59

10.

Morera's Theorem

10.1 In Section 9 it was proved that iff is left monogenic in an open set Q then for each closed interval I contained in Q

Ia

(10.1)

daf = 0.

I

Indeed this is nothing else but Cauchy's Theorem (Remark 9.4). Now the converse will be proved; iff is continuous inn and (10.1) holds for all closed intervals I contained inn, then f is left monogenic in n. In classical function theory this result, known as Morera's Theorem, is proved rather easily handling primitives of holomorphic functions, notion which is intimately related to line integrals. Although in Clifford analysis the notion of primitive is available (see Section 19) it is of no use here since it is not linked to surface integrals. So Morera's Theorem is proved in another, rather complicated manner. 10.2 Let I(n) denote the set of all closed intervals contained in the open set n, and for each IE I(n) denote by o(I) the length of its diagonal. Lemma Iff is continuous in Q such that (10.1) holds for all IE I(n) then for all I E I(Q)

IaI

h daf = 0

for any function h E C1 (n;A) which is right monogenic in n. Proof Consider an arbitrary but fixed closed interval I =

m IT

i=O

[ai,bi] E I(Q)

and divide it into 2m+ 1 closed intervals Jj by taking the midpoints ci of [ai,bi] for all i = 0,1, ••• ,m. Then 2m+1 2m+1 AI= and so

60

J

aI

h do f =

.

I

J= 1

I

aJ J.

hda f =

·

I

J= 1

AJ. J

2m+1 !Ar 10

I

;::

j

=1

Hence there ought to exist a j 0 (1 ;:: j 0 ~ 2m+ 1 ) for which IAJ.

Io

Jo

"' 2m1+1

Put ! 0 =I, 11 = J.

!Arlo·

Jo

for I.

After

q

and make the same construction for 11 as already done

subdivisions we arrive at Iq E

I(~)

for which

So a sequence (Iq)q~ of closed nested intervals is obtained the intersection of which is a point u. Now choose E > 0 arbitrarily. As f is continuous at u there exists an n(E) such that jx-uj < n(E) implies that jf(x) - f(u) j0

<

E

•

Choose q 1 sufficiently large in order that o(Iql) A Iql

J31 q

hdof =

far q I

: Jdl

J

3Iq

I

I

n(E).

Then

[h{x)-h(u)]dox f(x)+h(u) J 3Iql

dax

f(x)

[h{x-)h(u)]dox f(x) [h{x)-h(u)]do [f{x)-f(u)]+(J

q

<

X

1

As h is right monogenic in

~

31 ql

[h{x)-h(u)]do )f(u). X

we have that

J3Iql [h{x)-h(u)]do X = JIql [h{x)-h(u)]D X .dx

0

and so m

A1 = J I (xi-ui)3x 1.h[u+t(u-x)].dox.[f{x)-f(u)] ql 3Iql i=O As for some C > 0, jax_hj 0

;::

(0

<

t

<

1).

C in I for all i = 0.1, ••• ,m we arrive at

1

61

IAI

~

10

q'

n o(I) , V(al) (m+1).2 • ~ .C .c.~

and thus at

As c > 0 was chosen arbitrarily it is obtained that IA 1 10

faI 10.3

h do f

= 0.

c

Iff e c0 (n;A) is such that for all I e I(g)

Proposition

faI

= 0 or

dof = 0

then for any I e I(n) 0

f(x)

=J

ai

E(y-x)do f(y), for all x e I. x

0

Proof Take I E I(n) and x E I. Choose c > 0 arbitrarily. uous at x there exists an n(c) > 0 such that

As f is contin-

lf(x+o)- f(x)l 0 < c if lol < n.

E(y-x)do f(y) Y

= (J

ai

Y

k

Jai

0

j=1

Y

=J

aJ

k

+

I J J=

By Stokes's Theorem

E(y-x)do [f(y)-f(x)] Y

k

E(y-x)doyf(y) -

. 1 aJ .

62

Y

u JJ. be a partition of I'
J

I

(J aJ .

. 1

J=

J

c

B(x,R)

E(y-x)do [f(y)-f(x)]

E(y-x)do [f(y)-f(x)]. Y

ai

0

construct a closed cube J Then

E(y-x)do )\f(x) +

= f(x) + J Let I'
0

I;

TakeR> 0 such that R
E(y-x)day)f(x).

I

aJ.

E(y-x)do

0 for j

1,2, .•• ,k

Y

J

and by Lemma 10.2 r E(y-x)doyf(y) ; aJ.

0 for

1,2, ••• ,k.

j

J

Hence

fai E(y-x)doy[f(y)-f(x)]

faJ E(y-x)doy[f(y)-f(x)].

But

IaJ

I

E(y-x)do [f(y)-f(x)]lo y

~

2~ ~ - - . m.E.V(aJ) wm+1 s

22n+m+1(m+ 1)

.=...__ _..:..:.:.;..~

.E

C".E or

if a

I

As £

>

E(y-x)doy[f(y)- f(x)JI 0

~

C".£.

0 was chosen arbitrarily the desired result follows.

c

10.4 Theorem (Morera) A function f is left monogenic in the open set n if and only if f is continuous in n and

Ia

dof

=0

(10.1)

I

for all I E I(n). Proof The necessary condition follows from Cauchy's Theorem. Conversely suppose that (10.1) holds for all IE I(n). Take x* En arbitrarily; then there exists an I* E I(n) such that x* E I*. We know from Proposition 10.3 that f(x)

Iai*

E(y-x)do f(x), for all x E I*. x

63

Differentiation under the integral sign being allowed we have at once that f 0 0 is left monogenic in I*. So for x* we get an open neighbourhood I* in which f is left monogenic. Hence f is left monogenic in n. c 10.5

Corollary A function f is harmonic in Q if and only if f E C1 (Q;A) and

IaI

do Of

0

for all I E I(n). Notice that, as doD nOdS = ~ dS, this corollary can be looked upon as a ageneralization to A-valued fun8tions of the well known theorem of Koebe and its converse.

n

10.6 Theorem (Painlev~) Let be open in Rm and Q be open in Rm+ 1 such that Q n Rm = If f is left monogenic in ~ and moreover continuous in Q, then f is left monogenic in Q.

n.

Proof For any closed interval I contained in ~ we have

Ia

do f

=0

I

since f is left monogenic in ~. in Q of the following form:

Now consider a closed interval I contained

n.

where i is a closed interval contained in I

£

=i

x

For all s

[O,s].

Then by Cauchy's Theorem and the continuity of f

Iai do f = Iai do f =

I

[f(s+x)-f(O+x)JdS +

i

64

do f

= lim

s~o+

do f

aix[O,£]

and as f is uniformly continuous in

Jai

I

I

ai

do f

= 0.

it follows that

E

]O,a 0 ] we put

An analogous reasoning may be followed for all other closed intervals contained in D and intersecting D. We thus obtain that ~

Ia

da f

~ 0

I

for all IE I(n).

Conclusion by Morera's Theorem.

c

10.7 Corollary Iff is left monogenic in the open connected set D cRm+ 1 and vanishes in the open set A c D n {x 0 = a 0 } ~ ¢then f is identically zero in D. ~

Proof Let A be an open connected subset of D such that A n {x 0 = a 0 } = Aand such that A'A consists of two disjoint connected open sets A1 and A2 • Define in A a function g as follows: f(x) X E A1 g(x) = { f(x) = 0, x E h . 0

,

X

E

A2

Clearly g is left monogenic in A'A and continuous in A. By the Painlev~ Theorem g is left monogenic in A and hence real-analytic in A. So g is identically zero in A and this means that f = 0 in A1 • As f is real-analytic in Q it follows that f = 0 in n. c 11. 11.1

Taylor series Spherical harmonics

11.1.1 Let Sk denote a real-valued spherical harmonic of order k, i.e. a real valued homogeneous polynomial of degree k satisfying Laplace's equation. By means of spherical coordinates X

= rw,

w E Sm

(see also 8.7), Sk can be written in the form Sk(x) = r

k

Sk(w).

Sk(w) is called a surface spherical harmonic of order k. If Hk(Sm~) denotes the space of all real-valued surface spherical harmonics of order k, then

65

={

(2k+m-1)r(k+m-1) r(k+1 )r(m~ '

k

1

~

1

k = 0

~l(m, k )+M(m,k-1)

where M(m,k) denotes the number of coefficients in a homogeneous polynomial of degree k in m variables (see [15]}. 11.1.2

If Lk,m+ 1 denotes the Legendre polynomial of degree k in mm+ 1 then ( n) 1k r-z (--z) -~ (1-t 2 r(k~)

)

2-m -z-

which is known as the formula of Rodrigues. the estimate

k k+(m-2) 2 7.1< (1-t ) 2 dt d

The Legendre polynomials satisfy

1Lk,m+ 1(t)i ~ 1 for -1 ~ t ~ 1. A formula for the derivatives of the Legendre polynomials reads

Lk-j,2j+m+1(t)

N(m+1,k) N(2J+m+1,k-J)

r(~)

( ')

2-j Lk~m+1 ( t).

r(jlm21)

Next if w* denotes a fixed point on the unit sphere Sm then

with m

<w,w*>

= L

i=O is the unique surface spherical harmonic of order k satisfying the conditions (i)

(ii)

66

Sk(w*)

=

1;

Sk(Aw) = Sk(w) for each orthogonal transformation A leaving w* unaltered.

11.1.3 Each surface spherical harmonic Sk(w) is an eigenfunction of the Laplace-Beltrami operator 6~+ 1 with eigenvalue (-k)(k+m+1), i.e.

For further information on spherical harmonics we refer to [15], [23] and [17]. 11.2

Inner spherical monogenics

11.2.1 Let Pk be a homogeneous polynomial of degree k which is left monogenic in Rm+ 1, i.e. Pk(x)

= I eAPk,A{x) A

where the Pk,A are real-valued homogeneous polynomials of degree k in the variables x0 ,x 1, ••• ,xn and DPk ~ 0 in Rm+ 1• On the analogy of the spherical harmonics, such a Pk may be called a left inner spherical monogenic of order k. Clearly each inner spherical monogenic is an A-valued spherical harmonic of the same order. Moreover a straightforward computation shows that rPk

(-k)Pk

and ~

rPk

(k+m)Pk

which means that the left inner spherical monogenics of order k are eigenin R~+ 1 with respective eigenvalues {-k) functions of the operators r and and (k+m). The restriction to the unit sphere Sm of a (left) inner spherical monogenic Pk(x), notation Pk(w), is called a (left) surface inner spherical monogenic. As Pk(w) is in particular an A-valued surface spherical harmonic of order k, it is an eigenfunction of the Laplace-Beltrami operator 6~+ 1 with eigenvalue (-k)(k+m-1), i.e.

r

11.2.2 The set of all left inner spherical monogenics of order k becomes a right A-module under the classical laws of addition and multiplication by

67

k

Clifford scalars. It is denoted by P(r)• In order to construct a basis for P~r) the following hypercomplex variables are introduced: z1 = x1e0

-

x0 e1 , 1 = 1, 2 , ••• , m,

which are left and right monogenic in ~m+ 1 • Observe that in the case where n = m = 1 the hypercomplex variable

can be identified with the complex variable z = x + iy thinking of x1 as the real part and x0 as the imaginary part of z 1 and replacing -e 1 by i. Next for any combination (1 1, •••• 1 k) of k elements out of {1,2, ••• ,m} repetitions being allowed, we put

and V1

1 (x) = -k\ 2 1... k . 1T(11 •••• ,1 k)

z1

1

••• z1

k

where the sum runs over all distinguishable permutations of all of (1 1, ••• ,1k). E {1,2, ••• ,m}k, 11.2.3 Proposition The polynomials v1 1 , (1 1, ••• ,1~) 1• • • k " are both left and right inner spherical monogenics of order k. Proof Clearly the polynomials v1 are homogeneous of order k. 1 1···1 k have for all x E ~m+

Next we

--

m

DV1

1

1 (X) = TI 2 1···k 1Tn 1•••• ,1k> m

= k1!

2

1T(1 1 , ••• ,1 k)

[eo<\(, ••• , ) + 2 ei 3x/, 1···l k>] 0 1 k i=1

k =

1

TI 2 j=1

= a. 68

1T

0

2

1·····

[-e z 1 ••• z 1 e z ••• z +e z ••• z z 0 1 j -1 1j 1j +1 1k 1j 11 1j -1 1j + 1 1k ]

)

...

,

k

An analogous computation shows that V D = 0 in mm+ 1• 11 ••• 1k

c

11.2.4 Remark By Proposition 9.2 the following orthogonality relations follow at once:

f

aB(0,R)

v1

1 {x) do vs s (x) = 0 1··· k x 1··· t

k for all k, s E l'l, (1 1, ••• ,1k) E {1, ••• ,m}, (s 1, ••• ,st) E {1, ••• ,m} t and R > 0.

11.2.5 Proposition Any left inner spherical monogenic Pk of order k may be written as

where the sum runs over all possible combinations (1 1, ••• ,1k) of k elements out of {1, ••• ,m} repetitions being allowed. Proof The proof is given by induction on k. trivial. Fork= 1 we have to show that

For k

0 the assertion is

m

m

By Euler's formula and taking into account that DP 1 = 0 we have at once m P1(x) = xo ax P1(x) + I x.1 aX. P1(x) 0 1 i =1 m m = xo[op1 - I eiax.P 1(x)] + I X; ax.P 1(x) 1 1 i =1 i =1 m

Now assume the stated formula to be true for k; again by Euler's formula m

(k+1)Pk+ 1(x) = x0 ax Pk+ 1(x) + 0

I i=1

xi ax.Pk+ 1(x) 1

and as DPk+ 1 = 0, 69

m

L zi

(k+1) Pk+ 1(x) =

ax.Pk+l(x).

i =1

1

so by the hypothesis made m

1

pk+1(x) = l<+T

L zi

i=1 m

1

(k+1J!

L

(11····· 1k)

m

v1

1 (x).ax ••• ax (ax pk+1(x)) 11 1k i l''' k

m

L

I

I

i=1

11=1

1k=1

ziz 1 ••• z1 ax ••• a~ ax.Pk+ 1(x) 1 k 11 k 1 c

=

k 11.2.6 Proposition A basis for the right A-module P(r) of the left inner spherical monogenics of order k is given by

{ v1

1

1• • • k

: (1 1, ••• ,1k)

E

{1, ••• ,m}k}.

Proof In view of the Propositions 11.2.3 and 11.2.5 it suffices to prove that the polynomials v1 1 are right A-linear independent. So let 1• • • k

I

(1 1 • • • • • 1k )

v1

1 (x).>..1 1 = 1''' k 1''' k

o in 1Rm+1.

If si denotes the number of times that i appears in the sequence (i = 1, ••• ,m), then this equality can be rewritten as I

1

51

5m

I X1 • • • X

5 1 • • • • 5 m•

m

A1

(1 1, ••• ,1k)

1

1• • • k

for all x E IRm+l,

+ [terms containing x0 ] = 0

from which it easily follows that all >.. 1 1 must be zero. 1• • • k

c

11.2. 7 Remarks (i) ( i i)

70

The dimension of P~r) is 2n M(m,k). The inner spherical monogenics V

11 ... lk

all take their values in

11.3 Taylor series 11.3.1 Now the local behaviour of a monogenic function near a point is investigated. The aim is to expand a monogenic function locally into a socalled Taylor series; it turns out to be so that the polynomials v1 1 1• • • k

introduced above play an analogous role as the powers of the complex variable z do in the classical Taylor series for holomorphic functions. 11.3.2 Assume that the function f is left monogenic in an open set n cRm+ 1• Without loss of generality it may be supposed that Q contains the origin. We already remarked in 8.6 that f is an A-valued analytic function inn. So there exists an open neighbourhood A of the origin in which each component fA may be developed into its Taylor series m m fA(x) = L TI L xl ••• x1 '\ ••• ax fA(O) L 1k 1k=0 1 k 11 k=O 11=o which together with all its derived series, all of them considered as a multiple power series in the real variables x0 ,x 1, ••• ,xm, converges normally in A, i.e. if j denotes a single summation index the terms of the multiple power series being ordered in an arbitrary manner, then for each compact subset Kc/\

JI

if inf(J,J') ~ + oo. Hence for the function f itself one gets m m 1 f(x) = L TI L L x1 1••• x1 k a ••• ax f(O) X 11 1k 11=o k=O 1k=O 00

...

(11.1)

where again the right hand side and all of its derived series considered as multiple power series in the real variables x0 ,x 1, ••• ,xm, converge normally in /\. Bracketing terms together we arrive at 00

L Pkf(x) k=O where Pkf stands for the homogeneous polynomial of degree

(11.2)

f(x) =

~

71

m

m (11.3)

f( 0).

As for the convergence of the series in (11.2) and all its derived series it is evident that considered as series of polynomials they are still normally convergent in A, i.e. for each compact subset K c A 00

I k=O is a convergent numerical series. 11.3.3 Lemma Iff is left monogenic in~ containing the or1g1n then for each k E ~the polynomial Pkf given by (11.3) is a left inner spherical monogenic of order k. Proof As D(Pkf) is a homogeneous polynomial of degree (k-1) it can be written as

We have to prove that all A are zero. Take an arbitrary sequence s 0 s 1••• sm (1 1, ••• ,1k_ 1 ) E {0,1, ••• ,m}k-1 and consider the differential operator ak-1

where ti represents the number of times that occurs in the sequence <1 1•••• ,1k_ 1 ) (i 0,1, ••• ,m). Then in~ we have ••• ax Df = o 11 1k-1 which, after restriction to the area of convergence A of the expansion (11.2) off, yields ax

0 = ax

••• ax 11

72

D(Pkf) 1k-1

+

[homogeneous terms of degree q

~

1]

or 0 = t 0 ! t 1! .•. tm! At t

t

o 1··· m

+[homogeneous terms of degree q G 1].

At the origin this equality reduces to

As the sequence (1 1 , ••• ,1k_ 1 ) was arbitrarily chosen, all coefficients A are indeed zero; hence D(Pkf) = 0. c So···sm 11.3.4 Theorem Iff is left monogenic in an open set n containing the origin then there exists an open neighbourhood A of the origin in which f can be developed into a normally convergent series of left inner spherical monogenics f (x )

I( k=O

L

(1 1 , ••• ,lk)

V1

1 (x ) • ax 1 ••• ax 1 f ( o)) , 1 k

(11.4)

1• • • k

any derived series being normally convergent in A to the corresponding derivative of f. Proof In view of the expansion (11.2), Lemma 11.3.3 and Proposition 11.2.5 it suffices to prove that

which is evident from the definition itself of the polynomials Pkf. 11.3.5 of is as in

c

Remarks

(i) Up to now it is not at all certain that the region of convergence A the Taylor series (11.2) or (11.4), which besides are totally identical, maximal, i.e. a ball centered at the origin and with radius d(OJRm+ 1,n), would probably be expected. That this is still the case will be proved 11.5. (ii) Next to the Taylor series (11.2) or (11.4) also the series 00

f(x) =

(11.4') 73

where the terms are not bracketed together, will be considered. In fact it is a re-arrangement of the series (11.1), and when speaking of the Taylor series about the origin (11.4') converging as a multiple power series in A, it is meant that the multiple power series (11.1) converges normally on the compact subsets of A. For the moment, given a monogenic function f in n containing the origin, the domain of convergence of (11.4') considered as a multiple power series has not been determined yet. It will be shown in 11.5.3 (iii) that this domain of convergence is smaller than the domain of convergence of the corresponding Taylor series (11.4) considered as a series of left inner spherical monogenics. In the sequel (11.4') will be referred to as the multiple Taylor series of f about 0. 11.3.6 Theorem (Converse of Theorem 11.3.4). spherical monogenics

I k=O

Pk(x) =

i ( k=O

I

(1 1, ••• ,lk)

If the series of left inner

(11.5)

V (x) A \ 11 ••• lk 11 ••• lkJ

converges normally on the compact subsets of the open set A which contains the origin, then it represents a left monogenic function f in A. Moreover the given series is precisely the Taylor series about the origin of f in A. Proof From Weierstrass's Theorem (Theorem 9.11) we know that the given series (11.5) converges in A to a left monogenic function f, and that any series of derivatives will converge uniformly on the compact subsets of A to the corresponding derivative of f:

• ••a xs

t

vl 1 ••• 1k (x).Al 1••• 1k ) •

Now as f is left monogenic in A there exists an open neighbourhood of the origin A' c A in which

I

00

k=O

(

I
aX ..• aX v1 51

converges uniformly on the compact subsets of A'. 74

5t

1••• 1k

{x).a x •.• ax f(D)JI~ 1

Hence the series

1

1

k

~L. (

aX

L

a

V11 ••. 1 k(x}[>..11 ••• 1 k-"0 X

••• X

51

k=O (1 1, ••• ,1k)

5t

"

f(O}J)

••• 0 X

11

1k

converges uniformly on the compact subsets of A' to the null function for any combination (s 1, ••• ,st) E {1, ••• ,m} t • As eo if (s1, ••• ,st) = (11, ••• ,lk)

a

••• a

xst

><s 1

v1

1 (O)

{

1• • • k

0

otherwise

it follows that >..

=Cl

s1 ••• st

xs

••• Cl

1

xs

f(O) t

for all (s 1, ••• ,st) E {1, ••• ,m}t. This implies that the given series (11.5) c coincides with the Taylor series of f in A' and hence in A too. 11.3.7

Proposition

If the series

00

L

L

V1

k=O (1 1, ••• , 1k)

1 (x) >..1 1 1"""k 1"""k,

considered as a multiple power series, converges normally on the compact subsets of the open set A which contains the origin, then it represents a left monogenic function f in A. Moreover the given series is precisely the multiple Taylor series about the origin of f in A. Proof Clearly the sum of the given series is an A-valued analytic function in A and, termwise differentiation being allowed, we get immediately that Df = 0 in A. Now as f is left monogenic in A, there exists an open neighbourhood of the origin A' c A in which f may be developed into its multiple Taylor series 00

f(x> =

L

L

v1

k=O (1 1, ••• , 1 k)

1 (x)

1• • • k

a x1

••• a 1

x1

f(O). k

Hence the multiple power series 00

L

L

k=O (1 1, ••• ,1k) 75

converges normally in A' to the null function. On the same line of argument as in the proof of Theorem 11.3.6 it follows that "11"" .lk = ax

11

••• ax

1k

f(O)

for all (1 1 , ••• ,1k) E {1, ••• ,m}k, whence the given series coincides with the multiple Taylor series about 0 of f in A' and hence in A too. c 11.3.8 Remark Analogous results are valid for the local behaviour of a monogenic function f near a point a I 0. It is obtained that if f is left monogenic in an open set n then for each point a E n there exists a neighbourhood Aa of a, contained in n, such that in Aa f may be developed into a unique normally convergent Taylor series of left inner spherical monogenics 00

f(x) =

I k=O

P f(a) (x)

k

where

m 1 P f(a)(x) =rr k

I 11 =0

m

I k=0

(x(a 1 ) ••• (x 1 -a 1 )ax ••• ax 1 f(a) 1 1 k k 11 k

V~a) 1 ( x). ax ••• a f (a) 1""" k 11 x1 k

with (a) k z (a) 1 ••• z 1 , (1 1 , ••• , 1k) E {1,2, ••• ,m}

1

k

and

11.3.9 Theorem Let n be an open connected set and let f be left monogenic in n. Then the following are equivalent statements: ( i)

(ii)

76

=0

in n; there exists a point a En such that f

for all (1 1, ••• ,1 k) E {1, ••• ,m} k and k E lN. ~

Clearly (i) implies (ii). A= {y E n:ax

••• a 1 x1 1

So let

f (y )

= 0,

V (1

k

1 , ••• ,1 k ) E { 1 , ••• , m} , k E lN }•

k

From the hypothesis of (ii) we know that A I ~. We will show that A is both open and closed in Q; by the connectedness of Q it will follow that A must be n and so f = 0 in n. To see that A is closed in n let y E An n and let (y.) .SN be a sequence in A such that y =lim y .• As each a ••• a f(x) J J j-><x> J x1 x1 1

k

is continuous it follows that f(y) = lim a ••• a f(yJ.) = o, x1 x1 . 1k J->oo 1 k for all k E lN and all (1 1, ••• ,1k) E {1, ••• ,m} k • Soy E A is closed in Q. To see that A is open in n, let y E A and let R > 0 be such that B(y,R) c n. Then in lx-yl < R ax

11

f(x)

••• ax

=

~ k=O

(

L (l 1 , •••

,1k)

0

and, consequently, B(y,R) proof. c 11.4

c

A.

Thus A is open in n and this completes the

Expansions of the Cauchy kernel

11.4.1 In order to solve the problem on the radius of convergence for the Taylor series posed in 11.3.5, the Cauchy kernel E(y-x)

y-x --"--m-+. .1. . , ly-xl

YFx

is developed into a series of inner spherical monogenics in x, depending UROn y, which for each R > 0 converges normally in x E B(O,R) and in y E Rm+ 1,B(O,R). 11.4.2 Starting point is a known series expansion for the potential (m (see [15])

>

1)

77

00

( k-tm-2\

1 Iy-x 1m-1

L \

k

k=O

!x lk

) !Yim+k-1

Lk,m+1(<~, !~!>)

(11.6)

or 00

1 Iy-x 1m-1

L k=O

k 1 i.:lL<xv} k! ' y !Yim-1

(11.7)

which converges, together with all possible series of derivatives, normally in lxl < R < IYI for each R > 0. Each of the terms in (11.6) is a spherical harmonic in x and moreover is harmonic in y E Rm+ 1 since ~ 1 is the fundaJy I o mental solution of the Laplacian ~m+ 1 and the corresponding terms in (11.6) and (11.7) may be identified. The action of the operator Ox on (11.6) and (11.7) leads to the following expansions of the Cauchy kernel in lx! < R < !YI for each R > 0: 00

E(y-x) =

{1<+m-1 \ 1 ( \k+1 ) !Yim+k Kk,m+1,y x)

L k=O

( 11 .8)

or E{y-x)

(11.9)

(11.10)

(11.11) where we have put -

1

KO,m+1 ,y(x) - m-1

y 1-;J

(1

1 °x - x Jk+1 Lk+1,m+1 ( and

78

>)

w1

1••• 1k

(y)

for (1 1, ••• ,1k) E {1,2, ••• ,m} k , k E ~ 0 • 11.4.3 Clearly Kk,m+ 1,y(x), k E ~. is a left and right inner spherical monogenic of order k in x, and it is easily seen that there exists a constant c > 0 such that m

2n/2

1Ko,m+1,y(x)lo

=~

1Kk,m+1,y(x)lo

~ Cm(1+k2)1xlk, k E ~o·

(11.12)

, (1 1, ••• ,1k) E {1,2, ••• ,m} k , k E Jll, are left and Next all functions w1 1 • • • 1k right monogenic in R~+1 with 1i m W1

X-+oo

1 (X) = 0

1• • • k

and are homogeneous of order -(m+k). Hence identifying the homogeneous parts of (11.6) and (11.7) it is obtained that also the functions 1 (k+m-1'\ 1 wm+ 1 k+1 } IYim+k

K (x) k,m+1,y v1

(11, ••• ,1k)

1••• 1k

(x) .w

11 ••• 1k

(y)

are left and right monogenic in y E R~+ 1 with limit value zero at homogeneous of order -(m+k).

oo,

and are

11.4.4 Let us have a closer look at the structure of the functions w1 1 ••• 1'k defined for all (1 1, ••• ,1k) E {1, ••• ,m}k and k EIN. By definition we have i n Rm+ 1( m > 1) 0

79

(-1)k+1 wm+1 H (x) -[ 11···1k - D lxlm+Zk-1 1

=

m-1

o[ ax

••• ax 11

1 lxlm-1 1k

]

]

where H1 is a real-valued homogeneous polynomial of degree k. 1 • • •1 k

w11 ••• 1k only

takes its values in sp{e 0 ,e 1, ••• ,em}. H (x) 11 ••• 1k g (x) = 11••• 1k lxlm+Zk-1

Clearly

Next the function

1 1· t · homogeneous of order - ( m+ k - 1 ) an d as W · monogen1c · 1n · ,...,m+ 1s ~0 11 • • • 1k 1s follows that g1 1 is harmonic in ~m0 + 1 • Now

1... k

f),

g

m+1 11 ••• 1k

X E

(x)

~m+1 0

,

which means that H1 1 is a spherical harmonic of order k. 1••• k we have that H1 ···1 (x)

w

11 ••• 1k

'x)

= DCI.

. , x) =

-~ r· .1 k

o[ lx 1lm+2-1k

Summarizing

]

where g1 is a real-valued harmonic function in Rm0 +1 homogeneous of 1 •••1k order -(k+m-1) 1 while H1 1 is a real-valued spherical harmonic of order , ••• k k. Furthermore 1 (x) = ·m1Zk+ 1 [ lx I2 DH 1 1 (x)-(m+2k-1 )xH 1 1 (x)] 1"""k lxl 1"""k 1"""k (1, •••• ,,k) Hk+1 = --._.---.-lxlm+Zk+1

w1

(11·····1 k)

where Hk+ 1 is a harmonic homogeneous polynomial of degree (k+1) taking values in sp {e 0 ,e 1, ••• ,em}' which is not monogenic since 80

and

11.4.5

Finally notice the following orthogonality relations:

f

W (x) da W (x) = 0 11 ••• lk x s1 ••• st

as(6,R)

for all k, t E J.l, (1 1, .•• ,lk) E {1, ••• ,m} k , (s 1, ••. ,st) E {1, ••• ,m} t and R > 0. Indeed, by Proposition 9.2 this integral is independent of the radius R and so it equals

J

1i m

R~oo aB(O,R)

W

1 1··· 1 k

( x) da

J R-(k+m) W

lim

R~oo

Sm

W

s1 ••• st

X

11••• lk

( x)

(w) Rm da R-(t+m)W (w) w s 1 ••• st

11.5 The radius of convergence of the Taylor series 0

11.5.1 Let f be left monogenic in B(O,R). We saw in 11.3 that there exists an open neighbourhood of the origin A in which f may be developed into its Taylor series (see (11.2)) 00

f(x) =

L

Pkf(x)

k=O where for each k E J.l (see 11.3))

is a left inner spherical monogenic. This Taylor series and all possible derived series converge normally in A.

81

11.5.2 By Cauchy's Integral Formula (Corollary 9.6) we have for any O
I

f(x)

0

E(y-x)dcry f(y), for all x E B(O,R').

aB(O,R') Hence by means of the kernel expansions (11.8), (11.9) and (11.10) we obtain the development 00

f(x)

= I Pk f(x)

(11.13)

k=O which converges normally in B(O,R) and where we have put 0

( Pkf x)

1

= wm+1

(k+m-1) 1 k+1 R'm+k

r

asc6,R')

Kk,m+ 1 ,y(x) day f(y)

(11.14)

(11.15)

So by the uniqueness of the Taylor expansion both series (11.2) and (11.13) coincide in A and hence in B(O,R) too, i.e. 0

0

Pkf(x)

= Pkf(x) in B(O,R).

This means that theo original Taylor series (11.2) and all derived series converge normally in B(O,R). Moreover in view of (11.12) the following Cauchy estimate for the left inner spherical monogenics Pkf' k E ~ 0 • is obtained: (11.17) where the constant C > 0 only depends upon the dimension m and where R' may be chosen arbitrarily in ]O,R[. 11.5.3 Remarks (i) 82

The expansion (11.2) together with (11.15) constitutes a natural

generalization of the Taylor expansion

where d k 1 [(zdu) u].g(u)du J ;iB( ,R')

for a holomorphic function g of one complex variable.

~ K (y) is left inner spherical Iu I +1 1,m+ 1,u monogenic of order 1 in y. Substituting this function for f in formula (11.14) yields the orthogonality relation (ii)

We saw in 11.4.2 that

Kl,m+1,u(x) Iu lm+1

0

_

kl -

1

w1 m+

(k+m-1\ k+1 ) in~

(iii) In 11.3 we also saw that Taylor series (see (11.4' ))

J aB( ,R)

f may be expanded into its multiple

00

L

f(x) =

L

f ( o) 1 ( x ) ax ••• ax 1"""k 11 lk

V1

k=O (1 1, ••• ,lk)

which, considered as a multiple power series, converges normally in it is well known (see [11]) that '"' ( )_k , _-_1 L. k.'

Iy-x lm-1

<x v

• y

>k

1

lyl m- 1 •

~-

Now

(11.7')

k=O this time being considered as a multiple power series in the real variables x0 ,x 1, ••• ,xm• converges normally for lxl < (/Z-1)IYI· This leads to m m ( -1 ) k E(y) (11.9') E(y-x) = L ---r! L x1 ••• x1 ay ••• a L 1 k 11 y\ k=O 11 =0 1 k=O 00

00

L

L

k=O (11, ••• ,1K)

v

(y) 1 (x) w1 11""" k 1""" lk

(11.10')

83

which, considered as a multiple_power series in the real variables x0 ,x 1, •• ,x~ converges normally for Jxl < (/2-1)JyJ. So by Cauchy's Integral Formula we arrive at 00

f

f(x)

wl ••• 1 (y)day f(y) (11.13') 1 k

as(b,R')

which, still considered as a multiple power series, converges normally in B(0,(/2-1)R). From the uniqueness of the multiple Taylor series expansion (Proposition 11.3.7) it follows that (11.4') and (11.13') coincide in A and 0 hence in B(0,(~-1)R) too. This means that the original multiple Taylor series (11.4') converges normally in B(O,(iZ-1)R). This domain of convergence is indeed smaller than the domain of convergence of the corresponding Taylor series (11.2) considered as a series of left inner spherical monogenics. 0

1~

Laurent series and pointwise singularities

12.1

Outer spherical monogenics

Let the function g:aB(O,R) function~ven in 1Rm+1.-...aB(O,R) by 12.1.1

Lemma

f(x)

=

f

~A

be continuous and consider the

E(y-x)day g(y), Jx I I R.

as(b,R) Then (i) (ii)

f is left monogenic in §(O,R); f is left monogenic in 1Rm+ 1-...8(0,R) with lim f(x) x~

= 0 and admits

in 1Rm+ 1-...8(0,R) the following normally converging series expansion: 00

f(x)

= - L Qkg(x) k=O

where wl

with 84

1 (x) 1"""1(

J.ll

1 • 1"""k

k E I'J,

I

V1

aB(O,R)

1 (y)doy g(y).

1" •• k

Differentiation under the integral sign being allowed we have at once that f is left monogenic as well in B(O,R) as in Rm+ 1,§(0,R). Also it is easily checked that lim f(x) = 0. Reverting to the expansion (11.11) of the Cauchy kerne 1

~

E(x-y) =

I( k=O

~

w1

(1 1 , •••

,lk)

which converges normally in in Rm+ 1,§(0,R)

IYI

00

(

~

f(x) = -

\

k=O

1 (x)

1" •• k

< p <

v1

1 (y))

1" •• k

lxl for each

p >

~

\

W (x) ~ 11···\ (1 1 , ••• ,\)

0, it is found that

11 ••• lk}

where the constants involved are given by

I

~11 ••• 1 k

v1

aB(O,R)

1.. • 1 k

(y)doy g(y).

[]

Remarks

12.1.2

(i)

Notice that for each k E ~the function

is homogeneous of order -(k+m) and left monogenic in R~+ 1 with lim Qkg~x)

0.

X+oo

o

(ii) Let f be left monogenic in B(O,R). We already saw that f can be expanded in B(O,R) into its Taylor series (11.5) 00

f(x)

=

~

( V1 ] (X ) \1 1 )• ~ ~ 1 ••• k 1 ••• k k=O (1 1 , ••• ,lk)

Pkf(x)

k=O

Observe that the coefficients are also given by \1

1 1• • • k

=a Xl

••• a 1

X1

k

f(O)

r

J

W1 ••• 1 ( y) doy f (y) 1 aB(O,R') k

( 12. 1 )

85

for any 0

<

R'

<

R.

(iii)

Substituting V (y) for f(y) in the above formula (12.1) we get s 1••• st the orthogonality relations (12.2)

In an analogous way, handling right monogenic functions, we obtain e 0 if (s 1 , ••• ,st) = (1 v (y)da w1 (y) = otherwise. s 1 • • • st Y 1• • • 1k aB(O,R)

I

{0

12.1.3 Theorem

1 , ••• ,lk)

(12.3)

Let f be left monogenic in Rm+ 1,B(O,R) with lim f(x) =A.

m+1 x~ Then in R 'B(O,R), f admits the following normally convergent expansion: 00

f(x)

= A + L Qkf(x)

k=O where for each k E ~ W

11 ••• \

I

(x)

v1

aB(O,R')

1

•••l

(y)day f(y)

(12.4)

k

with R' > R, the summation running over all possible combinations (1 1 , ••• ,lk) out of {1,2, ••• ,m} repetitions being allowed. Proof By Cauchy's Integral Formula outside a ball (Proposition 9.8) we have for any x E Rm+ 1,B(O,R) f(x) = A -

I

E(y-x)day f(y)

aB(O,R") where R" is suitably chosen such that R < R" follows at once from Lemma 12.1.1. c 12.1.4

<

lxl.

The desired result then

Remarks

(i) Notice that the radius R' of the ball involved in the computation of the constants

86

I

aB(O,R')

v1 (y)doy f(y) 1 • • •1 k

may be chosen arbitrarily in so far R' > R, since v1 is right monogenic 1•• • 1 k in Rm+ 1 and f is left monogenic in Rm+ 1,B(O,R). (ii) By means of the expression (11.8) for the Cauchy kernel the homogeneous functions Qkf may also be expressed by Qkf(x) = _1_ (k+m-1) 1 wm+1 k+1 lx lm+k with R'

>

R.

I

Kk,m+1,x(y)doy f(y)

(12.5)

as (o, R' )

In view of (11.12) they also satisfy the Cauchy estimate (12.6)

where the constant C > 0 only depends on m, and R' may be chosen arbitrarily in ] R,+oo[. (iii) We saw in 11.4.3 that the function _1_ (1 +m-1) 1 K (u) wm+ 1 1 +1 IYim+l l,m+1,y is left monogenic in y E R~+ 1 with limit value zero at"" and moreover equals

I (s 1 , ••• ,s 1 )

W

~

1

••• s 1

(y).V s ••• s (u). 1 1

Substituting this function for f(y) in the formulae (12.4) and (12.5) gives rise to

I

I

r

wt

t (x) j vt .• •t (y)doyWs s (y)Vs s (u; k 1 · · · -~ r· · 1 (t 1 , ••• ,tk)(s 1 , ••• ,s 1 ) 1• • • k aB(O,R) 1

I

_ 1 (k+m-1)(1+m-1\ - wr-1 k+1 1 +1 ) m+ aB(O,R) In view of the orthogonality relations (12.3) the left hand side reduces to

87

eSc

=

) (

t1, ••• ,tk , s1, ••• ,sl

)

}:

(t1, .•. ,tk)

wt1 ••• tk (x) vt, ••• tk (u)

0( t 1• • • • • t k ) ' ( s 1' • • • ' s 1 ) ( k+m-1 ) 1 \k1 wm+ 1 IxI m+k +

k,m+ 1,x (u).

K

Consequently the following orthogonality relation holds:

= ____ 1 (k+m-1\ w m+ 1

J

k+1

[ aB( ,R)

12.1.5 In view of the considerations made above it is quite natural to introduce the following definition. Definition A function Qk which is homogeneous of order -(m+k) and left (right) monogenic in R~+ 1 is called a left (right) outer spherical monogenic of order k. Observe that any outer spherical monogenic has limit value zero at oo. As an example we mention the functions w1 1 , (1 1, ••• ,lk) E {1, ••• ,m}k, 1• • • k

all of which are left and right outer spherical monogenics of order k. If f is left monogenic in Rm+ 1,B(O,R) then the function Qkf is a left outer spherical monogenic of order k. The right A-module of left outer spherical monok genies of order k is denoted by Q(r)" 12.1.6 Proposition A basis for the right A-module Q~r) of the left outer spherical monogenics of order k is given by {Wl

l

1• • • k

and dim Qk

: (1 1, ••• ,lk)

E

k {1,2, ••• ,m}}

= 2nM(m,k).

k Proof Let Qk be an arbitrary element of Q(r)" have in Rm+ 1 that

Then by Theorem 12.1.3 we

0

Qk(x)

=

~ S=O

(

L (1 1 , ••• ,ls)

and hence, putting x = rw, 88

Wl1··· ls (x)

~11 ••• ls)

from which it apparently follows that the series on the right hand side only contains the term s = k and so

I (1 1 ' .•• '1 k) To prove the right A-linear independence in Rm+ 1 of the set {W 0

k

11 ••• lk

(1 1 , •.• ,lk) E {1,2, ••• ,m} }, let

I (11, ... ,lk)

W (x) ~ 11 ••• lk 11 ••• lk

0 for all x E Rm0 +1•

Take an arbitrary combination (s 1 , ••• ,sk) E {1,2, .•• ,m}k. (1

I

1 )

(

1•···• k

J aB(O,R)

Then

Vs ••• sk (x)do x W1 ••• 1k (x)\} ~ 1 •• .1k 1 1 1

0

for any R > 0, which in view of the orthogonality relations (12.3) turns into ~

s 1••• sk

=

o.

As the sequence (s 1 , ••. ,sk) was arbitrarily chosen it follows that all constants ~l 1 are zero. o 1• • • k 12.1.7 Let Qk be a left outer spherical monogenic of order k; then by 11.4.4 Qk can be expressed as

or Q k

D[

Hk

l

\x\m+2k-1J

or still

Where g is an A-valued harmonic homogeneous function of order -(m+k-1) in 89

Rm+ 1, Hk is an A-valued harmonic homogeneous polynomial of degree k and Hk+l is an A-valued harmonic homogeneous polynomial of degree (k+1) satisfying x.Hk+ 1(x) (m+2k+1) - - - JxJ2

D Hk+ 1(x) Moreover as

and

the left outer spherical monogenics of order k are eigenfunctions of the in R~+l with respective eigenvalues (k+m) and (-k). operators rand

r

12.1.8 The restriction to the unit sphere Sm of a (left) outer spherical monogenic Qk(x), notation Qk(w), is called a (left) surface outer spherical monogenic. From the considerations made in 12.1.8 it follows that

which means that Qk(w) is an A-valued surface spherical harmonic of order (k+1) and hence an eigenfunction of the Laplace-Beltrami operator ~~+ 1 with eigenvalue -(k+1)(k+m), i.e.

12.2 Laurent series 12.7..1 Here the behaviour of a function which is monogenic lar domain is investigated. The aim is to obtain a Laurent where the inner and outer spherical monogenics will play an as the positive and negative powers of the complex variable classical case.

in an open annuseries expansion analogous role z do in the 0

12.2.2 Theorem Let f be left monogenic in the annular domain G = B(O,R 2 B(O,R 1 ) (0 < R1 < R2 ). Then in G the function f may be expanded into a unique Laurent series

90

00

00

k=O

k=O

)'

where both series converge normally in B(O,R 2 ), respectively Rm+ 1,B(O,R 1 ). The left inner spherical monogenics are given by

with

J

W11 ••• lk(y)do/(y), (l 1 , ••• ,lk) E {1, ••• ,m}k

aB(O,R) while the left outer spherical monogenics are given by

with lJ

l = ( V1 ••• l (y)doy f(y), (1 1 , ••• ,lk) E {1, ••• ,m}k 11··· k aB(b,R) 1 k

the radius R being arbitrarily chosen in ]R 1 ,R 2 [. Proof Take x E G arbitrarily; then there exist R~ and R~ (R~ < R~) such that G' = B(O,R~),B(O,R') c G and x E G'. By Cauchy's Integral Formula

--0

-

f(x) =

J

aG'

E(y-x)do

Y

f(y)

or

with

I

aB(O,R~)

and f2(x) = -

I aB(O,R')

Conclusion by Lemma 12.1.1, Theorem 11.3.6 and Proposition 12.1.7.

o

12.2.3 Remark Assume that f is left monogenic in the annular domain G = §(O,R 2 ),B(O,R 1 ) (0 < R1 < R2 ). Then we just sa~ that f may be expanded 91

into a Laurent series of left inner and outer spherical monogenics which cono m+1 verge normally in respectively B(O.R 2 ) and m 'B(O.R 1 ). Just as in the case of the Taylor series expansion (see 11.5.3) it is possible to expand f into a so-called multiple Laurent series, where the homogeneous terms are not bracketed together. and where the domain of convergence is severely altered. In addition the annular domain has to satisfy a supplementary condition, which says that there exists an open subset 0 G' = B(O,R~ )'B(O.RU with R1 < R~ < R~ < R2 • such that ( /2"+1 )R~ < ( /Z-1 )R~. The expansion of the potential

'I --rr(-1 , k k

1

Iy-x lm-1

=

k=O

1

lxlm-1 •

which, considered as a multiple power series in the real variables y0 .y 1, •••ym, converges normally in IY I < ( /Z-1) lx I, leads to m

00

-E(y-x) =

l: k=O

which, still considered as a multiple power series, converges normally in IYI < (t~-1)lxl and which can be re-arranged into 00

-E(y-x) =

I

L

k=O (1 1, ••• ,lk)

wl

1 (x).Vl

1""" k

So for

I

aB(O,R~)

we get

This finally results into

92

E(y-x)doy f(y)

1 (y).

1""" k

00

f(x)

I k=O

I

vl1 ••• ll<(x) >q1 •• .lk

I

W (x) 11 ••• 11<

n1•••• ,lk>

00

+

I

k=O (1 I' ... ,11<)

lJ

11 ••• lk

where the first series converges normally in B(O,l2-1)R 2 ) and the second in Rm+ 1,B(/Z+1)R1 ). It can be shown that this multiple Laurent series is unique. 12.2.4

Remark The Laurent series expansion 00

00

f(x) = I Pk (x) k=O

I

+

Qk f(x)

k=O with

and k

Q f(x) = (-1) __1__ k ~ wm+1 appears as a direct generalization of the classical Laurent series expansion 00

g(z) =

I

k=O

k

azz +

00

I

1

bk ~ k=O z

with

and

of a function g(z) which is holomorphic in an annular domain.

93

12.3 Singular points 12.3.1

Definitions

(i) A point a E 1Rm+ 1 is called a left monogenic point of the function f if and on 1y if there exists an open neighbourhood r1 of a in which f is 1eft a monogenic. (ii) A point a E 1Rm+ 1 is called a left sing11lar point of the function f if and only if for any R > 0, f t M(r)(B(a,R);A). (iii) A point a E 1Rm+ 1 is called an icolated left c·ingular· point of the function f if it is 0 a left singular point of f for which there exists R > 0 such that f E M(r)(B(a,R),{a};A). 12.3.2 Let a E 1Rm+ 1 be an isolated left singular point of the function f; then in an annular domain centered at a f may be developed into a Laurent series

k=O

k=O

Definitions (i)

The isolated left singular point a off is called a left pole of order p if the second series in the Laurent expansion of f about a breaks off from k ~ p. This breaking off second series is called the singular part of f at a. (ii) The isolated left singular point a off is called a left essential singular point if the second series does not break off. 12.3.3 Definition A function f is called (left) meromorphic in an open set Q if there exists a subset S c Q such that (i) (ii) (iii)

S has no accumulation point in Q; f is (left) monogenic in ~S; f has a (left) pole at each point of S.

Notice that condition (i) implies that no compact subset of Q can contain an infinity of points of S; this means that S is at most countable. So if f is left meromorphic in 1Rm+ 1 there are three possibilities accordingly as the set S of left poles of f is empty, finite or countable.

94

12.3.4 Consider a function f which is left meromorphic in Rm+ 1 If the set of left poles is empty then the function f is called left entire. Theorem

If f is left entire then f admits the expansion

I (

f(x)

k=O

v1

L

(1 1 , ••• , 1 k)

1 (x).a

1""" k

~

f(o>)

••• a

~

1

k

converging normally in the whole space Rm+ 1 • Proof The Taylor series expansion off about the origin is valid in the -m+1 whole spaceR • c 12.3.5 Consider a function f which is left meromorphic in Rm+ 1 and suppose that the set of the left poles off is the finite set {a 1, ••• ,aj}. If pi denotes the order of the left pole ai and Gi(x) its singular part then j

L

f -

Gi

i =1

So f admits in Rm+ 1 the following expansion:

is left entire.

I ((1

f ( x)

k=O

L

vl 1 ••• 1 k ( x) • "1 1 ••• 1

1 , ••• ,lk)

J

. p.-1 J

1

(

I I .

+

1

(a.)

I

=1 k=0 (1 1 , ••• , 1 k)

(.)

)

w.1 1 1 Cx>.~l1 "" 1···k

1 •

1"""k

12.3.6 For meromorphic functions with a countable set of poles we prove Mittag-Leffler's Theorem. Theorem Let {a.}.ElN be a sequence of distinct points in Rm+ 1 without an 1 1 accumulation point in Rm+ 1 , such that with each point ai there corresponds a natural number pi and a function p.-1 G.(x)

=

1L· (

1

L

\

k=O

(1 1 , ••• ,lk)

w(ai) 11 ••• lk

(x)~(i)

)

11···lk.

Then there is at least one left meromorphic funct:on f in Rm+ 1 the left poles 95

of which are exactly the points a;• off at a; is G;(x).

E ~. and such that the singular part

Proof Order the points a; in such a way that their respective Euclidean norms form a non-decreasing sequence

o~

Ia 1 I ~ Ia2 I ~ • • •

For each i E ~a suitable polynomial Z; will be constructed such that the series 00

is the desired left meromorphic function. If possibly a 1 should be the origin then take z1 = 0. For each a. the singular part G; is left monogenic in 1 Rm+1 '{a;}· Take a and a' to be fixed and such that 0
G;(x)

0

= L PkGi(x) in B;. k=O

Hence for each E > 0 there exists a natural number N;(E) such that

N; sup IG;(x) - L x E Bi(O,a'ala; I) k=O

PkGi(x)

10

-S 21

or, putting

N; Z;(x)

L

PkGi(x),

k=O such that supiG-(x)-Z.(x)l <-S. 1 1 0 21 xEB!1 Take R > 0 arbitrarily and put R' = a:' • There are at most a finite number of points a; in the ball B(O,R'), say

Observe that if aj ElRm+l,B(O,R') then lajl 96

>

R' or a'alajl

>

R which means

that B(O,R) Put

c

Bj.

Now define two functions f1 and f 2 in the following way.

t

f 1(x) =

L (Gi(x)-Zi(x));

i=1 0 clearly this function f1 is left meromorphic in B(O,R) with left poles at a 1 , ••• ,at' (t' ~ t) and corresponding singular parts Gi(i = 1, •.• ,t'). Next put 00

L

(Gi(x)-Zi(x)),

i=t+1 where it is known that for each i

>

t

sup !Gi(x)-Zi(x)\ 0 XEB(O,R) 00

00

L (Gi-Zi) will converge normally on 2 i=t+1 i=t+1 B(O,R); so by 0 the Weierstrass Theorem (Theorem 9.11) f 2 (x) will be left monogenic in B(O,R). Hence the function As

L

~converges the series

00

fl + f2 =

L

(Gi-Zi)

i =1 0

is left meromorphic in B(O,R) with left poles at a 1 , ••• ,at, and corresponding singular parts Gi(i = 1, ••• ,t'). Now let R->-+oo to obtain the desired function. c 12.3.7 Remark The structure of the most general left meromorphic function f in Rm+ 1 with prescribed poles and singular parts is

where h is left entire. 12.3.8 Definition Let f be left monogenic in the open set n except for the left singular point a E n. This point a is said to be a removable left sin9ularity of f if there exists a left monogenic function h in n such that h(x) = f(x) for all x E ~{a}. To determine whether or not a left singular point is removable the following criterion is available. 97

12.3.9 Theorem Let n be an open set containing the origin and let f be left monogenic in ~{0}. The isolated left singularity at the origin is removable if and only if there exist R > 0 and M > 0 such that for all xE{y:O
00

L

f(x) =

I

Pkf(x) +

k=O

Qkf(x).

k=O

In view of (12.6)

~ M.c(k~~11)(1+k2)(1:ll)m+k

IQkf(x)lo

and as R1 may be chosen arbitrarily small, it is obtained that Qkf(x) = 0 for all k E ~. and so

~L (

00

f(x)

L

k=O (1 1, ••• ,lk)

v1 1""" 1k (x).,\1 1""" 1k ')

0

for all x E B(O,R 2 ),{0}. f(x), h(x)

Define an extension hoff by ~{0}

X

E

X

= 0

= { .\0

,

then it is clear that h is left monogenic in n. 12.3.10

Proposition C > 0, 1, s E ~.

Iff is left monogenic in m~+ 1 and satisfies for some

lf(x)l 0 ~ C(1 + 1x1 1 + then in m~+ 1 98

o

~+s), for all x # 0 lxl

f(x)

s

1

I

Pk f(x) +

k=O

I

Qk f(x).

k=O

Proof Let 00

f(x) =

00

I

Pkf(x) +

k=O

I

Qkf(x)

k=O

be the Laurent ~eries expansion of f about the origin. hypothesis made the following estimates hold: IP kf(x)l o

~

1 2 )(~)k c·(k+mk-1 \(1+k } R

~-

c·(k+m-1\(1+k2)( R )m+k(1+Rl+R-m-s) k-1 ) TxT

<

Then in view of the

(1+R1+R-m-s)

and IQ f(x)l k 0

for all R E ]O,+oo(.

Hence

lim !Pkf(x)j 0 = 0 if k > 1 R-->+oo and lim 1Qkf(x)l 0 =0ifk>s R-+0+ which reduces the Laurent expansion in R~+ 1 to 1 s f(x) = L Pkf(x) + L Qkf(x). c k=O k=O 12.3.11 Corollary (Liouville) If f is left entire and satisfies jf(x) 10 for all x E Rm+ 1 then f must be a constant function in Rm+ 1•

;;;

~

As f is left entire the Laurent series expansion about the origin of f reduces to the Taylor series expansion. With 1 = 0 the above Proposition Yields f(x) = P0 f(x) = f(O) for all x E Rm+1 • c 12.3.12 Corollary

Iff is left entire and lim f(x) exists, then f must be

a constant function in Rm+ 1• 99

M

12.3.13 Remark In the classical theory of holomorphic functions of one complex variable it is known that if f is a meromorphic function then lim f(z) = z-+a

oo

for each pole a of f; it is even so that putting f(a)

= oo

at each pole a, the meromorphic function f : [ . . . [ u {oo} becomes continuous. In Clifford analysis (m > 1) this is not anymore the case: a meromorphic function must not "explode" at each of its poles. This is illustrated by the following example. Take the function w1 with all 1 1.(i = 1, ••• ,k) 1 • • • 1k different. Then in the expression W1 l (x) 1"""k

=

lxl

m!2k+1 (ixi 2 5 H1 l (x)-(m+2k-1)i H1 l (x)) 1"""k 1"""k

the harmonic homogeneous polynomials H1

1 and 5H 1

x

,xm).

(k-1) respectively, only depend upon

w1

1 • • • 1k

(x 0 ,0) =

1""" k = (x 1, •••

1 of degree k and

1""" k

So for

x = 0:

o

which implies that when approaching the pole 0 along the x0 -axis the function w1 1 has limit value zero. 1• • • k

12.4

Residue theory

12.4.1 Assume that f has an isolated left singular point at the origin. first term in the second Laurent series of f about 0 reads

The

-

X

llo·

Definition The constant - 1- ll is called the residue of f at the left sinwm+1 o llo gular point 0; notation: Res 0f = - - • It follows from Theorem 12.2.2 that wm+1

J

day

f(y).

aB(O,R) However in the case of a pole we can prove 100

12.4.2 then

Proposition

Let 0 be a left pole of order {p+1) of the function f;

~

As 0 is a left pole of order {p+1) off we have in a suitable annular domain about the origin: p

00

f(x)

= I Pkf(x) + I Qkf(x) k=O

k=O

fdx)

Now 1 dp p+m-1 lim -p! --p [ixl xf1(x)] = 0 x-+0 d Ixi and

while, for 1

~

k

~

p, and putting x = lxlw, w E Sm,

lxlp+m-1x Qkf(x) = lxlp+m-1xlxl-(k+m)Qkf(w) lxlp-kw Qkf(w) and hence lim -~ ~p [ixlp+m- 1x Qkf(x)] x-+0 p. dlxl =lim . x-+0

1 pr.

dp p-k --p [ixl wQkf(w)] dlx I

=0 Which yields the desired result.

c

101

12.4.3 Theorem LetS be an (m+1)-dimensional compact differentiable oriented manifold-with-boundary contained in an open set r1. and let f be left monogenic in~ except for a finite number a 1, ••. ,at of isolated left singular points, 0 such that a.~ as, j = 1, ••• ,t, and a. , ••• ,a. E s, {i 1, ••• ,ik} c {1, ••• ,t}. J 11 1k Then k

I

daf

as

=~

+

1

Res a. f. 1s s=1 t

t..

Proof TakeRs such that B(a. ,R) c S for all s = 1, ••• ,k and B(a. ,R) n 1s s 1s s B(aj ,Rs,) =¢for all s,s' = 1, ••• ,k. Then f is left monogenic in s ~

k

z

S=1

B(ai ,Rs) and applying Cauchy's Theorem we find s k

Ias

dcrf

I

= L

daf

s=1 aB(a. ,R ) 1s s k

= wm+ 1

L Resa. s=1

f.

c

1s

13. Adjoint monogenic functions 13.1 A spherical transformation is defined turning left monogenic functions in B(O,R) into right monogenic functions in Rm+ 1,B(0,1/R) and vice versa. This spherical transformation resembles the transformation f(z)

+

f(.!.).!.

zz

from classical function theory, but only as far as the outlook is concerned, both transformations being essentially different from each other. The spherical transformation is only aefined for m > 1 and may not be considered as a generalization of the above cited classical one. 13.2 As a starting point the function 1

IY

1

1

-~~m- 1 .1ulm- 1 llulyl~lm- 1

(m

>

1)

is considered; it is harmonic in both variables u and y as long as y 1 u --.

This means that the function

.!.u

=

s(y,u) is left monogenic in y and right monogenic in u as long as y 1 ~· function s(y,u) can be rewritten as Du 1 1 s(y,u) = E(y - -) 1 1-m' u lulm-

1 y fu

which gives rise to the following expansions. we get s(y,u) =

~

_1_ (k+m-1\

k~O wm+ 1

This

1

K

k+1 } IYim+k

If IY I

the series being normally convergent in lu IIYI

u

>

1

-u- or Iu IIY I > 1

1 Du lulm-1 • 1-m

(1)

k,m+1,y

>

( 13. 1)

1, or still

I

s(y,u) =

k=O ( 1 1, .•• , 1 k) On the contrary ifM> IYI or luiiYI -s(y,u) =

I k=O

<

1 we get

_!_(y) 1 Du _1_ (k+m-1\lulm+k K k,m+1,u lulm-1 1-m wm+ 1 \ k+1 }

( 13.3)

or -s(y,u)

(13.4)

both expansions being normally convergent in luiiYI 0 13.3 Assume that f is left monogenic in B(O,R). monogenic in B(O,R) and the function

1

sf(u) = f(-) G

1 -:::--r-

Iu Im-1

< 1.

Then f(x) becomes right

Du 1-m

turns out to be right monogenic in ~m+ 1 ,B(O,*) with lim sf(u) = 0.

This

U-+oo

function sf is called the sphericaZ transform of f. It can be put into integral form by proceeding as follows. By Cauchy's Integral Formula

103

I

f'<x>

f(.Y)day E(y-x)

aB(O,R-d 1

and thus in I u I

>

R'

I

f{.Y)doy E(y-~)

I aB(O,R-E:)

f{.Y)doy s(y,u)

sf(u) =


5u

1

I u 1m-1 1-m

or sf(u)

where E:

>

0 is suitably chosen.

13.4 Let us give two examples of spherically transformed functions. 13.4.1

For the constant function f(x) = e0 we have

13.4.2 Let Pk be a left inner spherical monogenic of order k; then sPk(u) = (Pk =

(*)

(i5 k(u)

5

lul~-1)1-~ 1

lulm+2k-1

),5u

-m

5 = Pk(u).( - ( ) = Pk u.

1 ) ---1 u lulm+Zk-1 -m

luI

um+2k+1

m+2k-1 m-1

which is clearly a right outer spherical monogenic of order k. 13.5 Making use of the expansion (13.2) for s(y,u) it is obtained that sf(u) =

I

L

(

I

f{Y)daywl

k=0(1 1, ••• ,lk) ClB(O,R-E:) 1

(v

1 (y) \ 1 1 (.!) 1""".k 1 1··· ku _

°u)

Iu Im-1 1-m • 104

But as

I

I

r\

(l1' ... ,lk) aB(O,R-d

L L (l 1, •.•

a

I

((-1)m

(1 1, ••• ,lk)

B f'(y)aaywl, ••• l (y)\vl (,) , u k ) , ••• lk iulm-1 1-m

f(,Y)day-wl

aB(O,R-E)

(\

J

f{Y)day(-1)kw 1

,lk) aB(O,R-E)

(~) ~-1

1 CY))vl 1 1··· k 1··· k u lui

10-um

B (y)\v (1) 1 u 1••• 1 k ) 11••• 1k u Iu Im-1 1-m

this expansion for sf(u) in lui >*reduces to sf(u) =

L

sPkf(u)

k=O which is obviously the Laurent series expansion of sf(u) in mm+l,B(O,*) 13.6 Assume that g is left monogenic in ffim+l,B(O,R) with lim g(x)

= o.

X-><x>

Then g(x) is right monogenic in mm+l,B(O,R) with lim g(x) X-><x> function

= 0, and the

°u

1 1 sg(u) = g(-) 1 1-m ii !ulm-

is right monogenic in B(O,*). It is called the spherical transform of g. To put it in an integral form we proceed as follows. By Cauchy's Integral Formula outside a ball

9(x)

-I

g{.Y)doy E(y-x)

aB(O,R+E) 1 and thus in Iu I < R sg(u)

J

g(Y)day E(y-~)

aB(O,R+E)

I

1 !ulm-1

Bu 1-m

g(y)day s(y,u)

aB(O,R+E) 105

where E > 0 is suitably chosen. 13.7 Let us give two examples of spherical transforms of left monogenic functions outside a ball.

;:~ e0 •

13.7.1

For g(x) = W0 (x) = E(x) we get sg(u) = - 1-

13.7.2

Let Qk be a left outer spherical monogenic of order k; then

~+1

B

sQk(u) = (Qk(i)

lul~-1)1-~

_ - ( )-I m+2k-1 m+2k+1 - 0k u u u 1 1-m which is clearly a right inner spherical monogenic of order k. 13.8 Making use of the expansion (13.4) for s(y,u) it is obtained that sg(u)

i

=

r

( J

9(Y)dcry\ ••• 1, (y)(w], ... J,

k=O(l 1, ••• ,lk) aB(O,R+E)

(~) 1•1~-1 ,n:'.}

But as 1 1 °u sQkg(u) = Qkg(-_) 1 -1u lulm-m

( \

J

(1 1, ••• ,1 k) aB(O,R+t:)

(\

l:

J

(1 1, ••• , lk) ClB(O,R+E)

-

-

g(y)aayvl

\(1 1 °u 1 ;w..1 1 -=-> 1 -11··· k - 1··· k u lulm-m

k (y)\(_ 1)kw 1 1 °u 9G>doy<-n vl1 ••• lk J 11 ... lklulm-11-iii'

this expansion for sg(u) in lui <*reduces to 00

sg(u) =

l:

~Qkg(u)

k=O 1 which is obviously the Taylor series expansion of sg 1n B(O,R). •

106

0

13.9 (0 <

Definition Iff is left monogenic in the annular domain B(O,R 2 ),B(O,R 1 ) R1 < R2 ) then the spherical transform off is defined by sf(u)

1 (f (-_)

u lui

°u

1 ) m-1 1-m·

" 1 1 Notice that sf is then right monogenic in B(O,~),B(O,R;).

13.10

Lemma

0

Iff is left monogenic in B(O,R) then the series

00

L

m-1 m+Zk-1 sPkf(u)

k=O converges to a right monogenic function in Rm+ 1,8(o,{> with limit value zero at oo. 0

Proof If f is left monogenic in B(O,R) then sf is right monogenic in 1 Rm+ ,sco.{> with lim sf(u) = 0, and its Laurent series u--

L

sf(u)

sPkf(u)

k=O converges normally in Rm+ 1,8(o,{>. Hence the cited series will a fortiori converge normally to a right monogenic function in Rm+ 1,8(o,{> with limit value zero at oo. c 13.11 Lemma the series

If g is left monogenic in Rm+ 1,B(O,R) with lim g(x)

0 then

k=O converges to a right monogenic function in B(O,{). ~

Analogous to the proof of the preceding Lemma 13.10.

c

13.12 Iff is left monogenic in the annular domain B(O,R 2 ),B(O,R 1 ) then~ can be expanded into its Laurent series

00

L k=O

00

Pkf(x) +

L

Qkf(x)

k=O

107

both series being normally convergent in B(O,R 2 ) and Rm+ 1,B(O,R 1 ) respectively. By Lemmas 13.10 and 13.11 the next definition is meaningful.

"'

f(u)

m-1

oo

= I m+2K-1 k=O

is called the adjoint of f. "' 0 N~tice that f € M(l)(B(0,1/R 1 ),B(0,1/R 2 )) and that equivalent expressions for f(u) are ~

00

I

f(u) =

-

Pkf(u)

k=O

00

u + l ulm+2k+1

I

Qkf(u)u lulm+2k-1

k=O

00

I

"'

Pkf(u) +

k=O

14.

I

00

"'

Qkf.

k=O

A product of monogenic functions

14.1 An open rectangular parallelepiped in Rm with center at the origin and edges 2p = (2p 1, ••• ,2pm) will be denoted by Cm(O,p). By Cm+ 1(0,p) we denote the open set in Rm+ 1 given by -+

Cm+ 1(0,p) =

{X

m+1 ER :

X~

+

x~ < p~,

1,2, ••• ,m}.

Notice that cm+ 1(0,p) n Rm = cm(O,o) and that Cm+ 1(0,p) is in fact a kind of curved pyramid with basis Cm(O,p). 14.2 Theorem Let f(x) be an A-valued analytic function such that its Taylor series about 0 converges as a multiple power series in x 1, ••• ,xm inC (O,p). Then there exists a function f* which is left monogenic in Cm+ 1 (o,2-m~2p) such that f*1Cm(0,2-m/ 2p) = f.

108

-

proof

It is given that ....

I

f(xl =

m

1

00

m

x, ... x, ),,

I

IT

k=O

1

11=1

k

1 1 ••• k

1vith A.11··· lk =ax

f(O)

••• ax 11

\

converges as a multiple power series in Cm(O,p).

This means that for every

-.-P. such that 0 < p~ < P· for all i = 1, ... ,m 1

m

I

L

11=1

1k=1

1

I n k=M

1

m

N

if inf(M,N) .... +

p,' ••• p,' I Al 1

k

1

I ....

1 ••• k 0

0

Now call for each N E ~

oo.

N

I

SN(x)

v11 ••• 1k(x)

I

k=O (11·····\) N m m 1

...

I TI I

k=O

11 =1

I

lk=1

>-,1 ••• 1k

z, ••• z, A. . 1 k 11 ••• 1k'

then

;;

sup L I IV 1 1 (x)l 0 IA. 1 ••• >-. 1 10 EC 1• • • k 1 k x m+ 1 k =M ( 1 1 , ••• , 1 k ) N m m

;;

I xECm+1 k=M

m

N

L ••• L lk1

L •• • L

IT k=M

Jr

1 =1 1

lk=1

pl ... pl IAl

1 =1 1 k

lz, 1 10 • •• 1z, k 10 1>-, 1 ••• 1k I0

k

I0

••• 1

1

k

which tends to zero if inf(M,N) .... + oo. By Weierstrass's Theorem (Theorem 9.11) there ought to exist a function f* E M(r)(Cm+ 1(o,2-m/ 2p);A) such that f*{x) = lim SN(x) N-+oo

109

uniformly on each compact subset of C 1(o,2-m/ 2p). + -m/2+ m+ that f*(O+x) = f(x) in cm(0,2 p). c

Moreover it is clear

n

14.3 Definition If cRm is open then an open neighbourhood n of Din Rm+ 1 is said to be x 0 -nor'1Ti.al i f for each x E ~2 the 1 ine segment {x+te 0 } n \~ + is connected and contains just one point in n. + m 14.4 Proposition Let~ cR be open and let f be an A-valued analytic function in Then there exist a maximal x0 -normal open neighbourhood Q of 1 in Rm+ and a unique f* E M(r)(st;A) such that f*(O+x) = f(x).

n.

n

n;

Proof Let y be an arbitrary point in then there exists p* such that f(x) may be developed into its Taylor series in Cm(y,p*). So by the above Theorem 14.2 there exists a function f~ E M(r)(Cm+ 1(y,p);A) such that f;(o+x) = f(x) in Cm+ 1(y,p) n Rm

= Cm(y,p), with p = 2-m/ 2p*. Moreover if = f; in cm+ 1(y,p) n cm+ 1(i,p'). Indeed, o in cm(y,p) n cm(z,p') and as the function

cm+ 1(y,p) n cm+ 1(z,p') 1 ¢then f; (f;- fi)(O,x)

= f(x)-f(x)

f;- fi is left monogenic in Cm+ 1(y,p) n Cm+ 1(y,p') it follows that f~-ff vanishes in this connected set. So it is possible to gather the extensions to one left monogenic extension of f in the x0 -normal neighbourhood +u+ cm+ 1cYl yHl

of

+

s-2.

Next, if Q1 and Q2 are two x -normal neighbourhoods of Q and f~ , f~ 0 ''1 >12 are the corresponding left monogenic extensions of f, then Q1 u Q2 is again an x0 -normal neighbourhood of~- As f5 1 ID 1 n nz and f5 2 IS11 n nz are left monogenic extensions of f(x) in n 1 n nz, it follows that

which means that both extensions may be gathered to one left monogenic extension in n1 u rlz. Hence the union of all x0 -normal neighbourhoods of Q in which there exists a left monogenic extension of f is the maximal x0 -normal neighbourhood of Q with a left monogenic extension. Clearly both this neighbourhood and the extension are unique. c 110

14.5 Definition Let S1 clRm be open and let f be an A-valued analytic func. + tion 1n Q. The maximal left monogenic extension f* of f, as constructed in proposition 14.4, is called the left Cauchy-Kowalew3ki exten3ion (C-K extension) of f. 14.6 Corollary Let f be an A-valued analytic function in the whole of IRm. Then its (left) C-K extension is a (left) entire function. 14.7 Remark Iff and g are A-valued analytic functions in IRm and A E A then the left C-K extension of f + gA is given by (f+gA)* = f* + g*A· 14.8 Theorem Let f be an A-valued analytic function in~ clRm. function f* given by 2k+1 oo k [ X k ] f*(x) I (-1) 5 (Zk+ 1) 1 ~mf(x) k=O

Then the

is left monogenic in a neighbourhood Q of S1 in 1Rm+ 1 and satisfies f*(O+x) f(x) in S1. If moreover f is analytic in the whole of IRm then f* is left entire.

n;

Proof Let K be a compact subset of CK and AK' depending upon K, such that

then there exist positive constants

sup ~~~f(x)l 0 ~ cK(2k)!A~ XEK

111

In this way it is easily seen that the given series and all derived series will converge normally in

- , _1_[=() IrK IrK

0

Kx]-

U

OGo

Kc::;Q

This means that f* E E(r)(n;A).

Clearly f*(O+x) 2k+1 k [ xo k J (-1) ~+ 1 (Zk+,J! ~f(x)

00

of*(x) = ~ k=O 00

( -1)k

~

2k-1 xo (2k-1)! ll~f(x) +

k=1

f(x) and in n

00

l:

( -1)k

2k+1 xo 1 (2k+1)! ll~+ f(x)

k=O

o. In the particular case where f is analytic in the whole of Rm then for any compact set K c: Rm and any >. > 0 there exists a cK,>. > 0 such that la!Akf(x) I

sup

X€K

XIll

~

CK ,(2k+IBI)!>.k. 'A

So the cited series and all derived series will converge normally in Rm+ 1• c 14.9 Remark It follows from the above theorem that if f is a real-valued analytic function inn c:Rm, then its C-K extension takes values in sp{e0 ,e 1, ••• ,em} (see also 11.2.7(ii) and Theorem 11.3.4.). 14.10 Proposition Let f be an A-valued analytic function inn c:Rm and let f* E M(r)(Q;A) be its left C-K extension. Then for any multi-index s E ~m. a!f*(x) = [asf(x)J* in n.

X

X

-+

Proof The function a~f is analytic in n and possesses a left C-K extension ~* in a certain x~-normal neighbourhood Q' of Q. On the other hand the

fu~ction

a!f* is left monogenic in Q and its restriction to n is easily shown X

-+

to be precisely a~f. In view of the uniqueness of the C-K extension, Q c: Q' X and the statement follows. c 14.11 112

Now assume f and g to be left entire functions; then their restrictions

fiRm and giRm are A-valued analytic functions in Rm.

so the product

remains analytic in Rm. It therefore possesses a left entire C-K extension · ex t ens1on · · denote d by in Rm+ 1• Th 1s 1s

and called the Cauchy-Kowalewski product (C-K product) of the left entire functions f and g. Of course an analogous definition holds for right entire functions; the corresponding notation is GR. 14.12 (i) (ii) (iii) ( i v)

14. 13

Remark The C-K product shows the following properties. The C-K product is associative. If fiRm. giRm = giRm. fiRm then fG Lg 1elf = fQL1 =f. m+1 M(r)(R ;A) ,+,GL is a real algebra.

gGLf.

Examp 1es

14.13.1 As the C-K extension of a Clifford number A is A itself, it is clear that AQL~ = AGR~ = A~, YA,~ E A. 14.13.2 The C-K exter.sion of X; (i = 1, ••• ,m) is z; = xieo- x0 e;; the C-K extension of x.x. is ~(z.z. + z.z.) if I j or z~ if i = j; this means that 1

zielzj

J

c.:

1

1

J

J

1

Zi@RZj = zr(zizj + Z/i)

1

Vij(x) if i f j

and z 1.Glz 1. = z.GRz. .. (x) if i = j. 1 1 = z~1 = 2! v11 As the left- and right C-K product of Z; and zj coincide, the subscripts L and R may be dropped. Moreover the C-K product of z; and zj is commutativ~ (see 14.12(ii)) and so we arrive at n1 nm n 1 ! ••• n ! v1 z 1 G •.• G zm 1 ( x) m 1" •• k where n; stands for the number of times that i appears in (1 1, ••• ,lk). 113

14.13.3 Further we have

where ni and ni are the number of times that i appears in (1 1, ••• ,lk) and (s 1, ••• ,st) respectively. 14.13.4

Iff and g are left entire functions with Taylor expansions

f(x) = ~L k=O

(

\' L

(1 1, ••• , 1k)

vl

1 (x)ax

1• • • k

1

••• ax 1

f(o))

1 k

and

vs ••• st (x) axs ••• axs g(o)) 1

g(x)

1

t

then their left C-K product is expanded in the following manner: m , oo f IT (ni+ni\lv fGLg(x) = L L L l. ni JJ 11 ••• lks 1••• st(x) k,t=O (1 1, ••• ,lk)(s 1, ••• ,st) 1=1

a

xl

••• a

xl

1

f(O).a k

xs

1

••• a xs

g(O) t

14.4 Remark Assume that f is analytic in the whole of Rm; then it admits in Rm a Taylor expansion which can be written in the following forms: m m 1 f(x) = I TI I k=O 11=1 ....

00

00

k=O (1 1, ••• , 1 k)

1 A ••• x1 k 11••• 1k l nlxl n1···· m· 1

00

k=O n1+••• +nrn=k

n1 •I • • • nm·I

n1 n m,* x1 • • • xm /\n

n

1• • • m

where ni stands for the number of times that i appears in the combination (1 1, ••• ,1k) and the meaning of the An* n is obvious. We already know from 1• • • m

Corollary 14.6 that the left C-K extension f* of f is a left entire function. Its Taylor expansion about the origin holds in the whole of Rm+ 1 and reads 114

f*(x)

m

1

00

I

I

TI

k=O

m

1 1=1

v1 (x) ~ I 1 •• •1 k k=O (1 1 , ••. ,1 k)

k=O (1 1 , ••• ,lk)

)I

11 •••1 k

n1! ... nm!

~ k=O

14.15 Proposition (Leibniz's Rule) then for any multi-index RE ~m

1~( feL g) X

I ( ! ) a~feL

=

++

o~

B

a

X

Iff and g are left entire functions

a!-C:g. X

Proof In view of Proposition 14.10 and Remark 14.7 we have consecutively, denoting the C-K extension by *,

a~

(fGLg) =

+

r!) ...,af"'"'L

I

c;c:s \a 15. 15.1

l
a+ X

+

+

"~-ag. a~

CJ

X

Elementary monogenic functions For>.. E JR 0 the lR-valued functions ±A2x~

e

J, j = 1 , ••• ,m

are analytic in lRm.

Their left and right entire C-K extensions are denoted

by exp(±>.. 2 z~).

J

j

1, ••. ,m, 115

and admit the following Taylor expansion:

I

exp(±\2z~) = J

k=O

~ (±\z)k z~k. ":

J

Their left and right C-K product is the C-K extension of the function

and can be expanded into

~

G exp(±\2z~)

j= 1

I

=

k\ (±\z)k (zl

+ •••

+\2z2 This function generalizes e, \ E R0 , z E 15.2

+

z~)kG

~.

The real valued functions x. e

••• ,m

1 ,i=1,2,

Their left and right C-K extensions are denoted by

all are analytic in Rm.

exp(zi), i = 1, .•• ,m and admit as Taylor expansion: exp(z.) 1

1

= I

k

r.rZ .• II.: 1

k=O

The C-K product of those entire functions exp(zi), viz. m

exp(x) = n G exp(z.) 1

i =1

is the entire extension of the analytic function m

exp(x)

= n exp(x.) 1

i =1

m

e

IT

i=1

and can be developed into 00

exp(x) =

I

k=O

116

•

k=O

J

1

kT (z 1 + ••• + zm)

k0

X 1+ •••• +X

m

00

l:

l:

n

k=O n1+••• +nm=k 00

I

n

I

1 • • • • m•

1

l:u k=O 00

This exponential function which clearly generalizes the function ez, z E t, satisfies a lot of properties quite similar to those of this classical ez. It follows out of its definition that exp(x) is left and right entire; moreover it has no zeros and exp(O) = e0 • Next for all x and y in Rm+l, exp(x+y) = exp(x).exp(y) = exp(y).exp(x) relation which can be proved by multiplying the corresponding Taylor expansions. Now m

00

l:

kT

k=O but as

m

2q

l:

2q+1

1, •••• ,lk=1

we have

Hence

and so Jexp(x)J 0 While exp(x) = e 0 for x =

(~n,o), k E

z. 117

This means that exp(x) is periodic with real period 2n.

vm

15.3 We define the A-valued functions exp(x.e.) = e cos x. +e. sin x., j = 1, ••• ,m. JJ 0 J J J They are analytic in mm with Taylor expansion J J

1 k k T."Tx. e .• 11.: J J

00

L

exp(x.e.) =

k=O

Their left C-K extensions are 00

L ~ z~e~,

exp(zjej) =

j

1, ••• ,m.

k=O Next consider the A-valued analytic function ~ m E(x) = IT exp(x.e.) j=1 J J 00

I

1 k1!. •• km!

I

k=O k1+••• +km=k m m 1 00

I TI I

k=O

... I

11 =1

1 k=1

X

k1 km k1 km ••• xm e 1 ••• em x1

k1 km x 1 ••• xl e 1 ••• em 1 k

where in the last expansion k.1 stands for the number of times that i appears . in the combination (1 1, ••• ,lk). Its left C-K extension E(x) is precisely the left C-K product of the functions exp(zjej): m

E(x) =

IT

j=1

~L

exp(zJ.eJ.)

k k k1 z 1~ ~ z m e1 1: m 1 k1!. •• km! k=O k1+ ••• +k r.l =k m m k1 km 1 1: TI 2 1: zl ••• zl e 1 ••• em 1k=1 1 k k=O 11 =1 00

00

I

00

118

...

00

I

I

k=O (11, ••• ,lk)

v11 ••• 1k(x)

k1 km e 1 ••• em

1 k I k I zl 1 k=O (11, ••• ,lk) 1 · • • • m·

I

I

G ••• G Z l

k

e

k1

km ••• em

In the particular case where m=1 this exponential function reduces to exp(z 1e 1 ) which can be identified with e-iz, z E [. The functions cos(tjzj) and sin(tjzj) (j=1, ••• ,m) are defined as the left and right C-K extensions of the analytic functions cos(t.x.) and . ~ J m+1 sin(tjxj) respectively, where tj,xj E R for J = 1, ••• ,m. So 1n R 15.4

00

cos(t.z.) J J =

I

k=O

(-1)k 2k 2k "'k"l t. z. \C. I'. J: J J

and 00

sin(t.z.) = J J

( -1) k

I

t2k+1) 1

(2k+1) zj •

k=O

Next we define the A-valued analytic functions exp(t.x.e.) = e cos(t.x.) +e. sin(tJ.xJ.) JJJ 0 JJ J 1 I kf .

00

k=O

kkk.J = 1, •.• ,m. t.x.e., J J J

Their left C-K extensions are denoted by exp(tjzjej)' j

1, ••• ,m.

Clearly

exp(t.z.e.) = e cos(t.z.) +e. sin(t.z.) J J J 0 J J J J J 00

I

1 kkk. --k1 t.z.e., J = 1, ••• ,m. .

J J J

k=O Now define the left entire function E(!,x), t E Rm, x E Rm+ 1, called the ~xponential fun~tion, to be the left C-K extension of m

E(t,x) = n exp(t.x.e.) j=1 J J J

119

00

I

I

k=O k1+••• +km=k Clearly m

E(t,x)

rr 9L exp(t.z.e.)

j=1

J J J

k1 z1 e . . .

00

I

I

G>

k k1 km zmm e 1 ••• em

k=O k1+••• +km=k 00

I

k1 km 2 •••••• 2 m••• m(x)e 1 ••• em

I

.......

k=O k1+••• +km=k

k

m

Observe that in the particular case where m = 1

and E(t,x) = exp(t 1z 1e 1 ) = exp(t 1(x 1e0 -x 0 e 1 )e 1 ) which may be identified with e-itx and e-itz (t E R, x E R, z e [) respectively. Now we derive several expressions for this exponential function which will be used in the Laplace and Fourier transformation (see Sections 30 and 31). Introducing the notation cos(t.z.) if s.

H (t.z.) s. J J J

{

J J

0

J

sin(t.z.) i f sJ. = 1 J J

the exponential function can be written as E(t ,x)

120

But by Theorem 14.7 we have that

m

+

.... I ejtj cos(t 1x1) .•• sin(tjxj) .•• cos(tmxm)sh(l~lxo) j=1 ltl

and so

with

and

I

1 -s -s s s m ... IC-t 1 ) 1••• (-tm) max~···ax: I e}j cos(t 1x1) ••• sm=0 j=1 s1 sm sin(tjxj) ••• cos(tmxm)e 1 ••. em •

Notice that K and L are analytic functions in (!,x) E Rm x Rm while the functions ch(jtjx ) and sh(l!lxo) are, of course, real-valued and analytic in

It I

0

(!,x 0 ) E Rm

x

R.

Introducing the notation

{ H ( t .x.) = s j ,] J

cos(tjx) if sj = 0 sin(t.x.) if sJ. = 1 J J

a small computation leads to the expression 1

E(t,x)

1

I ··· I

sm=0

Hs (t1x1) ••• Hs m(t mXm) Ls ••• smct,xo) 1 1

Where 121

m

s1

sm + 1-s. L J s 1 ••• sm (t,x 0 )=e 1 ••• em ch(Jtlx 0 )+ L eJ.tJ.(-1) j=1

is an analytic function in (t,x 0 ) estimates

e sm sh ( 1 t 1 x0 m ltl E Rm x R which moreover satisfies the

i

for some k E ~ depending on E ~m and some C s ~ > 0 continuously + m s1···m'"j depending upon t E R • Now expressing the hyperbolic functions by means of exponentials it is + m m+1 obtained that for t E R and x E R

!

+

->

E(t,x) = B(t,;)e-ltJxo - A(t,;)eltlxo where

and

++

are bounded analytic functions in (t,x) sed as + ....

B(t,x) and + +

-A(t,x) with +

B (t) s 1••• sm and

122

E

1

Rm+ xRm, which also can be expreso

.... _ 1[ s 1 sm As s ( t ) - "'~" e 1 ••• em + 1""" m c. By the way, restricting to x0 fo 11 owing forms:

m

.

.

tj 1-sj s 1 1-sj s 01 J --:;- ( -1 ) e 1 ••• e . • •• e • J ltl J m

Le.

J =1

= 0 the analytic function E(t,x) yields the

m -+ ....

E(t,x)

= n exp(t.x.e.) J J J j=1

s(f,x) - ACf,x)

Kct,:x, 1

As a fi~al decomposition of the exponential function E(t,x) we mention the following. Putting

and

one gets E(t,x)

= E2 (t,x) - E1(t,x)

where E1 and E2 are analytic in (t,x) E R~ x Rm+ 1• As

and

....

-It I ( E1 +E 2 ) .... .... -21tiE 2 +1tiE .... ....

-21tiE~-ItiE

.

-+

m

-+

....

lt follows that for each t E R0 fixed E1(t,x) and E2 (t,x) are left entire functions. They are called the lower and upper parts of the exponential .... function E(t,x). 123

0

-+-

15.5 In an analogous way the conjugate exponential function E(t,x) is defined as to be the right C-K extension of the analytic function -+- -+-

0

E(t,x)

t

So for

1

= --

(2n)m

1

exp(-t.x.e.).

IT

J J J

j=m

E Rm and x E Rm+ 1 we have

E("t,x)

1

=--

(2n)m

IT

®R

exp (-tjejzj)

j=m

For this conjugate exponential function the following decompositions are obtained:

o o -+-+ m m where K and L are analytic functions in (t,x) E R x R • Next we have

1

0

where

L

s 1... sm

1

-+is analytic in (t,x 0 ) E Rm X R, and also -+-

E(t,x) = set,;) e-ltlxo _ ~(t,~)e It Ixo 0

-+-

The functions -+0

-+-

E1(t,x)

P.ct,lr)e It Ixo

~z (i,x)

8ct,~)e-1tlxo

and

-+-

are for each fixed tERm right entire functions, and are called the lower 0 -+and upper part respectively of E(t,x). 0 -+Notice that the easiest way to obtain those decompositions of E(t,x) is by handling the relation between the exponential function and the conjugate 0

124

exponential function expressed by 1

O-+

--+

E(t,x) = ------m E(t,-x) (2n) and the corresponding relations 1

o_.._,.

--r-+

E(t,x) = ------ E(l.,X), (2n)m 0-+-+

B(t,x)

1

--+-+

1

_-+-+

- ------m A(t,x) (2n)

and c-+-+

A(t,x) = - ------m B(t,x). (2n) 15.6 In illustration of the above decomposition formulae of the exponential function we treat of the two particular cases m = 1 and m = 2. 15.6.1

Form E(t 1 ,x)

1 it is obtained that

exp(t 1 z 1 e 1 ) e0 cos(t 1 z 1 ) + e 1 sin(t 1 z 1 ) K(t 1 ,x 1 )ch(lt 1 ix 0 ) + L(t 1 ,x 1 ) sh( lt 1 ixo) ltll

with

and L(t 1 ,x 1 ) = t 1 e 1 sin(t 1 x 1 ) + t 1 e0 cos(t 1 x1 ). Next we also have

and E2 ( t 10 x)

=

~( 1 - ~)(e 0 cos ( t 1 x1 )+e 1 s in( t 1 x1 ))e -1 t 1 1Xo.

itll This decomposition is, of course, exactly the one for e-itz, z E t:

125

e-itz

= ~(1

+

riJ>
~(1

-

~)(cos(tx)-i sin(tx))e-ltly.

+

Form = 2 the following formulae are obtained:

15.6.2

i eltlxo[e cos(tlxl)cos(t x ·-~

-E

(t,x) =

2

0

2

)(1 +

tl:t 2 ) Itl

and

+

ele 2 sin(t 1 x1 )sin(t 2 x2 )(1 - t 1 ;t 2 ~.

it! A

15.7 Now we construct some elementary monogenic functions by means of series of powers of special inner spherical monogenics of order one. 15.7.1

A left inner spherical monogenic of order 1 is of the form m

P1 (x) =

l:

Vl (X) Al

1=1 m

l:

zl :>.1 •

1=1 In the particula~ case where :>.1 spherical monogenic 126

a1

Em

we obtain the left and right inner

m

I

p(x,a)

z1a1

1=1

where the notationma stands as well for them-tuple (a 1, ••• ,am) E Rm as for

I

the 1-vector a=

1=1

e.a. E A,.

Notice that the function p(x,a) takes its

J J

values in sp{e 0 ,e 1, ••• ,em}. 15.7.2 Let us calculate an estimate for the Clifford norm of p(x,a).

As

m

p(x,a) =

I

(x1 e0

-

x0 e 1 )a 1

1=1

where <x,a> denotes the inner product in Rm, we find

m

_- 2 m;2[<~x,a~>2

+

x02 L\ j=1

15.7.3 Clearly the function p ( x,a~)kG>

is a left and right inner spherical monogenic of order k, taking its values in sp{e 0 ,e 1, ••• ,em}. Now we prove that

P(x' ~a)k Indeed,

p ( x,a~)kG> • m

~

p(x,a)

k

m

I

zl,"""z\ ali ••• alk

lk=1

k!

vl

1 (x) al ••• al •

1... k

1

k

127

But V.l

-i,-

1 ••• 1 ( x) = rtT zl Q • • • 1· • • k 1· nm · 1

zl

9

k

where ni stands for the number of times that i appears in (1 1 , ••• ,lk), and so, using 14.13.2, p(x,a)k =

I (l,. •..• lk)

n

k! I

n

zl

I

1• • • • m•

Q

•••

1

Q zl

k

a 1 ••• a 1 1 k

m

I

=

l,. •••• lk=1

zl

(!) •••

1

® zl

k

al ••• al • 1 k

This results finally into -+ k -+ k® p(x,a) =( m I z1 a 1)kQ = p(x,a) • 1=1

So we may conclude that the k-th power of the function p(x,a) is a left and right inner spherical monogenic of order k. 15.7.4 So we are led to consider series of the form

Y p(x,a)k "k

( 15 .1)

k=O which, if the coefficients >.k are chosen to be real, will represent left and right monogenic functions in their respective regions of convergence. As far as the investigation of the convergence of (15.1) is concerned we shall consider (15.1) as well as a series of spherical monogenics and as a multiple power series in the real variables x0 ,x 1, ••• ,xm. Proposition expansion

If the function f of one complex variable with Taylor series

k=O is holomorphic in the open disc {~ E [: and right inner spherical monogenics 128

1~1 <

p}, then the series of left

00

I lcklp(x,a)k k=O converges normally in the region given by

(15.2)

(A)

to a left and right monogenic function. However if the series (15.2) is considered as a multiple power series in the real variables x0 ,x 1, ••• ,xm then it converges absolutely in the smaller region given by m

Ixo I( \ aJ)~\1/2 + L

j =1

Ix1 I Ia 1 I

+ ••• +

Ixm I Iam I

< P

(B)

Proof (i) Let K be an arbitrary compact subset of the region (A); then there exists p' < p such that for all x E K + -+

<x,a>

2

+

2 -+ 2 xo Ia I <

2

(C)

P' •

Hence 00

L

SUP I

lckl p(x,a)klo

k=O xEK 00

:;;

2-mk/2

I

lckl

k=O 00

I

lckl

k=O

sup lp(x,a>l~ xE(C)

-+ -+ 2 sup [<x,a> xE(C)

+

x~I;I2Jk/2

00

I k=O the latter numerical series being convergent due to the holomorphy of f in

kl

< p.

( i i)

As

p(x,a-+)k

-+ -+ (<x,a>e 0

-

x0 -+a) k 129

the series (15.2), now considered as a multiple power series in the real variables x0 ,x 1, ••• ,xm, reads: k 1_ k0 k0 k1 k k k k 1••• ama_,. 0 • 'L. lc k I 'L. ( 1) m k0 ! ••• km! xo x 1 • · •xm a 1 m k=O k0 +••• +km=k oo

But as

a..2

_,. _,.

(153) •

_,. 2

(ala)=- lal, we have

(-1)slal 2s a

if

k

2s

if

k = 2s+1

k 0

which implies that a

is in sp{eo,e1, ••• ,em}; moreover

So investigating the absolute convergence of (15.3) leads to the series k k1 k k1 k k lx 0 1 °lx 1 1 ••• 1xm1 mla 1 1 ···laml mlal 0 •

00

2 k=O

(15.4)

It is clear that if x belongs to the region (B) then

or

L

k1 k

I

•

k

I

IX 0 I

k0

k

k1

k1

k

Ia I 0 IX 1 I Ia 1 I ... Ixm I mIam I

ko+••• +k m=k o· • • • m· which means that (15.4) is convergent in (B).

k

k

m< p

c

15.7.5 Remark The regions (A) and (B) introduced in the above proposition are optimal. Geometrically spoken they can be interpreted as a cylinder and a pyramid respectively.

130

15.7.6 The function ~

e r;; --

1

TI

L

r;;

k

k=O is holomorphic in the whole of [.

So the function

00

L ~

(z1a1

+ ••• +

k=O is left and right entire.

zmam)k It is denoted by

exp(x ,a). For a= (1,1, ••• ,1) E Rm the function exp(x), already introduced in 15.2 is obtained. Notice that

L kT1

00

exp(x,a) =

( <x,a>e -+ -+

0 -

x0 -+)k a

k=O

15.7.7 The expansion 00

I

1-r;;

holds in

r;;

k

k=O

lsi

<

1.

Hence the series

00

L

(z 1a 1 +

+

z a )k mm

k=O converges normally as a series of left and right inner spherical monogenics in the regi o)n

to the left and right monogenic function

131

~ ~ ) ( 1 - <x,a> e0

-

x0 ~a

~ ~ )2 ~ 2 (1- <x,a> + x02 1a1

which obviously is defined in

However the convergence of

L

00

~->-

(<x,a>e 0

-

~k

x0 a)

k=O considered as a multiple power series in the real variables x0 ,x 1, ••• ,xm only holds in the smaller region

15.8 Now we construct a generalization of the function ~u/~:, (u,d

E

a:

x 0: 0 •

15.8. 1 Consider the following functions appearing in the integral expressions of the inner and outer spherical monogenics constituting the Laurent series: qk(u,y)

(-1)k

k

-

= ~ --~Y~(~m+~1 m

L 1 k=0

u 1 • ··~ dyl •.. 'dyl E(y) 1 k 1 k

V (u) W (y) 11"""lk 11"""1k

I (11''""'lk) _ (k+m-1) 1 k+1 IYim+k -

w

11"""lk

(y) v

11···lk

( u)

K (u) k,m+1,y •

we know that those functions, which take their values in sp{e 0 ,e 1, ••• ,em}' 132

are left and right inner spherical monogenics in the variable u, while they are also left and right outer spherical monogenics in the variable y. Also recall that qk(u,y) is a direct generalization of the classical expression (-1)

k

K!

(u d )k 1 ~ z;;

where u and z;; are complex variables. for the Clifford norm of qk: Jqo(u,y) lo

By (11.12) we have at once an estimate

2n/2

=

IYim

and jqk(u,y)Jo

C (1+k2) (k+m-1) - m k+1 5

k

lui~

?

'

k E ~o·

15.8.2 So we are led to consider series of the form 00

L >..kqk(u,y) k=O which, if the coefficients >..k are chosen to be real, represent left and right monogenic functions in both variables u and y separately in a certain region, and which moreover take their values in sp{e0 ,e 1 , ••• ,em}. Proposition expansion

If the

f~nction

f of one complex variable with Taylor series

00

f ( z;;)

I

k=O is holomorphic in the open disc {z;; E C

Jz;;J < R}, then the series

00

L

JckJ qk(u,y)

k=O converges normally in the region {(u,y) E Rm+ 1

x

R~+ 1 : JuJ

< R

Jyj}

to a left and right monogenic function in u and y separately.

133

Proof Obviously

I

lckl lqk(u,y) 10

k=O -:;

em

00

1

lckl vm(k) l~lk' y

I

IYim k=O

where vm(k) is a polynomial of degree m in k. Hence the series converges normally in {(u,y) E 1Rm+ 1 x 1R~+ 1 : ~~~ < R}. c Now take the entire function

15.8.3

00

1

TI

k z:;.

;

then

is holomorphic in u E [ and z:;. E [ 0 separately. By the above Proposition 15.8.2 this is immediately generalized to the function 00

H(u,y) =

L

1

TI qk(u,y)

k=O which is thus left and right monogenic in u E 1Rm+ 1 and y E 1R~+ 1 separately with lim H(u,y) = 0, and takes its values in sp{e 0 ,e 1 , ••• ,em}. y->«>

15.8.4 A straightforward computation shows that H(u,y) satisfies the equations lul 0,1, ••• ,m (15.5) d H(u,y) H(r TuT,y)dr 0, 0 Ui while

J

-

H(O,y) = qo(O,y) = IY~m+1 • Conversely the function H(u,y) i~ completely determined by those equations and that initial value. Indeed it can be proved that Proposition Let f(u,y) be an A-valued analytic function in (u,y)ERm+l xlRm+l 0 134

which is left monogenic in u and right monogenic in y, and satisfying the equations 0

and the initial condition

-

y

f(O,y) Then f(u,y)

IY lm+l •

= H(u,y).

15.8.5 Remark Denoting by r u the spherical Cauchy-Riemann operator acting on the variable u, we have, as H(u,y) is left and right monogenic in u, that

and H(u,y)(ar

"'* r r u )w

1 +-

0,

whence r uH = Hr"'*u = - r

arH.

So for the function G(u,y)

r Iu I

1

= TUT Jo

u

H(r TUT,y)dr

appearing in the equations (15.5), we have

I lui 0

=-

(-r a H)dr r

H + G

or (1 - ru)G(u,y)

H(u,y)

and analogously G(u,y)(1 - "'* ru) = H(u,y). It then follows that the function G(u,y) is left and right monogenic in 135

u E Rm* 1; notice also that the equations (15.5) yield the left and right monogenicity of G(u,y) with respect to the variable y. 15.8.6 Now we give an estimate for the function H(u,y). Proposition Let 1u1 > IYI and let s E ~. . Rm+1 x Rm+1 such that 1n 0

Then there exists a constant Cs 1

~ Cs(1 + 1~1 )s+m - - e y IYim

>

I _Yu I

Proof As

and the left inner spherical monogenics are eigenfunctions of the spherical Cauchy-Riemann operator, we have

and hence, using the Cauchy estimate (11.17) 1(1- r )sH(u y)l u ' ~ C'

~

C'

< =

C ~ (1+k2)(1+k)s (k+m-1) lulk k~O k! \ k+1 ~

( +k)s+m ~k L 1

oo

k=O

k!

IYim+

m+s ( oo ) 1 d Iu Im+s L k\ 1~1 k IYim dlulm+s \ k=O 1

( m+s

:;; C' --m \ L cm:s) IY I j=O J :;; C _1_ ( 1 + 1-Yu I) m+s s IYim

136

(m+s)! 1-yulm+s-j) eiYI (m+s-j)!

J~l.

o

0

Notes to Chapter 2 In his book [Di] Dinghas writes: "Die Grundlegung der Funktionentheorie einer komplexer Ver~nderlichen als einer zusammenh~ngenden und selbstst~ndigen Disziplin is im wesentlichen das Werk von Cauchy, Riemann und Weierstrass". In this sentence three distinct approaches to holomorphic function theory of one complex variable are implicitly mentioned, namely the Weierstrass approach based on power series, the Cauchy approach based on complex differentiability and the Riemann approach based on the so-called Cauchy-Riemann equations. As is well known these three approaches are equivalent. Generalizations of the theory of holomorphic functions of a complex variable have ever since been developed along one of these three lines. In [L] P. Lounesto has beautifully analyzed how Clifford algebra can be used to generalize the Cauchy-Riemann operator. Let us briefly recall the three possibilities he mentions. Taken= 2, consider the Clifford algebra C(v 2 , 0 ) and introduce the operator e 1 + e 2 a~ • If one considers mappings f from c1 into c1, i.e. f: xe 1 + ye 2 ~ ue 1 + ve 2 satisfying the condition (e 1 a~+ e 2 a~) (ue 1 + ve 2 ) = 0 (i), then conformal maps of c1 are obtained unless the derivative of the map is zero. On the other hand, if mappings f from c1 into c0 ~ C~ are considered, i.e. f: xe 1+ye 2 ~ ue 0+ve 12 • satisfying the condition (e 1 ax+ e 2 a~)(ue 0 + ve 12 ) = 0 (ii), then the classical Cauchy-Riemann equations are obtained. Finally if spinor valued mappings f on c1 are considered. i.e. f:xe 1 + ye 2 ~ uf 1 + vf 2 , where f 1 = ~1 (e 0 + e 1) and f 2 = ~1 (e 2 - e 12 )

:x

:x

satisfying the condition (e 1 + e 2 a~)(uf 1 + vf 2 ) = 0 (iii), then it comes out that (iii) is equivalent to (i). Clifford algebras constructed over an m-dimensional real orthogonal vector space (m > 2) thus seem to be very appropriate to define differential operators in higher dimension which generalize the Cauchy-Riemann operator. Moreover these operators satisfy nice symmetry conditions. Note that if we put V = R6,m and if Y = R~,m + R6,m (= R + V) is the orthogonal space provided with the quadratic form X= xo +X~ (xlx) =X~ - 2, then the generalized Cauchy-Riemann operator D introduced in Section 8 is invariant under the group Spoin(Y) = Spoin(O,m) where Spoin (O,m) ~Spin (m+1,0). In this context also the paper [SW] by Stein-Weiss should be mentioned. in which systems of partial differential equations with constant coefficients are considered generalizing the classical

x

137

Cauchy-Riemann equations and being invariant under irreducible representations of the rotation group SO(n). Observe that Fueter, when building up a theory of regular functions of a quaternionic variable, already generalized the Cauchy-Riemann operator by introducing the operator D = a~+ i a~+ j a~+ k a~ (see [F]). As was demonstrated by Sudbery in [S] the definition proposed by Fueter of regular functions of a quaternionic variable, namely as null-solutions of the equation Df = 0, was the only plausible one to get a class of functions which would generalize the class of holomorphic functions of a complex variable in an appropriate way. Indeed, if a function of a quaternionic variable is called regular in an open subset n of R4 if it can be represented by a quaternionic power series about each point of n, then all ~-valued analytic functions in n are obtained, clearly a too large class when compared with the situation in the plane. On the other hand, if a function f:n +~is said to be regular in n if it admits a quaternionic derivative at each point of n, then only constant and linear functions remain, clearly a too small class of functions when again compared with the situation in the plane. Analogous statements hold in the Clifford algebra setting so that again only the Riemann approach is left. This situation clearly differs from the one in holomorphic function theory of several complex variables since there Hartogs's Theorem ensures the equivalence of the Weierstrass and Riemann approaches. Fueter and his students developed quaternionic analysis up to the fifties; a complete bibliography of their work may be found in [Hae 1], while a survey of the basic results of this theory is given by Deavours in [Dea]. More recently Sudbery (loc. cit.) and GHrsey-Tze in [GT] again payed attention to quaternionic analysis. Fueter's students Bosshard and Nef started a hypercomplex function theory in the framework of Clifford algebras in respectively [B] and [N], but only in the middle of the sixties this approach was taken up again independently by Iftimie in [I], Hestenes in [Hes], Delanghe in [Del 1] and [D~ 2] and Hile in [Hi]. More or less recently similar investigations were started by Goldschmidt in [G], Lounesto in [L] and Ryan in [Ry 1]. The latter author uses in [Ry 2] r.omplex Clifford algebras to study nullsolutions of a complex extension of the Cauchy-Riemann equations introduced in Section 8. Needless to say that Cauchy's Integral Formula plays a central role in function theorY· 138

It was shown by Habetha in [Hab] that, if one wishes to generalize cl ass·;cal function theory by considering algebra-valued functions in such a way that a 'simple' Cauchy Formula still holds then one has to restrict to the algebra of complex numbers, the algebra of quaternions or a Clifford algebra. Observe that Cauchy's Integral Formula (Theorem 5.9) is valid for open bounded domains in Rm+ 1 having sufficiently smooth boundary while in the theory of several complex variables it is normally given for special domains, namely polydiscs. Attempts to overcome this problem led to the integral formulae of BochnerMartinelli and Bergmann-Weil, whereby either the holomorphy of the kernel was lost or the applicability was again restricted to special domains (see [BT 1]. Only some ten years ago, to be more precise in 1970, Henkin and Ramirez succeeded in constructing an integral representation formula having a holomorphic kernel and this for functions which are holomorphic on a pseudoconvex domain (see [Hen] and [Ra]). Hence integral formulae for the solutions to the a-equation on such domains could be obtained (see e.g. [C]). Finally note that, due to the non commutativity of the Clifford algebra, the pointwise product of monogenic functions is not anymore monogenic. The product established in Section 14, though not being easy to handle, enables us to construct some kernel functions which are of basic importance in transform analysis (see Chapter 5). In the case of regular functions of a quaternionic variable such a product was already introduced by Haefeli in [Ha]. Bibliography [BJ

P. Bosshard, Die Cliffordschen Zahlen, ihre Algebra und ihre Funktionentheorie (Thesis, Universitat ZUrich, 1940). [BT 1] H. Behnke and P. Thullen, Theorie der Funktionen mehrerer komplexen Veranderlichen, Zweite Auflage (Springer Verlag, Berlin, 1970). [C] P. Charpentier, Formules explicites pour les solutions minimales de l'~quat;on au= f dans la boule et dans le polydisque de [n, Ann. Inst. Fourier 30 (1980), 121-154. [Dea] C.A. Deavours, The 4uaternion calculus, Amer. Math. Monthly 80 (1973) 995-1008. [Del 1]R. Delanghe, On regular-analytic functions with values in a Clifford algebra, Math. Ann. 185 (1970) 91-111. [Del 2] , On the singularities of functions with values in a Clifford algebra, Math. Ann. 196 (1972) 293-319. 139

[Di] A. Dinghas, Einf6hrung in die Cauchy-Wierstrass'sche Funktionentheorie (Bibliographisches Institut, Mannheim, 1968). R. Fueter, Die Funktionentheorie der Differentialgleichungen ~u = 0 [F] und ~~u = 0 mit vier reellen Variablen, Comment. Math. Helv.7 (1934) 307-330. [G] B. Goldschmidt, Verallgemeinerte analytische Vektoren in mn (Thesis, Universit!t Halle, 1980). [GT] F. G6rsey and H.C. Tze, Complex and quaternionic analyticity in chiral and gauge theories I, Annals of Physias 128 (1980) 29-130. [Hab] K. Habetha, Eine Bemerkung zur Funktionentheorie in Algebren, In: Function Theoretic Methods for Partial Differential Equations, Darmstadt 1976, Lecture Notes in Mathematics 561 (Springer-Verlag, Berlin, 1976) 502-509. [Hae 1]H. Haefeli, Hyperkomplexe Differentiale, Comment. Math. Helv. 20 (1947) 382-420. [Hae 2] , I funzionali lineari delle funzioni analitiche di una variabile quaternionale, Rend. Aaaad. Naz. dei XL (4) 2 (1952) 65-110. [Hen] G. Henkin, Integral representation of functions in strictly pseudoconvex domains and applications to the a-problem, Math. Sb. 82 (1970) 300-308, Math. USSR Sb. 11 l1970) 273-281. [Hes] D. Hestenes, Multivector functions, J. Math. Anal. Appl. 24 (1968) 467-473. G. Hile, Hypercomplex function theory applied to partial differential [Hi] equations (Thesis, Indiana University, 1972). [I] V. Iftimie, Fonctions hypercomplexes, Bull. Soa. Sai. Math. R.S. Roumanie 9 (1965) 279-332. P. Lounesto, Spinor valued regular functions in hypercomplex analysis [L] (Thesis, Helsinki University of Technology, 1979). [N] W. Nef, Die Funktionentheorie der partiellen Differentialgleichungen zweiter Ordnung (Hyperkomplexe Funktionentheorie), Bull. Soa. Fribourgeoise Sa. Nat. 37 (1944) 348-375. E. Ramirez De Arellano, Ein divisionsproblem und Randintegraldarstel[Ra] lungen in der komplexen Analysis, Math. Ann. 184 (1970) 172-187. [Ry 1] J. Ryan, Clifford analysis with generalized elliptic and quasielliptic functions, to appear in Appliaable Analysis.

140

[RY 2] J. Ryan, Complexified Clifford Analysis, preprint. [S] A. Sudbery, Quaternionic analysis, Math. Proc. Camb. Phil. Soc. 85 (1979) 199-225. [SW] E.M. Stein and G. Weiss, Generalization of the Cauchy-Riemann equations and representations of the rotation group, Amer. J. Math. 90 (1968) 163-196.

141

3 Spaces of monogenic functions

In this chapter we study the space M(r)(Q;A) of null solutions of the generalized Cauchy-Riemann operator D (§16) and characterize its dual and bidual (§§22 and 23). These characterizations rely heavily upon Runge type theorems (§18) \'lhich are obtained by using properties of the so-called regular solutions at infinity of D (§17). Just as in holomorphic function theory, the Runge approximation theorem plays a basic role in solving the equation Of = g where g belongs to some given class of functions or distributions (319) which in its turn enables us to introduce the notion of a primitive of a monogenic function (§20). In our case also a Mittag-Leffler type theorem may be derived from it (§21). Finally in Section 24 Hilbert modules with reproducing kernel are introduced which consist of square integrable monogenic functions and which generalize the classical HL 2 - and H2 -spaces. Further notice that only properties of spaces of left monogenic functions are proved, the passage to the:right monogenic case being straightforward. 16.

The space M(r)(Q;A)

16.1 Let Q be an open subset of Rm+ 1 ; then we called M(r)(Q;A) the unitary right A-module consisting of all left monogenic functions in Q and analogously M(l )(Q;A) the unitary left A-module the elements of which are right monogenic functions in n. Furthermore let Harm(Q;A) denote the unitary bi-A-module of all A-valued harmonic functions in n and let Harm(~;R) stand for the space of real valued harmonic functions in~. Then, considered as real vector spaces, Harm(n;A)

=

u

Harm(\l;R).

AEPN

Obviously M(r)~n;A) is a submodule of Harm(r)(n;A), which in its turn is a submodule of C(r)(n;A). Hence both of these spaces may be endowed with the c(r)(n;A)-topology defined in Section 3. 142

If Harm('.,;lR) also is provided with the topology of uniform compact convergence then clearly the c(r) (c;-1;A) - and the product topology on Harm(r) (q;A) are equivalent. As Harm(:J;JR) is a real Schwartz space, the same is true for Harm(:;;A) and M(r)(~2;A), Finally it is well known that Harm(;2;JR) is a Frechet space, whence Harm(r)(J;A) becomes a right Frechet A-module. In view of the Weierstrass Theorem stated in 9.11 we immediately get 16.2 Theorem M{r)(~;A) is closed in Harm(r)(~;A) and so it is a right Frechet A-module. Notice that from the above results it follows that M(r)(a;A) is a real Frichet-Schwartz space which in its turn implies 16.3 Theorem (Montel) Let B be a subset of t·1(r)('.?;A). if and only if B is closed and bounded.

Then B is compact

16.4 We close this section by considering the case where ~ = Rm+ 1• r4(r)(Rm+ 1 ;A) is called the ;;p:1UC of Zeft r'ntir>c funr~tion;; in 1Rm+ 1• By Theorem 12.3.5 any f E M(r)(Rm+ 1;A) admits the expansion 00

f( X)

=

where for each k E IN, ), 1 1 = _a_ 1••• k ax 1 in the whole spaceRm+1 • If Q= {V 11

E A, converging normally ... ~ f ax 1 x- 0 1

_

k k 1 : k E IN, (1 1 , ••• ,lk)E{1, ... ,m} .},

1' .. k

then by P(r) the right A-span of Q is meant; it is called the

spac!e of Zeft

inncn• Dpher>icaZ mcnogenicr{,

Clearly P~~~· the right A-module of left inner spherical monogenics of order k (see 11.2.2.), is a submodule of P(r)' We thus have 16,5 Theorem Each left entire function may be approximated by left inner spherical monogenics uniformly on every compact subset K of Rm+ 1, i.e. P(r) is dense in t4(r)(Rm+ 1;A). 17.

The space

M(r)(~;A)

17.1 In this section we shall deal with a submodule of M(r)(g;A), namely M(r)(n;A) consisting of those elements in M(r)(n;A) which are regular at infinity with respect to the funda~ental solution E of D. Let E denote the fundamental solution of D (see 8.9) given by 143

E(x) = - 1-

~+1

X

lx lm+l

, X

f: 0.

Recall that DE = ED = 8 * (Rm+1 ;A) as for E E V(r) * (lRm+1 ;A). F1nally . as well for E E V(l) let us recall that M(r)(n;A) may be identified with a submodule of 1 )(n;A).

v(

17.2 Definitions Let n be an open subset of 1Rm+ 1 such that 1Rm+ 1,n = K is compact and let f E M(r)(n;A) (resp. f E M(l )(n;A)). Then * (lRm+1 ;A ) ( resp. T E V(r) * (lRm+1 ;A ) ) 1s . called a left (resp. ( 1.) T E V(l) right) asymptotic extension off if there exists a compact set K* c1Rm+ 1 such that K c K* and Tis an extension of fj(~m+l\K*). (ii) f is called regular at infinity with respect toE if and only if any left (resp. right) asymptotic extension T of f satisfies the condition E•DT = T (resp. (TD)*E = T). 17.3 Definitions * (lRm+ 1;A ) ( resp.~ T E V(r) * (lRm+ 1;A ) ) for wh1ch . DT ( resp. TO) has ( 1. ) T E V(l) compact support is said to be regular at infinity with respect to E if E.DT=T (resp. (TD)•E = T). (ii) T E V(l)(G;A) (resp. T E VCr)(n;A)) is said to be a solution of Din n if DT = 0 (resp. TD = 0) in n. 17.4 Remarks 17.4.1

Take f E M(r)(n;A) c VCl)(i"l;A). 0 =

I

J

<j>(u)dcr/(u)

a(supp)

Then for any

+

I

I

)(n;A) ,

<j>(x).Df(x)dx

supp (<j>D) (x). f(x )dx,

supp which implies that > = -D> = 0. 144

E V(l

(<j>D)(x).f(x)dx

supp

=

<1>

Hence the D-derivative off, both as a function and as a left A-distribution, vanishes in D. This indicates that the Definition 17.3 (ii) has been suitably chosen. DD = ~m+ 1 e 0 , it follows immediately from the general theory 17.4.2 Since DO on elliptic operators (see e.g. [10]) and the Proposition 2.14 that any solution in Vtl)(D;A) (resp. v(r)(D;A)) of DT = 0 (resp. TD = 0) is a E(D;A)function. Hence any distributional solution in ~ of DT = 0 (resp. TD = O) is a left (resp. right) monogenic function in D. The proofs of the properties listed in the following proposition are left to the reader. 17.5

Proposition

(i) Each T E E{l)(Rm+ 1;A) (resp. E{r)(Rm+ 1 ;A)) is a solution of Din Rm+ 1,supp T which is regular at infinity with respect to E. (ii) E is regular at infinity with respect to E. (iii) Let T1 and T2 be solutions of Din the complement of some compact set K cRm+ 1 and coincide there. Then, if T1 is regular at infinity with respect to E, so is T2 • . ) LetKcRm+ 1 becompact,letT EV(l)R * ( m+ 1 ;A)(resp.V(r/R * m+ 1 ;A)) (1v 1 be a solution of D in Rm+ 1,K which is regular at infinity with respect to E m+1 * m+1 * and let T2 E E(l )(R ;A) (resp. E(r)(R ;A)). Then T1 * T2 (resp. T2 * T1 ) is a solution of D which is regular at infinity with respect to E. In part. * m+1 ;A) (resp. V(r)(R * m+1 ;A)) of the formS= E*T (resp. 1cular each S E V(l)(R * (Rm+1 ;A) (resp. E(r) * (Rm+1 ;A) is a solution of D which is T*E) where T E E(l) regular at infinity with respect to E.

1~.6 Proposition Let (Tj)jEJN be a sequence in v(1 )(Rm+ 1 ;A) (resp.

V(r)(Rm+ 1 ;A)) consisting of solutions of D outside some compact set K, all of them being regular at infinity with respect to E. Then if (T.) .EJN conm+1 * m+1 J J . * ;A) (resp. V(r)b(IR ;A)), Tis also a solution of verges toT 1n V(l)b(R D in Rm+ 1,K which is regular at infinity with respect to E. ~

Consider for instance the case of left A-distributions. Use the con. V(l * )b (Rm+ 1 ;A ) to show that DT = 0 1n . Rm+ 1'K. of the operator D 1n Apply furthermore Proposition 17.5 and the fact that the convolution is separately continuous in Vb(Rm+ 1 ;JR) (see [16]) to prove that E*DTJ..... E*DT in v*(l)b(Rm+1 ;A). o t 1nu1ty · ·

145

17.7 Definition Let r2 be an open subset of 1Rm+ 1 such that K = 1Rm+ 1,n is compact; then we call M(r)(r~;A) (resp. M(l)(n;A)) the set of left (right) monogenic functions in n which are regular at infinity with respect to E. It should be noted that M(r)(r~;A) (resp. M(l )(n;A)) is a submodule of M(r)(n;A) (resp. M(l )(n;A)). We have even more, namely 17.8 Theorem M(r)(r~;A) (resp. M(l )(Q;A)) is closed in M(r)(n;A) (resp. M(l )(n;A)) and hence a right (resp. left) Fr,chet A-module.

Proof Let for instance (fj)jSN be a sequence in M(r)(n;A) which converges to f in M(r)(Q;A). 0 Let furthermore H be a fixed compact subset in 1Rm+ 1 such that 1Rm+ 1,n = K c H and let l/i E E(IRm+ 1 ;IR) with 0 ::: l/i <' 1, lji(x) = 1 in 1Rm+ 1,H and lji(x) = 0 in an open subset w of 1Rm+ 1 satisfying K c w c H. Then the functions gj = fjlji' j E ~. all belong to E(r)(IRm+ 1 ;A) and coincide with fj on 1Rm+ 1,H. Hence each g. is regular at infinity \'lith respect to E. Now J m+1 let K be an arbitrary compact subset of IR • Then

s~ p

I (f r ( x)- f s ( x ) ) lji ( x) I0

xEKn(IRm+ 1'W) ~

sup 1fr(x)-fs(x)l 0 • 1 xEKn(IRm+ 'W)

But as K n (1Rm+ 1'W) is a compact subset of sup xEKn(IRm+ 1'W)

!.l,

we have that

~ 0 if inf(r,s) If r (x)-f.(x)l ~ o

~ ~.

Hence (gj)jEN is a Cauchy sequence in c(r)(IRm+ 1 ;A) and so it converges to some g E cyr)(IRm+ 1 ;A). But as in 1Rm+ 1,H, gj = fj for all j E ~. Propositior 17.6 tells us that g is regular at infinity with respect to E. Furthermore, as g =fin 1Rm+ 1,H we have that any T E v(1 )(1Rm+ 1 ;A) which is an asymptotic extension off, coincides with g in 1Rm+ 1,H so that in view of Proposition 17.5 T is regular at _infinity with respect to E. Hence f E M(r)(n;A). c

146

17.9 Remark Notice that the only left (resp. right) entire function which is regular at infinity with respect to E is the zero function. This means m+1 "" m+1 that M(r)(JR ;A)= {0} (resp. M(l )(JR ;A)= {0}). 17.10 We shall now construct a remarkable example of a monogenic function v1hich is regular at infinity with respect to E namely the Cauchy transfol'!Tl .;f a~; A-vaLued Measure in 1Rm+ 1 with compact support. Let IJ be an A-valued measure in lRm+l having compact support. Then, as E E L1°c(lRm+ 1 ;A), the convolutions E*IJ and ~*E, taken respectively in 1 ) (1Rm+ 1 ;A) and V~r) (JR::~ ;A) are meaningfu I and they yi.eld functions which surely belong to L1OC(JR ;A), namely

o(

J

E*p(x)

E(x-t)dp(t)

1Rm+1 and

As IJ has compact support, in view of Proposition 17.5 E*p and p*E are solutions of D in 1Rm+ 1,suppp which are regular at infinity with respect to E. m+1 suppp;A ) wh1le . p*E E M(l - )(JR m+1 \suppp;A). It thus follows that E*pE M(r)(JR 17.11 Theorem Let IJ be an A-valued measure in lRm+l having compact support. Then p = 0 if and only if E* IJ = 0 (p*E = 0) a.e. in 1Rm+ 1• Proof We for instance prove the theorem for the case E*p. Obviously IJ = 0 implies that E*p= 0 a.e. inlRm+ 1• Conversely, suppose that E*p= 0 a.e. in Rm+ 1. Then, as E is a fundamental solution of D, we have that for any : E V ( l ) (lRm+ 1 ; A)

~(x) = f

~m+1

(¢D)(y).E(x-y)dy.

Hence, using Fubini's theorem, we get that for all ¢ E V(l) (lRm+l ;A)

J

[J

J

(¢D)(y)

(¢D)(y)E(x-y)dy]dp(x)

1Rm+1 lRm+l 1Rm+1

[J1Rm+1 E(x-y)dp(x)]dy 147

= -J

1Rm+1

=

-J

Rm+1

=

(~D)(y)[J

1Rm+1

E(y-x)d~(x)]dy

(~D)(y) E*~(y)dy

0.

As the class of functions obtained by restricting to supp IJ the elements of V(l)(1Rm+ 1 ;A) is dense in CCl) (suppi.J;A) (see e.g. [16]) we finally obtain that f(x)d)J(X) = 0 for all f E c~) (supp)J;A). Consequently~ = 0. c 1Rm+1

I

To close this section we give some interesting applications of the notion of regularity at infinity with respect to E to the representation of monogenic functions on and off compact subsets in 1Rm+ 1• They are all formulated in terms of left monogenic functions. 17.12 Theorem (Representation formula on a compact set). Let n be an open subset of 1Rm+ 1 and let K c:nbe compact. Let furthermore f E M(r)(n;A) and ~ E V(n;A) with ~(x) = 1 in an open neighbourhood w~ of K which is contained in n. Then for any x E K f{x)

=J

E(x-t)D(f;)(t)dt.

1Rm+1 Proof Obviously f~ E V(r)(n;A) so that, as a left A-distribution with compact support, it is regular at infinity with respect toE, i.e. E*D(f~) = f~ or

f~(x) = I

1Rm+1

E(x-t)D(f~)(t)dt.

Take x in K to obtain the desired result. c As a corollary we get Cauchy's Integral Formula already established in Section 9. 17.13 Corollary Let n be an open subset of lRm+1 , let f E M(r)(n;A) and let K be a compact subset of n. Furthermore assume that K is a closed n-neigho n bourhood of K such that K c: K c: K c: n. Then for any x E K f(x)

=

IaK

n

148

n

E(t-x)dcrtf(t).

n

17.14 Theorem (Representation formula outside a compact set). Let K be a compact subset of m'"+ 1• Let furthermore f E M(r)(Rm+ 1,K;A) and ~EE(mm+ 1 ;R) with q1(x) = 0 in an open neighbourhood w of K and ~(x) = 1 in m'"+ 1,H where o ~ m+1 H is compact and K c w~ c H. Then for any x E R 'H f(x) = J E(x-t)D(f~)(t)dt. mm+1 Proof Since f~ E E(r)(Rm+ 1;A) with suppf~ cRm+ 1 'w~, the associated left A-distribution f~ is an asymptotic extension of f to H, f being itself left monogenic in m'"+ 1,H and regular at infinity with respect to E. Hence E*D(f~) = f~ so that

f~(x) = J

Rm+1

E(x-t)D(f~)(t)dt. .

Take x in mm+ 1 ~ to obtain the desired result. c As a corollary we get Cauchy's Integral formula outside a compact set. m+1 'K;A) 17.15 Corollary Let K be a compact subset of Rm+1 , let f E M(r)(m and let K be a closed n-neighbourhood of K. Then for any x E Rm+1,K n

n

f(x) = -J

a~
E(t-x)dotf(t).

17.16 Remark It follows from the above Corollary 17.15 that if f E M(r)(mm+ 1,K;A) then lim f(x) = 0. X-+oo

17.17 Conversely we can state Proposition Iff E C1 (mm+ 1;A) is left monogenic in mm+ 1 ~(0,R) for some R > 0 with lim f(x) = 0, then f is regular at infinity with respect to E. X-+oo

Proof As f is left monogenic in Rm+ 1 ~(0,R) for some R > 0 we have for each ;-;mm+1 E*Df(x)

f

E(x-t) Df(t)dt

Rm+1

J0R J m w(war s

+ ~a )f(x+rw)dr dw w

149

1 =---

um+1

N0\'1

I( sm

I

f(x+Rw)-f(x) )d,J.)- - 1- IR dr rf(x+rw)dw. wm+1 0 r sm

it is easy to check that r I

ff(X+rw)dw

Jsm

=0

while lim R++oo

J

(f(x+RuJ)-f(x))dw

Sm

=

-wm+ 1f(x).

Hence E*Df(x)

= f(x) for all x E Rm+ 1

which means that f is regular at infinity with respect to E. 18.

c

Runge type theorems

18.1 In the present section several Runge type theorems will be proved, namely: (i) Let K be a compact suEset of Rm+ 1 the complement of which is connected and let f be monogenic in an open neighbourhood of K. Then f may be uniformly approximated on K by monogenic polynomials. (ii) Let K be a compact subset of Rm+ 1 and letS be a countable subset of Rm+ 1,K having one point in each bounded component of Rm+ 1,K. Then each f which is monogenic in an open neighbourhood of K can be uniformly approximated on K by 'rational functions' having their poles in S. (iii) Let K and n be respectively compact and open subsets of Rm+ 1 with K c n such that n contains no relatively compact component of ~K. Furthermore let f be mo~ogenic in an open neighbourhood w of K with w c n. Then f can be approximated uniformly on K by elements of M(r)(n;A). ( i v) Let n be an ope:1 subset of Rm+ 1 and 1et ex. be a subset of Rm+ 1 ~ having one point in each bounded component of Rm+ 1 ~. Then a monogenic function f in Q may be approximated uniformly on each compact subset K of n by 'rational functions' having their poles in a. (v) Let K be a compact subset of Rm+1 and let ex. be a subset of K having one point in each component of K. Then a monogenic function- in Rm+ 1,K which 150

is regular at infinity with respect to E may be approximated uniformly on each compact subset H of Rm+l,K by 'rational functions' having their poles in a. When r·eferring in the sequel to 'The Runge Approximation Theorem', we in fact mean version (iii). For convenience we again only treat of left monogenic functions. 18.A The case M(r)(K;A) 18.2 Let K be a compact subset of Rm+ 1• Then we define M(r)(K;A) to be the set of those functions f for which there exists an open neighbourhood wf of K such that f E M(r)(wf;A). It is clear from this definition that the considered neighbourhood depends on the function chosen and that for the classical addition and scalar multiplication M(r)(K;A) becomes a unitary right A-module. In what follows, when speaking of subsets of M(r)(K;A) which are 'uniformly dense' in M(r)(K;A) then this density property should be understood in the sense of the norm induced by c(r)(K;A) on M(r)(K;A). 18.3 A central role is played by Lemma Let K be a compact subset of Rm+1 and let ~ be an A-valued measure in Rm+ 1 having its support contained in K. Then ;d~(x)f(x) = 0 for all f E M(r)(K;A) if and only if ~*E = 0 in Rm+ 1,K. Proof As to the necessary condition, let a E Rm+ 1,K and put f(x) Then f E M(r)(K;A) so that ~*E(a)

=

Jd~(x)

E(a-x)

E(a-x).

0,

or \I*E vanishes in Rm+ 1,K. Conversely let f E M(r)(K;A). Then there exists an open neighbourhood w of K such that f E M(r)(w;A). If w1 is an open subset satisfying K c w1 c w and cp E V(u.;;IR) with cp(x) = 1 in w1 , then fcp E V(w;A) so that, considering ~ is a right A-distribution, we have on the one hand that (p,f¢>

= Jd~(x)fcp(x) = Jd~(x)f(x)

While on the other hand

151

0.

Consequently

;d~(x)f(x)

= 0.

c

18.4 Theorem (Runge (i)). Let K be a compact subset of Rm+ 1 the complement of which is connected and let f E M(r~(K;A). Then f can be approximated uniformly on K by functions in M(r)(R + 1 ;A), i.e. M(r)(Rm+ 1 ;A) is uniformly dense in M(r)(K;A).

Proof Let B(O,R 1 ) = B1 be an open ball in Rm+ 1 such that K c B1 • Then obviously M(r) (B 1;A) is a submodule of M(r) (K;A). ~le now claim that every function f which is left monogenic in an open neighbourhood of K may be approximated uniformly on K by elements of M(r)(B 1;A), i.e. M(r)(B 1 ;A) is uniformly dense in M(r)(K;A). By means of the Hahn-Banach and Riesz representation theorems, it clearly suffices to show that each A-valued measure ~ in Rm+ 1 supported on K which annihilates M(r)(B 1;A) is also zero on M(r)(K;A). Let~ be such a measure. Then for each hE M(r)(B 1 ;A) we have by assumption that fd~(x)h(x) = 0 so that, in virtue of Lemma 18.3. ~*E = 0 in Rm+ 1,B 1• Since ~*E is an analytic A-valued function in Rm+ 1,K and Rm+ 1,K is connected, ~*E = 0 in Rm+ 1,K, which implies again by Lemma 18.3, that /d~(x)f(x) = 0 for all f EM ~(K;A). Next we prove that M(r)(R~r ;A) is uniformly dense in M(r)(K;A). 0 To this end consider a sequence of closed balls ]j = Bj(O;Rj) such that_K c B1, Rj < Rj+ 1 , j = 1 ,2, ••• , and Rj t"" and put for convenience K = B0 • By the preceding result, for each j E ~. M(r)(Bj+ 1 ;A) is uniformly dense in M(r)(Bj;A). Now take f = f 0 E M(r)(K;A). Then again by the.first step of the proof, given£> 0, a sequence (f.) "eN may be found such that J J fj E M(r)(Bj;A) and

152

sup jf.(x)-f. 1(x)j 0 ;,-fo:.. J J+ 2J+• xEB. J

Fix j E ~and consider the sequence (f.J+ k)ook=0

c

M( r )(B.;A). J

Then for s

<

t

sup jfs(x)-ft(x) lo ;, sup lfs(x)-ft(x) j0 xEBj xEBs ~sup

lfs(x)-fs+ 1(x)+fs+ 1(x)- ••. +ft_ 1(x)-ft(x) j0

xEBs lfs(x)-fs+ 1(x)

10

+••. + sup xEBt_ 1

t

l:

j=s+1

1

2J .

Hence (fj+k);=U is a Cauchy sequence in M(r)(~j;A) and so it converges to say g. in M(r)(~.;A). Moreover, as (f. k)ook-O is a subsequence of J J J+ (f._ 1+k);=O' the latter being convergent in M(r)(B._ 1 ;A) to gJ._ 1 , we have J · £~ m+ 1 oo R J m+ 1 . that g.jH. 1 =g. 1. AslR = u H., for each x ElR there ex1sts a J·-1 J J Jleast index j ~ 1 such that x E ~~- In this way a function g in 1Rm+ 1 may be defined by putting Q(x) = gj(x), ~ E 1Rm+ 1 , and· clearly g E M(r)(1Rm+ 1;A). 0

Finally, since for each x E K jf(x)-g(x)j 0 = =

~imjf(x)-fj(x)j 0

J-+oo

~imjf(x)-f 1 (x)+f 1 (x)- ••. +f._ 1(x)-fJ.(x)j 0

J-+oo

J ~ ~im [jf(x)-f 1(x)j 0 +••• +jf._ 1(x)-fJ.(x)j 0 ] J-+oo . J J 1 ;, limd z: 5 ) = E:, j-- s=1 2

we get that supjf(x)-g(x)j 0 xEK

:;;

E:.

c

153

18.5 Corollat·y Let K be a compact subset of lRm+l with connected complement. Then P(r) is uniformly dense in M(r)(K;A). 18.6 Before proceeding we introduce some notations. the set of outer spherical monogenics

If a E lR m+1 , call R(a)

where [au 1 .•• au 1 E(x-u}]u=a 1

k

(-1 )kax 1 ••• ax 1 E(x-a) 1

k

( -1 ) k+ 1'Clx 1 .•• ax 1 E(a-x). 1

k

If S = {a; : i E ~ 0 } is a finite or countable subset of lRm+ 1 , put R(S) = U R(a;) and R(r)(S) = spAR(S), the right A-span of R(S). Then i :;-1

clearly ~(r)(S) is a right A-module consisting of left meromorphic functions having their poles in S. Now let K be an arbitrary compact subset of lRm+ 1 . an d 1et no, 01 , n 2 , ••• be t he- components of lRm+ 1,K, no be1ng unbounded. Let furthermore S = {a; : i ~ 1} be a subset of lRm+ 1,K which contains one point in each bounded component ni' namely S n n; ={a;}, i = 1,2, •..• Then we have 18.7 Theorem (Runge (ii)) Let K be a compact subset of lRm+ 1 and letS be a subset of lRm+ 1,K having one point in each bounded component of lRm+ 1,K. Then each function which is left monogenic in a neighbourhood of K can be approximated uniformly on K by functions in M(r)(lRm+ 1 ;A) ~ R(r)(S), i.e. M(r)(lRm+ 1 ;A) ~ R(r)(S) is uniformly dense in M(r)(K;A). Proof In view of the Hahn-Banach and Riesz representation theorems, it clearly suffices to show that if ~ is an A-valued measure in lRm+ 1 having compact support contained in K such that Jd~(x)h(x) = 0 for each hE M(r)(lRm+ 1 ;A) ~ R(r)(S), then Jd~(x)f(x) = 0 for any f E M(r)(K;A}. Now let ~ be such a measure, consider ~*E and take i ? 1 arbitrarily. Then for k each k E~ and (1 1 , .•• ,1k) E {1, ••. ,m}, 154

P*E(a.) 1

\1* ~)X I ••• ()Xl 1

. Rm+1 ,supp\1 As Jl*E is analytic 1n

~

k

E(a.) 1

Rm+1 'K and ··.1 is connected, we get that

\l*E

= 0 in \·m+ 1 . Ca 11 K0 = R . . . .::1 0 and take f' E M( r) ( K0 ;A). Then, as K0 1s a compact set with connected complement, in virtue of Theorem 18.4., a sequence (g.) 'eN may be found in r~( )(Rm+ 1 ;A) such that sup lf'(x)-g.(x)l __,. 0 if j 0 r XEK J Hence o

i,:.

1 im J dJl(x)g .(x) = 0 JrdJl(x) f' (x) = j-~ J

which implies, by means of Lemma 18.3, that \l*E = 0 in Rm+ 1,K 0 = r·2 0 • We thus have proved that \l*E = 0 in Rm+ 1,K from which it follows, taking account of Lemma 18.3 again, that J1 annihilates r~(r)(K;A). c 18.8 Corollary Let K be a compact subset of Rm+1 and let S be a subset of Rm+ 1,K having one point in each bounded component of Rm+ 1,K. Then P(r) ~ R(r)(~ is uniformly dense in M(r)(K;A). Notice that a very useful refinement of the foregoing theorem may be given if K satisfies some supplementary condition as will be clear from 18.9 Theorem (Runge (iii); The Runge Approximation Theorem) Let K c ~be compact such that S}....K has no component which is relatively compact in ~~. Then every function which is left monogenic in a neighbourhood of K can be approximated uniformly on K by elements of t-1(r)(J;A), i.e. M(r)(.\l;A) is uniformly dense in M(r)(K;A). From the assumption on ~~K it follows that for each bounded component n (Rm+ 1......:r·) t- 4>- • Now t a ke a i E r,'"i n (Rm+ 1--.....:~r) c r, i ..: and put S = {ai : i :<; 1}. Then in view of Theorem 18.7, r~( r) (1Rm+ 1 ;A) ~ R( r)(S) is dense in M(r)(K;A). But, as each element of r~(r)(Rm+ 1 ;A) ~ R(r)(S) in fact belongs to r~(r)(~;A), the conclusion follows immediately. c Proof

r~-1 (i ? 1) of Rm+ 1,K, f2.1

18.10 Remark

Let

K. = {x E J

[I

\2 :

cRm+ 1 be open and consider for each lxl ~

j

j ':i'

1

and d(x,lRm+ 1.....n) ;;: ],. • J

Then it is well known that 155

(i) (ii)

any bounded component of mm+t,Kj meets mm+ 1,n; R'Kj has no component which is relatively compact inn.

The second property of course implies that M(r)(n;A) is uniformly dense in M(r)(Kj;A). 18B The case M(r)(rt;A) 18.11 Let n be an open subset of mm+1 and let~ be a subset of having one point in each bounded component of mm+ 1,n. Then for or countable subsetS of ~we put R(S) = u R(a). Furthermore aES R(~) = S~~ R(S) and we call R(r)(~) = spAR(~). the right A-span

mm+1 'n any finite we put of

R(~).

Then clearly R~r~(~) consists of left meromorphic functions having their poles in~ em+ '0.. 18.12 Theorem (Runge (iv)) Let n be an open subset of mm+ 1 and let~ be a subset of mm+ 1 ~ having one point in each bounded component of mm+t,n. Then every left monogenic function in n may be approximated uniformly on each compact subset K of n by elements of M(r)(mm+t;A) ~ R(r)(~). i.e. M(r)(mm+t;A) ~ R(r)(~) is dense in M(r)(n;A). Proof Consider the compact exhaustion Kj = {x En : Jxl

.

~ J

(Kj)j"~

m+1 and d(x,m 'n)

~

1 of n with

1 J},

,..,m+ 1'··· r< 1n • to 1 · t s componen t s no. j nj1 , nl, j ... w1· th noj th e unan d decomposeJI\ J m+1 m+1 j bounded one. As each bounded . component of m ,K.J meets m ,n, n.1 n ~. # ~ . for all i = 1,2, .... Take ai E ni n ~. i = 1,2, •.. , and call Sj = {ai: i = 1,2, ... }. Now let f E M(r)(n;A). Then clearly for each j EN, f E M(r)(Kj;A). By Theorem 18.7 M(r)(mm+ 1 ;A)~ R(r)(Sj) is uniformly dense in M(r)(Kj;A) so that, given s > 0, there exists hj E M(r)(mm+ 1;A) ~ R(r)(Sj) such that pK (f-h.) =sup j J xEKj As hj E M(r) (mm+1

156

;A)~

Jf(x)-hj(x)J0 ~

s.

R(r) ( ~). the proof is complete.

c

18.13 Corollary Let n be an open subset of Rm+ 1 and let z: be a subset of Rm+ 1,n having one point in each bounded component of Rm+ 1,n. Then P(r) ~ R(r)(E) is dense in M(r)(n;A). 18.14

From the proof of Theorem 18.12 we immediately get

Corollary Let n be an open subset of Rm+ 1 such that Rm+ 1,n is connected and unbounded.· Then M(r)(Rm+ 1 ;A) is dense in M(r)(n;A) and so P(r) is dense in M(r)(n;A). 18.15 Now let n be simply connected, i.e. n is connected and it is the union of a sequence of compact sets (HJ.) ·~·such that for all j8N, Rm+ 1,H. 0 J<:.111 J is connected and Hj c Hj+ 1• Then again from the proof of Theore.n 18.12 and taking account of Theorem 18.4 we obtain. Corollary Let n be a simply connected open subset of Rm+ 1. Then M(r)(Rm+ 1 ;A) is dense in M(r)(n;A). Hence P(r) is dense in M(r)(n;A). 18.16 Notice also that if n satisfies the conditions stated in Corollary 18.13 or 18.15 then M(r)(n;A) is separable. 18.17 We close this subsection with Theorem (Hartogs-Rosenthal) Let K be a compact subset of Rm+ 1 with Lebesgue measure zero and let S be a subset of Rm+ 1,K having one point in each bounded component ofRm+ 1,K. Then M(r)(Rm+ 1 ;A) ~ R~r)(S) is dense in c(r)(K;A). If moreoverRm+ 1,K is connected, then M(r)(Rm+ ;A) is dense in c(r)(K;A) and hence P(r) is dense in CCr)(K;A). Proof Let ~ be an A-valued measure in Rm+ 1 with support contained in K such that~ annihilates M(r)(Rm+ 1;A) ~ R(r)(S). Then Theorem 18.7 implies that ~ is zero on M(r)(K;A). Hence, in virtue of Lemma 18.3, ~*E = 0 in Rm+ 1,K so that, by the assumption of K, ~*E = 0 a.e. in Rm+ 1 It thus follows from Theorem 17.11 that~= 0. The Hahn-Banach and Riesz representation theorems then complete the.proof. c m+1 'K;A) 18.C The case M(r)(R 18.18 From Theorem 18.12 we already know that M(r)(Rm+1

;A)~

R(r)(Z:) is

dense in M(r)(n;A) and that and unbounded or where n is M(r)(rc;A). Now let n = 1Rm+ 1-....K where bounded. It will be proved 18.19

1n th e par t 1cu 1ar cases where IRm+ 1'0 1s connected simply connected, M(r) (lRm+1 ;A) is dense in 0

0

0

K is a compact subset of 1Rm+ 1• Then 1Rm+ 1-....r2 is that in this case R(r)(L} is dense in M(r)(si.;A).

We start with

Lemma Let K be a compact subset of 1Rm+ 1 , 1et ,.u be an open bounded neighbourhood of K such that each component of ul meets K. Furthermore 1et ].J be an A-valued measure having compact support contained in 1Rm+ 1'u,. Then -· m+1 !d].J(x)f(x) = 0 for all f € M(r)(IR -....K;A) if and only if ].J*E = 0 in t,l. Proof Take a ].J*E(a)

K and consider the function E(a-.)

€

€

m+1 M(r)(IR -....K;A).

Then

!d].J(x)E(a-x) 0.

If w

= u w1 is the decomposition of w into its components, take for each o

iElN i € ~. ai € wi n K. Then ].J*E(ai) = 0 so that, wi being connected and ].J*E being an A-valued analytic function in 1Rm+ 1-....supp ].J ::::> r.u, ].J*E = 0 in each wi, whence ].J*E = 0 in w. This proves the necessary condition. As to the sufficient condition, choose f E: M(r)(IRm+ 1-....K;A) arbitrarily and let K be a suitable compact neighbourhood of K which is still contained in n w. Then, in view of Cauchy's representation formula outside a compact set (see Corollary 17.15) for each x E: 1Rm+ 1-....K n

f(x)

r

=- j

Hence

aK

E(t-x)dotf(t). n

J

!d).J(x)f(x)

supp].J

- r

JaK

d].J(X) JaK E(t-x)dotf(t) n [Jd].J(X)E(t-x}]dotf(t)

n

-J

).J*E(t)dotf(t)

aK

0.

158

n

[J

18.20 Theorem (Runge (v)) Let K be a compact subset of Rm+ 1 and let L be a subset of K having one point in each component of K. Then R(r)(L) is . m+1 dense 1n M(r)(R '-K;A).

Proof Obviously R(r)(L) is a submodule of M(r)(Rm+ 1,K;A) since each of its elements is a right A-linear combination of functions having the form lE(x-ai)

ax 1

1

••• ax 1

k

Using the Hahn-Banach theorem, it clearly suffices to prove that if T E c(;)ORm+ 1,K;A) annihilates R(r)(L) then it vanishes on M(r)(Rm+ 1,K;A) too. To this end, let again (Kj)j~ 1 be the compact exhaustion of n = Rm+ 1,K considered so far and let T E co(r)(Rm+ 1,K;A) be such that it annihilates R(r)(L). Furthermore assume that T is bounded by PK. and choose j E ~ large enough such that both K and {x E Rm+ 1 : d(x,K) ~ 1} ~re contained in m+1 J {x E R : !xi <j}. Then an A-valued measure~ supported on K. may be associated with T such that for all f E c(r)(Rm+ 1,K;A) (see Seciion 3) =

Jd~(x)f(x).

Now let Rm+ 1,K. = nj u n1 u n{ u ••• be the decomposition of Rm+ 1,K. into J . J . 0 its components, n~ heing the unbounded one. Then K c w where w = U n~ i;;1 1 is a bounded open neighbourhood of K such that each of its components meets K. Hence, for each~ = 1,2, ••. , n1 n L ~~so that, taking a1 E n1 n z, i = 1,2, .•• , S ={a~ : i = 1,2, ••• } c L. As ~*E is an A-valued analytic function in Rm+ 1,supp ~. for each i ~ 1 there exists an open neighbourhood ~ . of aJ1: in which ~*E admits a Taylor development with coefficients a~

1

(a~) 1

0. 159

j so th a t , a.1 . Hence ~*E = 0 in w = u ~~.

Consequen tl y

in ~J1:.

~* E

-

• = 0 1n

for all f E M(r){IR 19.

m+1

~

. >1

'K;A).

1~

1

j c 1Rm+ 1'supp~ be1ng · connecte d ,

~i

~* E

=0

In view of Lemma 18.19 f d~{x)f{x) = 0

c

The equation Df = g

19.1 In this section we study the equation Df = g where g is an element of a given class of functions or distributions, namely E{r)(~;A), 1 ){n;A) and * { m+ 1 ) m+ 1 . S(l) lR ;A , n clR be1ng open. Of course, as DD = DD = ~+ 1 e 0 we may immediately obtain the existence of a solution f belonging to the considered class {see e.g. [20]}. Nevertheless in each of the cases we shall give a direct construction whereby we want to draw special attention to the pure function theoretic method developed in the case of s(1 )(1Rm+ 1 ;A). We also wish to point out that a same procedure may be followed to study the equation DJf =gin nJ clRM-J open. Hereby J = {i 1 , ••• ,ij}, 0 ~ j < m, is a proper subset of {0,1 , ••• ,m} = M such that J F {1 , ••• ,m}. Furthermore L { ) M-J and xJ iE~ xiei is then identified with the element xi iE~ of lR

v(

A

DJ stands for DJ = Di.

L

e. 1 In such a way D may be

while if 0

~

iE~

--0--. If J {i}, i EM, then DJ is denoted by axi written as D Moreover if 0 E J, then

J, then

D}iJ = DJDJ = e 0 t.~,

~ being the Laplacian in IR~. ~

In the special case where J = {0}, we put

= t.m·

Finally frequent use will be made of the following sequences of subsets of n. For j G 1 we put wj = {X E n: I>: I

1

<

j and d{x ,1Rm+ 1,n)

> ..... } J

~

j and d(x ,1Rm+ 1,n)

> .... } • = J

and as usual, Kj = {X E :1: lxJ

160

1

19.A.

The case E(r)(n;A)

19.2 Theorem Let g E E(r)(n;A). Of = g.

Then there exists f e: E(r)(n;A) such that

Proof Put G1 = w2 , Gj = wj+ 1'Wj_ 1 , j ~ 2. Then (Gj)j~ 1 is a locally finite open covering of n. Let (~.) .> 1 be a partition of unity subordinate to J J= (G.).> 1 with~- E V(G.~). Then g = E ~-gin E( )(n;A). Now call for J J= J J j=l J r each j, gj = E*~jg. Then gj e: E(r)(lRm+ ;A), Dgj = ~jg and hence, if j ~ 3, Dg.J = 0 in w-J- 1 or g.J e: M( r )(K.J- 2 ;A). In virtue of Theorem 18.9 there exists h. E M(r)(n;A) such that sup l(g.-h.){x)j 0 ~ J...J. Consequently the series J xe:K. J J 2 J-2 00

L

gl + gz +

j~3

g.-h. J

(19.1)

J

converges in c(r)(n;A) to an eleme~t f. We claim that fEE(r)(n;A). Indeed, consider an arbitrary closed ball B(a,r) contained inn. Then fork sufficiently large, B(a r) c ~k and so g1 + g2 + ~ (gj-hj) E E(r)(B(a,r);A), j=3

while, using the Weierstrass theorem (Theorem 9.11) 00

L (gj-hj) E M(r)(B(a,r);A). j=k+1 Hence f E E(r)(B(a,r);A). M~reover, for each k 0

L

Df(x) =

j~1

19.B.

The case

Dgj(x) = ~ jg(x) = g(x). j=1

~

1 and x e: wk

c

v( 1 ,(n;A)

* 19.3 Theorem LetT e: V(l)(n;A). DS = T.

Then there exists

* )(n;A) such that s e: V(l

Proof Put again G1 = w2 , G.= w. 1 ~. 1 , j ~ 2, and consider a partition of ----J J+ Junity (~j)J~ 1 subordinate to the locally finite open covering (Gj)j.:;: 1 of n, with ~j e: V(Gj ;JR). Then in v(n (n;A) 00

T=

L

~/·

j=1

161

* )Ui;A), sj E V(l * )(Q;A) Call for each j? 1, Sj = E*
~

21. 1

We claim that the series 51 + 52 + L (Sj-hj) j~3

converges in v(1 ) ,s (n;A). Indeed, let <1> E V(l) (~t;A) and let p least index for which supp<J> c Kp_ 2• Then, putting

>

3 be the

1

Hl = 51 + 52 +

L

(Sj-hj),

j=3

* we have that H1 E V(l

)(r~;A)

and 1

p

L <sk-hk,<J>>

= <S 1+s 2 ,q,> + As for each k

f;

*

L

+

k=1

<sk-hk,<J>>.

k=p+1

p + 1, I<Sk-hk,¢'>1 0 ~~ sup lct>(x) 10 , the sequence (H 1>1;;: 3

converges in v( 1 ),s(n;A).

Call

2

XHI

s = 51 + 52 + L (Sj-hj}. j~3

Then OS= ct>1T +

cj> 2

T+

L

00

O(Sj-hj)

j~3

L ct>} = r.

[]

j=1

19.C. · The case s *0 ) (lRm+1 ;A) 19.4 Theorem Let g E c(r)(1Rm+ 1 ;A) be of slow growth in 1Rm+ 1• Then there exists f E c(r)(1Rm+ 1;A), being of slow growth in 1Rm+ 1 too, such that Of= g in 1Rm+ 1• Moreover, if g has the growth factor 1, then f can be chosen in such a way that it has the grmtth factor 21 + 4. Proof Let g E c(r)(1Rm+ 1 ;A) be of slow growth in 1Rm+ 1 with ig(x)i 0 ~c*(1+1xl for some 1 E ~ and c* > 0. 162

f

Consider the sequence of open balls (Bk)k, 1, all of them being centred at the origin and having respective radius k~ Take a partition of unity (¢j)j~ 1 in V(Rm+ 1 ;R) subordinate to the locally finite open covering (Gj)j~ 1 m+1 o R oo of R , where G1 = B2 , Gj = tij+ 1'Bj_ 1, j ~ 2. Then g = I ¢jg in j=1 o m+1 . for a 11 j :;c 1 , the sequence of c(r)(R ;A). Not1ce that, as 0 ~ ¢j :: constants (C.) .. 1 with J J>

cJ. =

lgJ. (X) I0'

sup m+1 xElR

gJ.

¢J· g'

satisfies estimates of the type

Put for each j f . (X) J

Then Dfj

~

1,

= E*g . (X ) J

= gj and sup xElRm+1

Now, as for j ~ 3, Dfj

Js.J+ 1

= 0 in Bj_ 1• fj admits a Taylor development about

0

co

f.(x) J

=

I

s=O

P f.(x) s J

with

where C' depends upon the dimension m and R is chosen arbitrarily in ]O,j-1[ (see 11.5.2). As for any E > 0 there exists C > 0 such that for all s E ~ E

C'(S+m+ 1)(1+S 2 S+l

)

~ CE (1+E)S,

we get that 163

IPs f.(x)l J 0

C ((1+dlx1)s (1+J.)l+1

5

-

E:

J-1

Call for each k ~ 3, 1+2 hk(x) = 2 P5 fk(x). s=O m+1 Then hk E M(r) (~ ;A) and for all x E ~m+l for which (1+e:)lxl have that co

) 1 +1

S=l+3

<

k-1, we

lx I )s ( Q+e:) k-1

~c~ ((1+dlx1)1+3 (k-1)1+"! 1_(1+dlxl

k-1

Hence the series co

f1 + f2 +

2

(fk-hk)

k=3 converges in c(r)ORm+ 1;A), say to f, and obviously Of= gin distributional sense. Now we claim that f is ~f slow growth in ~m+ 1 • Let p > 1, take 0 < e: < {and call [(1+2e:)p] the least natural number which is greater than or equal to (1+2e:)p. Then for any x E ~m+1 [(1+2e:)p]+2 lf(x) lo ~ I (f1+f2)(x) lo + 2 lfk(x) lo + lhk(x) lo k=3 . co

2

+

k=[(1+2e:)p]+3 Moreover there exist suitable positive constants Ci, of p such that (1)

sup l(f 1+f 2 )(x)l 0 xElRm+1 [ ( 1+2e:) p]+2

(2)

2 k=3

164

~

c1;

1, ••• ,4, independent

[(1+2c)p]+2 (3)

L

k=3

sup lhk(x) lo ~ C3(1+p)21+4;. XEBP 00

( 4)

sup xEBP

L

: : c4'f1+p)l+3 •

k=[(1+2£)p]+3

Hereby BP stands for the closed ball centred at the origin with radius p. The above inequalities thus yield that for some constant C' > 0, independent of p, sup lf(x)l 0 ~ C'(1+p) 21 +4 • xEBP This of course implies that f is of slow growth in mm+ 1 and that its growth factor is 21+4. c * m+1 19.5 Theorem LetT E s(l)(m ;A). such that DS = T.

Then there exists

s

* (mm+1 ;A), a E ~ m+1 and g, cont1nuous . Proof As T E S(l) and in mm+ 1, can be found such that T = aag (Theorem 5.4). In 19.4 there exists f E c0( )(mm+ 1 ;A) which is of slow growth r * m+1 Putting S = aaf, then clearly S E s(l)(m ;A) and DS = T.

* )(mm+1 ;A) E s(l

of slow growth virtue of Theorem such that Df = g. c

19.6 Proposition Let g E c(r)(n;A) where n is an open and relatively como (-n,A) such that Df = g in n. pact subset of mm+1 Then there exists f E C(r) Proof Extend g with zero outside n. Df = (DE)*g = g in n. c 20,

Then f

= E*g E C0( r) (mm+ 1·A) and •

Primitives of monogenic functions

.

20.1 If n is an open subset of the complex plane and f E O(n), then primitives off inn may be constructed in a trivial way by using line integrals. In the case of monogenic functions the problem of primitivation is more comPlicated since such an elementary procedure is not available. Nevertheless it will be shown that for a wide class of open subsets of mm+ 1, primitives of monogenic functions do exist. 165

20.2 Let \lclRm+l be open and let i E {0,1, •.• ,m} = M. Then r2 is said to be xi-normal if for each x E rt the set {x+sei : s E lR} n p is connected and has exactly one point in common with the hyperplane {X; = 0}. Furthermore r1 is called norma! with rn;pect to the origin if 1:2 is xi-normal for all O,l, ••. ,m. Further we introduce the following notations: lRm+l + lRm+l lRm+l I

{X

E lRm+l

{X

E lRm+l

{X

E lRm+l

xo

>

0};

xo

<

0};

X

I 0}

0

lRm+l u lR~+l; + lR~+l = {X E lRm+l : X = 0 }. 0

Obviously JR~+l may be identified with lRm. 20.3 Lemma Let~ clRm+l be open and normal with respect to the origin. Furthermore let g E M(r)(n;A). Then there exists f E M(r)(~;A) such that

_a_ f = g in n. ClX 0

Proof

Consider the function X

h(x 0 +x) =

fo

0

g(s+x)ds.

Then clearly h is defined in~ and a~ 0 h

= g in ~. Moreover hE E(r)(~;A) and

an arbitrary solution of a~ 0 u =gin E(r)(~;A) can be written in the form u(x 0 +x) = h(x 0 +x) + ~(x) m+l where ~ E E(r) (1:2 n JR= ;A). Now we claim that~ can be chosen in such a way that h + Indeed, as in 1:2 X

X

= _a_(J ClXO

0

g(s+x)ds) +

Q

= g(O+x) + o 0 ~(xl, 166

~

E

M(r)(~;A).

J0°D g(s+x)ds + o 0 ~(x) 0

the condition D(h+~) = 0 in g is equivalent with o 0 ~(i) = -g(O+;) in ~ n R=+ 1• But, as -g(O+x) E E(r)(J n R:+ 1;A), in view of Theorem 19.2 there ought to exist I}! E E(r)(n n R~+1;A) such that D0 ,J!(X) = -g(O,x). c Obviously, if :2J c Rr+-J is open and normal ~Jith respect to the origin, g E M(r)J('-1J;A) with M(r)J(:-:,J;A) = {f E C1 (:lJ;A) : D/ = 0 in SiJ} and i EM---J, an analogous reasoning as made in the above lemma leads to the existence of () . f E M(r)J ( C?J;A ) such that ax-:- f = g 1n ~lJ• 20.4

1

Theorem Let nJ cRM--J be open and normal with respect to the origin; let g E M(r)J(r2J;A) and let a E JNM--J. Then there exists f E M(r)J(c:iJ;A) such that aaf = g in nJ. 20.5

21.

Mittag-Leffler theorems

21.1 In 12.3.6 a classical version of the Mittag-Leffler theorem has been proved. In this section it will be shown that this result may be obtained as a corollary to another form of the Mittag-Leffler theorem. 21.2 Theorem (Mittag-Leffler) Let (gi)iEI be an open covering of the open set g cRm+ 1 and let gi,kEM(r)(gi n gk;A), i, k E I, satisfy the conditions gi,k

= -gk,i

and gi,k + gk,l + gl,i

= 0 in ni n nk n n1 •

Then there exist functions gi E M(r)(gi;A) such that for all i,k E I gi,k = gk- gi in gin nk. ~

Let (ws)sElN be a locally finite countable open covering of Q which is finer than (n.) "EI and let (cp s ) sElN be a partition of unity in V(~i;R) subor1 1 dinate to the covering (ws)sElN" Furthermore, for each s E JN, take is E I such that suppcp c g. • Then, whatever k E I may be, 1s s hk =

L

cps gi ,k E E(r) (nk ;A)

sEJJ

While for all k,

s 1

EI 167

hk -hl

I

cps ( gi

sEJl

I

s'

k - g. 1) 1s'

cps g 1, k

sEJl = g 1,k

in r2k n rll"

As gl,k E M(r)(rlk n r~ 1 ;A), D(hk-hl) = 0 in rlk n r~ 1 • h E E(r)(ri;A) may be defined in r2 by putting h(x) = Dhi(x),

X

Consequently a function

E Qi.

In virtue of Theorem 19.2 there exists u E E(r)(ri;A) such that Du = h in r2. Put for each i E I, g.1 = h.-u and g.1, k = gk-g.1 1 in Q.; 1 then g.1 EM( r )(n.:A) 1 21.3 Corollary Let r~ 1 and r~ 2 be open subsets of Rm+1 such that r~ 1 n n2 F cp. Furthermore let f E M(r)(n1 n r~2 ;A). Then there exist f 1 E M(r)(n 1 ;A) and f 2 E M(r)(r~ 2 ;A) such that f = f 1 + f 2 in n1 n n2 •

Proof Put gn n -"1•"1 21.2. c Another direct application of Theorem 21.2 yields

-f and apply Theorem

Theorem (Poincare). Let n cRm+ 1 be open and let I 1 and I 2 be closed m-dimensional surfaces of ciass C1 contained in r2. Furthermore let f E M(r)(~ (I1 u I 2 );A). Then there exist f1 E M(r)(~L 1 ;A) and f 2 E M(r)(~I 2 ;A) such that f = f1 + f 2 in ~(I 1 U I 2 ). 21.5

Finally we get

Theorem (Mittag-Leffler) LetS= {a; : i = 1,2, ••• } be a subset of Rm+ 1 without any limit point; let (m(i) : i = 1,2, ••• ) be a sequence of natural numbers and let (G; : i = 1,2, ••• ) be a sequence of functions belonging to R(r)(S) such that for each i = 1,2, ••• , m(i)

I k=O 168

Then there exists a function f which is left meromorphic in ffim+ 1 and admits a; as poles and G; as corresponding principal parts. Proof. Put D0 = ffim+ 1,s; then D0 is open in ffim+ 1 • Moreover, for each i ~ 1, let D1- be an open ball centred at a. such that D. n D. = ¢ if i # j. Then 1 1 1 J (D 0 , D1, D2 , ••• ) is an open covering of ffim+ and Dj n Dk F ¢for j ~ k implies that j = 0 and that Dj n Dk = Dk'{ak} if k F 0. Put for each i c 1, g0 ,; = gi,o = -G; and g0 , 0 = 0; then g0 ,i E M(r)(D 0 n Di;A) = M(r)(D;'{ai};A) and the sequence (g 0 ,i)i~o satisfies the conditions of Theorem 21.2. Hence there exists gi E M(r)(D;;A), i ~ 0, such that

Clearly g0 is the desired left meromorphic function. 22.

c

The dual of M(r)(D;A)

22.1 One way of characterizing the dual of M(r)(D;A) might run as follows. As M(r)(D;A) is a submodule of Harm(r)(D;A), using Hahn-Banach's theorem, M(r)(D;A) is completely described once Harm(r)(D;A) is known. But, considered as a real vector space Harm(D;A)

=

IT

Harm(D;ffi)

AEPN so that, Harm' (D;ffi) being well known (see e.g. [19]), Harm' (D;A) and hence * Harm * (r)(D;A) are obtained. This way of determining M(r)(D;A) is of course rather formal and the spirit of this section consists exactly in describing explicitly the dual of M(r)(D;A). 22.2 Let D be an open subset of ffim+ 1 and let (Kj)j~ 1 be the c~mpact exhaustion of D considered so far (see 18.10). Furthermore letT E M(r)(D;A). Then there exist C > 0 and j E ~ such that JJ 0

;;

CpK. (f) for all f E M(r)(Q;A). J

As M(r)(D;A) is a submodule of c(r)(Kj;A), endowed with the norm llfllj = sup jf(x)j 0 , f E c(r)(Kj;A), xEKj 169

we have that in view of the Hahn-Banach and Riesz representation theorems there exists an A-valued measure~ in Rm+l supported on Kj such that for all f

E M(r) (\i;A)

fd ~ (X )f (X).

Now let ¢ E V(0;R) with ¢(X) 1 on an open neighbourhood w¢ of Kj which is still contained in Q. Then by virtue of the representation formula of left monogenic functions on compact sets (see Theorem 17.12), for each x E Kj f(x) = JE(x-t)D(f¢)(t)dt, so that, using Fubini's theorem, we get:

Jd~(x)

!E(x-t)D(f¢)(t)dt

f[fdp(x)E(x-t)]D(f¢)(t)dt -J~*E(t)D(f¢)(t)dt.

If we put

then we have that, up to the constant (-1), Fj equals the Cauchy transform m+1 'Kj;A). On the analogy of classical function ~*E of~ and so Fj E M(l)(R theory, F. is called the indicatrix of Fantappie associated with Tin m+1 J R 'Kj. We have thus proved Let T E M(r)(ro;A) be bounded by PK. and let Fj be its .... . m+1 J associated indicatrix of Fantapp1e 1n lR 'Kj. Then, if¢ E V(D;R) with ¢(t) = 1 on some open neighbourhood w¢ of Kj with w¢ c ro, we have that for all f E M(r)Ll;A) 22.3

Proposition

= !Fj(t)D(f¢)(t)dt.

( 22. 1)

22.4 The converse of the foregoing proposition will now be shown. Denoting by ¢· the set of functions ¢ E V(ro;R) such that ¢(x) = 1 in some open neighbourho~d w¢ of Kj contained in ro, then, by taking u E M(l)(lRm+l, Kj;A) and¢ E ¢j fixed, a right A-linear functional Tu on M(r)(ro;A) is 170

defined by = Ju(t)D(frp)(t)dt. Notice that this definition of Tu does not depend on ¢ E ¢j. Indeed, if cjlp
u(t)D(f~)(t)dt

supptj;

=-

J

(uD)(t).(fijJ)(t)dt

supplj_c~Kj

0.

We now claim that Tu E M(r)(~;A), i.e. that Tu is bounded. To this end we proceed as follows. As u determines a right A-distribution Tu in Rm+l,Kj by putting for each lJ! E Vr ) (Rm+l,K. ;A) ,r

J

J

=

u(t)ojJ(t)dt,

Rm+1,K. J

we obtain that for any U E v{r)(Rm+l ;A) which extends u and any (Rm+l ;A) of the form ljJ = D(frp) with ¢ E '~'j and f E r~(r)(ri;A)
r

J

ljJ

E V(r)

u(t)D(f¢)(t)dt

Rm+1,K. J

Nou let t: > 0 be such that K = {x E st:d(x,Kj) ~ 2E} is contained in w¢. Then, as UD E v*( )(Rm+ 1 ;A) with supp UD c K., there ought to exist ME~ and r m+1 J C > 0 such that for all ljJ E V(r)(R ;A) (see e.g. [6]).

I I0

;';

c

sup sup l3o.ijJ(x) lo [ai<M xEK

c

sup sup [3a.f (X) I0. [a[:;;M xEK

But, as f is an A-valued harmonic function in S1, a suitable compact neighbourhood K of K, with K c w , may be found such that for some constant ll ll

0 [[ 0

"::

C* sup XEK

jf(x)io· ll

171

As = -

we finally obtain that I lo:;; c* sup lf(x) lo · xEK 11

or Tu E M{r)(n;A).

We thus have proved

22.5 1 Propositio~ Let ¢ E ~j where j E ~ is fixed. Then for any u E M(l) (~m+ 'Kj;A) the right A-linear functional Tu defined on M(r)(n;A) by = Ju(t)D(f¢)(t)dt

(22.2)

belongs to M{r)(n;A). 22.6 Proposition If F. E M(l)(~m+ 1 ,K.;A) is the indicatrix of Fantappie of * J m+1 J T E M(r)(n;A) then for all s E ~ , the function aSF. is the indicatrix of J Fantappie of the functional a8T defined by = (-1)1SI, vf E M(r)(n;A) Proof It is clear that also a8Fj E M(l)(~m* 1 ,Kj;A); then by the foregoing Proposition 22.5 it is the indicatrix of Fantappie of the functional T1 E M{r)(n;A) given by = jaSFj(t) D(f¢)(t) dt, v f E M(r)(n;A) where ¢ E V(n~) is such that ¢(x) = 1 in some open neighbourhood w¢ of Kj contained in n. So

By Leibniz's rule a8(f¢)

=

I y:;;B

172

(~)a 8 -Yt.aY¢

=

-~. L (~)aB-Yf.aYq,.

a13f.q,

Ys:S

yilO

Now for

y

I 0



(-1)1BI!Fj(t).D[(asf).q,](t)dt <-1)

Is I

for all f E M(r)(n;A). 22.7

c

Example Let a be a fixed point in n~ define T E M{r)(n;A) by = f(a),

V f E M(r)(n;A).

So

= !dlJ f where lJ is the Dirac measure oa at a. with this T is F(x)

The indicatrix of Fantappie associated

oa*E(x) =

E(a-x).

Hence the functional T1 given by

corresponds to the indicatrix F1 (x)

(-1)1 8 1

a~ E(a-x)

a~ E(x-a). 173

22.8 Other expressions for the formulae (22.1) and (22.2) will now be derived. * LetT E M(r)(~;A) be bounded by PK. and consider its associated indicatrix J

of Fantappie Fj. Fu.rthermore choose q., E cpj and let K be a suitable compact neighbourhood of Kj fot· which K c supp¢ c w¢. Then, applying Green's Identity, we have that for all f E M(r)(~;A)

!Fj(t)D(f¢)(t)dt (

- J

;)K

(22.3)

F .(t)datf(t). J

"J m+1 Conversely, fix j E ~and take¢ E cpj and u E M(l )(m 'Kj;A). Then from the preceding considerations it follows that T is bounded by pK , where K u n is a suitable compact neighbourhood of Kj. Hence the domain of fntegration in formula (22.2) may be restricted to supp¢'K so that, again using Green's T] Identity, for all f E M(r)(Q;A) =- JaK u(t)dotf(t).

(22.4)

n

The formula (22.3) expresses the relationship between T and its associated indicatrix of Fantappie in a more classical form, namely by a Cauchy type integral (see also [12] for the case of holomorphic functions). 22.9 Definition Let j E ~be fixed. Then we denote by LJ(.l) the following m+1 m+1 . subset of M(l)(m 'Kj;A): u E M(l)(m 'Kj;A) belongs to ql) 1f there exists cp E ¢j such that for all f E M(r)(K;A) Ju(t).D(f¢)(t)dt = 0. Obviously L~l) is a submo~ule of M(l)(mm+ 1,K.;A) while for any u E Ltl)'Tu=O. Notice also that if u E Ltl) then for each¢~ E ¢j Ju(t).D(f¢*)(t)dt = 0 and this for all f E

M(r)(~;A).

j

The meaning of L( l) will become clear from

22.10 Proposition Let j E ~be fixed. Then u E Ltl) if and only if there exists i 0 > j such that u = 0 in mm+ 1,K .• 10

174

Proof Suppose that u E M(l)(IRm+ 1,K.;A) vanishes outside some K. (i J 1 0 Taking ~ E ~io then clearly ~ E ~j whence for any f E M(r}(~;A) 0

-

Ju(t).D(f~)(t)dt

> jj.

=0

j

or u E L(l)• . Conversely let u E Ltl)• f E M(r) (n;A)

Then there exists ~ E ~j such that for all

ju(t).D(f~)(t)dt =

0.

Now call i 0 > j the least index such that . 1Rm+1 ' K.• u = 0 1n

supp~ c

0

Ki

Then we claim that o

10

Indeed, by the density of M(r)(n;A) in M(r)(Ki 0 ;A) we have that, given f* E M( )(Ki ;A), there exists a sequence (hs)s~' in M( )(n;A) which converges . r o . c.rn m+ 1 r o un1formly on Ki to f*. Hence for arb1trary a E ~ and H c Ki compact, 0 0 (aa.hs)se:IN converges uniformly on H to aaf*. Consequently Ju(t}.D(f*~)(t)dt

=0

for all f* E M(r)(Ki 0 ;A). Now take a E lRm+1 'Ki 0 ; then clearly E(.-a) E M(r)(Ki 0 ;A) so that Ju(t).D(E(t-a}<(J(t)}dt

=

0.

Furthermore let K be a compact subset of w

i/1

0

= Ju(t).D(E(t-a)~(t))dt = J(utji)(t).D(E(t-a)
so that, using Green's Identity, J((utji)D)(t).E(t-a)~(t)dt

=0

or J((utji)D)(t).E(t-a)dt

J((utji)D)(t).E(t-a)(1-

~(t))dt

= 0. 175

As in Rm+ 1,K (uw)D = uD = 0 and 1-$=0 in w$• the second term vanishes. As to the first term, notice that uw E E(r)(Rm+ 1;A) is an asymptotic extension of u so that, u being regular at infinity with respect to E,

o = (-1 )J{(uw)D)(t).E(t-a)dt ((uw)D•E)(a) (uw) (a) =

u(a).

As a E Rm+ 1,K.

has been taken arbitrarily, u = 0 in Rm+ 1,K. •

10

c

10

m+1 'Kj;A) be such that Tu = Tu • Then, taking 22.11 Now let u 1 ,u 2 E M(l)(R 1 2 $ E ~j fixed, for any f E M(r)(n;A)
-

Tu 2 ,f> J(u 1

-

u2 )(t).D(f$)(t)dt

=0 j

andsoul-u2EL(l)• _ _ Conversely, if T E M(r)(n;A) is bounded by PIS and Fj E M(l)(Rm+ 1,Kj;A) is its associated indicatrix of Fantappie, then in view of the foregoing cons~derations, any element of the form F.+h, hE LJ(. l)' also respresents T. As J m+1 J L(l) is a submodule of M(l)(R 'Kj;A), the quotient left A-module j ( m+1 j M(l) = M(l) R 'Kj;A)/L(l)

is well defined and it consists of elements m+1 j [u]j = u + L(l)' u E M(l)(R 'Kj;A). As for each j E ~.Ltl) clearly is a submodule of Lt;~ and any element of M(l)(Rm+ 1,Kj;A), when ~estricting it to Rm+ 1,Kj+ 1 belongs to M~ 1 )(Rm+ 1 , Kj+ 1;A), each [u]j E Mtl) may be considere~ as be~onging to M~i~ and this by means of the canonical injection Ij : Mtl) ~ Mt~~ where for uEM(l)(Rm+ 1, Kj ;A)

176

lj ( [u]j ) = u

+

j+1 L(l)

= [u]j+1" Consequently an increasing sequence M(l) c MCl) c ••• of left A-modules is obtained, the union of which detennines a left A-module denoted by m+1 . M(l}(lR -....r~;A), 1.e. m+1 M( 1) (lR -....mA) = u Mil). j ;;:1

Define J: M{r)(r~;A)-+ M(l)(IRm+l,Q;.A) in the following way: if T E M(r)(Q;A) is bounded by PK. and has Fj as associated indicatrix of Fantappie, then J

J(T) = [Fj]j.

It is clear that J is an algebraic isomorphism between those left A-modules. But there is more! First of all notice that for each j E ~. Lil) _is closed m+1 . 1n M(l)(IR -....Kj;A}. . Indeed, let (us)s~ be a ~equence in Ltl) converging to u in M(l)(IRm+l...._ Kj;A) and suppose that u ~ Ltl)" Then for~ E ¢j given, there ought to exist f E M(r)(Q;A) such that Ju(t)D(f~)(t)dt

I 0.

As for each s E ~ Jus(t)D(f~)(t)dt

= 0,

the integral being taken over the compact set argument yields Ju(t)D(f¢)(t)dt

supp~-....w~

clRm+1 -....Kj, a classical

= 0,

. a left Frechet .. which is clearly a contradiction. So, -Mrl )(IRm+1 -....K.;A) be1ng j . J A-module, M(l) may be endowed with the system of seminorms Pj = {pK K clRm+l...._K. compact}, where for each K J

inf pK(h). hE[u]j 177

In this way (At1 1 )'Pj) bec:omes a left Frechet A-module too. Furthermore, the . . t . j +1 canonical lnJeC 10n I j: MJ(l) + M(l) is continuous since for each K emm+ 1'Kj+ 1 compact,

<:

inf

pK(u+h)

hELt 1) pK([u]j) and this for all u E utl)' Hence the left A-module M(l)(mm•l,~;A) may be equipped with the inductive limit topology. As such we denote the associated system of seminorms by . m+l Pind and we wr1te M(l)(m 'D;A)ind' 22.12 Our main objective now is to show that the algebraic isomorphism J already established between r4(r)(S1;A) and M(l)(mm+ 1'S"l;A) is also topological when M{r)(~;A) is provided with the strong topology. To this end we proceed as follows. As was already noted in Section 16, M(r)(~;A) is a real (F-S)-space. Consequently M(r)b(~;A)*becomes a sequentially complete bornological space (see [4]). Denoting by Ej the left A-module consisting of those*T E M{r)(Q;A) which are bounded by PK.' then, if we define for each T E Ej, J

!ITIIj

=

• {f .

sup "IJ 0 , PK. (f}.,;_1} J

Ej, II-II· is turned into a left Banach A-module.

.]

Clearly for each j E ~. Ej*

*

is a submodule of Ej+ 1 and

* M(r)(S1;A)

= u

j~1

* Ej.

As moreover the canonical injection from E~ II II into E~ 1 II II is * J' • j J+ ' j+ 1 continuous, M(r)(~;A) may be endowed with the inductive limit topology. The corresponding system of seminorms is denoted by P~nd and we write M(r)(~;A)ind' 0

178

* Moreover the strong topology on M(r)(n;A) is weaker than the inductive limit topology just defined. In the following proposition we point out that these two topologies are in fact equivalent on M~r)(n;A). 22.13 Proposition If in each of the underlying cases M(r)(n;A) is conside~ as a real convex space, then the following topological isomorphisms hold:

Consequently

Proof Let us first recall that in view of Proposition ?..14 the function e:M~r)(n;A)~(r)(n;A), with e(T)=Te T, TEM{r)(n;A), is an isomorphism between these real vector spaces. Using ProBosition 2.10 for all TEM(r)(n;A) and fEM(r) (n.!t> <e T,f> 2-n L eA . AEPN

* We now claim that e is the desired topological isomorphism between M(r)(n;Nind and M(r)b(n;A). Indeed, for any B c M(r)(n;A) bounded and T E M{r)(n;A)

* But the strong topology on M(r)(n;A) being weaker than the inductive limit one, there ought to exist C > 0 and q E Pind such that p6 (T) ~ Cq(T) for all T E M{r)(n;A). Hence for any T E M{r)(n;A). p6(e(T))

~

Cle0 1 q(T) 0

Which implies that e is continuous. A classical corollary to the closed graph theorem then yields that e- 1 is also continuous, whence (i) is proved. In order to prove (ii) notice that from the arguments used in (i), we 179

* already have that the same e:M(r)b(n;A) ~ M(r)b(n;A) is continuous so that only the continuity of e- 1 is left to show. To this end, take an arbitrary T E M(r)(n;A). Then -n

-1

l<e T,f>l;;; 2

L leAl

A

l. o

For each B c M(r)(n;A) bounded, put BA

= BeA' AEPN, and call s*

Then s* is bounded in M(r)(n;A) and sup l<e- 1r,f>l;;; 2n 12 sup Il fEB gEB* which means that e- 1 is continuous. 22.14

Obviously (i) and (ii)

imply (iii).

o

Using the above proposition we finally get

Theorem J is a topological isomorphism between M(r)b(n;A) and "'M(l )(Rm+1 ' n;A)ind" Proof From the considerations preceding Proposition 22.9 we already know that J is an algebraic isomorphism. To prove its continuity, it suffices to show that its restriction to each E~ ll !I . is continuous. To this end, * J, • J take T E Ej arbitrarily and consider J T = [Fj]j E M~l)" Then for any q E Pind 0

=c <

=

0

J

(sup inf hE[F j] j tEK

cJ sup IF/Y)I 0 yEK 0

;;; c. sup I J

yEK

;;; c}(Kj) ;;;

f

d~(x)E(y-x) 10

supp~

sup IE(w)lo wEKj-K

C( Kj ;K)

where (c.).~,, (p.) ·,~,are the sequences of constants and seminorms determJ J<:.~• J Jg• 1 ining the seminorm q E P. d' K is the compact subset of Rm+ ,K. defining 1n

180

J

p.

E

P.,

while C(K.;K) > 0 depends on q. Hence J is continuous. As both _J m+1 and 11( 1 )(lR '-n;A)ind are inductive limits of Frechet spaces, the closed graph theorem implies that J- 1 is also continuous. o ~

J r~(r)(\?,;A)ind

22.15 On the analogy of the classical cases of holomorphic functions of one or several complex variables, we call ~1(r)(~1;A) the space of analytic functionaL~ in g. By the previous theorem this space may thus be identified with a space of right monogenic functions in 1Rm+ 1,n. 23.

The bidual of M(r)(Q;A)

m+1 23.1 In this section we shall prove that the strong dual of r-1( 1 ) (lR '-~2;A)ind is topologically isomorphic to M(r)(n;A), which means that M(r)(n;A) is reflexive. Let f E M(r)(n;A) and associate with it the left-A-linear functional Tf on M(l)(lRm+ 1'-si;A) in the following way. Take T E M(r)(n;A) and consider its indicatrix of Fantappie [Fj]j. Then by the isomorphism J, T = J- 1[Fj]j. Now put J J



=

=

JF.(t)D(f¢)(t)dt. J

Then we claim that Tf is bounded ~n M(l )(1Rm+ 1,n;A)ind" Indeed, take any decomposition [Fk]k of [F] E r.\ 1 ) (JRm+kn;A), choose uk E [Fk]k and set u =

d)

I

(k)

uk and Tk

J

_,

[Fk]k.

I
lIo ;;

I (k)

I ( k)

I

(k)

Then [FkJk>lo

llo I Ia·

If for each k occurring in the above decomposition, ¢k E ¢k is taken arbitrarily but fixed~ constants Ck > 0 can be found such that

181

Ilo

~

ck

inf ukE[Fk]k

= ck i\<[FkJkl and so inf [F]=

L L [Fk]k

Ckpk([Fk]k).

(k)

( k)

Then we first prove 23.2

Proposition

I is an algebraic isomorphism between the right A-modules ( M(r)(ll;A) and M(l) lRm+1 '-\l;A ) • ~*

Proof It is clear that I is right A-linear. To show that I is injective, let f*E M(r)(ll;A) be such that I(f) = 0. Then of course = 0 for all T E M(r)(ll;A). Take a E \l arbitrarily, call j the least index such that a E K. and conm+1 J sider the function E(a-.). !hen E(a-.) E M(l)(JR 'Kj;A). Since by assumption = 0 for all T E M(r)(ll;A), whatever~ E ¢j may be JE(a-x)D(f¢)(x)dx

= 0.

Fix ¢ E ¢j and apply Green's Identity to a suitable compact neighbourhood K~ of Kj which is still contained in w¢. Then

I

E(a-x)do f(x) = 0

ClK

X ~

whence, by Cauchy's Integral Formula, f(a) I is injective. To prove the surjectivity rv* m+1 1" E M( l) (lR '-ll;A). Then we must find f E a E \l and call again j E lli the least index 182

Consequently f = 0 in 0 or of I we proceed as follows. Let M(r)(\l;A) such thatlT =lff. Take such that a E K.. Then obviously

= 0.

J

~

~1

E(a-.) E M(l)(R put f(a)

j

'Kj;A) and [E{a-.)]j E M(l)'

Hence it is meaningful to

= .

Using straightforward arguments it may be shown that f E for any a E ~ m+1

E(r)(~;A)

and that

As lis left A-linear, Df(a) and so f E

= = 0 M(r)(~;A).



We now claim that l

= lf. Notice that in any case

= 0.

(23.1)

Furth~rmore, fix k ~ 1 and consider the restriction of Tf-T to M~ 1

If [u]k E M(l) and hE [u]k, by Theorem 18.20 a sequence (hs)seN may be found in M(l)(Rm+1,Kk;A) which converges to h. Recall that each hs is a right A-linear combination of functions of the form )'

aPE(x-ai) 3x 1 ••• ax 1 1

From (23.1) lf-T = 0 on 23.3

p

it thus follows that for all s E ~. = 0 so that also = 0. Consequently lf-l = 0 on each M~ 1 )'which implies that m+1 M(l)(R "Q;A). c

As a final step we prove

Theorem

is a topological isomorphism between

M(r)(~;A)

-* m+1 '~;A). and M(l)b(R

m+1 '~;A)ind' Then by Theorem 22.10 B* = J -1 B Let B be bounded in M(l)(R is bounded in M(r)b(n;A). But as M(r)(n;A) is bornological, B* is equicontinuous and henGe contained in the polar of a semiball in M(r)(Q;A), say B* c b~K.(r) (see [4]). Hence, for any f E M(r)(Q;A)

~

J

183

= PB('Jrf) = sup lio [u]EB =sup Ii 0 TEB* s sup TEbll

Il 0 (r)

PK.

s

1

r

J

PK. (f), J

which means that I is continuous. To show that I- 1 is also continuous, first notice that M(r)(~;A) considered as a real convex space, is evaluable and has representable seminorms. Hence its natural system of seminorms is equivalent with the system {n 8 :BcM(r)b(n;A) bounded} (see [4]) where for each B n 8 (f)

=sup jJ. TEB

Consequently, for any j E ~. there exist C > 0 and B c M(r)b(n;A) bounded such that for all f E M(r)(~;A) PK. (f)

s Cn8 (f).

J

In virtue of Proposition 22.9 (ii), B* = e- 1(8) is bounded in M(r)b(Q;A) and * is bounded 1n . M(l)(~ m+1 so, using Theorem 22.10 J(B) ~;A)ind· We thus obtain that

s Cn 8 (f)

c

sup

[Fj] lJ(B*)

= Cp J ( B*) (11" f). 24. 24.1 184

IJ

Hilbert modules with reproducing kernel 2

In this section the classical HL 2 - and H -spaces and their corresponding

Bergman and SzegB kernels will be generalized to Euclidean space and this in the setting of monogenic function theory. 24.2 Defi~ition Let H(r) be a unitary right Hilbert A-module consisting of A-valued functions defined on some set F. Then a function K:F x F ~A is called a PcpPoducing krPn8l of H(r) if for any fixed t E F K(.,t) E H(r)

(i)

f(t) = (K(.,t),f) for all f E H(r)·

(ii)

In this case H(r) is said to be a unitary right Hilbert A-module with reproducing kernel. 24.3 Theorem (Aronszajn-Bergman) Let H(r) be a unitary right Hilbert Amodule consisting of functions defined on some set F. Then H(r) possesses a reproducing kernel if and only if for any t E F there exists a constant C(t) > 0 such that lf(t) Proof

10

<

C(t) llfll for all f

E

H(r)"

If H(r) has a reproducing kernel K then for any t E F and f E H(r) f(t) = (K(.,t),f)

so that lf(t)J 0

= <

I(K(.,t),f)l 0 IIK(.,t)ll. llfll·

Hence the necessary condition is fulfilled. As to the sufficient condition, take t E F and consider the right A-linear functional Tt on H(r) defined by = f(t) for all f E H(r)• Then by assumption

185

In view of the Riesz representation theorem, there exists a unique element ht E H(r) such that f(t) = = (ht,f) for all f E H(r)• Put

= ht(x), (t,x) E F

K(x,t)

x

F.

Then obviously K is a reproducing kernel of H(r)•

c

24.4 Now assume that H(r) is a unitary right Hilbert A-module admitting two different reproducing kernels K and K'. Then for some y E F, K(x,y) # K'(x,y) and so 0

<

IJK(.,y)-K'(.,y)ll2

Te (K-K' ,K-K') 0

Te (K,K-K')-Te (K',K-K') 0

0

0,

which yields a contradiction. Consequently, if H(r) admits a reproducing kernel then it is necessarily unique. 24.5 In what follows we suppose that of 1Rm+1.

~

is a relatively compact open subset

Definition Call ML 2 (r)(~;A) = M(r)(n;A) n L 2 (r)(~;A), i.e. ML 2 (r)(g;A) consists of those left monogenic functions in n which are square integrable in

n. Clearly ML 2 (r)(~;A) is a submodule of L 2 (r)(~;A) whence it may be endowed with the inner product and norm defined on L 2 (r)(~;A) (see also Section 7). It will be shown that in this way ML 2 (r)(g;A) is a right Hilbert A-module with reproducing kernel. Theorem Let t E such that 24.6

lf(t) 10

186

:;:;

~

be fixed.

C(t) llfll for all f

Then a constant C(t)

E ML 2 (r)(~;A).

>

0 may be found

Proof Call R(t) = d(tJRm+l~) and consider the closed ball B(t,r) with 0 < r = R(t)-s. By virtue of the Mean Value Theorem (Theorem 9.7)

I

f(t) = m+; r

Vm+ 1

r

m+ 1v m+l

f(u)du

B( t, r)

or f(t)

I

J

eA

A

fA(u)du.

B(t,r)

Consequently, applying the Cauchy-Schwarz inequality

I

L(

2

fA(u)du)

A B(t,r)

s

r

m+1v

L m+l

J

fA(u)du

A B(t,r)

I

lf(u)\~

du

B(t,r)

I

Q

rm+1v Letting s

+

\f(t)

\f(u) \ 2 du 0

m+l

0+ we get 10

~

C(t) \\fl\

where

24.7

Corollary

Let H c

pH(f) =sup \f(x)\ XEH

Q

0

be compact.

Then for all f E MLz(r)(Q;A)

s CH \lf\1

187

24.8 Theorem ducing kernel.

ML 2 (r)(~;A)

is a unitary right Hilbert A-module with repro-

Proof Taking account of the Theorems 24.3 and 24.6 it clearly suffices to show that ML 2 (r)(0;A) is complete. Let therefore (f i ) i €lN be a Cauchy sequence in ML 2 ( r) ( r:; A). Then there exists f E L2 (r)(r!;A) such that (fi)iE:lN converges to fin L2 (r)( ;A). But in view of Corollary 24.7 (fi)iE:lN is also a Cauchy sequence in M(r)(1;A) and hence it converges to f* in M( r) (c_;A). Clearly f* = f a.e. in ,, so that f* E ML 2 (r)(rz;A) and (fi)iE:lN converges to f* in ML 2 (r)(r;A). o 24.9

Remark

By the proof of the above Theorem 24.8 the Hilbert module

ML 2 (r)(u;A) is closed in L2 (r)(a;A). In general Mlz(r)('2;A) is not closed in M(r)( ,;A). Indeed, if a E Sm then E(x-a) E M(r)(s(0,1);A). By Runge's Theorem (Corollary 18.14) there exists a sequence of inner spherical monogenics converging to E(x-a) in M£r)(B(0,1 );A). All those inner spherical monogenics belong to Mlz(r) (B(0,1 );A) but obviously E(x-a) does not. Nevertheless we can state: If the sequence (fj)jE:lN of functions fj belonging to Mlz(r)('.;A) converges to f in M( ) C1;A) and possesses a subsequence (f. ) for which ( 11 f. Ill r Jk kE:lN Jk kE:lN is bounded, then f E

ML 2 (r)(~;A).

Indeed (f. ) has a subsequence which J k kE:lN converges weakly in 'z(r)('?;A) and hence pointwise, necessarily to f. So f E Lz(r)(];A). 24.10 Observe that, if the reproducing kernel of ML 2 (r)(~;A) is denoted by B(x,t), (x,t) E ~ x ~~.then for each f E Mlz(r)CI;A) and t E r,, f(t) 24.11

=

(B(.,t),f)

Theorem

=

L

ML 2 (r)(~I;A)

B(x,t) f(x)dx.

is separable.

Proof Consider a sequence of points (yi)iE:lN which is dense in' and associate with it the countable subset Q c ML 2 (r)(~;A) defined by Q

188

= {B(.,yi)' i

E lN}.

Then we claim that Q is total in orthogonal to Q, then

ML 2 (r)(r~;A).

Indeed, if hE ML 2 (r)(S"2;A) is

h(yi) = (B(.,y 1 ),h) = 0 for all i ElL Hence, by the continuity of h in

S"l,

h = 0 in r2.

o

24.12 Now let for any r > 0, 88r denote the boundary of the open (m+1)dimensional sphere 8r centred at the origin with radius r and consider the unitary right A-module L2(r)(88r;A) of A-valued square integrable functions on 88r, where Lebesgue measure is taken on 88r. 24.13 Definition Let R > 0; then ML 2 (r)(88R;A) consists of those elements in M(r)(8R;A) such that lim r-+R

<

I

I f 12

°

88 r

dS

< +

00 •

It will be shown that provided with the inner product

I

(f,g)R = 1 im r-+R

<

fgdS 8Br

and norm llfiiR=lim J lfl~ dS, r-+R 88 < r ML 2(r)(8BR;A) is a unitary right Hilbert A-module with reproducing kernel. 24.14

Theorem

For each t E 8R there exists C(t)

> 0

such that for all

f E ML 2 (r)(88R;A)

~

Lett E BRand take 0 < r < R such that t E Br. Integral Formula, for any f E ML 2 (r)(88R;A) f(t) = - 1wm+1

I

88

ii-t

r

lu-tlm+l

do

u

In view of Cauchy's

f(u)

189

so that, as dau f(t)

= ndS = (*)dS

= -1-

r~+1

on aBr'

J

u-t u f(u)dS. aBr iu-tlm+1

Hence, putting h(u) obtain:

(u-t)uf(u) and using the Cauchy-Schwarz inequality, we

lf(t)l~ ~

-r2 w...;.2_

m+1

(JaB r

ds \ r r ju-t12m+2} \JaB

r

Call d = d(t,IRm+ 1,Br); then on aBr' d <: iu-tj "2R whence ihl~ ~ 4R 2 r 2 jfl~· Consequently

so that, by passing to the limit for r

-

~

<

R,

where

24.15

Corollary Let H c BR be compact.

Then for each f E ML 2(r)(aBR;A)

where CH =

4Rm+ 2 1 R2m+2 with RH = d(H,IRm+ 'BR). wm+1 H

24.16 Next observe that the space Hh(aBR) consisting of those real valued harmonic functions f in BR for which

190

is a real Hilbert space when endowed with the inner product (see [22]) (f,g)R =lim r-+R

<

r

j

as

fg dS. r

As each component fA off E ML 2 (r)(aBR;A) belongs to Hh(aBR) we find that for all f,g E ML 2 (r)(aBR;A) and A E A, with f = L fAeA' g = L g8e 8 and A

B

Hence 1 im

r-+R

<

J

as

r

which means that ML 2 (r)(aBR;A) is a unitary right A-module. By analogous arguments it may be proved that for any f,g E ML 2 (r)(aBR;A) the expression (f,g)R = lim r-+R

<

J

as

fgdS r

is meaningful and that for all f,g,h E ML 2(r)(aBR;A) and A E A (f,g+h)R = (f,g)R + (f,h)R' (f,gA)R = (f,g)RA and

Moreover, as ,

is a (bounded) real linear functional on A, for each eo f E ML 2(r)(aBR;A) and A E A

while

191

and in view of the inequality stated in Corollary 24.15

Hence ML 2 (r)(aBR,A) is an inner product space. 24.17

Finally we get

Theorem ML 2(r)(aBR;A) is a unitary right Hilbert A-module with reproducing kernel. Proof Taking account of the Theorems 24.3 and 24.14 it clearly suffices to prove the completeness of ML 2 (r)(aBR;A). Let therefore (fi)i~ be a Cauchy sequence in ML 2(r)(aBR;A). Then for all s > 0, there ought to exist N(s) E~ such that llfj-fki!R ~ s if j,k ~ N(d. By the Corollary 24.15 (fi)i~ is a Cauchy sequence in M(r)(BR;A) so that, by the completeness of this space, there exists f E M(r)(BR;A) t~ which (fi)ieN converges. Now take 0 < r < R. Then, as for each~ E H (aBR) (see [22])

Jasr

lfj-fkl~dS ~

llfj-fkiiR

~

sz

so that, by passing to the limit for j JaBr

lf-fkl~dS ~

s 2 for all k r

Again using the property that ( J we find that aBr lim r~R

~

~ oo ,

N(s). !f-fki~dS)O
is increasing when r

t

J~s lf-fkl 02 dS ~ s 2 if k ~ N(s), a

r

< whence f-fk E ML 2(r)(aBR;A). Consequently f E ML 2(r)(aBR;A) while from the above reasoning it follows that (fi)i~ converges in norm to f in 192

R,

Notes to Chapter 3 The results in Section 16 are similar to those of the space O(n) of holomorphic functions in nor, even more generally, to those of the space of nullsolutions of a hypo-elliptic differential operator with constant coefficients (see [Ho 1]}. The original Montel Theorem concerning locally uniform bounded sequences of holomorphic functions (see [Mo]} gave rise to a special class of locally convex spaces, the so-called Frechet-Montel spaces (see e.g. [K1]}. As to the spirit of Section 17, it should be noted that in the more general context of elliptic differential operators with constant coefficients, the notion of a regular solution at infinity with respect to a fundamental solution, which in fact replaces the vanishing at infinity for holomorphic functions, was introduced by Grothendieck in [Gro1] and elaborated afterwards by Chauveheid in [Ch]. The Cauchy transform of a measure in the plane having compact support provides a useful example of such a function. As is well known, this Cauchy transform plays a basic role in approximation theory (see [Z]}. The Runge type theorems occurring in Section 18 are very close to the situation in one complex variable (see [Ho2]}. The problem of compact approximation of holomorphic functions in a domain by rational functions was solved initially by Runge in 1885 (see [Ru]}. Refinements dealing with polynomial approximation were obtained soon afterwards. The latter form of the Runge Theorem - polynomial approximation - gave rise to the introduction of so-called Runge domains in the theory of several complex variables (see [Ho2]}. The problem of approximating any solution f of P(D)f = 0, P(D) being a differential operator with constant coefficients, by nullsolutions of a special type was completely solved by 14algrange in [Mal1]. Note that his results already include the Behnke-Stein Theorem for holomorphic functions on a connected non compact Riemann surface (see also [BS]}. The problem of solving the equation P(D)f = g was treated in full generality by Malgrange and Ehrenpreis (see [Mal1], [Mal3], [E1], [E2], [E3]} and this, among other things, for the cases E(n) and V'(n). As to the case S'(Rn), which corresponds to the so-called division problem of any tempered distribution by a polynomial, it was first solved by H6rmander in [Ho3] While a still more general problem, namely the division of a distribution by 193

an analytic function, was treated by bojasiewicz (see [to]}. As to the special case of the operator~ we also refer to Martineau [Mar]. The methods az we have developed in Section 19 for solving the equation Df = g when g E E(r)(n;A) or g E v( 1)(n;A) are therefore standard. For the case g E S( 1)(Rm+ 1;A) we have made explicit use of function theoretic results occurring in Chapter 2. Note that Theorem 19.4, relating the growth condition of the solution to the right hand side of the equation Df = g, implies that for every bounded set Bin s*(Rm+ 1;A) there exists a bounded set a in s*(Rm+ 1 ;A) such that B c Da. Note also that the solution of the problem in the E(r)(n;A)-context gives rise to the construction of primitives of monogenic functions (Section 20). The form of the Mittag-Leffler Theorem as presented in Section 21 has now become standard for the case of holomorphic functions (see [Ho2]}. It is a strengthened form of the classical MittagLeffler Theorem. Its analogue in several complex variables is called the additive or first Cousin problem (see [BT2], [Ho2], [Mal2]}. For an abstract version of the Mittag-Leffler Theorem and its applications see also [Pe]. Duality in the theory of holomorphic functions was tackled in K6the's paper [K2] in which he showed that if V is an arbitrary proper open subset of the Riemann sphere 0 and if H(V) denotes the space of locally holomorphic functions on V, then its dual may b~ identified with H(U) the space of locally holomorphic functions on u = ~V. The identification between H'(V) and H(U) is obtained via the indicatrix of Fantappie of an analytic functional in V. The importance of this result of course lies in the fact that in this way linear functionals may be identified with holomorphic functions. Almost simultaneously Grothendieck developed in [Gro2]a duality theory for vector valued holomorphic functions defined on a proper open subset of 0. In fact by both authors' general results concerning duality theory for topological vector spaces were interpreted in the case of spaces of holomorphic functions. In this contexte Silva's papers [Si1] and [Si2] and the work [D] of Dias should also be mentioned, while for the historical notion of an analytic functional in the sense of Fantappie, we refer to Pelligrino's paper [P]. Tillmann worked out a duality theory for harmonic functions in the n-dimensional Euclidean space (n ~ 3) and for analytic functions on Riemann surfaces in respectively [T1] and [T2]. The latter case was treated differently by Gauthier and Rubel in [GR]. As to the case of several complex variables we refer to Lelong's work [Le] and Braun's thesis [Br], while in the infinite 194

dimensional case a duality theory was given by Dwyer in [Ow]. In [Ch] Chauveheid described completely the dual of the space of nullsolutions of an arbitrary elliptic differential equation with constant coefficients. If should be recalled that our proofs concerning the dual and bidual of M(r)(D;A) (Sections 22 and 23) rely heavily upon Runge type theorems established in Section 18, whereas in classical function theory the Runge approximation theorem appears to be a simple corollary to duality (see [RT]). The study of classes of functions which possess a reproducing kernel was put into an abstract Hilbert space setting by Aronszajn in [A]. The usefulness of the notion of a reproducing kernel in holomorphic function theory of one or several complex variables and in the solution of boundary value problems of elliptic differential equations has been beautifully exposed in a lot of works by Bergmann (see [Bel], [Be2]) and Bergmann-Schiffer ([BeS]). For a general introduction to the theory of reproducing kernels we refer to [Me] and [Hil]. It should be noticed that in Section 24 we have restricted ourselves to proving the existence of reproducing kernels for the spaces ML 2 (r)(D;A) ard No attention case where ~ is total subset of Hilbert modules Hile in [GH].

ML 2(r)(3BR;A). has thus been paid to the basis problem. However, in the open, bounded and connected, the existence of a countable ML 2(r)(D;A) may be shown (see [DB]). Finally note that with reproducing kernel were also studied by Gilbert and

Bibliography [A]

N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950) 337-404. [Bel] S. Bergmann, Sur les fonctions orthogonales de plusieurs variables complexes avec les applications a la th~orie des fonctions anal)tiques, M~morial des Sciences Math~matiques, Fasc. CVI(Gauthier-Villars, Paris, 1947). [Be2] , The kernel function and conformal mapping, Survey Math: Soc. 5 (1950). [BeS] S. Bergmann and M. Schiffer, Kernel functions and elliptic differential equations in mathematical physics (Academic Press, New York, 1953). R. Braun, Cauchy-Fantappi~ Formeln und Dualitat in der Funktion[Br] entheorie (Thesis, Gtlttingen, 1972). 195

[BS] H. Behnke and K. Stein, Entwicklung analytischer Funktionen auf Riemannschen Fl8chen, Math. Ann. 120 (1948) 430-461. [BT2] H. Behnke and P. Thullen, Theorie der Funktionen mehrerer komplexer Ver8nderlichen, Zweite Auflage (Springer-Verlag, Berlin, 1970). [Ch] P. Chauveheid, Application de 1 'analyse fonctionnelle ~ la theorie des fonctions analytiques (Th~se, Li~ge, 1973). [0] C. Dias, Espa~os vectoriais topologicos e sua aplica~ao nos espa~os funcionais analiticos, Sao Paulo (1951). [DB] R. Delanghe and F. Brackx, Hypercomplex function theory and Hilbert modules with reproducing kernel, ~oa. London Math. Soa. 37 (1978) 545576. [Ow] T. Dwyer, Dualite des espaces de fonctions enti~res en dimension infinie, Ann. Inst. Fourier 26, no 4 ( 1976) [E1] L. Ehrenpreis, Solutions of some problems of division I, Am. J. Math. 76 (1954) 883-903. [E2] , Solutions of some problems of division II, Am. J. Math. 78 (1956) 685-715. [E3] , Solutions of some problems of division III, Am. J. Math. 82 (1960) 522-588. [GH] R.P. Gilbert and G. Hile, Hilbert function modules with reproducing kernels, Nonlinear analysis 1 (1977) 135-150. [GR] P.H. Gauthier and L.A. Rubel, Holomorphic functionals on open Riemann surfaces, Can. J. Math. 28 (1976) 885-888. [Gro1] A. Grothendieck, Sur les espaces de solutions d'une classe generale d'equations aux derivees partielles, J. d'Analyse Math. 2 (1952/53) 243280. [Gro2] , Sur certains espaces de fonctions holomorphes, I, II, J. Reine Angew. Math. 192 (1953) 35-64, 77-95. [Hil] E. Hille, Introduction to the general theory of reproducing kernels, Roaky Mountain J. of Math. 2 (1972) 321-368. [Ho1] L. H6rmander, Linear partial differential operators (Springer-Verlag, Berlin, 1976). [Ho2] , An introduction to complex analysis in several variables (North-Holland, Amsterdam, 1973). [Ho3] , ·On the division of distributions by polynomials, Ark. Mat. 3 (1958) 555-568. 196

[K1] G. K6the, Topologische lineare R~ume I (Springer-Verlag, Berlin, 1960). [K2] , Dualit~t in der Funktionentheorie, J. Reine Angew. Math. 191 (1953) 30-49. P. Lelong, Fonctionnelles analytiques et fonctions entieres (n var[Le] iables) (Presses de l'Universite de Montreal, 1968). [to] S. tojasiewicz, Sur le probleme de division, Studia Math. 18 (1959) 87-136. [Mal1] B. Malgrange, Existence et approximation des solutions des equations aux d~riv~es partielles et des ~quations de convolution, Ann. Inst. Fourier 6 (1955/56) 271-355. [Mal2] Lectures on the theory of functions of several complex variables (Tata Institute of Fundamental Research, Bombay, 1958). [Mal3] - - - - Sur la propagation de la r~gularit~ des solutions des ~quations a coefficients constants, Bull. Math. Soc. Sci. Math. Phys. R.P. Roumanie 3 (1959) 433-440. [Mar] A. Martineau, Distributions et valeurs au bord des fonctions holomorphes, In: Theory of Distributions, Proc. Intern. Summer Institute, Inst. Gulbenkian de Ciencia, Lisboa (1964) 113-326. [Me] H. Meschkowski, Hilbertsche R~ume mit Kernfunktion (Springer-Verlag, Berlin, 1962). [Mo] P. Montel, Sur les suites infinies de fonctions, Ann. Sci. Ecole Norm. Sup. 24 (1907) 233-334. F. Pelligrino, Die analytische Funktionale und ihre Anwendungen, [P] Mat. Tidskrift B (1949) 31-62. [Pe] H.J. Petzsche, Some results of Mittag-Leffler type for vectorvalued functions and spaces of type A, in: Functional Analysis: Surveys and Recent Results II (North-Holland, Amsterdam, 1980) 183-204. [RT] L.A. Rubel and B.A. Taylor, Functional Analysis: proofs of some theorems in function theory, Amer. Math. Monthly 76 (1969) 483-488; Corrections, Amer. Math. Monthly 77 (1970) 58. [Ru] C. Runge, Zur Theorie der eindeutigen analytischen Funktionen, Acta, Math. 6 (1885) 229-244. [Si1] J.S. e Silva, As Funcqes analiticas e a analisa functional, Port. Math. 9 (1950) 1-130. [Si2] Sobre a topologica des espayos funcionais analiticos, Rev. Fac. Cienc. Lisboa 2, Ser. A1 (1950) 23-102. 197

[T1] H.G. Tillmann, DualitMt in der Potentialtheorie, Port. Math. 13 (1954) 55-86. [T2] , DualitMt in der Funktionentheorie auf Riemannschen FlMchen, J. Reine Angew. Math. 195 (1956) 76-101. [Z] L. Zalcman, Analytic capacity and rational approximation, Lecture Notes in Mathematics 50 (Springer-Verlag, Berlin, 1968).

198

4 Boundary values of monogenic functions and Fourier analysis on the sphere The present chapter is divided in two parts. In the first part we establish criteria under which a function f which is left monogenic in Q-..Q, n cJRm+ 1 being an x0 -normal open neighbourhood of~ cJRm, admits distributional boundary valuesxl~~+ f(x ± x0 ) in respectively v(l)s(~;A) and S(l)s(n;A) (§26 ). 0

In order to attack these boundary value problems we begin by studying distributional extension properties of harmonic functions (§25). Finally these boundary value results are applied to represent the spaces V{l)(n;A), V( 1 )(tl;A) and Ell)(JRm+ 1 ;A) by means of appropriate spaces of left monogenic functions in Q'n ( 27). In the second part it is shown how any A-valued function f which is either analytic or square integrable on the unit sphere Sm may be expanded into a series of surface spherical monogenics. In particular a decomposition of any surface spherical harmonic of order k into surface spherical monogenics is obtained, leading up to the representation of spaces of analytic functions, analytic functionals and L2-functions on Sm by means of spaces of sequences of surface spherical monogenics. Moreover the Bergman and Szegtl kernels, the existence of which has been proved in Section 24, are now calculated explicitly (§§28 and 29). 25. Distributional extension properties of harmonic functions 25.1 Let v cJRm+ 1 be open and U c V be open too. The aim of this section is to establish criteria under which an A-valued function f which is harmonic in U admits an extension in some space of distributions in V. First we consider the case of the distributional extensibility to s( 1 )(1Rm+ 1 ;A). 25.2 Definition Let U cJRm+ 1 be open and let wE V(JRm+ 1 ;JR) be a radial function such that supp wc B(0,1), w ~ 0 and m£ 1 w(x)dx = 1. Furthermore JR suppose that there exists a strictly positive continuous function r in U satisfying

199

r1kT ~ d(x,aU),

(i)

lr

(ii) (iii)

x E U;

is of slow growth in U;

r is of weak slow growth in U.

The the function ¢x:U

~ V(U~).

x E U, given by

¢x(y) = (r(x)r+ 1lJi(r(x)(y-x)),y E

u,

is called a harmonic testfield in U. Notice that, given lJi c V(ffim+ 1 ~). any open subset U of mm+ 1 admits a harmonic testfield. Indeed, if U = ffim+ 1 , choose r(x) = 1 in ffim+ 1 and if U~m+ 1 , take r(x) = 1 + d(x,1aU)' x E U. Notice also that if¢ is a harmonic testfield in U, then supp ¢x c B(x,~), x E u fixed, and that for any r\x, f E Harm(r)(U;A), f(x)

=J

<jJ

U

(y)f(y)dy,

X

X

E U.

* 25.3 Theorem Let U cffi m+1 be open and let f E Harm(r)(U;A) n S(l)(U;A). Then f is of weak slow growth in U. Proof

Let ¢ be a harmonic testfield in U. f(x) =

fu

Then for any x E u

¢x(y)f(y)dy.

As f E s( 1 )(U;A), there exist g, continuous and of slow growth in a E ~m+ 1 such that f = aag in U. Hence we have that f(x) = (-1) Ia I J aa¢ (y)g(y)dy Uy

X

so that If (x) Io ;'; Fa ( x )

J 1

B(x,rrxr)

Ig (y) Io dy

where both factors at the right hand side, namely F (x) a

200

=sup yEU

laa'" (y) 1 o Y ·'~'x

u,

and

and

are of weak slow growth in U.

Consequently f is of weak slow growth too.

o

25.4 Coro~~1 Let U cffim+1 be open and convex and let f E Harm(r)(U;A). Then the following are equivalent. (i)

(ii) (iii)

f E s(l)(U;A)

f is of weak slow growth in

u

f admits a distributional extension to

s( 1 )(ffim+ 1 ;A).

Proof ( i ) o=:., ( i i ) . See Theorem 25.3. (ii) -·.(iii). As U is convex Theorem 5.6 yields that f E hence f admits a distributional extension to SCl)(ffim+l;A) ( i i i ) = > ( i ) . Obvious. o

s( 1 )(U;A)

and

..,m+ 1 be an x0 -norma 1 open ne1g · hbour hoo d of th e open set 25 . 5 Nex t 1e t Q c~ ;~ cffim and let U = ~~+ = {x E Q:x 0 > 0}. Furthermore let f E CCr)(SI+;A). Then we say that lim f(x + x0 ) exists in v( 1 )s(~;A) if for each compact X ->0+ 0 0 interval K c C.', lim f(x + x0 ) exists in v( 1 )s(K;A). X ->0 :0 Notice that, as for each compact interval K c ~. oK > 0 may be found for which ]Q,,0+ 0 1 S 0

Using the 'principe du recollement' we obtain under the foregoing assumptions the existence of a unique distribution T E v( l)(~;A) such that T'G 1K = lim f ,_.x + x0 ) . It is ca 11 ed the die t1•ibu tiona l boundar>y va luc of f X ->0+ 0 in v( l)s(0.;A) and we write T = J+f =lim f(x + xo). Now assume that X ->0+ 0 0 = ]-b,b[x ~,b E]O,+m~, and that f E C?r)(Q+;A). If for any x0 E ]O,b[ fixed, f(x + x0 ) E s( 1 )(r,;A) and lim f(x+x 0 ) X -+0+ 0

201

exists in s(l)s(n;A), then this limit is called the distributional boundary Again we write a+f =lim f(x + x0 ). 0 Iff E C( r )(n - ;A), where n- = {x E n:x o < O}~O~O+ then it is clear what is meant by the existence of a-f = lim f(x- x ) in respectively v( 1 )s{n;A) xo~O+ o and s(l)s{n;A). Again a-f is called a dictributional boundary value off in v(l)s(n;A) or s(l)s(n;A) respectively. value off in s( 1)s(n;A).

25.6 The following theorem gives a sufficient condition under which f E c(r){n+;A) admits a distributional extension to 1 ){n;A).

v(

Theorem

Iff E c(r)(n+;A) is such that a+f =xl~+ f(x + x0 ) exists in 0

VCl)s(n;A), then f may be extended to a distribution in v( 1)(n;A). Proof Let K be a compact interval inn and let I = K X ]a,b[ be contained inn with a< 0
~o+

o

o

~

o

m

Theorem 4.10(2), there exist g E C( ){K x [O,b[;A) and a multi-index 1 E ~ o ~ r o such that in K x ]O,b[, f = alg. Now consider J = n n (K x J~,b[) and let h be defined in J by ,x 0 { g{x 0 ~;) ~

h{x)

g(O,x), x0

>

0

:;;

0. ~

Then h is continuous in J and hence T Furthermore for any ~ E V( 1 )(J;A) ~

= (-1) Ill

alh E v( 1 ){J;A) is an extension of f.

Li

a <J>(y)h{y)dy

b

fo {JK ~{yo,y)f(y

+ yo)dy)dyo

rO +

(25.1)

j_oo

which implies that Tis completely determined by fin J. Let K., i = 1 ,2, 1 be compact intervals inn, g.1 E C0( r )(K 1. x ]O,b[ ;A), i-1 E ~m be such that in o • Ki x ]O,b[ , f = a1 gi and let hi be defined in Ji = n n (Ki x J-oo,b[) by 0

202

h.{x) = { 1

gi(x 0 ,x),x 0

>

o

-+ g. ( 0 ,X), X

:~

0.

1

0

-+

li Then, putting Ti =a hi' = 1,2, we get by (25.1) that T1 = T2 in J 1 n J 2 • The foregoing construction together with the 'principe du recollement' imply the existence of a global extension off to o( 1 )(~;A). c 25.7 Corollary Let~-+ c~ m be open and let~ be an x0 -normal open neighbourhood of~. Furthermore let h E Harm(r)(~+;A) be such that a+h = lim h(x+x ) ....0+

X

. * Then h admits an extens1on to V(l)(n;A).

* exists in V(l)s(n;A).

0

0

25.8 The following result indicates that the extensibility of a harmonic function f in ~+ to v(* 1 )(n;A) implies that f satisfies locally some growth conditions. Theorem Iff E Harm(n+;A) admits an extension to v( 1 )(ro;A), then for any compact interval K c ~ there exist ~K > 0 and k E N such that sup O<X o~"'"K
sup xEK

+ oo.

-+

Proof Let K c ~be a compact interval. Choose a compact interval and o > 0 such that K' x [0,6] c n and K c K1 Furthermore, take 0 < ~K < o • If T E v( 1)(n;A) is a distributional extension off, then clearly * * B o T € v(l)(K' X [O,~];A) = s(l)(l\ 1 X ]0,6[ ;A) (see 5.5~. As v{l)(K'x]O,o[ ;A) is dense in S(!)(K' x ]O,f>[ ;A) and TIK' x ]0,6[ = fiK' x ]0,6[, we obtain that f E stn
0

sup sup O<xo::i ~K xEK 26.

k

-+

lx 0 f(x + x0 )

10

<

+ ""·

c

Distributional boundary values and canonical extensions of monogenic functions

26.1 Let Q c:~m be open and let r1 c:~m+ 1 be an x0 -normal open neighbourhood of Q. Furthermore let f E M(r)(~~;A). Then in this section conditions are stated under which f possesses distributional boundary values in v( 1 )s(Q;A) 203

*

A.

....

The case V(l)(n;A)

26.2

~~

We only formulate the boundary value problem for

Theorem

Suppose that f E M(r)(<\;A).

~+f =

(i)

lim

0

(~;A) S

*

0

( i i)

•

Then the following are equivalent

f(; + x ) exists in v*(l)

X -+0+

+

f admits a distributional extension to V(l)(q;A) .... For every compact i nterva 1 K c r2 there exist ~:K > 0 and k E

(iii)

]II

such that sup

....

xEK

Proof

(i) ==>(ii).

(ii) =,>(iii).

See Theorem 25.5.

As f E M(r)(ll+;A), f is harmonic in II+ and so Theorem 25.8

may be applied. (iii)

:..:=>

(i).

By the 'principe du recollement' it suffices to prove that -+--

for any compact interval K c 12,

lim X

Choose K and let ~K > 0, k E If(;+ x0 )

1

0

0

]II

-+0+

*

-+

0

f(x + x 0 ) exists in V(l)s(K,A).

Kx

and C > 0 be such that in

]O,cK[,

~ C(1 + x~k).

Furthermore put

....

f(x + s)ds. Then'\ h 1 0

=

f.

Moreover as f E M(r)(rl+;A), Dh 1

->-

=

0

f(t:K + x) E E(r)(K;A). 0

Find g 1 E E(r)(K;A) such that D0 g 1 (x) = -f(cK + x) inK (see also Theorem 19.2). Call f 1 = h 1 + g 1 • Then f 1 E M(r)(l( x]O,cK[ ;A). Proceeding in the same way, assume call

204

tha~

h 1 _ 1 , g 1 _ 1 and f 1 _ 1 have already been constructed,

let g1(x) be such that D0 g1 (x) = -f 1_1(EK + x) inn and put f 1 = h1 + g1 • Then 1n view of the growth condition satisfied by f, there ought to exist 10 E ~ such that f 1 is continuous in K x [O,EK[. Hence 0

0

1im

fl

x .....a+

(X+

X

0

0

= fl (X

)

°

+ 0)

0 0

exists in v(l)s(K;A).

But, as in K x ]O,EK[,

f

f 1 (x+o>. 0

* belongs to V( 1) (K;A). 0

c

26.3 Remark It should be noticed that the monogenicity is not necessary the weakest condition upon f in order to get some of the implications stated in Theorem 26.2. To this end we refer to the results proved in the previous section. 26.4 In what follows a relationship will be established between the existence of boundary values and the solution of the equation OS = T. Theorem * Assume that f E M( r )(n+ ;A) admits a distributional boundary value -'--'--....::._ a+f in V(l)s(~;A). Then there exists a unique distributional extension f* off in v( 1 )(n;A) such that of*

= a(x 0 )

~

a+f(x). .....

Moreover the support of f* is contained in n+ u n. ~

Let K c ~ be a compact interval. Following the proof of Theorem 26.2 there exists 1 E ~. where 1 may be chosen to be even, say 10 = 2s, and 0 0 0 f 1 E C(r)(K x [O,+oo[) n n;A) n M(r)((K x ]O,+oo[) n n;A) such that 0

0

0

1

a of

x 0 10

= ( -!J. ) s m

205

*

0

exists in V(l)(K~A). Next extend f 1

o

n n where

to yK in (K x lR)

0

* * 0 *" Then fK E v(l)(K xlR) n n;A),fki(K Moreover, if 10 K x [-o.o] c

n

= 2s,

f;.

= (-~)syK

X

. *0 ]O,:cx{) n n= f, Whlle fKi(K ]-oo,Q[)nn:o. in (K x JR) n n.

and let ¢ E V(K x ]-6,0[ ;lR).

* ~ = ( -1)
1 +1 0

I I

Takeo > 0 such that

Then, using Cauchy's Theorem

( ( D( - t.m ) s

Kx[O, o]

1 +1

(-1) 0

D( ( -llm)s <J>}yK)dx

Kx[O, o]

I

1 +1

(-1) o

do((-1\n)s<J>)yK

( Kx[O, o])

-n

1

<

=

«tf.

o

L<<- ~,

<x

s H"X + o )y K

+ o,

dx

¢(1 + O)>

=
;tf .¢>.

The same equality holds of course for any for any <J>E V((K x JR) n r1;A).


E V((K x JR) n n;JR) and hence also

But this implies that

(K

x lR) n n

(26. 1) whence

f~ is a distributional extension of f i(K

x lR)

n n+ to

(K x JR)

n n sat-

isfying (26.1) 0

Furthermore, if f 1 and f 2 are distributional extensions of fi(K x JR) n n 0 0 + to (K x lR) n nwhich satisfy DS = o(x 0 ) 0 a+f in (K x JR) n n. then f 1-f 2 is 0

left monogenic in (K x lR) n n

206

0

and vanishes in (K x JR) n n + so that f 1 = f 2 •

Consequently, if f* denotes the distributional extension of f to Q such that f * I(K" x R) n Q = fK* for each compact interval K c ;t "• then in Q Of* = o(x 0 ) ® a+f and by the foregoing, f* is unique.

D

26.5 From Theorem 26.4 it follo~s that iff E M(r)(Q\n;A) is such that the distributional boundary values a f lim f(x+x 0 ) and a-f lim f(x-x 0 ) X -+0+ X -+0+ 0 0 exist in V{!)s(Q;A) then, putting BVf = a+f-a-f, a unique distributional extension f of f to Q may be found satisfying in Q

It is called the canonical extension of f. 26.6 Remark In what precedes we assumed that for f E M(r)(Q\~;A) the boundary values a±f exist separately. Now suppose that for f E M(r)(Q\Q;A), 1 im X

0

-+0+

(f(x+x ) - f(x-x 0 )) 0

exists in v(l)s(~;A). Then we claim that this implies the existence of each of a+f and Indeed, ca 11 Q* = {x E Rm+l : x E Q and -x E Q}. Then the function h(x 0 ,x) f(x+x 0 ) - f(x-x 0 ) is harmonic in Q+* and by assumption lim h(x 0 ,x) exists X -+0+ 0 in v(l)s($1;A). Hence, in view of the Corollary 25.7, h(x 0 ,x) admits a distributional extension to V( 1)(Q;A) and so does of course Dh(x 0 ,x). But in

rt+

-+ so that, using Theorem 26.2 for every compact interval K c Q, EK k E lN may be found such that sup

O<x 0 ~EK

sup xEK

1

x~ a~f (x+x 0 ) o

10

<

+

>

0 and

oo.

Consequently f also satisfies estimates of the above type in the considered 207

* )(n;A). -+ domain whence, again by Theorem 26.2, a+f exists in V(l true for a-f.

The same is

* (-+n;A) The case s(l)

B.

-+

m

.

-+

26.7 Let n c R be open and convex and let n be of the form n = ]-b,b[ x n where b € ]O,+oo]. Then it follows from Theorem 25.3 that f € M(r) (n+;A) n s( 1 )(n+;A) if and only iff € M(r)(n+;A) and f is of weak slow growth in n+. Notice that the last condition upon f means that for some k € ~ and

c>6 (26.2)

(i )

if b is finite or that (i i )

if

b

(26.3)

= + oo,

26.8 Theorem Let f

€

M(r)(n+;A) n s( 1 )(n+;A).

Then a+f = xl~+ f(x+x 0 ) 0

Proof We only work out the proof forb<+ oo, the case b = +oo being similar. Call

Then a h1 = f. xo Moreover as f € M(r)(n+;A), Dh 1 = f(% + x) inn+ where, in view of the growth condition (26.2),

Hence there exists~ € s( 1 )(Rm;A) such that ~In= f(% + x). Furthermore, in virtue of Theorem 19.5, there exists

n. Consequently g1 E s(l)(n;A) n E(r)(n;A) may ~e found such that D(hl+g1)=0 inn+. Put f1 = h1+g1; then fl E M(r)(n+;A) n s(l)(n+;A). Now assume that we~have already ;onstructed h1_ 1 and g1 _1 such that fl-1 = hl-1 + gl-1 E M(r)(n+;A) n s(l)(n+;A). Furthermore put ....

h1 (x ,x) = 0

rXo

....

J

f 1 1(x+s)ds b/2 -

* (....n;A ) n E(r) (-+n;A ) for wh1ch D g (x) = -f _ ( b + ....x). and choose g 1 E S(l) 0 1 1 12 Then f 1 = h1 + g1 E M(r)(n+;A) n s{l)(n+;A). But for some 10 E lN, f 1 is 0 continuous in n+ u nand for some C* > 0 in n+ 0

....

Hence 1 im f 1 (x+x ) X

0

....0+

0

O

lim f(x+x) exists in s*(l) (n;A).

X ....0+ 0

0

S

26.9 Just as in the case of

o

v(1 )(n;A)

(see Theorem 26.4) we may prove

Theorem Let f E M(r)(n+;A) n s(1 )(n+;A). Then there exists a unique distributional extension f* off to srl){n;A) such that

Moreover the support of f* is contained in n+ 26.10 Now suppose that f 26.8 the boundary values

E

M(r)(n\n;A)

n

u

n.

s( 1)(n\n;A).

Then by Theorem

a±f = lim f(x+x ) X 0

....0+

0

exist in s(l)s(n;A) and, using Theorem 26.9, f admits a unique extension f* to s( 1 )(n;A), called the canonica~ extension off, for which Of* = o{x0 ) ® BVf where again BVf = a+f- a- f. 26.11

In Theorem 25.3 we have characterized the space M(r)(n;A)

n

* S(l)(n;A) 209

in terms of growth conditions. be proved.

If~= ~m+l the following finer result may

Theorem 8 * (~m+l ;A) we have (see Theorem 25.3) that Proof As f e: M(r) (~m+l ;A) n S(l) there exist C > 0 and k e: N such that jf(x)

10

s C(l + jxj)k.

But this implies that the Taylor series of f at the origin has to break off and so f e: P(r)" 0 27.

Representation of distributions by monogenic functions

v(

In this section we show how distributions in 1 ) (~;A), s( 1 ~ U1;A) and E( 1 )(~m;A) may be represented by monogenic functions in ~\~and R +l~m respectively, where ~and ~ satisfy the conditions of the previous section. 27.1

A. 27.2

The cases v( 1 )(~;A) and s( 1 )(~;A) Let us recall that if

fe:

M(,r )(~\~;A) has the boundary value

BVf = lim f(x+x ) - lim f(x-x ), xo~+ o xo~+ o then the canonical extension f* of f satisfies the equation

A converse of this result will now be formulated. 27.3 Theorem ;et T(x) e: v( 1 )(~;A) (resp. s( 1 )(~;A)) be given and let F be a solution in V(l)(~;A) (resp. s( 1 )(~;A)) of the equation

Then F is the canonicai extension of f = FI~\~"* V(l)s(~;A) (rcsp. S(l)s(~;A)).

210

e:

"* M(r) ( ~\~;A) and BVf = T in

Proof We only work out the proof for v(1 )s(~;A), the other case being treated analogously. First observe that as DF = 0 in ~\Q,f = FI~\Q indeed belongs to M(r)(~\Q;A) (see 17.4.2.). Moreover, as f has a distributional extension F to~. by means of Theorem 26.2 the boundary values lim f(x±x ) ...0+

X +

0

0

>

= a-f exist in-+v(1 )s(S'l;A). Let f* be the canonical extension off; then F-f* = 0 in ~\~ and so supp(F-f*) c n. Now let K c ~ be a compact interval and put ~K = (KY~) n ~. Then in ~K'F-f* may be written as 0

n

F-f* =

Y.

o(h)(x 0 ) e eh(x)

k=O

*

0

where eh c V(l)(K;A). Consequently, as Of* = o(x 0 ) e BVf in

~.

we have that in

~K

o(x0 ) e (T-BVf) = D(F-f*)

L

(o(h+l)(x0 ) e eh(x)+o(h)(x 0 ) e D0 eh(x))

O~h~n 0

from which it follows that en (x) Hence F = f* in ~K and as moreover o(x0 ) e (T-BVf)

o(x 0 ) e

0 in K.

o0 e0 (x)

0 0

we find that T = BVf in K. Using the 'principe du recollement', we get the desired result.

Q

* -+ * -+ 27.4 Theorem (Representation Theorem) LetT E v(l)(~;A) (resp. s(l)(~;A)). Then there exists f c M(r)(Q\n;A) such that BVf = T. Furthermore if g ~ M(r)(~\~;A) is such that BVg = T, then f-g is extendible to a function h

c M(r)(~;A).

~Put f = Flm~ where F is a solution in v(l)(~;A) (resp. s( 1 )(~;A)) of the equation OS= o(x) e T(x). Then Theorem 27.3 implies that BVf = T. 0 .... As to the second part, let g c M(r)(~\~;A) be such that BVg = T. Then, denoting by g* c V(l)(~;A) (resp. S( 1 )(~;A)) the canonical extension of g we get that D(F-g*) = 0 in ~ so that h=F-g*extends monogenically f-g to ~. 0

211

B.

The case E(l) (lRm;A)

27.5 The representation of E{l)(lRm;A) may be given in a much more direct way than in the cases V( 1l(n;A) and S(l)(0,;A) and this by using the so-called Cauchy tmnsform of T E E( 1) (lRm;A). A

27.6 Definition given by

LetT E E{ 1 )(lRm;A).

Then the Cauchy transform T of T is

+-

T(x)

~+1


I

U+X ) ' X E JRm+1~m. U+X- 1m+1

+

Notice that f is well defined since for

u E lRm

and x E JRm+ 1 ~m.

u+x U+X 1 m+1 m m+1 m E(l)(lR ;A). f4oreover one easily verifies that T E f·1(r)(lR ~;A). But there is more, namely I

+ -

E

A

27.7 Theorem LetT E E{ 1 )(lRm;A): Then Tmay be extended to a left monogenic function, still denoted by T, in lRm+ 1,supp T. Furthermore lim T(x)=O X->=

and for any~ E V(l)(lRm;A)

=lim

x +0+ 0

J lR

m

~(x)(T(x+x ) - T(x-x 0 ))dx. 0

Finally there exists kEN such that for all E sup

1

x~supp

k " (d(x,supp T+Bm(d)) T(x)

T+Bm( d d(x,suppT + Bm(d)

<

>

10

0 <

+

oo.

1

Proof We first prove that T may be extended to a left monogenic function in Rm+ 1,supp T. By Painlev~'s Theorem (see Theorem 10.6) it is sufficient to show that T admits a continuous extension to Rm+ 1,supp T. Let y E JRm,supp T and choose E > 0 such that B(y,E) clRm+ 1,supp T. Furthermore let a E V(lRm;lR) be such that a = 1 in supp T and a = 0 in B(y,E) n lRm. Then in B(y,E) the function f given by 1

-+

U+X

f(x) = - - wm+1 u lu+xl is continuous and its value does not depend upon 212

Moreover f coincides

" o m " with T in B(y,£)~ and hence T indeed admits a continuous extension to Rm+l,supp T.

Clearly lim T(x)

0 and we claim that in

X-+oo

svf = T = lim X

0

-+0+

v( 1 )(Rm;A)

(T(x+x ) - T(x-x )). 0

To this end observe that V(l)(Rm+l;A) given by

-

x

lxlm+l

0

T has the distributional extension Text in

*(o(x 0 )

@

T(x)).

Furthermore, as

using Theorem 27.4 we get that

1im

=

X -+0+ 0

whence for any >

4> E

V( 1) (lRm ;A)

= lim X -+0+ 0

T

Finally, as Tadmits the distributional extension t to v*( )(Rm+l ;A). it " ex 1 follows from Theorem 26.2 that T satisfies the stated growth conditions. c 27.8 Now we formulate a characterization for the Cauchy transform of distributions in E(1 )(Rm;A). Theorem Let K cRm be compact and let f E M(r) (Rm+1 'K;A) be such that lim f(x) = 0 and that for some k E ~ X-+oo

sup l(d(x,K))k f(x)i 0 < + oo. d(x,K)<1 xtK Then f admits a boundary value BVf in v( 1)s(1Rm;A). supp BVf c K and f = BVf.

Moreover BVf E E{1 )(lRm;~. 213

Proof In view of Theorem 26.2 BVf exists in v{l)s(Rm;A) and as f E M(r)(Rm+ 1, K;A), we have that for any~ E V(l )(Rm,K;A) lim xo....O+

J Rm,K

~(x)(f(x+x 0 )

-

f(x-x 0 ))dx

=o

whence BVfiRm,K = 0. Hence BVf E E{ 1 )(Rm;A) with supp BVf c K. By virtue of Theorem 27.7 BVf may be defined and, using Theorem 27.4, f- BVf admits a left monogenic extension to Rm+ 1• But, as lim (f(x) - BVf(x)) = 0, Liouville's Theorem yields X-+oo that f = BVf. c 27.9 Putting together the results of the conclude as follows: Let -+Q cRm be open, neighbourhood of~ and call M(r)V*(~~;A) f E M(r)(~~;A) such that for any compact and k E N for which

Theorems 27.4, 27.7 and 27.8 we may let ::2 cRm+1 be an x0 -normal open the space of all functions interval K c ~there exist cK > 0

sup 1x 0k f(x 0 ,x-+) 10 < + oo, O 0 and k E ~ for which sup

k

If ( x) Io ~ c( 1+ Ix I ) ( 1 +

a(x,1au) )k

m+1 Finally let M(r)E* denote the space of functions f E M(r) (R 'K;A) for some K cRm compact such that (i)

there exists k E ~for which sup l(d(x,K))kf(x) d(x,K)<1 x~K

and (ii)

lim f(x) X->oo

214

0.

10

<+co

Next consider the boundary value mappings, denoted by BV, which map the spaces M(r)V*(:z,0;A), M(r)S*(:.~;A) and M(r)E* respectively upon V( 1) (0.;A) ,S( l) (n;A) and E(ll(Rm;A). Then these mappings are left A-linear and surjective. Moreover the following algebraic isomorphisms hold. Theorem ( i)

(ii) (iii) 28.

v( l) (~;A)

~ M(r)V*(s-1'-n;A)/M(r) (12;A).

S(l)(n;A) ~ M(r)S*(7'~;A)/M(r)S*(~). E{l)(Rm;A) : M(r)E*.

Analytic functions and functionals on the unit sphere

28.1 In this section we treat of the following problem. Given an A-module E of functions or functionals on the unit sphere, determine an A-module ME consisting of monogenic functions which represents E. This leads up to the decomposition of the elements in E into a series of surface spherical monogenies satisfying certain estimates. 28.2 Lemma Let f be an A-valued harmonic function in B(0,1). f = f 1 + f 2 where Df 1 = 0 and Df 2 0 in B(0,1).

Then

Proof Let f E Harm(r)(B(0,1);A); then Df = g is le~t monogenic in B(0,1) so that, in view of Leffima 20.3, there exists hE M(r)(B(0,1);A) for which a h=g -h h h xo and so 02- = g = Of. Take f 1 = 'Z and f 2 = f--z. c 28.3 The proof of the following Painleve-type theorem will be omitted since it may be carried out as in 10.6. Let us just point out that when speaking of an m-dimeno·ional. injec:tive C -surfac:e L: in Rm+ 1 the range is meant of a C -injection from an open subset U of Rm into Rm+l 00

00

Theorem (Painleve). Let n cRm+ 1 be open and let L: be an m-dimensional injective C -surface in n such that D'-L is open. Furthermore let f E M(r)(D'-L:;A) n c(r)(n;~). Then f E M(r)(n;A). 28.4 We now state a Cauchy-Kowalewski type theorem for analytic functions on the unit sphere. Let us recall that an A-valued function is said to be analytic: on the unit sphere Sm if it is locally analytic on Sm. The set of m such functions clearly constitutes a bi-A-module which is denoted by a(S ;A). 215

Theorem (Cauchy-Kowalewski) Let f E a(r)(Sm;A). Then there exist 0 < R2 < 1 < R1 and a unique function f* which is left monogenic in B(O,R 1 ),B(O,R 2 ) such that f*ISm = f. Proof Take P E Sm and define a system of spherical coordinates in a neighbourhood n of P in Rm+ 1• Then in n, D may be written as D = war + 1 a (see r w also 8.9). Obviously g(w) = -~3 f(w) is analytic in Sm n n and as g does not depend w m m on the coordinate system chosen on S , g is well defined on the whole of S • By the Cauchy-Kowalewski theorem for the Laplacian (see e.g. [21]} there exist 0 < R2 < 1 < R1 and a unique A-valued harmonic function h in B(O,R 1 ) ' B(O,R 2 ) such that h!Sm = f and arhiSm(w) = -~awf(w) = g(w) = -~ ~awhiSm(w). We claim that h is the desired left monogenic extension f* of f. Indeed, let P and n be as before and chooser> 0 such that B(P,r) c B(O,R 1 ),B(O,R 2 ). Then, using Lemma 28.2, h = f 1 + f 2 in B(P,r) where Df 1 = 0 and Df 2 = 0. Put 0 y = Dh in B(P,r). Then o

1 -

m

o

m

y!B(P,r) n S = w(a r +wa w)h!B(P,r) n S = 0. r o

o

m

Consequently, as Dy = 0 in B(P,r) and as moreover y!B(P,r) n S = 0, we get that y = 0 in B(P,r). Since P E Sm has been taken arbitrarily we finally obtain that Dh = y = 0 in B(O,Rl),B(O,R2 ). c 28.5 By means of the foregoing theorem each f E a(r)(Sm;A) admits a unique (left) monogenic extension f* to a suitable open neighbourhood of Sm in Rm+ 1; it is called the Cauchy-Ko~aZe~ski extension of f. Of course; if g E M(r)(B(0,1+E)'B(1-E);A), 0 < E < 1, then g!Sm E a(r)(Sm;A) and the Cauchy-Kowalewski extension g* of g!Sm coincides with g. Now call

m o _ M(r)(S ;A)= ind lim M(r)(B(0,1+E),B(0,1-E);A); 1>£>0 then the previous considerations yield that a(r)(Sm;A) and M(r)(Sm;A) are isomorphic right A-modules. But, as each of the spaces M(r)(B(0,1+E)'B(0,1-E);A) may be provided with its natural topology (see 16.1), the inductive limit topology can be defined on M(r)(Sm;A) (see 2.15) and hence be carried over to ~(r)(Sm;A). Analogously a(l)(sm;A) is isomorphic to M(l)(Sm;A) =lim ind M(l)(B(0,1+E)'B(0,1-E);A) 1>£>0 216

from which it inherits a locally convex topology. *(m *(m . 28.6 The duals a(r) S ;A) and a(l) S ;A) are respect1vely called the spaces of right and left analytic functionals in Sm. If r\r)(lRm+ 1-.....sm;A) -+O denotes the right A-module of all left monogenic functions in lRm+ 1-.....sm which vanish at infinity, then in what follows it will be sketched how this space represents a( 1)(Sm;A). The details of the proof are omitted since they may be reconstructed by using arguments similar to those developed in Section 22. LetT E a( 1 )(Sm;A) and consider its Cauchy transform

1

A

T(x) = - -
x-w m+1>'

( 28. 1)

IX-wl

then f is left monogenic in lRm+ 1-.....sm and lim T(x)

0.

Furthermore

X-->oo A

T(x)

Xo (-PkT)(x) B( 0,1)

where for H clRm+ 1 , XH stands for the characteristic function of H. PkT and QkT are given by -PkT(x) = _1_ wm+ 1

Hereby

(k~m11) + w , ,w

and

(28.2)
Now Tis represented by Tin the following way. Let¢ E a(l)(Sm;A); then for some 0 < s < 1,¢ admits a Cauchy-Kowalewski extension to M(l)(i3(0,1+s)-..... B(0,1-s);A), still denoted by¢, such that for any 0 < n < s r

=

A

¢(x)doxT(x) -

J

3B(0,1+n) Conversely, if f

E.

I <

n

<

:JB(0,1-n)

f1(l )(lRm+ 1-.....sm;A) ~~o then Tf defined on a(l )(Sm;A) by

ZJB(0,1+n) 0

I

A

¢(x)dox T(x).

¢(x)doxf(x) -

I

¢(x)doxf(x),

()8(0,1-s)

1 being suitably chosen and depending upon ¢, is a left analytic 217

· Sm an d T"' f = f • consequently M(l) * ( Sm;A ) -·= a(l) * ( Sm;A ) is iso1n morphic to M(r)(lRm+ 1-.....sm;A) .... o· f·1oreover this isomorphism is continuous when 1 )(Sm;A) carries the strong m+1 m topology and ~1(r) (lR "-.S ;A) .... 0 is provided with the system of seminorms in. her1ted from ~4(r) (lR m+1 -.....s m;A). · 1 f unct1ona

M(

28.7 We now show how analytic functions on the unit sphere may be expanded into a series of surface spherical monogenics. Theorem Let f E a(r)(Sm;A). Then there exists a sequence (Pkf, Qkf)keN of surface spherical monogenics such that on Sm

k=O

k=O

Moreover the series

L

00

Pkf(x) +

l

Qkf(x) converges in an open neighbour-

k=O k=O hood of sm to the Cauchy-Kowalewski extension f* of f and there exist 1 :· and C ~ 0 such that for all k E ~

r :.

0

and

Conversely, if a sequence (Pk,Qk)k8N of surface spherical monogenics satisfies estimates of the above type, then 00

I Pk(v) + I Qk(r.,l! k=O k=O is analytic in sm. f(w) =

m

Proof Let f E a( )(S ;A). Then by Theorem 28.4 there exist L " r -1 m f* E M(r)(B(0,(1-2,:) )-.....B(0,1-2c);A) such that f*!S =f. If 00

f*(x) =

I

k=O

>

0 and

00

Pkf*(x) +

I

Qkf*{x)

k=O

is the Laurent expansion off*, then by the Cauchy estimates (see (11.17) and 218

(12.6)) there ought to exist C , 0 such that for all k

E .N

Hence the sequence (Pkf,Qkf)k8N with Pkf = Pkf*ISm and Qkf = Qkf*ISm satisfies the stated estimates. Conversely, let (Pk,Qk)k8N be a sequence of surface spherical monogenics fulfilling estimates of the prescribed type. Put for each k EN, Pk(x) X ) X • lXI k Pk (TXT and Qk ( x ) = 1x1 - ( k+m) Qk(TXf). Then 1t may be shown that 00

f*(x) =

00

I

pk(x) +

k=O

I

Qk(x)

k=O

is left monogenic in an annular neighbourhood of Sm.

I k=O iS analytiC ir. Sm.

"'

I

Pk(w) +

Hence

Qk(w)

k=O D

28.8 Now let Sk be an A-valued surface spherical harmonic of order k. clearly Sk E a(r)(Sm;A) and so, by Theorem 28.7, \(w) =

I

pl \(w) +

1=0

I

QlSk((ll) •

Then

(28.3)

1 =0

But from the sections 11.2.1 and 12.1.9 it follows that

and

Consequently, as ~+ 1 sk in (28.3) and for which Theorem

-k(k+m-1)Sk, all terms P1 Sk and Q1_1sk appearing f k vanish. Hence we obtain

Let Sk be an A-valued surface spherical harmonic of order k.

Then

i.e. each surface spherical harmonic of order k may be decomposed in a unique 219

way into the sum of a surface inner spherical monogenic of order k and a surface outer spherical monogenic of order (k-1). Notice too that we have (see also (11.14) and (12.5)): Pk\ () w -- -1- (k+m-1)J k+1 m Kk m+1 Wm+1 S '

I

o(ll

c,))do

I

W

Sk(w 1 )

and

these expressions give rise to the Cauchy estimates (see also (11.17) and (12.6)):

~~~m

IPk\(w) lo;;:

c(k~~1 1 )( 1 +k 2 ) ~~m

15kL,))

lo

and

(28.4)

Observe furthermore that the decomposition Sk = PkSk + Qk_ 1sk is consistent with the dimension 2nN(m+1,k) of the space of A-valued surface spherical harmonics of order k. In fact, by the Cauchy-Kowalewski theorem for analytic functions in Rm we obtain that the dimension of the A-module of (left) inner surface spherical monogenics of order k equals 2nM(m;k), whereas the dimension of the A-module of (left) outer surface spherical monogenics of order (k-1) is equal to 2nM(m;k-1). But, as is well known (see also 11.1) N(m+1,k) = M(m,k) + M(m,k-1). 28.9 Now let us recall some fundamental results, which have been obtained by several authors, concerning the decomposition of analytic functions and functionals on Sm by means of surface spherical harmonics, the functions or functionals under consideration being [-valued. (i)

f E a(Sm) if and only if f(w) =

L

Sk(c,J), (Sk)kElN

k=O being a sequence of surface spherical harmonics for which C may be found such that for all k E fi

220

~

0 and 0

< r. <

(see [18]). 00

(ii)

I

f E E(Sm) if and only if f(w)

skc,J. (sk)kEJJ

k=O being a sequence of surface spherical harmonics such that for all s e constant Cs > 0 may be found with

~

a

and this for all k E ~ (see [17]). 00

(iii)

T E V'(Sm) if and only if T

k=O spherical harmonics for which C > 0 and S > 0 may be found such that for all k E

~

sup 1\(w)l ;:: C(1+k)s. wesm (see [14]). 00

(iv)

T

t

a'(Sm) if and only if T =

I

Sk' (Sk)kEl'J being a sequence of

k=O surface spherical harmonics such that for all E > 0, there exists CE > 0 with

and this for all k E ~. (see [7]). Using Theorem 28.8 and the estimates (28.4) we get: (ii ') f E E(r)(Sm;A) if and only if there exists a sequence (Pk,Qk)keN of surface spherical monogenics such that 00

I (Pk(w) + Qk(w)) k=O and satisfying the condition: for all s E ~. Cs f(w) =

ks sup ( IPk(w) 10 , IOk(w) 10 ) wesm and this for all k E ~.

;::

>

0 may be found with

Cs

221

(iii') T E v(r)(Sm;A) if and only if a sequence,.(Pk,Qk)kEI'l of surface spherical monogenics may be found such that T = L (Pk + Qk) and satisfying k=O the condition: there exist C ~ 0 and s ~ 0 such that for all kEN sup (I Pk(t>l) wESm

J

0,

IQk(r,))

J

0)

-:

C( 1+k)s.

(iv') T E a(r)(Sm;A) if and only if a sequence)Pk,Qk)kEN of surface spherical monogenics may be found such that T L (Pk + Qk) and satisfying k=O the conditon: for all s > 0 there exists C 0 such that for all k E ~ f.

0>

It should be emphasized that if T E a( 1)(Sm;A) then its expansion T=

L (Pk+Qk)

k=O is in fact a direct consequence of the isomorphism between a(lj(Sm;A) and M(r) (1Rm+ 1,sm;A) _,. 0 which associates to T its Cauchy transform T given by (28.1). Hereby notice that the sequence (Pk,Qk)keN of surface spherical monogenics is determined by the expressions (28.2). Observing that Theorem 28.7 already provided a monogenic version of statement (i) we have thus obtained ~presentations of the spaces a(r)(Sm;A), E(r)(Sm;A), v(r)(Sm;A) and a{r)(Sm;A) by means of spaces of sequences (Pk,Ok\EJ.J of surface spherical monogenics satisfying appropriate estimates. Of course these representations are topological isomorphisms when the spaces under consideration are endowed with their natural locally convex topology. 28.10 To close this section we define an inner product on Sm between analytic functions. Definition

Let f,g E a(r)(Sm;A).

(f,g) m = - 1S wm+1 the

222

inne~ p~oduct

Jr

Then we call

mf';w)do g(w) S ul

between f and g.

28.11

Remarks

28.11.1 Let f, g E a(r)(Sm;A) and l~t for some R1 > 1 f* and g* be the Cauch Kowalewski extensions off and g to B(O,R 1 ),B(O,R~ 1 ). Then for any open neig bourhood S of the origin having C1 -boundary as c B(O,R 1 ),B(O,R -1 1 ), 0

(f,g) m = - 11J)m+1 s

J

as

-

(28.5)

f*(u)do g*(u). u

r1oreover, if 00

f(ul) =

l:

pk f(l.&.l) +

k=O

l:

Qkf(w)

k=O

then 00

'\,

f(u.l)

l:

'\,

00

Pk f(w) +

I

k=O I

Qk f{l.&.i)

k=O 00

00

l k=Ol:

'\,

l:

Pkf(w) +

lJkT(w)

k=O

r\-

f(w)~.

Consequently, as on Sm, do = wdS , dS w

(f,g) m = - 1wm+1 S

w

being the elementary surface element,

J m f(w)g(w)dS • S

w

It thus follows that
,(f,f)> ~ 0 while = 0 implies f = 0. eo eo Hence all the requirements formulated in 7(i) - 7(v) are clearly satisfied and so ( , )Sm is indeed an inner product on a(r)(Sm;A). 28.11.2

If f,g E a(r)(Sm;A) admit the expansions 00

f(w) =

l:

00

Pk f(w) +

k=O.

l:

Qkf(w)

k=O

and 00

g(w)

l: k=O

00

Pkg(ul) +

l:

Qkg(w),

k=O 223

then the orthogonality relations between spherical monogenics (see 11.2.4 and 11.4.5.) imply that for any k 1 1,

Moreover, as for any pair Pk and Q1 of inner and outer spherical monogenics, k, lE.tl,

lim

0

R+oo

and

we obtain, using (28.5), that for all k, 1 E .tJ

Hence (f,g) m

s

29.

L2-Functions on the unit sphere- the Szeg6 and Bergmann kernels

29.1 In this section it is proved that each function f E L2(r)(Sm;A) may be expanded into a series of surface spherical monogenics. Moreover, using the Cauchy transform in L2 (r)(Sm;A), an orthogonal decomposition of this space is obtained. Finally the Szeg6 and Bergman kernels corresponding to the unit ball in Rm+ 1 are calculated explicitly. 29.2 Let f,g E L2 (r)(Sm;A) {see also 7.14) and define the inne~ p~oduct between them by 1

r

-

(f,g) m = - - Jsm ~(w)g{w)dSw' S wm+1 dS again being the elementary surface element. w

224

Furthermore put

Jsm <<e 0 , f{w)g(w)>dS w.

J m lf{w) lo2 dS w •

s

29.3 Theorem Let f E L2(r)(S m;A). surface spherical monogenics:

Then f may be expanded into a series of

00

I

f(w)

(Pkf(w) + Qkf(w)).

k=O r~oreover

Finally, if g E L2 (r)(Sm;A) admits the expansion 00

I

g{w) =

(Pkg{w) + Qkg{w)),

k=O then (f,g) m

s

Proof

Iff E L2 (r)(Sm;A) call for each k E fi

I

1 (k+m- 1\ Pkf(x) -- -w1 k+1 J m Kk m+ 1 w(x)do wf(w) ' m+ 1 ' S ' '

and

I

x E 1Rm+ 1 ,

Q f(x) = _1_ (k+m-1\ Kk,m+1 ,x(w) do f(w) x k wm+ 1 k+1 } 5m lxlm+k w ' Furthermore define rrk, rr-(k+ 1):L 2 (r)(Sm;A)

7

€

1Rm+1. o

L2(r)(Sm;A), k ~ 0, by

rrkf(w) = Pkf(w) and rr-(k+ 1 )f(w) = Qkf(w),

wE sm.

225

m

Then for each s E l, ns is a bounded right A-linear operator on L2 (r)(S ;A) satisfying n~ = ns· Moreover for all f,g E L 2 (r)(Sm;A) and k ~ 0

Jsm "Pkf(w)g(,d)dS.)

(J

1- (k+m-1\\ k 1 ;f(w' )do- ,Kk , (<.J)}g(r,J)dS JSm Sm ''lm+ 1 + ''I ,m+ 1 •''I .,,

Jsm f(w')Pkg(w')dS w, (f,nkg) m·

s

Analogously for all f,g E L2 (r)(Sm;A) and k ~ 0 (n_(k+1 /,g) m = (f ,n_(k+1 )g) m·

s

s

Finally, whenever s # s', s,s' E Z, we have that nsns' = 0. Hence (ns:s E l) is a family of mutually orthogonal projection operators in L2 (r)(Sm;A) and so m for any f E L2 (r)(S ;A)

Consequently

But, iff is considered k=O

as an element of

a( 1)(Sm;A), co

(see also (28.2)) so that

~

i<=O

implies that

226

w

then

I

(Pkf+Qkf) converges to f in

a( 1 ) (Sm;A)

k=O

(Pkf(w)+Qkf(w))

=

f(,,i) in L2(r)(Sm;A).

This

and that, using the orthogonality relations between spherical monogenics, (f,g) m

0

s

29.4

'"\,

Remark · - - As (f,g)

fC,,),~,,

f(,,,)

r

sm

J sm

'''m+1

then for all f,g E L2 (r)(Sm;A)

'f' (,JJ) do

g (1,1) 1.1]

so that the inner product in L2 (r)(Sm;A) generalizes the one defined in a(r)(Sm;A). 29.5

Now let f E L2 (r)(Sm;A) have the expansion f(,>J)

( Pk f(uJ) + Qk f(,u))

2:

k=O and define its Cauchy tr•ancfor>m by f(x) = -1tum+1

fsm

x-w do f(,.D), m+1 w IX-r.DI

(see also (28.1)). ~ m+1 m ThenfEM(r)(lR . . . s;A),lim

~

f(x)

x E lRm+ 1,sm.

Oand

X->=

f(x)

I

(x m+ 1 _ (-Pkf)(x)). Qkf(x)-xo lR '-B(0,1) B(0,1)

k=O

Furthermore f(lxlul) E L2 (r)(Sm;A) if 1x1 I 1 and a straightforward calculation shows that lim f(w(1±c)) exists in L2 (r)(Sm;A). Consequently the operators E~O+ 'f+ and rr_, glVen by ~

n+ f(~) = lim f(w(1+r_:)) r_:-+0+ 00

L

Qk f(uJ)

k=O -1

I f\ )f(w)

k=-oo and 227

-lim f(w(1-d) £-+0+

11_f(w)

co Pkf(w)

I k=O co <

I

11s)f(w)

k=O are well defined on L2(r)(Sm;A). Moreover we have that 11+ and 11_ are projection operators in L2(r)(Sm;A) satisfying the relations

and 11+ + 11_ = 1 . Hence, the following orthogonal decomposition of L2(r)(Sm;A) is obtained: Theorem L2(r)(Sm;A) = 11+L 2(r)(Sm;A) EDl. 11_L 2(r)(Sm;A). 29.6 Now we arrive at the explicit calculation of the Szegtl and Bergman kernels, the existence of which has been proved in Section 24. We first deal with the Szegtl kernel. 29.6.1 Let us recall that ML 2(r)(Sm;A) stands for the space of functions f E M(r)(~(0,1);A) such that lim r-+1

Ism

Jf(rw)

1~

dSw

<

+co.

<

For convenience we modify slightly the expressions of the inner product and norm on ML 2(r)(Sm;A) in t~e following way (see also Section 24). For all f,g E ML 2(r)(S ;A) we put (f,g)

m---

1,S

wm+1

1 im

r-+1

<

228

I

m f(rw)g(rw)dS

S'

w

and wm+1

lim J if(rw)i 2 dS. 0 sm w

r~1

< Calling for each f E ML 2 (Sm;A) and 0

<

r

1, fr(w) = f(rw), wE Sm, we have

<

Theorem Let f E ML 2 (r)(Sm;A). Then the generalized sequence (fr)rE]0, 1[ converges in L2 (r)(Sm;A) when r ~ 1. If g(w) =lim fr(w), then

29.6.2

<

in L2 (r)(Sm;A), g(w) =

~

r~1

Pkf(w) and f =-gin B(0,1).

k=O g E L2 (r)(S ;A) has the expansion g(w)

<

Conversely, if 00

00

m

L

Pkg(w)' then f(x)

k=O m 8(0,1). belongs to ML 2 (r)(S ;A) and f = - g in A

0

Proof We first prove the converse statement. 0 immediately get that for each x E 8(0,1)

Taking account of 29.5 we

----

00

L Pkg(x) k=O is well defined. mMoreover f ~ M(r)(B(0,1);A) and ~1~ fr E L2(r)(Sm;A). Hence f E ML 2 (r) (S ;A) and in 8(0,1), f =-g. < Now let f E ML 2 (r)(Sm;A) admit in B(0,1) the Taylor series expansion about the origin: f(x) =

00

L Pkf(x).

f(x) =

k=O Then clearly fr E L2 (r)(Sm;A) may be expanded into surface spherical monogenies by (see Theorem 29.3) 00

fr(w) =

L

Pkfr(w)

with

Pkfr(w)

k=O Consequently, using Theorem 29.3 and the fact that f E ML 2 (r)(Sm;A), we get that 1 im

< +

00

r~1

<

229

or 00

L

k=O Hence, again by Theorem 29.3 the function g given by 00

L

g(w) =

Pkf(w)

k=O limfr(w) r-+1

<

~elongs to L2 (r)(Sm;A) and in view of 29.5 we have that in B(0,1), f(x)

-g(x).

o

29.6.3 Remarks (1) By means of the foregoing theorem we know that for each f E ML 2 (r)(Sm;A), lim1 fr(t;l) exists in L2 (r)(Sm;A). In the sequel we still r-+ < m write f(w) = lim fr(w). If f,g E ML 2 (r)(S ;A) we thus have: r-+1

<

(f,g)

1 ,S

m=

I < ,I -

1r.um+1

-

lim m f(rw)g(rw)dS r-+1 S-w

m f(w)g(w)dS S w

= --

m+1

(;1

(f,g)

(29.6)

m

s

and llf!l 2 m = llfll 52m• 1 ,s

(2)

Iff E ML 2 (r)(Sm;A) and t E B(0,1) then . f(t) = -1- 1 1m wm+1 r-+1

< -

-,- I

- wm+l

I

sm

;;;-i

s~ lw-tlm+1

'V

(K(.,t),f) m

s

230

r~-t lrw-tlm+1

d a f( rw )

da f((li)

w

w

( 29. 7)

where

'\,

.

29.6.4 Theorem The Szeg6 kernel S(u,t) is given by S(u,t) = K(x,t)(u) where ~(x,t) = x-t stands for the right monogenic function in x, depending on •. m+1 I

x- ~ 1

the parameter t E ~(0,1). 0

.0

1

Proof FortE B(0,1) fixed, S(u,t) is left monogenic in B(O,TfT). Hence, as a function of u, S(u,t) E ML 2 (r)(Sm;A). In view of the uniqueness of thereproducing kernel it suffices to prove that for each f E ML 2(r)(Sm;A) f(t) = (S(.,t),f)

1 ,s

m·

But in view of (29.6) and (29.7) f(t)

'\,

= K(.,t),f) m

s

(S(.,t),f) m

s

= (S(.,t),f)

1 ,s

m·

c

29.7 We now end up this section with the determination of the Bergmann kernel in the case of the unit ball. 29.7.1 Let us recall that ML 2(r)(B(0,1);A) 0 iS the right Hilbert A-module consisting of the left monogenic functions in B(0,1) which at the same time belong to L2(r)(B(0,1);A). If(,) denotes the inner product in L2(r)(B(0,1);A), then for convenience we put for all f,g E ML 2 (r)(B(0,1);A) 1 (f,g)B =---(f,g). wm+1

We have

J

f(x)g(x)dx

s
J~

(J 5m fr(w)gr(w)dsw)rmdr 231

0

0

29,7.2 Now let f E ML 2 (r)(B(0,1);A). Then f E M(r)(B(0,1);A) and so it admits a Taylor series expansion about tne origin: 00

f(x)

~

I

Pkf(x).

k=O 0

We now claim that this series still converges i~ L2 (r) (B(O, 1) ;A) t~ f. Indeed, define for each k E :N the operator nk t4L 2 (r)(B(0,1);A) .... ML 2 (r)(B(0,1);A) by nkf(x)

= Pkf(x) 1 ( k+m-1 \ -_ wm+ 1 k+ 1 )

J

() () o ) 1 Kk,m+ 1 ,y x dayf y , x E 8(0,1 • aB(O,-z)

Then clearly rrk is a right A-linear operator. Moreover it is bounded. Indeed, a straightforward calculation yields the existence of Ck > 0 such that for all X E B(0,1)

so that, taking account of Theorem 24.6,

and hence

By the uniqueness of the Taylor series we also have that rrk = rrk and, as for k # 1, (rrkf,rr 1g)Sm = 0, for all f,g E ML 2(r)(B(0,1);A), we get that (rrkf,rr1 g) 8 =

1 k+1+m( ) r rrkf,n 1g smdr

Jor

= o. 232

Finally for all f,g E ML 2 (r)(B(0,1);A) k E ~.

(nkf,g)B = (f,nkg) 8 ,

Consequent!y (rrk)keN is a se~uence of mutually orthogonal projection operators in ML 2(r)(B(0,1);A) whence k~O Pkf converges in the L2-sense to f. It thus follows that 00

(f,g}B =

L

(Pkf'Pkg)B

k=O

Y J~ k=O

(29.8)

r2k+m(Pkf, Pkg) mdr S

0

0

and hence that f E M(r)(B(0,1);A) belongs to L2(r)(B(0,1);A) if and only if (29.9) 00

As we have already seen, f

E

ML 2(r)(Sm;A) if and only if

Consequently, for such f 00

L (2k+m+1)

-1

k=O

I

k=O

11Pkf11 2 m S

< + oo.

11/Zk+m+l Pk fW m < + oo S

and so, taking account of (29.9),

y /2k+m+1

Pkf E ML 2(r)(B(0,1);A).

k=O H~nce it is meaningful to define the operator /m+1-2I: ML 2(r}(Sm;A)-+ML 2(r) (B(0,1);A) by 00

~r

f =

I

12k+m+1 Pkf.

k=O Clearly this operator is injective; but it is easily seen that it is also onto. Obviously the operator ~r conserves inner products. Finally, if 233

g E M(r)(~(O,R);A) where R > 1, then its Taylor series about the origin 00

g(x) =

I

Pkg(x)

k=O converges uniformly on the compact subsets of B(O,R). Moreover, taking account Qf the Cauchy estimates satisfied by Pkg, k E ~. (see 11.17) and repeating the arguments developed above, we get that 00

,A11+1-2r g(x) =

I

/Zk+m+1 Pkg(x)

k=O belongs to ML 2(r)(B(O,R');A), 0 < 1 < R'
I (2k+m+1) Pkg(x) k=O still belongs to ML 2(r)(B(O,R');A). (m+1-2r)g(x) =

(29.10)

0

29.7.3 Theorem The Bergmann kernel B(u,t) for the unit ball is given by B(u,t) = 5 u

u - t juj2

ITur-

lui

t

r+1

"' (m+1-2f)K(x,t)(u). Proof First observe that fortE B(0,1) fixed, B(u,t) is left monogenic in 1 0 B(O, TfT) so that B(~,t) E ML 2(r)(B(0,1);A). So we only have to show that for each f E ML 2(r)(B(0,1);A)

-0---

f(t) = (B(.,t),f) 8• For r

>

ltl we have that

I

X - t 1 dcrxf(x) f(t) = -1wm+ 1 aB{O,r) lx-tlm+

lim _r_ m J r-+1 wm+1 sm 234

<

r~-t dcr f (w) lrw-tlm+1 wr

m

lim _r_ J K(rx,t)(u.1)ruf CuddS r-+1 lllm+1 sm r ru

<

. \ m lim r m (-w -K(rx,t)(w),fr(w);

s

~1

<

m(~

\

(29.11)

lim r \K(rx,t)(w), fr(w); m r-+1 s

<

x-t is looked upon as a right monogenic function !x-t!m+1 in X and where f Cul) = f(rul)' E sm. ~ r o Since K(rx,t)(u) is left monogenic in B(O, I~[) with~> 1 we may consider its Taylor deve1opment about the origin wh1ch converges uniformly on -co r : B ,R ') , 1 < RI < TfT

where again K(x,t) =

(J)

'\,

00

L

K(rx,t)(u) =

Pk,r(u)

k=O ~

where Pk,r(u) = Pk K(rx,t)(u). ~ Hence, in view of (29.10), (m+1-2r)K(rx,t)(u) to ML 2(r)(B(0,1);A). But, in virtue of (29.8) 0

00

L

(2k+m+1)Pk,r(u) belongs

k=O

(<m+1-2r)K(rx,t)(u),f(ru)) 8 = (K(rx,tHuJ), fr(w))sm (29.12) Consequently, by means of (29.11) and (29.12) ~

m(

\

(29.13)

f(t) =lim r \(m+1-2r)K(rx,t)(u), f(ru>;s· r-+1

'<

Now

~

~

t

rm(m+1-2r)K(rx,t)(u) = (m+1-2r)K(x, ·r)(u) belongs to M(r)(B(O, I~I);A).

Hence, choosing r > ~ we obtain that in

0~

M(r)(B(O,

~);A) ~

~

rm(m+1-2r)K(rx,t)(u) -+ (m+1-2r)K(x,t)(u) if r-+ 1 0

so that also in L2(r)(B(0,1);A)

<

235

Furthermore, as llf(ru) 11 8 converges to llf(u) 11 8 when r<1 and as for each ¢ E V(l)d3(0,1);A) lim (fr,¢) 8 = (f,¢) 8 , r+1

<

0

we have that for all g E L2(r)(B(0,1);A) lim (f ,g)B = (f,g) 8 , r+1 r

<

0.

which implies that lim llfr-f11 8 r+1 < (29.13) we get:

Hence, taking the limit for r+1 in

<

0

0

As f E ML 2(r){B(0,1);A) and t E B(0,1) have been chosen arbitrarily, we finally obtain that '\,

B(u,t) = (m+1-2f)K(x,t)(u). Using a general result concerning adjoint monogenic functions (see Section 13) we also have: '\,

B(u,t) = (m+1-2f)K(x,t)(u) 00

L

'\,

(2k+m+1)PkK(x,t)(u)

k=O 00

L (2k+m+1){QkK(x,t)(u) k=O

I k=O

(2k+m+1)

1-m s{QkK(x,t))(u) 2k+m+1

00

L (1-m) s(QkK(x,t))(u) k=O

236

00

L

(1-m) s(

QkK(x,t))(u)

k=O

= (1-m) s(K(x,t)(u) -

u-

t

lulz

0u -----'--'--m-+-.-1 •

j{ur-

0

tl

1u1

29.7.4 As a final result we establish a relationship between the Bergmann and Szeg6 kernels. Lett E B(0,1) be fixed; then 0

S(u,t)

'\,

= K(x,t)(u)

where Pk,t(u) = PkS(u,t).

Furthermore

00

s(K(x,t))(u)

= L Pk,t(u) k=O

with Pk,t(u) = Pk s(K(x,t))(u).

Consequently

1-m

=~

=

k +m+1 2

~

1 lul(m+1)/2

c.

Jlul r(m+1)/2 P' (r u )dr 0 k,t TuT

whence Theorem In the case of the unit ball in Rm+ 1 the Bergmann and Szeg6 kern~ls are related by S( U, t) =

1 Jlul r(m+1)/2 B(r (m+1)/2

21 u1

o

~u 1u1

,t)dr.

237

Notes to Chapter 4 The problem of representing distributions by holomorphic functions goes back to K6the's paper [K3]. He investigated under which conditions a holomorphic function in [~ 1 defines a distribution on S1 • In [T3J Tillmann continued K6the's investigations to the case of unbounded domains in[; he thus obtained· a representation of distributions with compact support in R by means of holomorphic functions in NR. In [T4] and [T5] Tillmann represented the spaces VL (RN) and S'(RN) by means of subspaces of O(([~)N). In fact for the case p

N = 1 he obtained algebraic isomorphisms between respectively the spaces S'(R) and HS' and V'(R) and H0 ,. The procedure used in the plane could also be followed to represent S'(RN) by means of a space of holomorphic functions in ([~)N, but it appeared that the cases V' (R) and V' (RN) (N > 1) had to be treated differently. It was Vogt who gave in [V1] a complete description for the representation of V' (RN). f1eanwhile r1artineau studied in [t1ar] the boundary value problems connected with V'(R) and S'(R) in a more general framework. In fact he established a relationship between the existence of a distributional boundary value off E 0([~) and the extensibility off as a distribution in [, which in its turn could be characterized by means of growth conditions upon f. Moreover he put a link between the problem of representing V'(R) (resp. S'(R)) and the solution inV'(R 2 ) (resp. S'(R 2 )) of the equation a_S = T, T being given. In this way Martineau obtained the z algebraic isomorphisms already established by Tillmann. The question of course arose whether or not these isomorphisms were also topological when the spaces H0 , and HS' are provided with a suitable topological structure and when V'(R) and S'(R) carry the strong topology. This question was first answered by Konder in [Ko] while a refined study of the topology in H0 , was worked o~t by Vogt in [V2]. As was already mentioned, some problems arose with respect to the representation of V'(RN) (N > 1), especially concerning the surjectivity of the boundary value mapping. Vogt proved this surjectivity in [V1] by using the representation and boundary value theory for vector valued distributions and holomorphic functions. The case S'(RN;E),E being a locally convex space, was studied by r4eise in [t-1ei1], [Mei2] and [Mei3], while the cases S'(RN;E) and V'(RN;E) were investigated by Vogt in [V1], [V3], [V4]. The problem of the representation of distributions as boundary values of holomorphic functions 238

thus gave rise to intensive research during the last decades. As furtha· contributions to this theory we mention the papers by Luszczki - Zielesny, Carmichael and Vladimirov (see [LZ], [Ca1], [Ca2], [Ca3], [Ca4], [Ca5], [Vl1], [Vl2]). An excellent survey concerning its development up to 1977 may be found in [Mei4]. Applications of the theory of distributional boundary values of holomorphic functions to quantum field theory and network theory can be found in [BW] and [Bre]. It should be mentioned that the first investigations made in the beginning of the fifties were followed up only in the seventies by results concerning the representation of ultra-distributions and this in the works of Komatsu [Kom] and K6rner [Kor]. Recently Petzsche succeeded in giving a unified approach to the relationship between generalized functions (distributions, ultra-distributions, hyperfunctions) and boundary values of holomorphic functions, both in the one and several dimensional cases (see [Pe1]). Note that the concepts of hyperfunctions and microfunctions were worked out essentially by Sato, Kashiwara and Kawai (see e.g. [Sa], [SKK]). In the Sections 25-27 we deal with the problem of representing distributions in Rm by monogenic functions in Rm+ 1 ~m. Although the main idea goes back to the work of Martineau, our techniques are quite different. Note that the isomorphisms we finally obtain in 27.9 are of a pure algebraic nature. Nevertheless, just as in the case of one complex variable, it may be proved that they are also topological. The nature of our approach, namely to represent distributions in Rm as boundary values of functions in (m+1)-variables, should be compared with the one using several complex variables in order to distinguish its own character. In this context we also want to mention Langenbruch's papers [la1], [La2], [La3], [La4] and [la5], where distributions in Rm are represent~d as boundary values of nullsolutions of hypoelliptic differential equations; his methods again differ completely from our function theoretic ones. Finally note that ~yperfunctions may also be described in terms of monogenic functions as shown in Sommen's paper [So1]. It should be noted that hereby, just as in the case of one complex variable, the use of cohomology theory could be avoided. The theory of Fourier series has a long history and some basic questions already posed at the appearance of Fourier's work "Th~orie analytique de la chaleur" (1822) could only be answered by using Lebesgue's integration theory. The decomposition of a function f E L2 (S 1 ) , where the latter space may be 239

identified with L2 {[0,2rr]), may be found in most textbooks on real analysis (see e.g. [HS]). Also in the more dimensional case it is well known that the set of surface spherical harmonics forms a complete orthonormal system of eigenfunctions of the Laplace-Beltrami operator~*m+ 1 acting on the Hilbert space L2 (Sm) (see e.g. [Tr]). The problem of decomposing functions or functionals on Sm into a series of surface spherical harmonics, and hence the problem of generalizing the classical Fourier expansion for f E L2 (Sm), has been the subject of more recent research. As was already mentioned in Section 28, growth conditions upon the sequence (Sk)kEN of surface spherical harmonics have to be imposed in order to get series expansions of the functions or functionals under consideration. Such estimates were obtained by Seeley (see [Se1], [Se2]) for C and analytic functions, by Morimoto (see [Mor]) for distributions and by Hashizume-MinemuraOkamoto (see [HMO]) for analytic functionals. Furthermore Helgason showed in [Hel] that any harmonic function f in the unit ball admits a boundary value in a'(Sm), whence it may be decomposed into a series of surface spherical harmonics satisfying certain estimates. But also the converse is true, i.e. if a sequence (Sk)kEN of surface sphe~ical harmonics is given which satisfies the required estimates, then f(x) = L lxlk Sk(w) is harmonic in the unit k=O ball. The space of analytic functionals on Sm thus appears to be the largest space of functionals which may still be represented as boundary values of harmonic functions. The representation of a( 1 )(Sm;A) by M(r)(ffim+l,Sm;Al~a as sketched in 28.6 thus yields a refinement of Helgason's result. In this context it should be mentioned that Hashizume-Kowata-Minemura-Okamoto constructed in [HKMO] spaces of functionals on Sm which contain a'(Sm) as a proper subset. These spaces are connected with the Fourier-Borel transform (see also Chapter 5). Finally it should be no~ed that the representation of any f E L2 (r)(Sm;A) as the L2 -boundary value of f(w(1±E)) and the derived orthogonal decomposition of L2(r)(Sm;A) seem to be inherent in the function theory under consideration. Note also that the splitting of any surface spherical harmonic of order k into a sum of an inner and outer surface spherical monogenic, being themselves eigenfunctions of the spherical Cauchy-Riemann operator, has applications in the theory of electron spin (see [L]). Further investigations in this context have been carried out by Sommen in [So2]. The form of the Bergmann and Szeg6 kernels obtained in Section 29 for respectively the unit ball and sphere in ffim+l again illustrates the closeness 00 -

240

of monogenic function theory to the theory of holomorphic functions of one complex variable. Bibliography [Bre] H. Bremermann, Distributions, Complex Variables and Fourier Transforms (Addison-Wesley, New York, 1965). [BW] E.J. Beltrami, M.R. Wohlers, Distributions and the Boundary Values of Analytic Functions (Academic Press, New York, 1966). [Ca1] R.D. Carmichael, Distributional boundary values in VL , Rend. Sem. Math. di Padova 43 (1970) 35-53. p [Ca2] , Distributional boundary values in Vlp II, Rend. Sem. Math. di Padova 45 (1971) 249-277. [Ca3] , Distributional boundary values of functions analytic in tubular radial domains, Indiana u. Math. J 20 (1971) 843-853. [Ca4] , Representation of distributions with compact support, ManuscPipta Math. 11 (1974) 305-338. [CaS] , Distributional boundary values in the dual spaces of spaces of typeS, Pacific J. Math. 56 (1975) 385-422. [Hel] S. Helgason, Eigenspaces of the Laplacian: Integral representations and irreducibility, J. Functional Analysis 17 (1974) 328-353. [HKMO] M. Hashizume, A. Kowata, K. Minemura, K. Okamoto, An integral representation of the ~aplacian on Euclidean space, Hiroshima Math. J. 2 (1972) 535-545. [HMO] M. Hashizume, K. Minemura, K. Okamoto, Harmonic functions on Hermitian hyperbolic spaces, Hiroshima Math. J. 3 (1973) 81-108. [HS] E. Hewitt, K. Stromberg, Real and Abstract Analysis (Springer Verlag, Berlin, 1965). [H3] G. K6the, Die Randverteilungen analytischer Funktionen, Mach. z. 57 (1952) 13-33. [Ko] P.P. Kender, Funktionentheoretische Charakterisierung der Topologie in DistributionenrMume, Math. z. 123 (1971) 241-263. [Kom] H. Komatsu, Ultradistributions I. Structure theorems and a characterization, J. Fac. Sci. Tokyo 20 (1973) 25-105. [Kor] J. K6rner, Romieusche Ultradistributionen als Randverteilungen holomorpher Funktionen (Thesis, UniversitMt Kiel, 1975).

241

[LJ

P. Lounesto, Spinor valued regular functions in hypercomplex analysis (Thesis, Helsinki University of Technology, 1979). [La1] M. Langenbruch, Randverteilungen von Null6sungen hypoelliptischer Differentialgleichungen, ManusrJl'ipta Uath. 26 (1978) 17-35. [La2] P-Funktionale und Randwerte zu hypoelliptischen Differentialoperatoren, Math. Ann. 239 (1979) 55-74. [La3] , Fortsetzung von Randwerten zu hypoelliptischen Differentialoperatoren und partiellen Differentialgleichungen, J. Reine AngeUJ. f.1ath. 311/312 (1979) 57-79. [La4] , Darstellung von Distributionen endlicher Ordnung als Randwerte zu hypoelliptischen Differentialoperatoren, Math. Ann. 248 ( 1980) 1-1 7. [La5] -------------, Dualraum und Topologie der (lokal) langsam wachsenden Nu116sungen hypoe 11 i pti scher Differentia 1operatoren, f.1anuacripta Math. 32 ( 1980) 29-49. [LZ] Z. Luszczki, Z. Zielesny, Distributionen der RMume VL und Randverteilungen analytischer Funktionen, Colloq. Math. 8 (1961) 12~-131. [Mar] A. Martineau, Distributions et valeurs au bord des fonctions holomorphes, in Theory of Distributions, Proc. Intern. Summer Inst., Inst. Gulbenkian de Ciencia, Lisboa (1974) 113-326. [Mei1] R. Meise, Darstellung temperierter vektorwertiger Distributionen durch holomorphe Funktionen I, Math. Ann. 198 (1972) 147-159. [Mei2] , Darstellung temperierter vektorwertiger Distributionen durch holomorphe Funktionen II, Math. Ann. 198 (1972) 161-178. [Mei3] , Ra6me holomorpher Vektorfunktionen mit Wachstumbedingungen und topologische Tensorprodukte, Math. Ann. 199 (1972) 293-312. [14ei4] , Representation of distributions and ultradi stributions by holomorphic functions, in Functional Analysis: Surveys and Recent Results, Proc. Paderborn Conference on Functional Analysis (North-Holland, Amsterdam, 1977) 189-208. [Mor] M. Morimoto, Analytic functionals on the sphere and their FourierBorel transformations, to appear in a volume of the Banach Center Publication. [Pel] H.J. Petzsche, Verallgemeinerte Funktionen und Randwerte holomorpher Funktionen (Habilitation, UniversitMt D6sseldorf, 1981).

242

[Sa] M. Sa to, Theory of hyperfunctions, I and I I, J. FarJ. Sci. Univ. 1'-:;kyo 8 (1959/60) 139-193 and 387-436. [SKK] M. Sato, M. Kashiwara, T. Kawai, Hyperfunctions and pseudodifferential equations, in Lecture Notes in Mathematics 287 (Springer-Verlag, Berlin, 1973) 265-529. [Se1] R.T. Seeley, Spherical harmonics, Af.J:: :.Jortthly 73, part II, no 4 (1966) 115-121. [Se2] - - - - - , Eigenfunction expansions of analytic functions, Pr>oc. A/?l .. i". l·iath. Soc. 21 ( 1969) 734-738. [So1] F. Sommen, Hyperfunctions with values in a Clifford algebra, to appear in ,)'·::"!lor:. :Jt.·vin. [So2] Spherical monogenic functions and analytic functionals on the unit sphere, to appear in To~:;1o Jo;B•nal of 1-iath. [T1] H.G. Tillmann, Randverteilungen analytischer Funktionen und Distributionen, f.1ath. z. 59 (1953) 61-83. [T2] , Distributionen als Randverteilungen analytischer Funktionen I I, l-1cith. z. 76 ( 1961) 5-21. [T3] , Darstellung der Schwartzschen Distributionen durch analytische Funktionen, Math. z. 77 (1961) 106-124. H. Triebel, HBhere Analysis (VEB Deutscher Verlag der Wissenschaften, [Tr] Berlin, 1972). [Vl1] V.S. Vladimirov, On functions holomorphic in tubular cones, Izv. Akad. Nauk :JSSR Scr>. Matlz,;m. 27 ( 1963) 75-100. [V12] , On the construction of envelopes of holomorphy for regions of a special type and their applications, Amu•. Math. Soc. Tmnslation 48 (series 2) (1966) 107-150. [V1] D. Vogt, Distributionen auf dem ffiN als Randverteilungen holomorpher Funktionen, J. Reine Angcw. Matlt. 261 (1973) 134-145. [V2] ------ Randverteilungen holomorpher Funktionen und die Topologie von V', Nath. Ann. 196 (1972) 281-292. [V3] , Temperierter vektorwertiger Distributionen und langsam wachsende holomorphe Funktionen, !-lath. z. 132 (1973) 227-237. [V4] Vektorwertige Distributionen als Randverteilungen holomorpher Funktionen, f.Janu;;cr-ipta Math. 17 (1975) 267-290.

243

5 Transform analysis in Euclidean space

The first two sections of this last chapter are devoted to the Fourier transform and the generalized Laplace transform in higher dimension within the framework of monogenic functions and A-distributions. First we treat of the Fourier transform in S(ffim;A), V(ffim;A) and E(ffim;A) and their respective duals of A-distributions. This leads a.o. to a generalization of the GelfandShilov Z-space in one complex variable, which consists of all entire functions of the form f(z) =

JR

e-itz ¢(t)dt

for some¢ E V(R). Moreover an analogue of the classical Paley-Wiener-Schwar~ theorem is obtained (§30). Next the generalized Laplace transform of tempered A-distributions is studied, extending the complex Laplace transform im

z

> 0

im

z

<

LT(z)

0.

It turns out that forTE s(l)(ffim;A) its generalized Laplace transform LT is a left monogenic function in ffi~+l having its Fourier transform FT as s(l )boundary value for x0 -+ 0±. ~1oreover the following boundary value result is generalized: let OR([~) and OR([) be the spaces of holomorphic functions in respectively [~ and [ satisfying the respective estimates k, 1 € l'J,

c

> 0

and 1 € :D'l,

then the boundary value mapping 244

c>

0;

is surjective, bounded and open, while ker BV =OR([). In Section 32 we introduce a generalized version of the Fourier-Borel transform, which maps monogenic functions in annular domains into monogenic functions in the complement of the origin. This transform extends the transform o = o+ + o in the complex case, where T ~ 0

T being an analytic functional with carrier in an annular domain. the classical result

Moreover

Exp'([) being the class of Helgason's entire functionals, is generalized, 0 leading up to an integral form for the Laurent series expansion of a monogenic function in an annular domain. In the last section the Radon transform P:

O'(B(0,1))-. O(B(0,1))

given by 1

P(T)(u) = , ~

~

is generalized to higher dimensions using a monogenic version P(u,z),(u,z) E Rm+ 1 x [m of the function (1-u~)- 1 , (u,~) E [ x [. This results into an optimal version of the Cauchy-Kowalewski theorem (see 14.2) concerning the monogenic extension of an A-valued analytic function in some open subset of Rm 30.

Hypercomplex Fourier transforms

30.1 In this section we first introduce the Fourier transform of rapidly decreasing A-valued C -functions; the classical Fourier kernel e-itx,(t,x) E m m R x R is replaced here by the function E(t,x), (t,x) E R x R introduced in 15.4. 00

~-.

~~

245

Next we introduce the Fourier transform of A-valued testfunctions by means of the exponential function E(t,x), (t,x) E ~m x ~m+ 1 , introduced in 15.4, which itself is a natural generalization of e-itz, (t,z) E ~ x [and which fort fixed is monogenic in the whole of ~m+ 1 • This leads to a generalization of the Gel 'fand-Shilov Z-space in one complex variable. Finally the generalized Fourier transform of A-distributions with compact support is studied. A.

The Fourier transform in S(~m;A) and S*(~m;A)

30.2

Let ~ E S(r)(~m;A); then we define 1 , ••• ,m

and

The function F¢ is called the

Four•ier' tmncfor•m

of ¢·

30.3 Just as in the classical theory it is obtained that Fj is a topological automorphism of S(r)(~m;A) and

Theorem

F.-1 q,(t 1 , ••• ,t. 1,x.,t. 1 , ••• , t ) = -21 J JJ J+ m IT Proof

Let ¢

=

J+ooexp(-t.x.e.)~(t)dt -+ .• _ 00

J J J

1eA~A' where ¢A E S(~m;~) for all A E PN.

J

Then for each

component ¢A the classical Fourier inversion formula holds since exp(t.x.e.) J J J

= e0 cos(t.x.) + e.sin(t.x.). J J J J J

So for each A E PN, 1

~ Ti

J+ooexp(-t.x.e.)F.¢Adx. -co

J J J

J

J

1

[ F.~

J

LTI

J+ooexp(-t.x.e.)tAdt.Jl -oo

J J J

J

1,A(t).

As F. and F~ 1 are right A-linear the above inversion formula also holds for J J -1 ¢· Furthermore Fj and Fj are continuous. o 246

30.4

F- 1¢(x) 30.5

F is a topological automorphism of s(r)(Rm;A) and

Corollary

JRm E(t,;)~(t)dt. If we define for~ E s(l)(Rm;A)

Remark

= J+'~'q;(t)exp( t .x .e. )dt., -oo

J J J

J

j

1 , ••• ,m

and

then of course Fj{j S( l ) (Rm; A) • 30.6

1 , ••• ,m) and Fare topological automorphisms of

Definitions

(i) LetT E s(l)(Rm;A); its Fourier transform is the tempered left Adistribution FT given by = , ~ E S{l)(Rm;A). (ii)

LetS E s(r)(Rm;A); its Fourier transform is given by <SF,¢> = <S,F¢>, ¢ E S(r)(Rm;A).

. of both S(l)s * (Rm;A) It can be proved that F is a topological automorph1sm and S(l)b(Rm;A). 30.7

Introducing the r>eflection oper>ator>u Si,

1, ••• ,m, given by

S/ ("Jt) = f (X 1 , ••• , Xi _1 , -Xi , Xi+ 1 , ••• , Xm) where f stands for a function or an A-distribution in Rm, some calculation formulae for the Fourier transform can be stated: (i)

F(3t_f)("it) = -xiei s 1 ••• si_ 1 Ff(x) 1

247

(i i )

(iii )

F(e.f)(;) =e.1 s 1••• S.1- 1s.1+ 1••• 5m Ff(x) 1 ~

~

F(eiat_f)(x) =X; Si+ 1••• Sm Ff(x) 1

(iv) (v)

F(t;f)(x) = -ei aX; s 1••• S;_ 1 Ff(x) ~

~

F(t;e;f)(x) =ax. Si+ 1••• Sm Ff(x) 1

(vi)

~~

~

~

F(f(t+a))(x)=exp(-a 1x1e 1) ••• exp(-a 2x2e 2s 1) ••• exp(-amxmems 1••• Sm- 1)Ff(x)

where

B.

The generalized Fourier transform in V(Rm;A)

30.8 Let¢ E V(r)(Rm;A); then we define

f

~

~

~

F¢(x) = Rm E(t,x)¢(t)dt,

m+1 x ER •

It is clear that F¢ is left entire; moreover it is the unique left monogenic extension of F¢Cit)

=

~

~

~

E(t,x)¢(t)dt

which belongs to S(r)(Rm;A). Hence the following inversion formula is obtained:

30.9 Call Z(r)(m;A) the module of all functions F¢ where¢ runs through V(r)(Rm;A). Obviously Z(r)(m;A) is a submodule of M(r)(Rm+ 1;A). This module Z(r)(m;A) is now characterized by means of estimates in the following theorem, the proof of which is rather straightforward. Theorem Let¢ E V(r)(Rm;A) have its support contained in B(O,R). Then for each ~s E ~ m and s > 0 a constant C 6 > 0 may be found such that for all m+1 s '" X E R

248

s

30.10 Remark In virtue of Cauchy's Representation Theorem, for any E ~m. m+1 m a E~ and c > 0 there exists c~R.a,c > 0 such that for all ~ E V(r)(R ;A)

In the following theorem such estimates are used to determine completely the space Z(r)(m;A). 30.11 Theorem Let f E M(r)(Rm+ 1;A) be such that for a certain R > 0 and for any ! E ~m. a E ~m+ 1 and c > 0, there exists c~ > 0 such that f3,c:t.E

IXB aaf(x) I X

0

<: -

c~

f3,a,c

e(R+c) IXol.

Then there exists a unique testfunction ~ E V(r)(Rm;A) with supp~ such that f(x) = F¢(x).

c

B(O,R)

Proof In view of the stated estimates the restriction off to R:+ 1 , f(X), belongs to Sir)(Rm;A). So there exists a unique testfunction ~ E s(r)(Rm;A) such that f(x) = F~(x) or~ = F- 1f, or still

We now prove that the support of ~ is contained in B(O,R+c) for any s > 0. Choose s > 0 arbitrarily and take o > 0; applying Cauchy's Theorem yields

From the assumed estimates it follows that for a certain constant C > 0 ( R+c) IX I o 1+1XIm+1 o ~ m and as A is bounded for x E R we get lf(x)lo:;;:Ce

249

In an analogous way it can be shown that for any ,.,

0

I r m B(t,x)f(x)dxlo ~ C"e(R+c-ltl ),'.. J]R

Now iftElRm.....S(O,R+d then R+s-!tl < 0 and taking limits for,~++oo in the + above inequalities leads to ~(t) = 0. So supp~ c B(O,R) and it is possible to consider its generalized Fourier transform F¢(x). The function f-F¢ is left entire and it vanishes in JR~+ 1 • Hence it is identically zero in lRm+ 1 or f(x) = F¢(x). o 30.12 Now we are able to endow the module Z(r)(m;A) with a locally convex topology. Let k,s E ~and call Z(r),k,s(m,A) the submodule of Z(r)(m;A) consisting of those left entire functions f which are such that for each E ~m and a E ~m+ 1 there exists a constant C+ > 0 such that

e

s.a

+

lxB

(

a~f(x) lo ~ c!.ae k+s

-1)

lxol.

Provided with the proper system of seminorms {Pj,l :(j,l) E ~ 2 }, where sup e-(k+s lu.l:;:;l

-1

+

Z(r),k,s(m;A) becomes a right Frechet A-module. Z(r)(m;A) = lim ind lim proj kE:tl

sEJl

a~f(x)lo•

)lxollxB

Then we put

Z(r),k,s(m;A).

Notice that z(r)(m;A) is an inductive limit of right Frechet A-modules. 30.13 The following topological result may be proved along the same lines as in the classical case of one complex variable. Theorem The Fourier transform

is a topological isomorphism. 30.14 250

Denote by z(r)(m;A) the dual module of Z(r)(m;A), i.e. the left A-

module of bounded right A-linear functionals on z( )(m;A). By means of this . * m r dual module a Four1er transform on V(r)(R ;A) may be defined. Indeed, if T E Vtr)(Rm;A) then its Fourier transform is defined as to be the element of z(r)(m;A) which acts on functions f E Z(r)(m;A) as follows: = , for all f E Z(r)(m;A). It is clear that this transformation F is a topological isomorphism between * m;A) and Z(r) * (Rm;A ) • bot h prov1"d ed either with the weak* or strong V(r)(R topology. 30.15

If¢ E V( l)(ffim;A) then we define ¢F- 1 (x) = J ¢(t)E(t,x)dt. Rm

It is clear that ¢F- 1 is right entire. Call z(l)(m;A) the module of functions ¢F- 1 where ¢ runs through V(l)(Rm;A). This module Z(l)(m;A) may be characterized in the same way as was done for z(r)(m;A). Moreover a locally convex topology may be defined on it such that Z(l)(m;A) and z~ 1 )(m;A) are topologically isomorphic to v(l)(P.m;A) and v( 1 )(ffim·A) respectively. C.

. E(l)(ffi * m;A) The generalized Fourier trans f orm 1n

30.16 LetT E E( 1 )(ffim;A); then of course T E S(l)(ffim;A) and soFT is defined and belongs to s(1 ) (Rm;A); moreover for any ¢ E s(l) (ffim;A) holds




-+ -+

4-

-+

¢(x)E(t,x)dx>

f""\J

-+

-+

¢(x)T(x)dx where we have put

251

So if we consider the function "' T(x)

..... = , x E Rm+ 1 ,

we have at once that .....

FT(x}.

Hence it is quite natural to define 30.17 Definition function

* (Rm;A) is the The generalized Fourier transform ofT E E(l)

... m As T is left A-linear and bounded on E(l)(R ;A) and as E(t,x) is analytic in Rm x Rm+ 1 it is easy to show that FT is left entire; it is in fact the unique '\, ..... left monogenic extension of T(x). 30.18 Theorem LetT E E{ 1 )(Rm;A) and let R > 0 be such that supp T c B(O,R). Then there exist C > 0 and k E J~ such that for all x E Rm+ 1

Proof The desired inequality follows immediately from the definition of FT. 30.19 The converse of Theorem 30.18 may be regarded as the analogue of the famous Paley-Wiener-Schwartz Theorem. The proof, which is rather technical and may be developed analogously to the proof of Theorem 30.11, is omitted. Theorem (Paley-Wiener-Schwart~) Let f E M(r)(R m+1 ;A) and let R that for some C > 0 and k E ~

0 be such

. * m+1 ;A) with supp T c B(O,R) Then there ex1sts T E E(l)(R such that f =FT. 30.20 Let k E ~and let R > 0; call ~(r),k,R the module of all left entire functions f satisfying an estimate of the form

252

c

for some C > 0. Equipped with the norm

~

(r),k,R is a right Banach A-module.

Putting

~(r)(m;A) = li~.~nd ~(r),k,R

the following result is obtained. 30.21

Theorem The generalized Fourier transformation

* m;A) .... ~(r)(m;A) F:E(l),b(lR is a topological isomorphism. Proof Use Theorem 30.3 and 30.19 and an analogous reasoning as in the case of one complex variable. c 31.

The generalized Laplace transform in

s(1 )(1Rm;A)

31.1 First we introduce the Laplace transform ~L- 1 of a testfunction ~ E S(l)(lRm;A) vanishing in a closed ball centered at the origin. As ~L- 1 is right monogenic in lR~+ 1 we may investigate its S(l)-boundary value for xo .... 0±.

~F

-1 ....

(x)

= JRm

~(t)E(t,x)dt

....

= JlRm

~(t)[B(t,x)-A(t,x)]dt

+

0

....

....

o++

....

o-++

belongs to S(l)(lRm;A). Now take ~ E.S(l)(lRm;A) with ~ ~L- 1 in JR~+ 1 given by

-+

0 in Bm(O,R) and consider the function

253

¢L

-1

JIRm (x)

-+ ¢(t)B(t,;)e-ltixodt,

{J!Rm ¢(t)A(t,;)elt[xodt,

X

E

1Rm+1 •

1 . c 1ear th a t ¢L- 1 1s . 1n . 1Rm+ h 11 prove th at ¢L- 1 . r1g . ht monogen1c It 1s 1 ; we s a -1 admits ¢F as S(l )-boundary value. Thereto we need a lemma on estimates of ¢L- 1 , the proof of which is omitted since it is based on straightforward estimates.

e

31.3 Lemma Let¢ E S(l)(!Rm;A) vanish in Bm(O,R). Then for each E Nm, m~ > 0 such that for all a E Jl and s > 0 there exists a constant C-+ B,u.' r: m+1 1R X E f

31.4

Proposition 1im ¢L X +0+

-1

The boundary values

+ (x±x 0)

0

exist in S(l)s(!Rm;A) and BV¢L- 1(x) = lim X +0+ 0

Proof It follows from the previous Lemma 31.3 that for each S E Jlm and a E Jlm+ 1 there exists a constant C+ > 0 such that for all x0 E]0,1]

s.a.

-1 I+s X dCJ. ¢ L (X) I x o

Hence for each

254

Z E Nm

~

C+

s.a. •

and x~ 1 ), x~ 2 ) E ]0,1]

Hence

exists in S(l)s(lRm;A).

Analogously

exists in S(l)s(lRm;A) and moreover

converges to 4-

0

-+ -+

-+

¢(t)E(t,x)dt for x0

-+

0+.

¢F

-1

-~

(x)

o

31.5 A converse of Lemma 31.3 runs as follows. m+1 Lemma Let f E ~1(ll(lR# ;A) be such that there exists R > 0 for which, given any E ~m. a E ~m 1 and 0 < E < R, a constant C > 0 may be found such S,a,E that

s

Then there exists a testfunction that f ¢L- 1


E S(l)(lRm;A) with


0 in Bm(O,R) such

Proof From the given estimates and the proof of Proposition 31.4 it follows that the boundary values 1 im

fCx±x 0 )

X -rO+ 0

exist in S(l )s(IRm;A). -+

So put -+

¢(t) = [f(x+O) - f(x-O)]F; 255

then clearly~ E S(l)(Rm;A). Using Cauchy's Theorem it is easily shown that ~

~

4

0

(f(x±x 0 )F)(t) = 0 for t E Bm(O,R) which implies that

that~=

0 in Bm(O,R).

It then follows from Proposition 31.4

1 im X

0

->-0+

and as f and ~L- 1 both satisfy esti~ates of the above type, we obtain, using Liouville's Theorem, that f = ~L- 1 • o 31.6 Now letT E S~ )(Rm;A); then pered distribution. Our aim is to of FT; this is done by introducing which is a left monogenic function for x0 + 0±.

its Fourier transform FT is again a temconstruct representing monogenic functions the generalized Laplace transform LT ofT, in R;+ 1 having FT as S(l)-boundary value

0

31.7 First assume that T 0 i~ Bm(O,R). Introduce the real-valued C -functions a E (t), depending onE E]O,R[ such that 00

+

+

{

aE(t) =

-

0, t E B(O,R-E) +

m

-+

-+-+

o

1, t E R ,B(O,R-

E

~).

Then the functions -+-+

AE (t,x) = a E (t)A(t,x) and

+

and their derivatives with respect to t, are coo-functions of slow growth in Rm x lRm. Put +

E1 (t,x) ,E

and 256

__,.

E2 (t,x) ,E: then for X fixed, both functions belong to s(l )(Rm;A). definition makes sense. 0

LetT E s(1 )(~m;A) with T alized Laplace transfo~ is given by 31.8 Definition

__,.

f

LT(x)

,

1.

X

.....

,

X

So the following

0 in Bm(O,R).

Then its gener-

.Rm+1 E + .Rm+1 E - •

Notice that this definition is independent of E E]O,R[ and that LT is left monogenic in .R~+ 1 • r~oreover it can be proved in a straightforward manner that LT satisfies an estimate of the following type. 0

31.9 Theorem LetT E s( 1 )(Rm;A) be zero in Bm(O,R) and let E E]O,R[. there exist k, r E ~ and CE > 0 such that for x E ~~+ 1

31.10

Then

In view of the above estimate and Theorem 26.8 the boundary values LT(x±" 0 )

and lim [LT(x+x ) - LT(x-x )J

BVLT X

0

.....0+

0

0

exist in s( 1 )s(.Rm;A). In the following theorem the nature of those boundary values is examined; they turn out to be Fourier transforms.

* m;A) be zero in Bm(O,R). o 31.11 Theorem LetT E S(l)(.R ct> E S(l) (Rm;A)

Then for any

lim J cp(x)[LT(x+x 0 )-LT(x-x 0 )Jdx = >· x -+0+ .Rm 0

257

m

-~rs>_o_:f

-:,..

-+

Let¢ E S(l)(JR ;A); for x0 "0 fixed the function rp(x)LT(x+x 0 ) belongs to S(l)(lRm;A) and hence, using an approximation by Riemann-sums N ~

-..

-.. ~ ~ -JtJx ¢(X N)B (t,x N)e O;(K N)>. V>

F.

V>

V>

\;=0

Observe that the sequence of Riemann-sums N ~

+

~

¢(X

v.

-..-.. -ltJx N)B (t,x N)e I 0dK c

v.

v,

N), N

1 ,2' ...

( 31. 1)

\!=0

converges uniformly on the compact subsets to

:t even converges in E( 1) (lRm;A). r~oreover this sequence a~d all sequences of t-derivatives are uniformly bounded with respect to N and t. Hence the sequence (31.1) converges in S(l)(lRm;A), which leads to ->-->-

->-

J~ ¢(x)LT(x+x 0 )dx = . r~oreover'

as ->-

-.. -..-.. -JtJx Odx-.. -.. J m¢(x)B -.. (t,x)dx -..-..-.. JJRm ¢(x)B E (t,x)e JR E

In an analogous manner we get that

... r -.. -.. -.. -.. ! ¢(x)A (t,x):ix > JlRm ¢(x)LT(x-x o )dx =
1i m X

0

-.-0+

and hence ->->) JlRm¢(x)[LT(x+x o

1 im X -.-0+

->-

->-

- LT(x-x )]dx o

0

-jo.

-+-+

-jo.-jo.

-+

= "L

258

lRm

c

E.

+

-~

= . <

which yields BVLT

=

FT.

c

31.12 The converse of Theorem 31.9 runs as follows • .!_heorem Let f E r~( r) (JR~+ 1 ;A) be such that for some R > 0, k, r 0 < f: ..-: R, a constant CE: > 0 may be found such that in JR~+ 1

E

f'l and any

* (lRm;A ) such that Then there exists a unique tempered A-distri bution T E S(l) T is zero in Bm(O,R) and LT = f. Proof As for x0 E]- 1,0[U]0,1[

f admits an s( 1 )(JRm;A)-boundary value for x0 exists a unique T E s(1 )(JRm;A) such that FT

= BVf = lim X

0

+0+

+

0 (see 26.8).

Hence there

[f(x+x ) - f(x-x 0 )]. O

First we prove that TiBm(O,R) = 0. Take a testfunction ~ E V(l)(JRm~A) wit~ suppcp c: Bm(O,R) and take E: > 0 sufficiently small such that suppcp c: Bm(O,R-2,J Then


259

l

= lim x -+0+ IRm

~F

-1

-+

-+

0

But as

~F

-+

-+

(x)[f(x+x )-f(x-x )]dx. o

o

-1

E z(l)(m;A) there exists C > 0 such that 3 1 1 (R- 7d IXol I( 1+ Ix 12 ) r+m+
and so by Cauchy's Theorem for any x0 > 0 and 6

which, letting

>

0

+ oo, leads to

a-+

It can be shown analogously that

f

JIRm

~F- 1 ("it)f(x-x )d"it = O·for x0 o

>

0,

0

so that T = 0 in Bm(O,R). Now consider LT; we know from Theorem 31.11 that BV LT

= H = BVf,

and so the function LT-f has a left entire extension in 1Rm+ 1• Furthermore as LT - f E s( l)(IRm+ 1 ;A) it is a polynomial and as both LT and f satisfy an estimate of the given form, (LT-f)("it+x 0 ) -+ 0 if lx 0 1 -+ + oo. Hence LT = f. c 31.13 Now we return to the general case where no restrictions on the support ofT E s( 1)(1Rm;A) are imposed. If R > 0 and 0 < E
260

support contained in Bm(O,R- ~). From 30.17 we know that FT 2 (x) is the restriction to ffi~+ 1 of the left entire function FT 2 (x); so FT 2 (X) is the s( 1 )-boundary value of the function g E M(r)(ffi;+ 1 ;A) given by

and for which there exist C > 0 and l E ~ such that

s(

By Theorem 31.11 FT 1 (X) is the 1 )-boundary value of the function f = LT 1 E M(r)(ffim+ 1 ;A) for which, given 0 E ]O,R-s[ there exist C0 > 0 and k,r E ~ such that

Hence FT(x) = FT 1 (x) + FT 2 (x) is the m+1 LT(x) E ~1(r) (Rf ;A) given by

s( 1 )-boundary

value of the function

LT 1 (x) + ~FT(x), x E R:+ 1 LT(x) = { LT 1 (x) - ~FT(x). x E R~+ 1 and satisfying an inequality of the form j LT ( x)

I

o

~ C( 1+ - 1- ) ( 1 + 1 x2 lxolr

1)

k e RI Xo I • x E R~+ 1 ~

( 31.2)

for some C > 0 and k, r E ~. Conversely let the function hE t4(r)(R;+ 1 ;A) satisfy an estimate of type (31.2). Then h admits an s( l)(Rm;A)-boundary value which equals FT(x) for •a unique T E s( 1 )(Rm;A). So BVh(x) = FT(x) = BVLT(x). and it is easily shown that h-LT is a left entire function satisfying 261

(31.3) 31.14 TakeR> 0 to be fixed. Call M(r),R,k,s(R~+ 1 ;A) the module of left monogenic functions in R~+ 1 satisfying an estimate of type (31.2). Provided with the norm

= sup

xeR~+ 1 (1+ ---1---s)(1+lxl 2 )k lx 0 1 it is a right Banach A-module. Next consider the module lim ind t~( ) R k s(R~+ 1 ;A). k • sEJl

r ' ' '

r

Each function hE M(r),R(R~+ 1 ;A) admits a boundary value BVh in S~l)(Rm;A). So we can consider the boundary value mapping * (Rm;A) BV .. M(r),R (Rim+1 ;A ) ~ S(l) As it was shown in 31.13 that for each T E s( 1)(Rm;A) there exists a function g* E t\r) ,R(R~+ 1 ;A) such that BVg* = T, this boundary value mapping BV is surjective. Next, putting ~(r),R(m;A}

=lim kind ~ (r),k,R

where ~(r),k,R is the right A-module of left entire functions satisfying an estimate of type (31.3), already introduced in 30.20, we have ker BV = ~(r),R(m;A). This leads to 31.15 Theorem (i)

The boundary value mapping

is bounded and open. 262

(1°1°)

1 * (IRmA) s(l) ; an d M(r),R (1Rm+ I ; A)/ ~{r),R (m;A)

are topologically isomorphic. 31.16 Remark Define

Let R > 0 be fixed and letT

E

1Rm+ 1

X E

1Rm+ 1

X E

1Rm+1

LT(x), x

{

0

,

E

s( 1)(1Rm;A) vanish in Bm(O,R).

+

and 0

L_T(x)

{

LT ( X )

'

+

x

,

E

IRm+ 1•

1 d Bot h f unct1ons are 1e ft monogen1c 1n IRm+ 1 an 0

0

0

Moreover, just as LT does, they both satisfy an estimate of the form appearing in Theorem 31.9. Then by Theorem 31.12 there exist unique tempered diso( O,R ) and such that tributions P+T, P_T E S(* l) (lRm;A ) vanishing in B

and LP T = LT. Furthermore P!T = P+T,

P~T

P_T,

P+T + P T = T and P+ P- T

= P- P+T = 0.

Moreover it can be shown that

263

m j =1

s1 ••• s.J- 1t.)T. J

s(

This decomposition ofT E 1)(Rm;A) is illustrated as follows in the cases where m = 1 and m = 2. If m = 1 then it is an easy matter to check that

notice that here p_ is the restriction operator toR+. For m = 2, let + -+t = (t 1 ,t 2 ) E R2 and call e the polar angle of t. Then it is obtained that

-+-

P+Tt = ~ (1±cose)T(t 1 ,t 2 ) ± ~(sine)T(-t 1 ,t 2 )· In complex analysis the Laplace transform can be defined as follows. e.g. f be a continuous function of slow growth in R2 and put

Let

-+-

p±1,±1f = fi{t E IR 2

:

+t 1 > 0, :;t 2 > 0}.

Then for (o 1 ,o2 ) E {1 ,-1} 2 one can define

this function is holomorphic in transform which is defined by L 01•02

is holomorphic in

{[~) 2

f(~ 1 ,~ 2 )

{(~ 1 ,~ 2 )

if sgn

E [ 2 : sgn

Im~ 1.

Im~i

oi}.

So the Laplace

= o1.,

and can be split up into four parts:

which correspond to the Laplace transforms of the restriction of f to the 'octants'; hence one could say that this Laplace transform is of a 'cartesian nature'. In the hypercomplex approach here presented, the Laplace transform always consists of two parts Lf

= LP +f + LP·- f

where P± are the above introduced 'orientation operators' in the Euclidean 264

space, which rather have a 'spherical nature'. 'natures' coincide. 32.

Only in the case m

1 both

A generalized Fourier-Borel transform

32.1 In this section we study a generalized version of the Fourier-Borel transform of certain classes of analytic functionals with support in an annular domain or the complement of an annular domain in ~m+ 1 • The transform a = a+ + a_ under consideration reduces in the complex case to

o_

T being a complex analytic functional with carrier in an annular domain. In the present monogenic approach the function~ eu/z, (u,z) E [ x [ 0 is replaced by the function H(u,y), already introduced in 15.8.3~ which for y E ~~+ 1 fixed is left and right monogenic in y E ~~+ 1 with lim H(u,y) = 0. Y-><x>

Notice that first some modules of monogenic functions and analytic functionals in annular domains and the complement of annular domains are introduced, the images under o of which will be determined.

32.2 Let R1 and R2 be positive constants. The notation

First assume that 0

<

R2

<

Rt,

for the right A-module of left monogenic functions in the considered annular domain, is now abbreviated to ~·1 ( r) ( R10 R2

) •

We know from Theorem 12.2.2 that iff E f(x) =

k

L IX I k=O

M(r)(R 1 ~R 2 )

then

00

Pkf(w)

+

L

k=O 265

0

the first and the second series being normally convergent in B(O,R 1 ) and Rm+1 'B(O,R Moreover those left inner and outer spherical 2 ) respectively. monogenics satisfy the following estimates (11.17) and (12.6)

for all R' E]0,R 1 [ and R" E]R 2 ,+oo[ respectively. Hence by restricting to the unit sphere Sm it is obtained that for any 0 > 0 there exists c6 > 0 such that 1Pkf(w)l 0

~

C0(1+o)

k -k R1

( 32. 1)

IOkf(w) lo ~ C0(1+o)k R~. ')!

-1

-1

Notice that f E M(r)(R 1 ,R 2 ) if and only if t E M(l)(R 2 ,R 1 adjoint off (see 13.12) with Laurent expansion 'V

f(x) =

'V

)

where f is the

k 'V IXI-(k+m) Pkf(w) + 1: lXI Qkf(w) .... k=O k=O 00

'V

00

)

where 'V

Pkf(w) = Pkf(w) ~

and 'V

Qk f(w) = QkT(w) ~.

32.3

Now assume that Rl :;:; R2.

We keep the notation

for the right A-module of left monogenic functions in

with limit zero at infinity. Notice however that this notation is consistent with the one in 32.2 since for R1 > R2 266

in view of the Laurent series expansion of a function in M(r)(R 1 ,R 2 ). of course for f E M(r)(R 1 ,R 2 ) we have alternatively

Then

or f(x) =

~

I

1x1

k

.

0

PkfC,1) 1n B(O,R 1 )

k=O where again the left inner and outer surface spherical monogenics satisfy estimates of the form (32.1). -1 Observe that f E M(r)(R 1 ,R 2 ) if and only if ~t E M(l)(R -1 2 ,R 1 ) where now

"J

f(x) =

I

00

1x1

k ""

Qkf(w)

0

-1

in B(O,R 2

)

k=O or f(x) =

Y

IXI-(k+m) Pkf(w) in Rm+ 1 ,B(O,R~ 1 ).

k=O 32.4 Combining the considerations made in 32.2 and 32.3, to any f E M(r) (R 10 R2 ) a sequence (Pkf(w), Qkf((l]))kElN of left surface spherical monogenics may be associated satisfying the estimates (32.1). So call S(r)(R 1 ,R 2 ) the right A-module of the sequences s = (Pk(w), Qk(w))kElN Pk(~)

k

k

E P(r) and Qk(w) E Q(r) are such that for each ~ > 0 there exists c6 > 0 for which (32.1) hold. Then S(r)(R 1 ,R 2 ) is a right (FS)-A-module fqr the system of seminorms P = {pj : j E ~} where

where

k 1 -k IPkCu)l • Rz-k (1+ ~J) 1 -k IQk (ul ) 1 }. p.(s) =sup sup {Rl(1+ -J.) 0 0 J k€14 tuESm

267

Furthermore the mapping from M(r)(R 1 ,R 2 ) to S(r)(R 1 ,R 2 ) which maps a function f into its associated sequence of surface spherical monogenics, is a topological isomorphism. Moreover if (Pkf(u)), 0,~f(uJ\EINES~r)(R 1 ,R 2 ) is the sequence corresponding to f E M(r)(Rl'R 2 ) , then to f E H(l)(R2, R~1) the sequence (Qkf(w), P'kf( 0 1))kEINES(l)(R;1, R~ 1 ) is associated. m+1 32.5 As for each ~2 clR open, M(r)(~;A) and r~(l)(";A) are (FS)-A-modules when endowed with the topology of uniform convergence on the compact subsets of~. it is meaningful to introduce the following modules. Definition 1 i m i nd O
with

Of course those A-modules are equipped with the inductive limit topology. -1 -1 Note that f E M(r)(R 1 ,R 2 ) if and only iff E r~(l )(R 2 ,R 1 ). ru

-

32.6 In view of the topological isomorphism between M(r)(R 1 ,R 2 ) and S(r)(R 1 ,R 2 ) (see 32.4) it is evident to introduce the following modules. Definition

Endowed with the inductive limit topology, S(r)(R 1 ,R 2 ) is topologically isomorphic to M(r)(R 1 ,R 2 ). -1 -1 '\, '\, Observe that (Pk,Qk)kEINES(r)(R 1 , R2 ) if and only if {Qk,Pk\Ef'.JES(l)(R 2 ,R 1 ). 32.7 In the following proposition, the proof of which is straightforward, the module S(r)(R 1 ,R 2 ) is characterized by means of estimates. Proposition 268

A sequence of left surface spherical monogenics

belongs to S(r)(R 1 ,R 2 that

if and only if there exist 0

)

<

n

<

1 and C

>

0 such

:;: C(1-n)k R~k (32.2)

~ C(1-n)k R~ 32.8 It follows readily from the duality theory enunciated in the Sections 22 and 23 that M(r)(R1,R 2 ) and M(l)(R 2 ,Rl) form a dual pair; the same is true for M(l)(R1,R 2 ) and M(r)(R 2 ,Rl). By the topological isomorphisms introduced in 32.4 and 32.6 it is obtained that S(r)(R1,R 2 ) and S(l)(R 2 ,R 1) and also S(l)(R1,R 2 ) and S(r)(R 2 ,Rl) form dual pairs. Notice that those duality phenomena imply that for both M(r)(R1,R 2 ) and S(r)(R 1,R2) the inductive limit topology is equivalent to a (DFS)-topology. Notice also that for hE M(r)(R1,R 2 ) and h* E M(l)(R 2 ,RI), the duality between M(r)(R1,R 2 ) and M(l)(R 2 ,RI) is expressed as follows:

( i ) if R1

>

R2 then

I

=

a(B(Rl-n),B(R 2 +n)) (i i )

h*(u) do u h(u);

if R1 <; R2 then



=

where twice n

J

h*(u) do u h(u), a(B(R 2 +n)' B(Rl-n)) > 0

is chosen sufficiently small.

32.9 The duality between M(r)(R1,R 2 ) and M(l )(R 2 ,RI) may be regarded as an 'inner product dual1ty between M(~)(R1,R 2 ) and M(r)(R1-1 R2-1 ) in the following sense. -1 -1 '\, Take g E M(r)(R1,R 2 ) and f E M(r)(R1 , R2 ) ; then f E t1(l )(R 2 ,RI). Hence it is quite natural to define an inner product between f and g by putting •

1

-

'\,

(f,g)

.

269

Furthermore this duality may be expressed in terms of the associated sequence spaces. ForgE M(r)(R 1 ,R 2 ) the associated sequence (Pkg(,,J), Qkg(u.J\Ei'J is in -1 -1 . . ~(r)(R.:;R 2 ~i for f E M(r)(R 1 ~ R2 ) the sequence (Pkf(,D), Qkf( 0 J))k.Q-J 1s 1n S(r)(R 1 ,R 2 ), while for~ E M(l)(R 2 ,R 1 ) the associated sequence (Qkf(w), 'V PkfC<~))kEI'J is in S(l)(R 2 ,R 1 ). Now first assume that R1 < R2 ; then there exists n > 0 such that 'V f E M(l)(R 2 +n, R1 -n), which leads to the Laurent expansion f(x) =

I

I

~kf(x) +

k=O

Pkf(x)

k=O

" in B(O,R 2 +n),B(O,R 1 -n). Meanwhile forgE M(r)(R 1 ,R 2 ) holds

g(x) =

,,

L

Pkg(x)

in

B(O,R 1 )

Qkg(x)

i n IR

k=O or g(x) =

I

m+1 'B ( 0 , R2

) •

k=O of the estimates satisfied by the sequences (Pkg'Qkg)kEI'J and (Qkf, Pkf)kEI'J we arrive at I~ vie~

00

I k=O 00

=-

I

(Pkf,Pkg) +

I

(Qkf, Qkg)

k=O co

(Pkf,Pkg) m + I (Qkf'Qkg) m k=O S k=O S

where (.,.) m is the inner product on a(Sm;A) defined in 28.10.

s

Now if

'V

R1 > R2 there exists n > 0 such that f E M(l)(R 2 +n, R1 -n) where still R1 -n > R2 +n; this leads to the expansions

270

I

f(x)

~kf(x) in B(O,R 2 +n)

k=O or

L

"'

oo

f(x)

Pkf(x) in mm+1 'B(O,Rl-n).

"'

k=O ForgE M(r)(R 1 ,R 2 ) we now have

L

g(x) =

k=O

00

Pkg(x) +

L

Qkg{x) in B(O,R 1 ),B(O,R 2 ),

k=O

and a~ain,.,in view of the estimates satisfied by the sequences (Pkg,Qkg)keN and {Qkf,Pkf)kSN we arrive at <>0

(f,g) =

L k=O co

00

(Pkf, Pkg) +

L

(Qkf,Qkg)

k=O 00

(Pkf,Pkg) m - L (Qkf, Qkg) m· s s k=O k=O

L

32.10 It should be noticed that some modules of left monogenic functions and left analytic functionals, already introduced in the foregoing sections, fit into the framework set up about M(r)(R 1 ,R 2 ) or M(r)(Rl,R 2 ). 32.10.1

M(r)(1,1) =lim ind M(r)(1+c,1-c) is nothing else but M(r)(Sm;A) and 1>c>O it is isomorphic to a(r)(sm;A), the module of A-valued analytic functions on sm (see 28.5). 32.10.2 r"\r)(1,1) is the module of left monogenic functions in mm+ 1 ~m with limit zero at infinity, i.e. M(r)(mr.t+ 1,sm;A) -+O" It is isomorphic to the dual of M:1)(1,1~ = r1(1)(Sm;A) = a(l)(Sm;A), i.e. the module of left analytic functionals on S (see 28.6). 32.10.3 t4(r)(oo,O) = l~m kr~~ M(r)(R 1 ,R 2 ) coincides with M(r)(m~+ 1 ;A). over we have I• z

Mote-

Proposition M(r)(oo,O) is isomorphic to the right module of A-valued harmonic functions in mm+1. 271

Proof

Let h be an A-valued harmonic function in ffi 00

h(x) =

I

k=O

I

, and let

k

00

Skh(x) =

m+·l

Ix 1 \h(w)

k=O

be its expansion into spherical harmonics. Now in view of Theorem 28.8 we have the decomposition Skh(w) = Pk(w) + Qk_ 1(w) where Pk(w) and Qk(w) satisfy estimates of the form sup IPk(cu) lo :;; C (k+m-1\ ( 1+k2) sup 1\h(w)lo \ k+1 ) wESm wESm and sup IOk-1 (w) lo wESm

~

c(k+~-2)(1+k2) sup

1\h(w) lo

wESm

But as h is harmonic in ffim+ 1, the sequence (Skh\EJJ satisfies

(see (28.4)).

So, putting e(h)(x) k=O

k=1

1• th e f unc t 1on e h ) 1s 1e ft monogen1c 1n lRm+ 0 Conversely let f E M(r)(oo,O) and put 0

h(x) =

(

I

0

0

IXIk Pkf(w) +

k=O

l

0

lxlk Qk-1 f(w)

k=1 00

Then as the sequeilce (Pkf(uJ), Qkf(uJ))kEJIJ satisfies estimates of the form IPkf(w)lo:::; Cssk 272

and IQkf(w)lo ~ CEEk, · h lS · harmon1c · 1n · lRm+ 1 and clearly e(h)(x) = f ( x ) 1n . JRm+ 1• t he f unct1on 0

c

32.10.4 M(r)(O,~) =lim ind M( ) (E,!) consists of the germs of left monoE>O r E genic functions in a neighbourhood of the origin and the point infinity, while vanishing at the latter point. It is the dual of M(l)(oo,O) = M(l) (lR~+ 1 ;A) and thus represents all left analytic functionals with carrier in some annular domain.

32.11 Now we introduce some submodules of M(r)(oo,O) which consist of left monogenic functions in lR~+ 1 satisfying some growth conditions. 32.11.1 Definition Exp(r)(R1,R 2 ) is the submodule of M(r)(oo,O) consisting of those functions such that for each E > 0 there exists CE > 0 for which lf(x)l 32.11.2

o

~ C exp[{1+E)(J;l+_&_)]. ~1

E

lxl

Provided with the proper system of seminorms P = {pk

k E ~}where

Exp(r)(R1,R 2 ) becomes a right Frechet A-module. 32.11.3 A function f belonging to M(r)(oo,O)- and a fortiori belonging to Exp(r)(R1,R 2 ) - has a sequence representation (Pkf(w), Qkf(w))keNES(r)(oo,O) such that for any E > 0 there exists CE > 0 for which JIPkf{w)

k

10

LIQkf(w)lo

~ CE E

~

CE Ek.

The module Exp(r)(R1,R2 ) is now characterized in terms of its sequence space representation, by means of sharper estimates than the above ones; the proof is postponed to 32.22(i). 273

Theorem A function f E M(r)(oo.O) with sequence representation (Pkf(u,), Qkf(rll))kEJ.I E S(r)(oo,O) belongs to Exp(r)(Rl'R 2 ) if and only if for any r; there exists C > 0 such that

>

0

n

( 1+n) k JIPkf ((•) l Io .:: cn k! Rl-k 1 (1+!J)k k IQkf(<.:)lo <: cri k! R2. 'I

3~.11.4

We know that f E rvt1r)(".,O) if and only iff E r\ )(""•0). Now as IQkf(ul) I = IQkf(t,,) 1 and l~kf(,,,) 1 = IPkf(,,,) 1 , it is oblained that o o . 'l
Definition

Exp(r)(R 10 R2 ) =lim ind Exp(r)(R 1 +s, R2 - , ) .

This mod-

c·O

ule is equipped with the inductive limit topology. 32.12.2 In the following proposition Exp(r)(R 1 ,R 2 ) is characterized by means of estimates; the proof is straightforward. Proposition A function f belongs to Exp(r)(R 1 ,R 2 ) if and only iff E r4(r)("",O) and there exist C > 0 and 6 > 0 such that lf(x)

10

;:;

C

exp [(1-r;)

(Jtf + Vr)].

32.12.3 This module Exp(r)(R 10 R2 ) nay also be characterized in terms of the sequence space representation. The proof of this theorem is also postponed to 32.22. Theorem A function f E M(r){oo,O) with sequence representation (Pkf(w), Qkf(w))keN E S(r)(oo,O) belongs to Exp(r)(R 1 ,R2 ) if and only if there exist C > 0 and f > 0 such that ( {IPkf w)l 0

:;;;

IQk f( w ) Io-<

274

(1-f)k -k C-IT~ R1 c ( 1-o)

k!

Rk 2 •

C.

A generalized Fourier-Borel transform

32.13 In 15.8.3 we introduced the function H(u,y) as a generalization of ~ eu/z, (u,z) E [ x [ 0 • Recall that this function is left and right mono1 separate 1y, w1t . 1n · u E IRm+ 1 an d y E .,...m+ · h 11m · H( u,y ) = 0; moreover 1· t gen1c ~0 Y-+«>

takes its values in spR {e 0 ,e 1 , ••• ,em}· Now it will be used in the definition of a generalized Fourier-Borel transform. 32.14 LetT E M(r)(R 1 ,R 2 ); then T may be considered as a bounded linear functional on M(l )(R 2 ,R 1 ) =lim ind M(l) (R 2 +E, R1 -E). If R1 ~ R2 then E>O

clearly H(u,y) E M(l)(R 2 ,R 1 ) in both variables separately. to define in IR~+ 1 :

So it makes sense

o+(T)(y) =

J

H(u,y)dou T(u)

a[B(R2+n)~(Rl-n)J

=

f

aB(~2+n)

H(u,y) dou T(u)

where of course n has to be chosen sufficiently small. Clearly o+(T) is left monogenic in IR~+ 1 and lim o+(T)(y) = 0. Analogously we put in 1Rm+ 1 Y-+«>

o_(T)(y)



J

H(y,u)dou T(u)

a[B(R 2+n),B(Rl-n)] r

J H(y,u) dou T(u), aB(R 1 -n) which is a left entire function. On the other hand if R1 > R2 we introduce the functions

275

H(u,y),

1 ( 2R +R ), y E Rm+1 1 2 0

IU I

< ~

luI

>~

f 1

H+(u,y)

0

1 ( 2R 1+R 2 ), y E Rm+1 0

and u E Rm+1' IYI

0

f 1

H_(u ,y)

H( u ,y) ,

u

€

m+1 R • IY I

1

< ~(R 1 +2R 2 )

1

> ~(Rl+2R2).

Then clearly H+(u,y) E M(l)(R 2 ,R 1) with respect to the variable u, while H_(u,y) E M(l )(R 2 ,R 1) with respect toy. So we are able to define in R~+ 1 :

(

J

H+(u,y) dou T(u)

a[B(Rl-n),B(R2+n)]

J

H(u,y) dou T(u)

aB(R 2+n)

=-I

H(u,y) dou T(u) as{R1+R2) 2

which is a left monogenic function in R~+ 1 with lim o+(T)(y) = 0. y~ ogously we put in Rm+1': o_(T)(y) =

=

f

a[B(~l-n),B(R2+n)J

=I

H (y,u) dou T(u)

H(y,u) dou T(u) aB(Rl-n)

=

I

H(y,u)dou T(u)

aB( R1 +R2) 2

which is a left entire function. 276

Anal-

Now letT E M(r)(R 1 ,R 2) =lim ind M(r)(R 1 +E,R 2-c); then T can be

32.15

oO

looked upon as a bounded linear functional on M(l)(R 2 ,R 1 ) . If R1 < R2 then clearly H(u,y) E M(l )(R 2 ,R 1 ) in both variables u andy separately and so it is again meaningful to define in R~+ 1 : 0

+(T)(y) =

I

H(u,y) d0 u T(u)

a[B(R2-n)'S(Rl+n}J =

J H(u,y) d0 u T(u) aBCR2-n)

which is a left monogenic function in R~+l with lim 0 +(T)(y) ously we put in Rm+l Y~

0.

Analog-

o_(T)(y) =

H(y~u) dau T(u)

f a[B(R2-n,'B(R 1 +n)J =

- I H(y ~u) d0 u T(u) aB(R 1 +n)

which is a left entire function. On the other hand if R1 ~ R2 we define

r

H(u,y) dau T(u)

J

asu~2 -n)

=-

J

H(u,y) dau T(u)

aB(R1+R 2 ) 2

which is left monogenic in R~+l with lim o+(T}(y) = 0, and Y~

277

=

a_(T)(y)

r

H (y ,u) da T(u)

J a[B(R 1 +n),B(R 2 -n)J

r J

-

u

H(y,u) dau T(u)

aB(Rl+n)

r R R H(y,u) dau T(u). a~(~) 2 32.16 In view of the considerations made in 32.14 and 32.15 we define the generalized Fourier-Borel transform to be the map M

{ _(r)

a

M(r) given by a(T)

a (T) + a (T) + -

where a+ and a_ are to be taken in the sense of 32.13 and 32.14. So a transforms modules of monogenic functions in annular domains or the complement of annular domains into modules of monogenic functions in the complement of the origin. But there is more as shown in the following theorem. 32.17 Theorem The following inclusions hold: (i)

a(M(r)(R 1 ,R 2 ))

c

Exp{r)(R 1 ,R 2 );

(ii)

a(M(r)(R 1 ,R 2 ))

c

Exp(r)(R 1 ,R 2 ).

Proof It is clear that (ii) follows from (i) by taking inductive limits. So letT E M(r)(R 1 ,R2 ). If R1 ::;; R2 then for every n > 0 we have Ja+(T)(y)

J0

~ I·

J aB(R 2 +n)

278

H(u,y) dau T(u) I o

~

C sup JH(u,y) Ia· n 1u:=R2+n

Now by Proposition 15.8.6 we have JH(u,y) J0 when juJ

>

-;;

C(1+ ~~~~J)m ~ eJUJIJYI IYI

JYl, while for JYI

~

JuJ

JH(u,y) J0 -s C' ~n IYI Hence for all y E R~+ 1 holds Ja+(T)(y) lo :;; c~ e(R2+2n)/ IYI. In an analogous way it is obtained that

Consequently for each Rm+1

E

>

0 a constant C

>

0 may be found such that in

E

0

which means that a(T) E Exp(r)(R 1 ,R 2).

The case R1

>

R2 is treated similarly.1

32.18 Actually the inclusions stated in the above Theorem 32.17 are equalities between the considered function modules. This will be proved in 32.19, but first the images of M(r)(R 1 ,R 2 ) and M(r)(R 1 ,R 2 ) under the transformation a are characterized in terms of their associated sequence spaces. _Theorem Let f E ~4(r)(oo,O) = M(r)(JR~+ 1 ;A) have (Pkf(c,l),Qkf(w))kElN as sequence representation in S(r)(oo,O). Then (i) f E a(M(r)(R 1 ,R 2 )) if and only if for every n such that

>

0 there exists en> 0

279

{ (ii) that

IPkf(w)lo

:; c11

IQkf(w) lo

:; c

( 1+n)K

k!

-k Rl

( 1+~)k

k Rz;

k.

f E o(M(r)(R 1 ,R 2 )) if and only if there exist C

>

0 and 6 > 0 such

Proof Assume that R1 :;; Rz • Take T E M(r)(R 1 ,R 2 ) and put f(y) (IR~+ 1 ;A). Then

I

f(y) = aB( =

I

2

H(u,y) dau T(u) +n)

L ~!

aB(R 2 +n) k=O

I

o(T)(y)

E

M(r)

H(y,u) dou T(u)

aB(Rl-n) qk(u,y)dou T(u)aB( 00

L

r

t

00

I

1 TI qk(y,u) dou T(u)

-n) k=O

1

IT pk T(y)

k=O wm+1 PkT(w ) and Qkf ( w) = ~ ~+1 which means that Pkf(w) =- ~ QkT ( w). The statement (i) then follows from the sequence representation (PkT(w), QkT(w)) E S(r)(R 1 ,R2 ). The cases T E M(r)(R 1 ,R2 ) with R1 > R2 and T E M(r)(R 1 ,R 2 ) are treated along the same lines. c 32.19 Finally we show that a is a topological isomorphism between M(r)(R 10 R2 ) (resp. M(r)(R 1 ,R2 )) and Exp(r)(R 1 ,R2 ) (resp. Exp(r)(R 1 ,R2 )), all modules under consideration being endowed with their respective natural topologies. Theorem The following topological isomorphisms hold:

280

Proof It is clear that (ii) follows from (i) by taking induct·ive limits. Now by Theorem 32.17 it is already known that

In view of Theorem 32.18 we still have to prove the necessary condition of Theorem 32.11.3 in order to conclude that

m+1 So let therefore f E Exp(r) (R 10 R2 ) . i.e. f E t-1(r) (R0 ;A) and for every E: there exists C > 0 such that E: lf(x) lo:;;; CE: exp [(1+d

>

(~ + 1~1 )].

Expending f into its Laurent series f(x) =

I

lxlk Pkf(w) +

k=O

I

lxl-(k+m) Qkf(w)

k=O

where, independently of r E ]O,+oo[,

it is obtained that forE:

:; ; C'

E:

>

0 and for all r

>

r0

>

0

1 exp(1+E: ) -R r • (1+k) m --.. rll. 1

For all k E N such that r 0

<

~ l+E:

,

we ar~ive at

and, using Stirlings inequality _k_l_:;; kke-k, at l21il< 281

0

An analogous estimate holds for !Qkf(w) lo·

c

32.20 The isomorphism a established above is now carried over by transposition to an isomorphism between the respective dual modules. Thereto we define: Definition LetT E Exp(r)(R 2 ,R 1 ) (resp. T E IXP(r)(R 2 ,R 1 )); then its generalized Fourier-Borel transform is the function o(T) E M(l)(R 1 ,R2 ) (resp. o(T) E M(l)(R 1 ,R2 )) such that

32.21 Theorem (i) (ii) 32.22

=

In view of Theorem 32.19 it is obtained by transposition that The following topological isomorphisms hold: cr(Exp~r),b(R 2 ,R 1 ))= ~(lf(R 1 ,R 2 );

o(Exp(r),b(R 2 ,R 1 ))

=

t4(1)(R 10 R2 ) .

Remarks

(i) The sequence characterizations of the modules Exp(r)(R 1 ,R2 ) and Exp(r)(R 1 ,R 2 ), already stated in 32.11.3 and 32.12.3, now follow immediately from Theorems 32.19 and 32.18. (ii) Observe that Exp(r)(O,oo) is the generalization of Helgason's space of entire functionals in the complex plane (see [9]). Moreover holds o(Exp(r)(O,oo)) = t4(l )(oo,O) = f4(l )(lR~+ 1 ;A). (iii) The isomorphism established in Theorem 32.21 (ii) means that, mutatis mutandis, for any f E M(r)(R 10 R2 ) (R 1 > R2 ) there exists T E Exp(l)(R2 ,R 1 ) such that o(T) = f, or

282

When x E B(O,R 1 ),B(O,R 2 ) then g(u) particular case we get

= E(u-x) belongs to M(l )(R 2 ,R 1 ) and in thi:

= . Now

I



E(u-x)dcu f(u)

f(x)

;;[B( R1 -n)'B ( R2 +n>J

while

I

o(E(u-x)) (y)

J

E(u-x) dou H(u,y) -

aB(R 1-n)

H(x,y)

E(u-x)dou H(y,u)

;)B( R2+n)

H(y,x)

+

and hence f(x)

=
+

H(y,x)>,

0

-

x E B(O,R 1 ),B(O,R 2 ),

yielding, when~ is an A-valued measure representing T E Exp( 1)(R2 ,R 1 ), f(x) =

I

H(x,y)dpy

+

1Rm+1

r J

o

H(y,x) dpy' x E B(O,R 1 ),B(O,R 2 ).

1Rm+1

In terms of the Laurent series expansion of f this leads to

and

L

oo

k=O

Qkf(x) =

I

H(y,x)

d~y'

x E lRm+1 'B(O,R 2

).

1Rm+1

Those two integrals might be called the "Taylor and Law•ent integrals" of f E M(r)(R 1 ,R 2 ) (R 1 > R2 ). A same reasoning may be followed in the case where R1 s R2 leading up to the same expressions for the "Taylor and Laurent integrals".

283

33.

Connections with complex analysis- a generalized Radon transform

33.1 In [8] Hayman proved that every harmonic function, and hence every holomorphic function, in the open disc {r; E [ : lsi < R} admits a multiple Taylor series expansion about the origin converging absolutely in the domain {(x,y) E JIF : JxJ + IYI

<

R}.

In this section this result is generalized to the monogenic functions in the following sense : taking an analytic function f the multiple Taylor series of which converges absolutely in a domain of Rm we look for the optimal domain in Rm+ 1 where the multiple Taylor series of the monogenic extension of f converges absolutely. To this end a hypercomplex version of the Radon transform ~

P: O'(B(0,1))

0(8(0,1))

given by 1 u> P(T)(u) =
is defined. However it is necessary to leave the framework of the real Clifford algebra A used up to now in this book, and to handle the complex version of it, which will be denoted by Ac • 33.2

For u = u0 + u E Rm+ 1 , (z 1, ••• ,zm) E [m and {r; 1 •••• ,r;m) E [m we put m

m ~

z =

I j=1

e .z. "' J J

I

~

~



I

Jz J·12

j=1 m

I

~

X

j=1

m JzJ2 =

ej(xj+iy) = m

I

(xj+Yj),

j=1

r;jzj

j =1 and m



I j=1

284

ujzj

+ i.

.~

+ ly,

Next to the classical ball B( 0 , R) = {x E Rm+ 1 :

1

x1

<

R} ,

the following domains in Rm+ 1 and [m will be used constantly. (R 1 , ••• ,Rm), Rj > 0, j = 1, ••• ,m, we put B*(O,R) = (i

E

[m :

111

<

With

R=

R}; m

n (R)

{x

E

Rm+ 1 : 1x0 1 1R1 +

L j

Rj 1xj 1 < 1};

=1

where n.(R.) = {X E Rm+1 J

J

{X E Rm+1

n*(R)

33.3

-+ /\(R)

-+ {z E [m

'K
-+ {Z E [m

If Q* f =

c

lx 0 1 + IXj I < Rj}; -+ 1 . lxol + lXI < R} • lzj I
[m is a domain of holomorphy then we consider functions

L eAfA

: Q*-+ Ac

A

where each of the components fA : Q*-+ [ is holomorphic in Q*. The Ac-module of those functions is denoted by O{Q*;Ac); it may be considered as well as a left or as a right Ac-module, and it is provided with the usual topology of uniform convergence on the compact subsets of Q*, turning it into a Frechet Ac-module. The dual modules O(l)(Q*;Ac) and O(r)(Q+;Ac) consist of all left, respectively right, Ac-linear analytic functionals in Q*. For a relatively compact convex open neighbourhood Q* of the origin iA am we put lim ind O(l)((1+s)Q*;Ac). s>O

285

33.4 Just as was done in 15.7 for the real Clifford case, we consider the left and right inner spherical monogenic of order one: m

L

p(u,z) =

(u 1e0

-

u0 e1 )z 1

1-1 .... .... e 0

....

u0 z,

-

which for any u E Rm+1 clearly belongs to 0([ m;A c ). obtain

For its Clifford norm we

The function p(u,z)

k

.... kQ

= p(u,z)

is a left and right inner spherical monogenic of order k, taking c in sp[{e 0 ,e 1, ••• ,em} cA. Just as in 15.7.4 it is proved that Proposition

its values

If the function k

00

L

ckz:; '

ck E [ for all k E ~

k=O is holomorphic in the open disc {z:; E [ and right spherical monogenics F(u,z) =

[z:;[

<

p}, then the series of left

I

ck p(u,z)k k=O when considered as a multiple power series in the real variables u0 ,u 1 , ••. ,um, converges absolutely in the region given by [u 0

[

.... [z[

m +

L

[uj[ [zj[

< p

j=1

In the particular case where z = X E Rm this domain is the largest one in which the multiple Taylor series of F(u,z) about the origin converges absolutely.

286

Remark When m = 1 and by identifying e 1 with -i, the above left and right monogenic function F(u,z) reduces to f(iuz), (u,z) E [ x [.

33.5

1Rm+ 1 of the origin is called m+1 optimal with I'e::pect to absolute c'on-uel'gcn,H:; if for every u E lR 'S"2 there exists a left (or right) monogenic function f in ~ such that the multiple Taylor series expansion of f about the origin converges absolutely in Q but not in u. So by Proposition 33.4 all domains which are given by inequalities of the form 33.6

Definition An open neighbourhood

'> c:

m

->-

L

luol IRI +

lujiRj

1,

<

Rj

>

0

j=1 are optimal with respect to absolute convergence. 33.7 The following example of a function of type F(u,z) is essential in the construction of the generalized Radon transform. Starting with the well known Taylor series expansion 1

k

00

T=Z = l: s ,

lsi

<

k=O the above Proposition 33.4 leads to the function P(u,z)

= ----

1-p(u,z)

1--u 0 z m (1-) 2 +u~(

l:

zj)

j =1 with Taylor series expansion about the origin 00

P(u,z)

=

z:

-+ k p(u,z)

k=O which, considered as a multiple power series in u0 , ••• ,um, converges absolutely in the domain given by

Notice that P(u,z) is defined, holomorphic with respect to z and monogenic with respect to u in (1Rm+ 1 x [m)-....s where the set S is determined by the equations 287

{

-+

+ u~lxlz

(1-)z

-+-+

-+-+

(1-)

~-+

= z

-+

+ u~IYiz

-+-+ = u~<x,y>.

The function P(u,z) is thus a direct generalization of ---1--1-u~

(u,~) E [

x [.

33.8 Just as was done in 15.8 we may consider series of the form

where the coefficients ck, k E ~. are now chosen to be complex, the series thus representing left and right monogenic functions in both variables u and y separately in a certain region, and taking values in sp[{e 0 ,e 1 , ••• ,em}. This yields a generalization of the function

.! f(~), ~

~

(u,d E [

x [

0,

the function f being holomorphic in an open disc. Proposition If the function with Taylor series expansion

f(~)

We have in fact

is holomorphic in the open disc

{~€[: l~l
00

f(d

=

2 k=O

then the series 00

¢(u,y)

=

2

ckqk(u,y)

k=O converges normally in the region

to a left and right monogenic function in both u and y separately. 33.9 The above two monogenic generalizations as presented in 33.4 and 33.8 are linked to each other in the following way. 288

Proposition f( rJ

Let the function 00

2

=

k=O be holomorphic in the open disc {s E [ regions:

2

00

F(u,i)

ck p(u,i)

lsi

~

p} and put in the appropriate

k

k=O and 00

q,(u,y) Then for sufficiently small r > 0 and lui< rp we have F(u,z)

¢(u,y)

(

day

P(y,i).

asc6,r) Proof Take all lzjl < R, j = 1, ••• ,m, for some R > 0; then P(y,i) is left and right monogenic in the region U given by m

+

IYo I Iz I

+

1

2 IYj I Izj I < R" j=1

Chooser> 0 small enough such that B(O,r) cU. Taking into account that ¢(u,y) is left and right monogenic in both variables in the region lui
(

aB (

6, r)

¢(u,y)

day

P(y,i)

is a function of u and z which is left and right monogenic in lui < pr. function is nothing else but

This

00

2

ck PkP(u,z)

k=O

289

00

L ck p(u,i>

k

k=O F(u,z).

c

33.10 Now we define a generalized Radon transform. Definition Let D* Then we put

c

[m be a domain of holomorphy and let T E

o( 1 ) CJ*;Ac).

PT(u) = , u belonging to a suitable open subset r2 of 1Rm+ 1 depending on ::z*.So observe that if P(T) is well defined it is left monogenic in 12. Notice also that P transforms analytic functionals in complex domains into left monogenic functions. Furthermore P(TA) = P(T)>..

* ( g* ;Ac) and all >.. E Ac • for all T E O(l) Finally note that form = 1 and by the 'classical' identifications, P reduces to P( T)( u)

= • ( u •d z;,

E [

X

[

1-uz;,

which implies, by K6the's duality theory [12], that Pis then a topological isomorphism from O'(B(0,1)) to 0(8(0,1)). 33.11 Our aim now is to characterize P(O*(l)(n*;Ac)) in the particular cases .... 'V-+ where g* is either A(R) or A(R) or still B*(O,R). We first prove .... * - .... c ). Then P(T) is left monogenic in rr(R) and its Theorem LetT E O(l)(A(R);A multiple Taylor series expansion about the origin converges absolutely in .... rr(R).

* - .... c ). Proof Take T E O(l)(A(R);A that

290

Then for each

E >

0 there exists C

E

>

0 such

-+

c

m

->-

for all 1> E 0(1) (~(R+2t) ;A ) , where R + [~ stands for (R 1+E•· •• ,Rm+E) E R • Now let u E n(R) be fixed and choose~> 0 such that u E n(J + 2l ); then -+ -+ c -+ of course P(u,z) E O(l)(AlR + 2t);A ). So for all u E n(R} the function P(u,!) belongs to O(l)(f.(R);Ac) and P(T)(u) = z -+

is well defined. Hence P(T) is left monogenic in rr(R). -+ u E n(R) as above we have P(T)(u) =


z

00

z

-+-+

I

'f

With the same fixed

p(u.1}>

k=O -+k

- u0 z) > k=O z 00

I

l:

k=O k0 +

+ km=k

k! k I k I o · • • • m" -+


ko k 1 z1

k

u

0

0

k um m

km zm >.

( 33. 1 )

Now

whence ko

l:

+ ••• +

k! km=k m -+ -+

:; : CE:(Ju 0 J JR+E:I +

L

JujJ (Rj+d)

k

j=1

for some

c5

E:

> 0.

291

This means that the multiple Taylor series (33.1) converges absolutely in ->the point u. As u was chosen arbitrarily in rr(R) the multiple Taylor series ->expansion of P(T) about the origin will converge absolutely in rr(R). c 33.12 In a completely analogous way or OCl)(B*(R);Ac) respectively, then respectively, with a multiple Taylor verging absolutely in rr(R) anJ rr*(R) 33.13 In order to prove the result due to Martineau (see B*(O,R) and put respectively mapping P* : 0'(~*)-.. O'(Q*) p*(T)(i)

=


*

'\,

->-

c

it may be proved that if TEOQ) (t,(R) ;A )

P(T) is left monogenic in rr(RJ or rr*(R) series expansion about the origin conrespectively.

converse of Theorem 33.11 we need the following ->'\,->[13]). Let g* be either A(R) or A(R) or still '\,->->1 Q* equal to A(R) or A(RJ or B*(O,R- ). Then the given by

1

1-

>

is a topological isomorphism. If T E 0( 1 )(~*;Ac) then there exist TA E 0'(~*), A c {1 , ••• ,m}, such that T = L eATA in 0( 1 )(~*;Ac), and it is an easy matter to check that the transform~tion which maps Ton L eAP*(TA), and which is still denoted by P*, is a topological isomorphism fro~ the Frechet Ac-module O{l)(~*;Ac) onto the Frechet Ac-module O(r)(n*;Ac). 33.14 Theorem Let f be left monogenic in a neighbourhood of the origin such ..... '\,-+ that its multiple Taylor series converges absolutely in either rr(R) or rr(R) or still IT'ii(R). Then there exists T E OCl)(A(R),Ac), respectively T E 0(1 ) (A(R);Ac) or T E o( 1)(B*(R);Ac), such that P( T)

= f.

Proof Let f be left monogenic such that its Taylor series expansion about the origin converges absolutely in n(R). Then f(t) = f(x)IR~+ 1 admits an absolutely converging multiple Taylor series expansion in m

rr(R)JR:+ 1

=

{tERm :

L

Rj Jxj I

< 1}.

j =1

Hence its complex extension f(!) is holomorphic in A(R) and so by the con* )(A(R);A - . . . c ) such that siderations made in 33.13 there exists T E O(l 292

p*(T) (!)

f(t)

or

)

f(!)

whence .....

1 z 1- or sti 11

f(u)

. . . = IIR_m+1 f(u) z

-

in n(R) 1R:+ 1 • As the left monogenic extensions of analytic functions in open subsets of Rm+ 1 are unique (see Section 14), it follows that in n(R) .....

f(u) = z PT(u). The other cases are treated similarly.

c

..... "'n(R) ..... 33.15 Proposition The domains n(R), and n*(R) are optimal with respect to absolute conve1~ence.

.....

Proof We already observed in 33.6 that a domain such as n(R), which is given by the inequality ..... m !x 0 1 IRI + l: Rj 1xj1 < 1, Rj > 0, j=1 is

with respect to absolute convergence. Next n(R) is an intersection of optimal domains with respect to absolute convergence; so it is optimal itself. Finally as n*(R) can be written as opti~al

'\,+

n*(R) =

.....

n rr(R) ..... IRI =R

it follows that rr*(R) is optimal with respect to absolute convergence too. 293

~

-+ The above characterizations of P(O(l)(i).*;Ac)) for ;.2* ~qual to II(R), A(R) or B*(O,R), lead to an optimal version of the Cauchy-Kowalewski ex tension theorem stated in 14.2. 33~16

Theorem Let f(!) be an Ac-valued holomorphic function in 1\(R) (respectively A(R), B*(O,R)) and let f(;) be its Ac-valued analytic restriction to Rm. Then f(;> admits a unique left monogenic extension f*(x> to rr(R> (respectively rr(R), rr*(R- 1)) which is given by 2k+1 f*(x) = (-1)k D[ xo A~f(;}]. (2k+1)! k=O

I

The multiple Taylor series expansion about the or1g1n of f* converges absolutely in rr(R) (respectively rr(R), rr*(R- 1>> which is optimal with respect to absolute convergence. Proof The uniqueness of the Cauchy-Kowalewski extension was already proved in Section 14. -+ -+ c . As f(r;} E O(r)(A(R)!A-+), 1n view of t4artineau's result mentioned in 33.13, there exists T E 0( 1 )tA~R);Ac) such that P*(T) = f, or more explicitly 1

-+-+> 1- whence -+-+ 1-<x,z> or '\.-+ We know from Theorem 33.11 that P(T) is left monogenic in rr(R) with a multiple Taylor series expansion about the origin converging absolutely in the '\,-+ -+ '\.-+ optimal domain rr(R). So P(T) is the left monogenic extension of f(x) to rr(R). On the other hand it is easi1y verified that the given function f*(x) is left monogenic and extends f(x). By the uniqueness of the C-K extension it follows "'-+ that f*(x) = P(T)(x) in rr(R). c

294

33.17

In the particular case where m = 2 the following pictures may be drawn:

33.17.1

-+

c

-+

For f € O(!I(R);A ), f(x) E rv+

c

-+

a({x

m

E 1R: Jx.l J

<

R., J

.

J

= 1,2};Ac) and

f* E M(r)(n(R);A ):

33.17.2

rv-+

c

-+

For f E 0(/\(R);A ), f(x) E -+

-+

a({x

E lR

m

c

f* E M(r)(n(R);A ):

295

33.17.3

For f E O(B*(O,R);Ac). f(;) E a({~ E Rm

f* E M(r)(rr*(R- 1);Ac): Xo

296

Notes to Chapter 5 The Fourier transform F for functions can be extended to distributions in two ways. On the one hand, as Schwartz did (see [Sch]), a space of testfunctions vaster than V(mm), namely s(mm), is introduced such that the Fourier transform becomes a topological isomorphism on it~ which may then be carried over to the dual space S'(mm) by putting = for all T E S'(mm) and ¢ E S(mm). In this way the Fourier transform is defined on the subspace S'(mm) of V'(mm)~ this restriction being essentially due to the fact that for any¢ E V(mm)~ F¢ is not in V(mm)~ unless¢ is identically zero, as follows from the famous Paley-Wiener Theorem. On the other hand, as was done by Ehrenpreis and Gel 'fand-Shilov (see [E4] and [GS]), using the generalized Fourier transform F, the space V(mm) is mapped isomorphically onto the space Z of all entire functions of exponential type on [m so that, again by duality, FT may be defined forTE V'(mm) by putting = for all ¢ E z. In this way FT is not in V'(mm) but in Z'. In quantum field theory, boundary values in the S'-sense of holomorphic functions in tubular domains in [m are often needed. They appear to be Laplace transforms of S'-distributions with support in a cone of mm (see [Ca3] and [St Wi]). In Section 30 both approaches to the Fourier transform of A-distributions are worked out, and this by using the kernel E(t,x) and its monogenic extension E(t,x) introduced in Section 15.4. In Section 31 a generalized Laplace transform L of elements T E s( 1 )(mm;A) is introduced. It appears that LT is monogenic in m~+ 1 and that its s( 1 )-boundary value BVLT equals FT. For further results concerning these Fourier and Laplace transforms we refer to Sommen's paper [So3]. Note also that in [Mor] Morimoto obtained several isomorphisms between spaces of functionals on the unit sphere sm- 1 and spaces of eigenfu~ctions of the Laplacian ~· Indeed, using the operator P\:Tw~)>,w E Sm- 1, introduced by Hashizume-Minemura-Kowata-Okamoto in [HKMO] and which transforms functionals on Sm- 1 into solutions of the equation (~ + \ 2 )f = 0 (1), Morimoto proved that E'(Sm- 1 ) is isomorphic to the' space of solutions of (1) which are of slow growth and that a'(Sm- 1 ) is isomorphic to a space of solutions of (1) satisfying exponential growth conditions. Moreover in [HKMO] a space ~B(S m-1 ) of generalized hyperfunctions on the unit sphere was introduced and it was shown that P\(~(Sm- 1 )) coincides ~~

~

297

with the space of C -solutions of (1). In [Mor] also an interpretation is given of B(Sm- 1) and this by using a theory of analytic functionals on m m {z = (z 1 , ••• ,zm) E 0: : I zj = 1}. j=1 . * (Sm-1 ;A ) , FT (+) + Sm-1 , Now 1f we put for each T E E(l) x = ,wE uJ m-1 then F, just as P1 , becomes an isomorphism between E(l)(S ;A) and the space of (A-valued) solutions of (Lm+1)f = 0 which are of slow growth in Rm. Moreover, defining forTE s( 1 )(Rm;A), P+T and P_T as in Section 31.16, it may be proved that F is an isomorphism of P±E1 1 )(Sm- 1;A) to the spaces of functions in Rm which are of slow growth and satisfy respectively the equations m m m ( I e. + 1)f = 0. Hence, as (1- I ej a~.)(1 + I ej a~.> j =1 J ax j j =1 J j =1 J

_a

= ~+1, a refinement is obtained of t4orimoto's result for the case E'(Sm- 1). Observe too that when m = 2, the entire functionals in 0:'{0} introduced by "' . Helgason in [Hel] represent the space B(S 1 ). The theory of analytic functionals and their Fourier-Borel transforms is of fundamental importance in solving existence and approximation problems for the equation P(Dz)u = h, h being holomorphic in an open convex subset ~ of O:m and P(Dz) being a differential polynomial in az , ••• ,az • The Fourier-Borel transform itself was 1 m introduced by Martineau in his basic work [Mar1]. To fix the ideas, let T be an analytic functional in 0: and call FT(z) = . Then F transforms analytic functionals into entire functions of exponential type, where the growth conditions of FT are determined by the carrier ofT, and conversely if f is an entire function of exponential type, then f = FT for some analytic functional T in 0: having convex carrier. In [Mor1] Morimoto extended the Fourier-Borel transform to analytic functionals having non convex carrier. LetT E 0'(~) where~= B(O,R 1 ),B(O,R2 ), and put F,T(u,v) = ..(uz+! )>, z z (u,v) E 0: x 0:. Then in this way a transform from 0'(0.) to spaces of solutions of the equation at2:v = A. 2 F is obtained. Morimoto also showed that for each T, FA.T is completely determined by its restrictions to the coordinate axes, i.e. F(u,O) and F(O,v), and hence by 1\

298

..uz)>

F(u,O)

..f)>

F(O,v)

and so also by o+T


eu/z>

a_T



...!_

u. z

1 F T(-1 D) 1 z. •

=z =

T F1 (u)(O ,z).

Putting a = a+ - a_, this a-transform leads to a generalization of Helgason's enti1·e functionals already mentioned, and thus to a representation of B(Sm- 1) in the higher dimensional case. Note that in [Mor2] Morimoto also extended the Fourier-Borel transform to analytic functionals having non compact carrier, while in [Mor3] analytic functionals on the Lie sphere are considered. In Section 32 a generalized version of the a-transform cited above is introduced and this by using the kernel H(u,y) defined in Section 15.8.3. In this connection also the paper [So4] of Sommen should be mentioned. If T E O'(B(0,1)) and PT(z) = , then Kllthe's duality theory (see also u -1-uz Chapter 3) implies that Pis an isomorphism from O'(B(0,1)) to O(B(0,1)). In [Mar2] Martineau generalized this result in the following way. Let K c [m be absolutely convex and compact let K6 be the polar of K and define for each 6 1 T E O~K), PToE O(K ) by PT(z) = >. Then an isomorphism from O'(K) to O(K 6 ) is obtained Consider, as we did in Section 33, for each T E o( 1 )(~*;Ac), PT(u) = . Then the transformation P enables us to get results concerning z the multiple series convergence of monogenic extensions of (A-valued) analytic functions. These results may be compared with those obtained by Hayman in the case of the unit disc (see [Hay]). In the several dimensional case Siciak tackled an analogous problem using a holomorphic extension ·of a harmonic function in the unit ball (see [Sic]). Finally note that in [So5] Sommen calculated the entire functional which represents the Cauchy kernel; this led up to a representation of monogenic functions in a strip by means of holomorphic functions of several complex variables. I

0

299

Bibliography [Ca3] R.D. Carmichael, Distributional boundary values of functions analytic in tubular radial domains, Indiana Univ. Math. J. 20 (1971) 843-853. L. Ehrenpreis, Analytic functions and the Fourier transform of dis[E4] tributions, I, Ann. of Math. 63 (1956) 129-159; II, Trans. Amer. Math. Soc. 89 (1958) 450-483. [GS] I.M. Gel 'fand and G.E. Shilov, Generalized functions, II, Spaces of fundamental and generalized functions (Academic Press, New York, 1968). [HKMO] M. Hashizume, A. Kowata, K. r4inemura, K. Okamoto, An integral representation of the Laplacian on Euclidean space, Hiroshima Math. J. 2 (1972) 535-545. [Hay] W.K. Hayman, Power series expansions for harmonic functions, Bull. London Math. Soc. 2 (1970) 152-158. S. Helgason, Eigenspaces of the Laplacian : Integral representations [Hel] and irreducibility, J. Functional Analysis 17 (1974) 328-353. [Mar1] A. l~artineau, Sur les fonctionnelles analytiques et la transformation de Fourier-Borel, J. d'Anal. Math. 11 (1963) 1-164. [Mar2] , Equations differentielles d'ordre infini, Bull. Soc. Math. France 95 (1967) 109-154. [Mor] M. Morimoto, Analytic functionals on the sphere and their FourierBorel transformations, to appear in a volume of the Banach Center Publications. [Mor1] - - - - , A generalization of the Fourier-Borel transformation for the analytic functionals with non-convex carrier, Tokyo J. Math 2 (1979) 301-322. [Mor2] , Analytic functionals with non-compact carrier, Tokyo J. Math. 1 (1978) 77-103. [r4or3] , Analytic functional s on the Lie sphere, Tokyo J. Math. 3 (1980) 1-35. [Sch] L. Schwartz, Theorie des distributions (Hermann, Paris, 1966). [Sic] J. Siciak, Holomorphic continuation of harmonic functions, Ann. Pol. Math. 29 (1974) 67-73. [So3] F. Sommen, Hypercomplex Fourier and Laplace Transforms II, to appear. [So4] -------,A generalized version of the Fourier-Borel transform, to appear in Bull. Soc. Roy. Sci de Liege. 300

[So5] F. Sommen, Some connections between Clifford analysis and complex analysis, to appear in Complex Variables : Theory and Application. [StWi] R.F. Streater and A.S. Wightman, PCT, Spin and Statistics, and All That (Benjamin, New York, 1964).

301

References

[1]

W. Ambrose,

Structure theorems for a special class of Banach algebras,

Trans. Amer. Math. Soc. 57 (1945) 364-386.

[2]

F.F. Bonsall and A.W. Goldie, Algebras which represent their linear functionals, Proc. Cambridge Philos. Soc. 49 (1953) 1-14. [3] H.G. Garnir, Problemes aux limites pour les ~quations aux d~riv~es partielles de la physique II (Cours de troisi~me cycle de 1 'Univ. de Li~ge, 1974-75). [4] H.G. Garnir, M. De Wilde and J. Schmets, Analyse fonctionnelle, tome I (Birkh~user Verlag, Basel, 1968). [5] ----------, Analyse fonctionnelle, tome II (Birkh~user Verlag, Basel, 1972). [6] ----------,Analyse fonctionnelle, tome III (Birkh~user Verlag, Basel, 1973). [7] M. Hashizume, K. Minemura and K. Okamoto, Harmonic functions and Hermitian hyperbolic spaces, Hiroshima Math. J. 3 (1973) 81-108. [8] W.K. Hayman, Power series expansions for harmonic functions, Bull. London Math. Soc. 2(1970) 152-158. [9] S. Helgason, Eigenspaces of the Laplacian: integral representations and irreducibility, J. Funct. Anal. 17 (1974) 328-353. [10] L. H~rmander, Linear Partial Differential Operators {Springer-Verlag, Berlin, 1963). [11] O.D. Kellogg, Foundations of Potential Theory (Springer-Verlag, Berlin, 1929). [12] G. K~the, Dualit~t in der Funktionentheorie, J. Reine und Angew. Math. 191 (1953) 30-49. [13] A. Martineau, Sur les fonctionnelles analytiques et la transformation de Fourier-Borel, J. d'Anal. Math. 11 (1963) 1-164. [14] M. Morimoto, Analytic functionals on the sphere and their Fourier-Borel transformations (to appear in a volume of the Banach Center Publications). [15] C. MUller, Spherical Harmonics. Lecture Notes in Mathematics 17 {Springer-Verlag, Berlin, 1966). 302

[16] L. Schwartz, Theorie des distributions (Hermann, Paris, 1966). [17] R.T. Seeley, Spherical harmonics, A.M.;-;. f.Janthly 73, II 4 (1966) 115-121. [18] , Eigenfunction expansions of analytic functions, P1•oc. AmeP. Math. Soc. 21 (1969) 734-738. [19] H.G. Tillmann, DualiUt in der Potentialtheorie, Po1•t. Math. 13 (1954) 55-86. [20] F. Tr~ves, Linear partial differential equations with constant coefficients (Gordon and Breach, New York. 1966). [21] Basic linear partial differential equations (Academic Press, New York, 1975). [22] A. Paulik, Optimale Approximation linearer Funktionale auf analytischen und harmonischen Funktionen (Dissertation, Universit8t G6ttingen, 1975). [23] N. Vilenkin, Special functions and the theory of group representations (Translations of Math. Monographs 22, A.M.S., Providence, 1968).

303

Author index

The roman numbers refer to the Notes at the end of the respective chapters W. Ambrose 1.12 N. Aronszajn III M.F. Atiyah I H. Behnke II, III E.J. Beltrami IV S. Bergmann III F.F. Bonsall 1.12 R. Bosshard II R. Bott I F.F. Brackx III R. Braun III H. Bremermann IV R.D. Carmichael IV, V P. Charpentier II P. Chauveheid III C. Chevalley I W.K. Clifford I A. Crumeyrolle I C.A. Deavours II R. Delanghe II, III M. De Wilde 2.4, 3.2, 17.11, 22.4, I C. Dias III A. Dinghas II T. Dwyer III J.D. Edmonds, jr. L. Ehrenpreis III, v. R. Fueter 304

I, I I

H.G. Garnir 2.4, 3.2, 8.3, 17.11, 22.4, 23.3, I P.M. Gauthier III I .M. Gel' fand V J.W. Gibbs R.P. Gilbert III A.W. Goldie 1.12 B. Goldschmidt II A. Grothendieck III F. GUrsey I, II K. Habetha I I H. Haefel i II P.R. Halmos W.R. Hamilton M. Hashizume 28.9, IV, v W.K. Hayman 33. 1 ' v o. Heaviside I s. Helgason 32.22, IV, v G. Henkin II D. Hestenes I , I I E. Hewitt IV G. Hile I I , III E. Hille I I I L. H6rmander 17.4.2, III V. Iftimie K. Imaeda

II

M.A. Jafarizadeh M. Jimbo I

Yu. N. Kafiev I M. Kashiwara IV T. Kawai IV O.D. Kellogg 11.5.3 H. Komatsu IV P.P. Konder IV J. K6rner IV G. K6the 22.8, I I I , IV A. Kowata IV, V M. P. S. P.

z.

Langenbruch IV Lelong III -L.ojasiewicz III Lounesto II, IV Luszczki IV

B. Malgrange III A. Martineau 33.13, III, IV, V J . C. Max we 11 I R. Meise IV H. Meschkowski III K. Minemura 28.9, IV, V T. Miwa I P. Montel III M. Morimoto 28.9, IV, V c. M!Jller 11.1.1, 11.1.3, 11.4.2 W. Nef

II

K. Okamoto 28.9, IV, V A. Paulik 24.16, 24.17 F. Pelligrino III H.J. Petzsche III, IV I.R. Porteous E. Ramirez II M. Riesz I L.A. Rubel III C. Runge III

J. Ryan

I, II

M. Sa to I, IV P.P. Saworotnow R. Schapiro I M. Schiffer III J. Schmets 2.4, 3.2, 17.11, 22.4, 23.3, I L. Schwartz 4.7, 5.3, 17.6, I, V R.T. Seeley 11.1.3, 28.9, IV G.E. Shi lov V J. Siciak V J.S. e Silva III M. Snyder I F. Sommen IV, V

v.

sou~ek

E.M. Stein II K. S~ein III R.F. Streater V K. Stromberg IV A. Sudberry II B.A. Taylor III P. Thullen II, III H.G. Tillmann 22.1, III, IV F. Tr~ves 19.1, 28.4 H. Triebel IV H.C. Tze I, II N. Vilenkin 11.1.3 V.S. Vladimirov IV D. Vogt IV G. Weiss II A.S. Wightman V E.B. Wilson I M.R. Wohlers IV Zalcman III Z. Zielesny IV L.

305

Adjoint function 13.12 algebra Clifford algebra 1.3, 33.1 Dirac algebra 1.7 H*-algebra 1.12 Pauli algebra 1.7 quaternion algebra 1.7 trace algebra 1.12 analytic function 8.4 analytic functional 22.15, 28.6 Aronszajn-Eergmann Theorem 24.3 asymptotic extension 17.2

Cauchy-Riemann operator 8.3 Cauchy-Schwarz inequality 7.5 convolution 4.6

Bergmann kernel 24.1, 29.7 Bessel's inequality 7.4 boundary value 25.5, 26.5 boundary value mapping 27.9 bounded set 2.7

Functional A-linear functional 2.3 bounded A-linear functional 2.6 fundamental solution 8.8 Fourier transform 30.2, 30.6, 30.8, 30.16, 31.11, 31.13 Fourier-Borel transform 32.16

Canonical extension 26.5 Cauchy's Integral Formula 9.6, 9.8 Cauchy kernel 8.8 Cauchy's Theorem 9.4 Cauchy transform of an A-measure 17.9 of a distribution with compact support 27.6 of an analytic functional 28.6 Cauchy-Kowalewski extension 14.5, 28.5, 33.16 Cauchy-Kowalewski product 14.11 Cauchy-Kowalewski Theorem 14.2, 28.4 306

Distributions 4.3, 5.2, 5.6, 6.2 dual algebraic dual 2.3 dual module 2.6 dual of M(r)(~;A) 22 bidual of M(r)(Q;A) 23 Entire function 12.3.5 exponential function 15.4, 15.5

Gelfand-Shilov Z-space

30.19

Hahn-Banach Theorem 2.11 harmonic harmonic function 8.4, 16.1 harmonic testfield 25.2 spherical harmonic 11.1.1 Hartogs-Rosenthal Theorem 18.17 Helgason's entire functiorials 32.22 hermitian conjugation 1.9 Hilbert A-module 7.2 Indicatrix of

Fantappi~

22.2

inductive limit 2.15 inner product on A 1.11 on L 2 (r)(H;A;~) 7.14 on ML 2 (r)(~;A) 24.5 on ML 2 (r)(aBR;A) 24.13 on a(r)(sm;A) 28.10 on L2 (r)(sm;A) 29.2 inversion 1.8 involution 1.10 Koebe's Theorem

10.5

Laplace operator 8.4, 8.7 Laplace transform 31.2, 31.8, 31.13 Laplace-Beltrami operator 8.7 Laurent integral 32.22 Laurent series 12.2 Legendre polynomial 11.1.2 Liouville's Theorem 12.3.11 Maximum Modulus Theorem 9.9 Mean Value Theorem 9.7 measure 3.2 meromorphic function 12.3.3 Mittag-Leffler's Theorem 12.3.6, 21.5 module unitary left A-module 2.2 Hilbert A-module 7.2 monogenic monogenic function 8.3 (surface) inner spherical monogenic 11.2.1, 16.4 (surface) outer spherical monogenic 12.1.6, 12.1.9 monogenic point 12.3.1 Montel Theorem 16.3 Morera's Theorem 10.4

Norm on A 1.11 normal normal convergence 11.3.1 x0 -normal 14.3 xi-normal 20.2 normal w.r.t. the origin 20.2 Optimal w.r.t. absolute convergence

33.6

p-vector

1.4 Painlev~ Theorem 10.6, 28.3 Paley-Wiener-Schwartz Theorem 30.19 Poincar~ Theorem 21.3 primitive 20.6 projection operator 7.13, 29.5 Radon transform 33.10 reflection operator 30.7 regular at infinity 17.2, 17.3 reproducing kernel 24.2 residue 12.4.1 Residue Theorem 12.4.3 Riesz representation theorem 3.3, 7.6 Rodriguez Formula 11.1.2 Runge Approximation Theorem 18.4, 18.7, 18.9, 18.12, 18.20 Scalar part 1.11 singular point 12.3 slow growth (functions) 5.5 spherical Cauchy-Riemann operator 8.7 spherical transform 13.3, 13.6, 13.9 strong topology 2.7 system of seminorms 2.4, 2.5, 2.15 Szegtl kernel 24.17, 29.6 Taylor integral 32.22 Taylor series 11.3.5, 11.5 tensor product of functions and distributions 4.6 307

trace

1.12

Weak slow growth 5.5 weak topology 2.7 Weierstrass Theorem 9.11

308