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