Microlocal Analysis and Complex Fourier Analysis

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Author: Kawai T. | Fujita K. | Kyoto Daigaku Suri Kaiseki Kenkyujo (Corporate Author). (eds.)

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Microiocai Analysis and Complex Fourier Analysis

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Microlocal Analysis and Complex Fourier Analysis

Editors

Takahiro Kawai Kyoto University, Japan

Keiko Fujita Saga University, Japan

| 8 j World Scientific "•

New Jersev Jersey • London • Sine Singapore • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: Suite 202,1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

MICROLOCAL ANALYSIS AND COMPLEX FOURIER ANALYSIS Copyright © 2002 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-238-161-9

This book is printed on acid-free paper.

Printed in Singapore by Uto-Print

PREFACE This is a collection of articles on microlocal analysis, complex Fourier analysis and related topics. The authors, (M. Morimoto excepted), and we, the editors, dedicate this volume to Professor Mitsuo MORIMOTO, who has made substantial contributions to these subjects. Some articles in this volume were read at the conference "Prospect of Generalized Functions", held from November 27 through November 30, 2001 at Research Institute for Mathematical Sciences, Kyoto University, which was organized by K. Fujita to celebrate Professor M. Morimoto's sixtieth birthday. May 2002 Takahiro Kawai Research Institute for Mathematical Sciences, Kyoto University Keiko Fujita Faculty of Culture and Education, Saga University

v

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CONTENTS

Preface

v

Vanishing of Stokes Curves T. Aoki, T. Koike and Y. Takei

1

Parabolic Equations with Singularity on the Boundary C.P. Arceo, J.M.L. Escaner IV, M. Otani and P.W. Sy

23

Residues: Analysis or Algebra? C.A. Berenstein

36

Moment Conditions for Pompeiu Problem Extended to General Radial Surfaces D.C. Chang and W. Eby

44

Heat Equation via Generalized Functions S.-Y. Chung

67

Bergman Transformation for Analytic Functionals on Some Balls K. Fujita

81

Explicit Construction of Fourier Hyperfunctions Supported at Infinity A. Kaneko

99

On Infra-Red Singularities Associated with QC Photons T. Kawai and H.P. Stapp

115

On the Linear Hull of Exponentials in Cn and Applications to Convolution Equations L.H. Khoi

135

Hyperfunctions and Kernel Method D. Kim

149

VII

VIII

Generalized Fourier Transformations: The Work of Bochner and Carleman Viewed in the Light of the Theories of Schwartz and Sato CO. Kiselman

166

The Effect of New Stokes Curves in the Exact Steepest Descent Method T. Koike and Y. Takei

186

Fourier's Hyperfunctions and Heaviside's Pseudo-Differential Operators H. Komatsu

200

Geometric Aspects of Large Deviations for Random Walks on a Crystal Lattice M. Kotani and T. Sunada

215

Boehmians on the Sphere and Zonal Spherical Functions M. Morimoto

224

A New Lax Pair for the Sixth Painleve Equation Associated with sb(8) M. Noumi and Y. Yamada

238

On a Generalization of the Laurent Expansion Y. Saburi

253

Domains of Convergence of Laplace Series J. Siciak

261

On the Singularites of Solutions of Nonlinear Partial Differential Equations in the Complex Domain H. Tahara

273

Exponential Polynomials and the Fourier-Borel Transforms of Algebraic Local Cohomology Classes S. Tajima

284

IX

The Reproducing Kernels of the Space of Harmonic Polynomials in the Case of Real Rank 1 R. Wada and Y. Agaoka

297

On the Lame Series Representation of Analytic Hyperfunctions on a Two-Dimensional Complex Manifold A.I. Zayed

317

VANISHING OF STOKES CURVES TAKASHI AOKI Department of Mathematics and Physics, The School of Science and Technology, Kinki University, Higashi-Osaka, 577-8502 Japan E-mail: [email protected] TATSUYA KOIKE Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto, 606-8502 Japan E-mail: [email protected] YOSHITSUGU TAKEI Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502 Japan E-mail: [email protected]

1

Introduction

No algorithm of describing the complete Stokes geometry is known for higher order ordinary differential equations with a large parameter. However, by virtue of the exact steepest descent method proposed in [3] it now becomes possible to determine whether a Stokes phenomenon for Borel resummed WKB solutions actually occurs at a given point or not with the aid of a computer. In this report, using the exact steepest descent method mainly, we discuss an interesting phenomenon that a Stokes curve emanating from an ordinary turning point may vanish (i.e., a Stokes phenomenon for Borel resummed WKB solutions no longer occurs on it) after crossing other Stokes curves. In the case of second order equations such a phenomenon of vanishing of a Stokes curve never happens (cf. [11]). In contrast with the second order case, since Stokes curves with different types often intersect, it happens to higher order equations that a Stokes curve may vanish or the structure of Stokes phenomena for WKB solutions occurring on it may change after it crosses other Stokes curves. For example, as is explained in [1, Section 2] (cf. [4, Remark 2.1] also), at a crossing point of three Stokes curves of type (j < k), (k < I) and (j < /) (see, e.g., [1] for the terminology frequently used in the exact WKB analysis), the structure of Stokes phenomena for WKB solutions occurring on the Stokes curve of type (j < I) ("non-adjacent Stokes

1

2

curve") changes in general. In particular, when turning points of an equation or an operator P in question are all simple, that is, when the Borel transform PB of P is with simple characteristics, (note that an operator with simple discriminant in the sense of [1, Definition 1.1] satisfies this condition,) the nonadjacent Stokes curve is expected to vanish after passing through a crossing point of three Stokes curves unless any degeneracy happens to P. (We will discuss a kind of non-apparent degeneracy in Section 3.) In our opinion the validity of this expectation is closely related to the existence of an algorithm of describing the complete Stokes geometry for higher order equations. The purpose of this report is to examine if a phenomenon of vanishing of a Stokes curve actually occurs to a Stokes curve emanating from an ordinary turning point (we sometimes call such a Stokes curve an "ordinary Stokes curve" to distinguish it from a new Stokes curve) by studying several concrete examples mainly with the aid of a computer. The report is organized as follows: In Section 2 we discuss an example whose turning points are all simple. For this example it can be confirmed that vanishing of an ordinary Stokes curve really occurs. Next we investigate in Section 3 the case where all turning points are simple but there is a kind of degeneracy that an ordinary turning point and a virtual turning point (which is new terminology for a new turning point of [1]) are merged. In this case vanishing of an ordinary Stokes curve does not occur; instead the structure of Stokes phenomena for WKB solutions on a non-adjacent Stokes curve (more precisely, the value of the Stokes coefficient describing Stokes phenomena) changes at a crossing point of Stokes curves. Section 4 is devoted to the study of an equation with double turning points, which appears in connection with the problem of non-adiabatic transition probabilities in quantum mechanics. In this case also we can conclude that vanishing of an ordinary Stokes curve does not occur in general. In Section 5 we give a summary and conclusions. 2

A n equation with simple turning points

In this section we study the following equation with a large parameter 77 > 0: P*l> = (js

+ 6(1 + x)r,2 j - + (2 - Aix)rA

ip = 0.

(1)

This is an equation obtained by putting A = it] in [1, Example 2.5]. The configuration and the type of Stokes curves of (1) is shown in Figure 1, where a wiggly line designates a cut which we have placed to define a characteristic root of (1) as single-valued analytic function. There are three turning points ao, a\ and 0,2 all of which are simple turning points. As is clear from Figure 1,

Figure 1. Stokes curves of (1).

Figure 2. Magnification of Figure 1 near the crossing point A.

a Stokes curve emanating from ao intersects another Stokes curve emanating from ai at an ordered crossing point B. Hence we have to add a new Stokes curve passing through B, which is also included in Figure 1. (In Figure 1 and

4

subsequent figures describing Stokes curves as well, a virtual turning point is designated by a small dot like an ordinary turning point and a broken line indicates that no Stokes phenomenon for WKB solutions occurs on the portion.) Note that the new Stokes curve thus added intersects with two ordinary Stokes curves again at A and a Stokes curve emanating from ao is non-adjacent there. Thus we can expect that the Stokes curve emanating from a0 may vanish after passing through the crossing point A. In what follows we will check this expectation by using two different methods. 2.1

Verification by the steepest descent method

As (1) is a Laplace type equation and the employment of the Laplace transformation ip = f exp(77££)Vid£ provides us with an integral representation of solutions of the form tp(x) = J expfo/Oc, 0)^ZT^

(2)

where

the ordinary steepest descent method is applicable to (1) (cf. [9], [10], [7]). We first use the steepest descent method to examine if the Stokes curve emanating from a0 vanishes after passing through A. Figure 3 illustrates the configuration of steepest descent paths of R e / (x, f) passing through saddle points of f(x,£) near the Stokes curve in question below the crossing point A, more precisely at x = XQ and x = X\ whose location is indicated in Figure 2. (In Figure 3 and subsequent Figures 4 and (a)

,

(b)

Figure 3. Steepest descent paths at x = xo (a) and x = x\ (b).

5 a small dot designates a saddle point and a larger dot designates a singular

5

point of the integrand of (2).) Figure 3 clearly shows that the configuration changes when one crosses the Stokes curve below A. This change of the configuration implies that a Stokes phenomenon for Borel resummed WKB solutions really occurs on the Stokes curve (cf. [7, Proposition 3]). On the other hand, above the crossing point A the configuration becomes a different one as is illustrated in Figure 4 and its magnification Figure 5 near the unique singular point in Figure 4. We can read from Figs. 4 and 5 the fact that a steepest descent path passing through the lowest saddle point flows into a singular point in both Figs. 4(a) and 4(b) (or 5(a) and 5(b)) and no change of the configuration occurs when one crosses the Stokes curve above A. Hence we can conclude that the Stokes curve in question vanishes after passing through the crossing point A. (a)

(b)

Figure 4. Steepest descent paths at x = xi (a) and x = X3 (b).

(a)

(b)

Figure 5. Magnification of Figure 4 near the singular point.

2.2

Verification by using the connection formula

Next we try to determine the connection formula on the Stokes curve in question by employing the reasoning used in [5] to confirm that it vanishes after passing through A.

6 Let ipj (j = 0,1) be WKB solutions of (1) with the good normalization in the sense that they satisfy the Airy type connection formula near the simple turning point a\ (cf. [1, Theorem 1.8]). We first consider the situation near the ordered crossing point B (Figure 6). It follows from the above normalization 7oi 1<0

7i2 7o2 2 < 1 2 <0

7i2 2<1

7(ii 1<0

Figure 6. Schematic illustration of Stokes curves near B.

that ipj (j — 0,1,2) should satisfy the following connection formula when crossing a Stokes curve 7 01 in the direction indicated by an arrow in Figure 6: i>o>—>• V'o + # i ,

^ I '—• 4>i, i>2'—> i>2-

(4)

Similarly, since 7 ^ is a Stokes curve emanating from the simple turning point ao, we can choose an appropriate normalization of ip2 so that ipj (j = 0,1,2) should satisfy ipo 1—• tpo,

4>i 1—y ipi + # 2 ,

1P2 <—> 4>2-

(5)

across 7 1 2 . On the other hand, the explicit form of the connection formula on 702 is different from that on 7 ^ as 7 ^ is non-adjacent at B. On the side 7 ^ the connection formula is trivial (i.e., no Stokes phenomenon occurs) since there exists a virtual turning point on this side (cf. [1, p. 77], [11, p. 244]), while the connection formula on 7 ^ should be of the form ^0 '

>4>0 + Clp2,

1pl> >1pl, V>2 '

>i>2

(6)

with some constant c. T h e constant c in (6) can be determined explicitly by the reasoning used in [5] in t h e following manner: We consider the analytic

7

continuation of ipo from Region I to Region II. If we continue it via the right side ("+" side) of B, we find by (4), (5) and (6) that ip0 should become ipo + iipi + (c — 1)^2, while Vo continued via the left side ("—" side) of B should be ipo + iipi i n Region II. Since these two resulting expressions must coincide (as the crossing point B is a regular point of (1)), we obtain c = 1. Having this result in mind, we next consider the situation near the crossing point A (Figure 7). It follows from our normalization of ipj (j = 0,1) that we

7oi 0 <1

7i2 2 <1

7o2

7i+2

7o+i

2<0

2< 1

0< 1

7o2 2<0

Figure 7. Schematic illustration of Stokes curves near A.

have t h e following connection formula on 7 0 1 : •00 '

> IpO,

Ipl '

>i>l+ # 0 ,

i>2 !

> i>2-

(7)

Furthermore, as JQ2 in Figure 7 is the same Stokes curve with 7 ^ in Figure 6 and (6) holds with c = 1 there, we find t h a t the connection formula IpO I

>1p0+

1p2,

Ipl'

>tpl,

1p2< • 1p2

(8)

holds on 7 ^ . Similarly, the same formula with (5) holds on the side j ^ 2 of the non-adjacent Stokes curve 7 ^ . Then the connection formula IpO '

> tpO,

Ipl * > Ipl + CTp2,

tp2 I

> 1p2.

(9)

on the opposite side 712 can be determined by the same reasoning as above. As a matter of fact, we can deduce c = 0 from the coincidence of the two possible analytic continuations of xp\ from Region I to Region II. We have thus verified that the Stokes curve in question, i.e., 7 ^ really vanishes.

8 3

A degenerate equation with simple turning points

In a similar manner to the preceding section we study the following concrete equation whose turning points are all simple in this section: / d3

, , d

Figure 8 shows the configuration and the type of Stokes curves of (10). 2<() 1
Figure 8. Stokes curves of (10). Equation (10) has two crossing points, which are denoted by A and B in Figure 8, of three ordinary Stokes curves. In what follows we take one of the crossing points (say, A) and examine whether vanishing of the non-adjacent Stokes curve (i.e., a Stokes curve emanating from ao) occurs there or not. 3.1

Verification by the exact steepest descent method

First we apply the exact steepest descent method to (10). That is, consider the Laplace transform of (10) d2

d

take a WKB solution r/>± = exp J {-r)x±(£) -I ) df of it, where x±(£) is a root of the equation £ 3 + x2£ + (2 — i) = 0 (i.e., the characteristic polynomial

9 0<1

2<1 0<2

0<2 2<1 0<1 Figure 9. Magnification of Figure 8 near the crossing point A.

of the original equation (10)) with respect to x, and investigate the exact steepest descent paths of its inverse Laplace transform

J'exp(T,xZ)4±dZ

= jexp

J n Ix£ - J

x±(t)dA

+ • • • I d£.

(12)

Figures 10 and 11 illustrate the configuration of exact steepest descent paths of (12) (and of Stokes curves of (11) which are necessary to draw exact steepest descent paths) at x = xo, • • • ,xs whose location is indicated in Figure 9. (In Figures 10, 11 and subsequent figures describing exact steepest descent paths a solid line and a small dot respectively designate a steepest descent path and a saddle point, while a broken line and a small circle respectively designate a Stokes curve and a turning point for Pip = 0.) In Figure 10 (i.e., across the Stokes curve in question above the crossing point A) we can observe a change of the configuration caused by the degeneracy that a steepest descent path a, which is obtained by repeated bifurcation from an ordinary steepest descent path passing through a saddle point £i, hits another saddle point £o- On the other hand, in Figure 11 (i.e., across the Stokes curve below A) another change of the configuration can be observed besides the above change; another bifurcated steepest descent path a hits £o simultaneously. This implies that the structure of Stokes phenomena for Borel resummed WKB solutions on the non-adjacent Stokes curve in question changes at the crossing point A, but vanishing of a Stokes curve does not occur there, although all turning points of (10) are simple.

10 (a)

(b)

Figure 10. Exact steepest descent paths at x = xo (a) and x = x\ (b). (a)

(b)

Figure 11. Exact steepest descent paths at x = £2 (a) and x = X3 (b).

3.2

Discussion by adding a small perturbation

To clarify the reason why vanishing of a Stokes curve does not occur for (10) at A, we add the following small perturbation to (10): P(tP = ( —

+ (x2 + ie)r]2— + (2 - i)r)3 j i> = 0

(e > 0 : small).

(13)

Then the configuration of Stokes curves is slightly perturbed as is shown in Figure 12. In Figure 12 the value of e is taken to be 0.1 and some relevant new Stokes curves are also included to resolve ordered crossing points. As is visualized in Figure 12 and subsequent Figure 13, this perturbation splits the Stokes curve for e = 0 in question emanating from a 0 into two Stokes

11

Figure 12. Stokes curves of (13).

curves when e > 0; one is an ordinary Stokes curve emanating from oo and the other is a new Stokes curve emanating from a virtual turning point b. (Both turning points a0 and b approach ao as e —> 0.) Consequently when e > 0 three crossing points Aj (j = 0,1,2) appear near the crossing point A for e = 0. Among them only A2 is an ordered crossing point of (13). We now examine whether a Stokes phenomenon occurs or not near these crossing points by using the exact steepest descent method.

0<1 0<1

Figure 13. Magnification of Figure 12 near {.Aj}j=o,i,2-

Let us take six points XQ, •••, X5 near the crossing points (cf. Figure 13) and draw the configuration of exact steepest descent paths at these points

12

Xj (j = 0 , . . . ,5); the resulting figures are Figures 14 and 15. (a)

(b)

There is no

(c)

Figure 14. Exact steepest descent paths at x = xo (a), x — x\ (b) and x = X2 (c).

(a)

(b)

(c)

Figure 15. Exact steepest descent paths at x = X3 (a), x = X4 (b) and x = X5 (c).

topological difference between the configuration of Fig. 14(a) and that of Fig. 14(b), while between Fig. 14(b) and Fig. 14(c) we can observe the same change of the configuration with that in Figure 10. The same change can be found between Fig. 15(b) and Fig. 15(c) also, and between Fig. 15(a) and Fig. 15(b) there occurs another change of the configuration which is the same with one of the changes observed in Figure 11. Comparison of this observation with that in the preceding subsection leads to the following conclusion: The Stokes curve for e = 0 in question has been split into two Stokes curves when e > 0. Between them an ordinary Stokes curve emanating from an passes through no ordered crossing point and the structure of the Stokes phenomenon for Borel resummed WKB solutions on

13

it is unchanged near the crossing points in question. On the other hand, a new Stokes curve emanating from b passes through an ordered crossing point Ai and there vanishing of a Stokes curve really occurs. The phenomenon observed at the crossing point A for e = 0 can be considered as confluence of these two phenomena. That is, the reason why vanishing of a Stokes curve does not occur for e = 0 is that (10) has a degeneracy that a turning point and a virtual turning point are merged. 4

A n equation with double turning points

To see what occurs when we drop the assumption that all turning points are simple we discuss the following 3 x 3 system in this section: d

'Po(x) 0 0 \ / 0 a 0 i a02' 0 pi{x) 0 + 7 T 1 / 2 55T 0 Q12 0 0 P2{x)J \ao2cFnr 0

i

ax

,

(14)

where po(x) = po = 1 + i,

pi(x) = I 1 -

p2(x)

=[2+-i)x.

(15)

System (14) has its origin in a non-adiabatic level crossing problem in quantum mechanics (in that context each pj (x) is assumed to be real) and is equivalent to a third-order single equation of the form Pip

d3

..

.

d2

d - (p0p! + pip2 + p2pa + • • •) r) -£-i

(16) (P0P1P2 + • • •) V ) ^ = 0

(where • • • denotes lower order terms with respect to 7?). Note that the turning points of an equation of this type are all double. The configuration of ordinary Stokes curves and relevant new Stokes curves of (14) (or equivalently of (16)) is described in Figure 16. At a crossing point A in Figure 16 an ordinary Stokes curve emanating from a turning point 002 is non-adjacent. In what follows we examine whether vanishing of a Stokes curve occurs at A with this ordinary Stokes curve. 4-1

Verification by the exact steepest descent method

We first apply the exact steepest descent method. In this case, since po(x) = po is a constant, there exists a characteristic root being independent of x.

14 2<0 KO

w

2<1 0<1

KO 1<2 0<2

Figure 16. Stokes curves of (14) (or equivalently (16)).

Hence we have to take into account exact steepest descent paths emanating from a singular point corresponding to a constant characteristic root together with those emanating from saddle points (cf. [6, Section 4]). Figures 18 and 19 describe the configuration of all such exact steepest descent p a t h s at x — XQ, x — x\ (i.e., below A) and at x = x?, x — xz (i.e., above A) respectively, where Xj (j = 0 , . . . , 3) is a point marked in Figure 17.

l<0

KO 2<12<0 Figure 17. Magnification of Figure 16 near the crossing point A.

15 (a)

(b)

Figure 18. Exact steepest descent paths at x = xo (a) and x = x\ (b).

(a)

(b)

Figure 19. Exact steepest descent paths at x = X2 (a) and x = x$ (b).

A change of the configuration can be observed in Figure 18; a steepest descent path a+ emanating from £o, a singular point corresponding to a constant characteristic root —ipo, hits a saddle point £2- The same change occurs also in Figure 19, but this time it is "two-ply", that is, not only a+ but also another steepest descent path, which a steepest descent path er_ emanating from £0 bifurcates at its crossing point with a Stokes curve passing through £0, hits £2 simultaneously. (The crossing point of
16

4-2

Discussion by adding a small perturbation

We next check the conclusion obtained in the preceding subsection by adding a small perturbation to (14) or (16) and reducing the problem to that for an equation without any multiple turning point. As a perturbed equation of (16) we consider the following here: PtiP= (P + e(i-l)n3)xP

= Q.

(17)

The configuration of ordinary Stokes curves and relevant new Stokes curves of (17) for e = 0.008 is given in Figure 20. The Stokes curve in question

Figure 20. Stokes curves of (17).

for e = 0 is split into two curves when e > 0; one is an ordinary Stokes curve emanating from an ordinary turning point <5o2 and the other is a new Stokes curve emanating from a virtual turning point b. The crossing point A accordingly splits into two crossing points AQ and Ai (cf. Figure 21). Taking six points xo, ... ,X5 near these crossing points (cf. Figure 21) and drawing the configuration of exact steepest descent paths at these points Xj (j = 0 , . . . , 5) with the aid of a computer, we obtain Figures 22 and 23. In Figure 22, a change occurs with a steepest descent path emanating from a saddle point £o between Fig. 22(a) and 22(b), but there is no change of the configuration between Fig. 22(b) and 22(c). (See Fig. 24 below for the configuration of exact steepest descent paths near £o; the configuration is

17 \x& £5

Figure 21. Magnification of Figure 20 near {^4j}j=o,i(a)

(b)

Figure 22. Exact steepest descent paths at x = xo (a), x = xi (b) and x = X2 (c). (a)

,\s^

(b)

A N X

(c)

Figure 23. Exact steepest descent paths at x = X3 (a), x = X4 (b) and x = xs (c).

not changed near | 0 between Fig. 22 and Fig. 23.) On the other hand, three figures in Figure 23 are mutually different; a steepest descent path cr0 ema-

18

(a)

(b)

.'70.-

,'7o„

Figure 24. Magnification of Figures 23 (a) and (b).

nating from £0 and another steepest descent path a^, which is obtained from a;, by bifurcation (cf. Figure 24; Ob itself is a bifurcated steepest descent path obtained from £r0), simultaneously hit a saddle point £2 between Figs. 23(a) and 23(b), and a steepest descent path which a steepest descent pathCTOemanating from £0 bifurcates hits £2 between Figs. 23(b) and 23(c). We should here notice that the change between Figs. 23(a) and 23(b) is an "apparent" one due to the following delicate cancellation: Locally near the saddle point £0 we consider an inverse Laplace transform J explrjxQip+dt, of an appropriate WKB solution t/>+. However, when one crosses a Stokes curve 70 in Figure 24, the Borel resummed WKB solution V>+ becomes i/>+ + ctp- with some constant c ("Stokes phenomenon"). Similarly, ^_ becomes tp- + cip+ with another constant c when one crosses a Stokes curve 71. Taking account of these connection formulas and of the configuration of steepest descent paths and Stokes curves, we find that globally we should consider

/ •> On

i

exp(r]x£)ip+dt; + c I exp(r)x£)ip-d£ + cc

/

exp(rjx£)ip+df;.

(18)

This is the core idea of the exact steepest descent method proposed in [3]. Since both Stokes curves 70 and 71 emanate from the same simple turning point in this case, we can normalize WKB solutions %j)± so that c = c = i is satisfied. Hence the integral along the steepest descent path a^ in (18) is cancelled. (Note that abb overlies GQ thanks to the same reasoning as in the preceding subsection.) We have thus verified that no Stokes phenomenon occurs between Figs. 23(a) and 23(b).

19 As a conclusion we can claim that when e > 0 vanishing of a Stokes curve occurs for both of the two Stokes curves that the Stokes curve in question for e = 0 is split into. Furthermore the Stokes phenomena on the Stokes curve for e = 0 can be considered as confluence of the Stokes phenomena on these two Stokes curves for e > 0. The reason why the structure of Stokes phenomena for WKB solutions changes but does not vanish at the crossing point A for e = 0 is that it has these two different origins for e > 0. 4-3

Verification by using the connection formula

Finally we further confirm the conclusion obtained in Section 4.1 by computing the connection formulas on the Stokes curve in question explicitly. We employ the same argument as in Section 2.2. Let us first consider the situation near the crossing point B (Figure 25). If we assume that ipj 7i2 7o2 2< 1 2< 0

7oi 0 <1

7oi 0< 1

7o2 2 <0

Figure 25. Schematic illustration of Stokes curves near B.

(j = 0,1,2) should satisfy the connection formula. ^0 '

> IpO,

^1

> i>l + CoiV'O,

i>2 '

> i>2

(19)

with some constant coi across 7 ^ and further assume that they satisfy •00 ' — > •tpo + c02ij>2,

ipi 1 — H ^ i ,

1P2'—>• ^ 2

(20)

with another constant C02 across 7 ^ , the same reasoning as in Section 2.2

20 verifies that they should satisfy Vo '

> Vu,

Vl '

> Vl + CoiC02^2,

V2 '

• V2

(21)

across 7 ^ . (Note that the connection formula is trivial on the side 7 ^ since there exists a virtual turning point on this side.) Then, again by the same reasoning near the crossing point A, we find that V>j U = 0,1,2) should satisfy the connection formula V>o '—> V>o + ctl>2, V>i '—•V'l.

^'—>• V>2

(22)

across 7 ^ with c = c 0 2+C01C01C02,

(23)

assuming that they should satisfy V>o '—• Vo + coi^i,

V"i '—'•V'l,

V"2 '—• V2

(24)

with a constant c0i across 7 ^ (cf. Figure 26). Now, since am is a double 702

7l+2

2 <0

2 <1

7l2

702

2<1

2<0

Figure 26. Schematic illustration of Stokes curves near A. turning point and we have a local reduction of (14) to a Weber type equation near a double turning point ([4, Appendix]), the connection formula near a 0 i should be described by that of the Weber equation. In particular, we can choose appropriate normalization of tpj so that c0i and c0i are respectively

21

given by C01 =

r ( - . + i/2)e'

+

-»

^ = WTW)+'"

(25)

(cf. [8, Section 3]). Here K is of the form K = a|a 0 i| 2 + b with some constants o ^ O and 6, and • • • denotes lower order terms with respect to r\. The other constant C02 has a similar expression also. Hence we obtain C = C 0 2 ( l +C01C01)

=

°»('-r(-

— -co2(e

H

+

ifflr(» + i/ ! o''" + - )

(26>

j,

which is equal neither to co2 nor to 0 for generic values of Q 0 I and «02 • We have thus verified that the value of the Stokes coefficient describing Stokes phenomena on the non-adjacent ordinary Stokes curve in question changes at A, but vanishing of a Stokes curve does not occur there in general. 5

Conclusions

As we have seen in Section 2, in the case of higher order ordinary differential equations an ordinary Stokes curve may vanish after crossing other Stokes curves. Such a phenomenon of vanishing of a Stokes curve is expected to occur quite generically for equations whose turning points are all simple. In contrast with the case of equations with simple turning points, as the examples in Sections 3 and 4 show, the Stokes coefficient describing Stokes phenomena for WKB solutions only changes its value (not necessarily vanishes) at a crossing point of Stokes curves when a double turning point exists or when there is a degeneracy that an ordinary turning point and a virtual turning point are merged: The confluence of turning points causes vanishing of a Stokes curve not to be observed in these cases. Both in the example of Section 2 and in the perturbed equations discussed in Sections 3.2 and 4.2 vanishing of a Stokes curve certainly occurs. As these examples are suggesting, we can expect that a non-adjacent Stokes curve should vanish after passing through a crossing point of three Stokes curves for an equation with simple turning points if any two (ordinary or virtual) turning points are not merged or connected by a Stokes curve. In the exact WKB theory for higher order equations it is one of the important problems to prove this expectation rigorously by analyzing the Riemann sheet structure of Borel transformed WKB solutions.

22

Acknowledgments The authors would like to express their sincere gratitude to Professor T. Kawai for many valuable discussions with him. This work is supported in part by JSPS Grant-in-Aid (No. 11440042 and No. 12640195 for T.A., No. 13740096 for T.K., and No. 11440042 and No. 13640167 for Y.T.). References [I] T. Aoki, T. Kawai and Y. Takei: New turning points in the exact WKB analysis for higher order ordinary differential equations. Analyse algebrique des perturbations singulieres, I; Methodes resurgentes, Hermann, 1994, pp. 69-84. [2] : On the exact WKB analysis for the third order ordinary differential equations with a large parameter. Asian J. Math., 2(1998), 625-640. [3] : On the exact steepest descent method — a new method for the description of Stokes curves. J. Math. Phys., 42(2001), 3691-3713. [4] : Exact WKB analysis of non-adiabatic transition probabilities for three levels. RIMS preprint (No. 1331), to appear in J. Phys. A. [5] H. L. Berk, W. M. Nevins and K. V. Roberts: New Stokes' line in WKB theory. J. Math. Phys., 23(1982), 988-1002. [6] T. Koike and Y. Takei: The effect of new Stokes curves in the exact steepest descent method. In this volume. [7] Y. Takei: Integral representation for ordinary differential equations of Laplace type and exact WKB analysis. RIMS Kokyuroku, No. 1168, 2000, pp. 80-92. [8] : An explicit description of the connection formula for the first Painleve equation. Toward the Exact WKB Analysis of Differential Equations, Linear or Non-Linear, Kyoto Univ. Press, 2000, pp. 271-296. [9] K. Uchiyama: On examples of Voros analysis of complex WKB theory. Analyse algebrique des perturbations singulieres, I, Methodes resurgentes, Hermann, 1994, pp. 104-109. [10] : Graphical illustration of Stokes phenomenon of integrals with saddles. Toward the Exact WKB Analysis of Differential Equations, Linear or Non-Linear, Kyoto Univ. Press, 2000, pp. 87-96. [II] A. Voros: The return of the quartic oscillator — The complex WKB method. Ann. Inst. Henri Poincare, 39(1983), 211-338.

PARABOLIC EQUATIONS WITH SINGULARITY ON THE BOUNDARY CARLENE P. ARCEO, JOSE MA. L, ESCANER IV Department of Mathematics, College of Science, University of the Philippines Diliman, Quezon City, Philippines Email: [email protected]; [email protected] MITSUHARU OTANI Department of Applied Physics, School of Science and Engineering, University 3-4-1, Ohkubo, Shinjyuuku, Tokyo 169-8555, Japan Email: [email protected]

Waseda

POLLY W. SY Department of Mathematics, College of Science, University of the Philippines Diliman, Quezon City, Philippines Email: [email protected] The existence and blowing up of solutions are discussed for the parabolic equation ut = ApU + ' . .. (1-1*1)" in the unit ball B = {x € HN : \x\ < 1} under the Dirichlet boundary condition ""\dB = 0) P > 2 and a > 0. The methodologies employed here include some approximation procedures, techniques used by Hashimoto and Otani ([2],[3]), and techniques used by Pujii and Ohta [1].

§1 Introduction In this paper, we study the existence and blowup problems of the following nonlinear parabolic equation with singularity. ut = Apu+ J l | . (E)

) a

, (M) e B x [0,T],

u(x,Q) = uo(x)

, x € B,

u{x,t) = 0

, (x,t) edBx

[O.T],

where p > 2, T > 0, a > 0, Apu = div(|Vu| p - 2 Vu) and B is the unit ball { i € R N : |z| < 1} in HN.

23

24

Several authors (see [1], [6], [7], [10] and [15]) have studied the existence and nonexistence of global solutions and blowing up of solutions of the following initial boundary value problem.

{

ut = Apu + \u\q~2u u(x,t)=0

,xeCl,t>0, ,x£dn,t>0,

u(x,0)=uo(x) , x £ n, where q,p > 2, and fi is a bounded domain in R ^ with smooth boundary dfl. The following results on (E)i are known: (I) When p > q, (E)i has a global solution for any u 0 6 WQ'P(Q); (II) When p < q, (E)i has a global solution if u0 e W01,p(fi) is sufficiently small and the solution of (E) x blows up in finite time if uo £ WQ'P(CI) is large enough. (III) When p = q, let X1 = inf{||Vu||£,/IMIk ! « e W o lj, (fi)\{0}}. If Ai > 1, then (E)i has a global solution for any u 0 € WQ'P(Q). Moreover, if Ai < 1, Fujii and Ohta [1] derived sufficient conditions on blowing-up of solutions of (E)i and studied the asymptotic behavior of solutions of (E)i. In 1997, S. Hashimoto and Otani [2] introduced a singularity into the elliptic equation and considered the following elliptic equation with singularity. -Au(x) = (E) a

u{x) > 0 ^ u(x) = 0

(1

_ u l 0 " / ? ( g ) . x e B, ,x€B, ,x€ dB,

where a > 0, f3 > 1, B = {x £ HN : \x\ < 1}, and K(-) is a given nonnegative continuous function on B. They established some results on the existence and nonexistence of positive classical solutions by employing methods from the theory of partial differential equations based on the variational method. Our study of equation (E) is motivated by their study of (E) 2 . Our main purpose here is to investigate the existence and blowup of solutions of (E) for the three cases: (1) p = q > 2 ; (2) p > 2 and 1 < q < 2; (3) 2 < p < q. As in the study of Hashimoto and Otani ([2],[3]) of (E) 2 , we employ the methodologies which involve some techniques from the theory of nonlinear partial differential equations, such as variational and approximation methods to establish the existence of solutions for (E). We also find the conditions for blowing up of solutions for (E) by using techniques similar to those of Fujii and Ohta [1], Tsutsumi [15] and Otani [9] in their study of (E)i.

25 §2 M a i n R e s u l t s Consider the following nonlinear parabolic problem. \u\q-2u (E)

Ut

=

ApU

+

(l-|g|)° '

u(x,0) =u0(x) u(x,t) = 0

( M )

€

B

X

[

°'T]'

, x E B, ,xedBx[0,T}.

Here, p > 2, T > 0, a > 0, A p u = div(|Vu| p - 2 Vu) and B is the unit ball {arelR : |a:| < 1} in R . Moreover, we aim for radially symmetric solutions of (E), that is, u(x) = u(r) where r = \x\. Define X = {ue WQ'P(B);U(X) is radially symmetric} with norm Mlwtf* = /

^
= CN

lUrWI"^-1*,

J

where

CN= f

IdS.

JdB

In the case that q = p > 2, our results for the existence of radially symmetric solutions to (E) and blowing up of solutions of (E) are stated in the following three theorems. 2(r> — l ) 2 -|- p Theorem 1 (Existence) Let p = q > 2, 0 < a < — and 2p Np P < WTTJ T + I if p < N, p < oo if p > N. Then for any u0 e X, there exists a positive number T < oo such that (E) has a radially symmetric strong solution u satisfying u £ C([0,T]; WQ'P(B)) n Wl'2(0,T;L2(B)) and 11

e L2(0,T;L2(B)).

Furthermore, if the initial data u0 e -X"\{0} is ( 1 - \x\) nonnegative in B, then the solution u is nonnegative in B for t>0. Set a

E(u(t)) = \\Vu(t)\\lP - JB

Theorem 2 (Blowup) Let p = q > 2,

{

^

{ ) a

0 < a <

dx.

(1)

^P

and p < 2p

Np 2(N-p) + 1 if p < N, p < oc if p > N. Assume that UQ € X satisfies E(uo) < 0. Then, the radially symmetric strong solution of (E) blows up in a finite time T.

26

Let Ai =

.JB\ f

inf uew^(B)\{o}

JB

' W\p

, where 0 < a < p.

(1 - M) a

(2)

dx

Note that Ai gives the first eigenvalue of f

A

,

\u\P~2u

and as is stated in Theorem 2.3 of [3], Ai is positive, simple and finite. Theorem 3 (Blowup)

Let p - q > 2, 0 < a <

0

and p <

Np + 1 if p < N, p < oo if p > N. Let Ai < 1. Assume that u0 G 2(N-p) X\{0} is nonnegative in B. Then, the radially symmetric strong solution of (E) blows up in a finite time T. Remark : When p = q > 2 and Ai > 1, the existence of global solutions for (E) is assured by applying Theorem 3.2 of Otani's paper [9]. WThen Ai = 1, we need to rely on Lemma 1 with s = p and /? = a (given in the next section). In the case that p > 2 and 1 < q < 2, we state our result in the following theorem for the global existence of solutions for (E). Theorem 4 (Global Existence) Let p > 2, 1 < q < 2, and a < . When u(x,0) = u0(x) £ X, then there exists a radi2p ally symmetric strong solution u(x, t) of (E) satisfying u G W 1 ' 2 (O i r;L 2 (B))nC([Q,r];W^ P (B)) and J " 1 ' " ' , " , €

L2(0,T;L2(B)).

In particular, for the case p = 2, u satisfies u G L2(0,T;H2(B)). Moreover, if the initial data satisfies Uo(x) >0a.e. in B, then u(x, t) > 0 a.e. in B for t > 0. Let J(u) = -a(u) p

b(u), q

27 where an

a(u) = ||u||5f)

b(u) = f

|u n

|'

d

dx.

Here we define the so-called potential depth d by d=

inf supJ(Au). «ex\{o}A>o

Furthermore, if 0 < a < — 2p

(which is weaker than 0 < a o

\ o(u) q — p fa(u)q\ qp \ b(u)P

^r

>i^c-^>o. pq Thus d can be well defined as a positive number. Define the stable set by W = { u a ; 0 < J(Xu)
W* = {u £ X • J(u) < d, j(u) = a{u) - b(u) > 0}.

In the case that 2 p, and 2{N - p)

'^~ ' + P . Then for any 2p UQ G X, there exists a positive number To = To(||uo||jf) such that (E) has a radially symmetric strong solution u(x, t) satisfying 1 < q < oo for N < p. Let 0 < a <

(g

~

L2(B)) and ( j " ! ' ^ e

(#) u e C([0,To]; W^mnW^iO,To;

L2(0,TQ;L2(B)).

Moreover, if UQ > 0 a.e. in B , t/ien u(a;,t) > 0 a.e. in B for t > 0. T h e o r e m 6 (Global Existence) Let all assumptions on q,p and a in Theorem 5 be satisfied. Then for any UQ G W ; the solution given in Theorem 5 can be continued globally up to [0,T]. In particular, (E) has a global solution u satisfying (#) with To replaced by T.

§3 Proof of Theorems To prove Theorem 1, we begin with an important lemma. L e m m a 1 Let 0 < s < p* with p* = Np/(N — p) for p < N, p* = oo for sfp _ i) p > N, and let 0 < /? < — + 1. Then for any n > 0, there exists a P constant Cv independent of e satisfying

L

(1 + eL\x\)Pdx

^ 'JlMIx + C f H I k .

Vu € X and VE > 0.

(3)

Proof : For a sufficiently small S > 0, we divide the integration IB (i+iU-\x\)^dx i n t 0 t w 0 P a r t s i n Bl = {x E B\l - 8 < \x\ < 1} and Bg — {x G B;0 < \x\ < 1 - 6}. Then it suffices to apply the inequality |u(ar)| < C7||u||x(l - N ) " ^ V " e X in Bj (cf. lemma 1 of [2]) and use Gagliardo-Nirenberg's inequality in Bg (see [4]). D Sketch of Proof of Theorem 1 The proof is divided into two parts: Part I : (Existence of solutions of approximate equations) sider the approximating problem (us)t = A„u e +

(E)j

vi

uc-(z,0) = u%{x) ue(x,t) = 0

' g'

s

-t-g — |X|;

-e\ue\-'ue

,(x,t)€Bx

We first con-

[0,oo),

,x£B, ,xedBx[0,oo),

29 where p > 2, a > 0, and s > p. Here UQ e L°°(B) n X are radially symmetric functions such that UQ ~* u o in -^ as e -¥ 0. Since this approximating problem contains no singularity, the existence of a global strong solution e L o o (0,T;L o o (B))nW 1 ' 2 (0,T;L 2 (B))nL o o (0,T; W^'P(B))

ue(x,t)

(VT > 0)

is assured by Corollary 3 of [13]. Then, by standard argument, it is easy to show that this solution ue is also radially symmetric and unique (cf. the proof of Theorem 5). Part I I : (A priori bounds, convergence and nonnegativity) We can easily establish the following estimates. \u \p~2u (i) By multiplying (u£)t = Apu£ + , _ , * - e\u£\8~2u£ by u£ and then applying Lemma 1 with s = p and /3 = a, there exist a To = T0(p, ||uo||) > 0 and a constant C > 0 independent of e such that ||U£||Z,~(O,TO;Z,2(B)) + 'lUelll(0,To;X) - ^' = Apue + — ^ -~- - e\u£\3~2u£ by {1 + e- \x\)a {u£)t and2 then2 applying Lemma 1 with s = p and ft — a, we obtain ll(w£)t||L (oT0£ (B))+ SUP IheWllx < C for some constant C independent *e[o,7b] of e. (iii) By virtue of the conditions a < {2(p - l) 2 +p}/2p, p < {Np/2(N p)} + 1, we can apply Lemma 1 with s = 2(p - 1) and /? = 2a to get (ii)

I

I P

-

By multiplying (ue)t

^

r,*1 i i u

< C for some constant C.

Estimates (i)-(iii) imply that Apu£ is also bounded in L2(0,T0;L2(B)). From these estimates, the fact that ue satisfies (E)jr and by using standard argument, we can prove that {ue} converges to the desired solution u as e tends to 0 and that, if UQ is nonnegative, then u is nonnegative (see [9],[11]).

•

\u\p~2u Proof of Theorem 2 By multiplying ut = Apu + — —n— by u and ut, (l~\x\)a we first obtain dt\\u\\i, =-2E(u); BtEiu) =-p\\ut\\h. We next prepare the following lemma.

and

(4) (5)

30

Lemma 2 Assume thatu0 £ WQ'P(B), and letTmax be the maximal time for the strong solution u(t) of (E) to exist. Then, we have

dt

a e

- - <e[o>T-)-

¥m-°

Proof : This lemma is proven by the direct calculation with the aid of (4), (5) and the fact that d\\u(t)\\L2/dt < \\du{t)/dt\\L2. U Put k = -E{UO)I\\U0\\PL2 > 0, then Lemma 2 implies that -E(u(t))/\\u(t)\fL2 > -E(u0)/\\u0\\pL2 = k, whence follows -E(u(t)) > p k\\u(t)\\ L2 for all t e (0,Tmax). Therefore (4) gives &||u(t)|£ 2 > 2fc||u(*)||£2 v i E (0,Tmax), which assures that Tmax < +oo. D To prove Theorem 3, we need the following two propositions and one lemma. For the moment, we assume all the assumptions in Theorem 3. Proposition 1 Let uo £ -X"\{0} and let Tmax be the maximal existence time of the strong solution u(t) of (E). Put 7 , = Km [E(u(t))/\\u{t)\\pL2]. Then *

rl

max

the following facts hold true. (i) If Tmax < 00, then 7* < 0 and

lim [-7.(p-2)(T m o a -t)]p^||u(t)||L2 = l.

(6)

(ii) If Tmax = 00 and 7* > 0, then Um[7.(p-2)t]5±*||u(t)||L' = l . (7) Proof : It suffices to repeat the same proof for Theorem D of [1] with obvious modifications. • Proposition 2 Let UQ £ -^\{0} and let Tmax be the maximal existence time of the strong solution u(t) of (E). Then for any sequence {tj} satisfying tj - • Tmax, there exist a subsequence {tjk} of {tj} and w £ WQ'P such that

-Apw

Proof :

M*JJIIL* \w\p~2w

- ^

w inWo'p(B),

and

= 7.10 inS)'(B), N I L » = 1.

Define a rescaled function u(x, r) by

lluWllz2

Jo

(8) (9)

31

If Tmax < oo, then (i) of Proposition 1 assures that ||u(i)||£7 2 = 0({Tmax t)~l). Hence T(Tmax) = oo. As for the case 7* > 0, we can deduce that Tmax = 00 and ||u(£)||£l = 0 ( £ - 1 ) near t = 00, which implies that r(oo) = 00. (See Remark 2.1 of [1].) Thus we get T(Tmax) = 00. Furthermore by differentiating directly u, we see that u(x, r) satisfies

UT{X,T)

=

APU(X,T)+

|U(

||^

"p

T )

,

+U(X,T)E{U(T))

T€[0,OO).

(10)

Multiplying (10) by uT and integrating over B, we obtain dT{E{u{T))) = -p\\uT{r)\\h Hence the fact that lim^.E(i2(T)) =

, r € [0,00).

lim [E(u(t))/\\u(t)\\pL2] = j * gives

||uT(T)||£adr
/ Jo

Moreover, since a < 00.

0
Then by setting iij{a) = U{TJ + a) for 0 < a < 1, we find that sup{||uj|| L< „ (0)1 . Wo i.j. (B)) + ||5 C T UJ||L2( 0 > 1 . L 2 ( B ) )} < C < 00, whence follows the 3 i.-i..;j>- 2 *

boundedness of ||Apuy||L2(o,i;£2(B)) and II (i-|z|)"' 1IL°°(O,I.:^2(B))(10) and the definition of Uj(o) directly give us d^Uj = Apiij + |" J | P [ a .|"a + £ ( " i ) f i i >

CT e

Moreover,

[°> !]•

Thus, by the standard arguments of convex analysis, we can show that there exists a subsequence iijk of iij, such that iijk(a) —> w(a) — w strongly in C([0,1];L 2 (B)) satisfying (8) and (9). • Lemma 3 Let \i < 1 and 7 > 0. Suppose that w £ X is nonnegative in lw\p~2w B, and satisfies -A„iu - 7- r-77— = jw in ID'(S). 77ien w = 0 in B. (1 - | z | ) a

32

Proof : Suppose that w ^ 0 in B. First we show that w € L°° by the "bootstrap" method as in [3]_and [12]. Then by using a theorem of Tolksdorf [14], we see that w £ C1,a{B). Thus the same argument as in the proof of Theorem II of Idogawa and Otani [5], we arrive at a contradiction. D Proof of Theorem 3 We shall prove this Theorem by contradiction. Let u{t) be a global solution of (E) such that u0 e WQ'P{B)\{0} is nonnegative in B. We already know from our existence results in Theorem 1 that if the initial data uo is nonnegative in B, then the solution u is nonnegative in B for t > 0. It can be shown from Theorem 2 that we have only to consider the case where 7, > 0, and from Proposition 2 that 3 {tj} and w £ WQ'P such that tj ->• 00 and

and ID—'

I

-\w

-

x

I

= ^w

in ©'(B), \\w\\L2 = l.

Since u > 0, then w > 0. Applying Lemma 3, we then conclude that w = 0, a contradiction to the assertion in (9) that ||W||L 2 = 1. Therefore, u(t) is not a global solution, i.e., u(t) blows up in finite time. • Sketch of Proof of Theorem 4 When p = 2, we divide the proof into two parts: Part I : (Existence of solutions of approximate equations) We first consider the approximating problem ( (ue)t(x,t) v

Je

J k E

where u

0

= Aue(x,t)

ue(x,t)=0 ue(x,0) = usa{x)

+ (Kl2 + g ) * ^ 0 M )

] ( M )

,

B x

[0)T],

,(x,t)£dBx[0,T], ,xeB,

are radially symmetric functions in HQ (B) 0 H2 (B) such that «Q ->

UQ as e -¥ 0 strongly in

HQ(B).

Since the last term of (E) 2 is C 1 -continuous and globally Lipshitz from L (B) to L2(B), it is well known that (E) 2 admits a unique solution 2

33

u£ £ C{[0,T];H^(B))f\C([0,T];L2(B)) (see, e.g., [11] ). Furthermore the uniqueness of solution assures that u£ is radially symmetric. Part I I : (A priori bounds, convergence and nonnegativity) We can easily show the following estimates: (i) By multiplying (E)2: by u£ and using Young's inequality, we can show that u£ is bounded in C([0,T];L2(B))nL2(0,T;X). (ii) By multiplying (E)f by (u£)t, we can show that (u£)t is bounded in L 2 (0,T;L 2 (J3)) and u£ is bounded in C([0,T];X). (iii) By virtue of assumptions on q and a, we can apply Lemma 1 with s = 2(q — 1) and 8 = 2a to observe that -!—^ p-r;—- is bounded in ( l + £ - |x|) a L2(0,T;L2(B)). It then follows that Au£ is bounded in L2(0,T0;L2(B)). From (i)-(iii), the fact that u£ satisfies (E) 2 and by using standard argument (see e.g., [9],[11]), we can prove that {ue} converges to the desired solution u as e tends to 0 and that, if Uo is nonnegative, then u is nonnegative. For the case p > 2, we consider the approximating problem 2

(us)t(x,t)

(E)«

= Apu£(x,t)

+ ^f+*^^\)"'

>(*>')

u£{x,t) = 0 u£(x,0) =ue0(x)

€ B x

M .

, (x,t) € dB x [0,T], ,xeB,

where a > 0 and 2 0, and u§ are radially symmetric functions in C2(B)D WQ'P(B) such that UQ ->• uo as e -> 0 strongly in Wo' p (B). The proof then follows the same procedure as in the case p — 2.

• Proof of Theorem 5 We apply Theorem II of [11], by choosing H = L2r{B) = {u £ L2{B);u is radially symmetric}, <^'(u) = y>(tt) = —

—r v

/

\pdx with Z % ) = X and B(t,u) = B(u) = J _ , Jta• To do - / |V«(z)|' P JB this, we must verify the boundedness condition for £(•)• By virtue of the assumptions on q and a, we can apply Lemma 1 with s = 2(q— 1) and 8 — 2a to obtain ||B(u)||^ 2 < C
,1U|'"2|1 (1 - \x\)a

in

L?(B)

a.e.

*6[0,T 0 ],

(11)

34

u 6 C(lO,T0];X)nW1'2(0,T0;L2r(B)),

| [

e

L2(0,T0;L2(B)).

To show that u solves (E) in L2(B), we first note that any test function w 6 L2(B) can be decomposed into w - wr +w^r with wr 6 L2T{B) and u;^ 6 (LKB))-*-, where (Z^C^))1" denotes the orthogonal complement of L2(B). On the other hand, since ut, Apu, \u\9~2u/(l - \x\)a belong to L2r(B), they are all orthogonal to {L^B))1-. Hence (11) implies that u solves (E) in L2(B). The nonnegativity of the solution can be verified by the usual comparison theorem. • Proof of Theorem 6 It is easy to see that for any u £ X\{0}, J(Xu) = \pa(u)/p-\qb(u)/q attains its maximum at A = 1 if and only if j(u) = a(u) b(u) = 0. Hence, by the definition of d, the set {u 6 X;j(u) = 0, J(u) < d} is empty. Therefore we deduce X d = {u € X; J{u) < d } = W u V ,

W n V = 0,

W = W , U {0}, (12)

W , = { u € X; j(u) > 0, J(u)
V = {u£ X; j(u) < 0, J(u) < d}.

On the other hand, by multiplying (E) by du/dt, we get ||u((i)||^ 2 + dJ(u(t))/dt = 0. Hence J(u(t)) is monotonically nonincreasing in t, so if u0 £ W , then u(t) € X d for a l H € [0,T m a x ). Furthermore, since a(u(t)) and b(u(t)) are continuous in t, by virtue of (12), we find that u(t) £ W for all t e [0,Tmax). Since u £ W implies ( i - ^)a(u) < d, we can obtain a priori bound for a(u(t)) for all t 6 [0,Tmax). Thus Theorem 5 assures that the solution can be continued globally. • Acknowledgment The authors wish to thank the referee for his careful reading and valuable comments. References 1. A. Fujii and M. Ohta, Aymptotic behavior of blow-up solutions of a parabolic equation with the p-Laplacian, Publ. RIMS, Kyoto University, 32(1996), 503-515. 2. S. Hashimoto and M. Otani, Elliptic equation with singularity on the boundary, Differential and Integral equation, 12(1999), 339-349. 3. S. Hashimoto and M. Otani, Sublinear elliptic equations and eigenvalue problems with singular coefficients, 2001, preprint. 4. S. Hashimoto and M. Otani, Ljusternik-Schnirelman theory for elliptic equations with singularity on the boundary, in preparation.

35

5. T. Idogawa and M. Otani, The first eigenvalues of some abstract elliptic operators, Funk. Ekva., 38(1987), 1-40. 6. H. Ishii, Aymptotic stability and blowing up of solutions of some nonlinear equations, J. Differential Equations, 26(1977), 291-319. 7. H. A. Levine and L.E. Payne, Nonexistence of global weak solutions of classes of nonlinear wave and parabolic equations, J. Math. Anal. Appl., 55(1976), 329-334. 8. M. Otani, An Introduction to Nonlinear Evolution Equations (in Japanese), The Fifth Seminar on Evolution Equations for Young Mathematicians, 1974. 9. M. Otani, On existence of strong solutions for

4- dii>l{u{t)) —

&ip2(u(t)) 9 f(t), J. Fac. Sci. Univ. Tokyo Sec. IA, 24(1977), 575605

\ 10. M. Otani, Existence and asymptotic stability of strong solutions of nonlinear evolution equations with a difference term of subdifferentials, Qualitative Theory of Differential Equations, North-Holland, Colloq. Math. Soc. Janos Bolya, 30(1981), 795-809. 11. M. Otani, Non-monotone perturbations for nonlinear parabolic equations associated with subdifferential operators, Cauchy problems, J. Differential Equations, 46(1982), 268-299. 12. M. Otani, Existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations, J. Functional Analysis, 76(1988), 140-159. 13. M. Otani, A priori estimates for some nonlinear parabolic equations via Lyapunov functions, Functional Analysis and Related Topics, Springer Lecture Note, No.l540(1991), pp.319-327, edited by H. Komatsu. 14. P. Tolksdorf, On the Dirichlet problem for quasilinear equations in domains with conical boundary points, Comm. in Partial Differential Equations, 8(1983), 773-817. 15. M. Tsutsumi, Existence and nonexistence of global solutions for nonlinear parabolic equations, Publ. RIMS, Kyoto University, 8(1972), 211-229.

RESIDUES: ANALYSIS O R ALGEBRA? IN MEMORIAM JIM KELLEHER CARLOS. A. BERENSTEIN Institute for Systems Research, University of Maryland, College Park, MD 20742, USA E-mail: [email protected] We discuss the key role the theory of residues plays in complexity problems in Algebra and constructive procedures in Algebraic Geometry.

It was Jim Kelleher's work 22 which was a crucial component of my early work 7 and it got me interested in complexity questions about Hilbert's Nullstellensatz and beyond. The motivation of the work described below is to bound the complexity of deciding whether the system of M polynomial equations in n variables, Pj{x)

= J2ajaxa=0,

j = l,...,M,

(1)

a

has a solution x € K , where the coefficients a,ja € A, an integral domain with quotient field K and integral closure K. The non-existence of a solution of the system (1) is equivalent (by Hilbert's Nullstellensatz) to the existence of polynomials qj 6 A[x], j = 1, • • •, M , and r 0 G A \ {0} such that r0 = pi{x)qi_{x) + • • • + pM{x)qM{x)-

(2)

In the engineering literature, this equation is known as the algebraic Bezout equation. It plays an important role in Systems Theory. The complexity question we discuss below is closely tied to the well known problem P vs. NP, as discussed in the work of Shub and Smale. 27 ' 26 . One measure of the complexity of solving (1) is to find a priori bounds for the degrees of the polynomials qj. The main result that follows from the work in the late 80's of Brownawell 2 , Heintz and its collaborators 14 , KoUar 2 0 , and others is that when we set D = maxdegpj

36

37

then (1) has no solutions if and only if there are polynomials qj solving equation (2), and have their degrees bounded by maxdeggj = i

0{Dn),

which is essentially the best possible estimate. More precisely, if we denote by Dj = degpj and assume D\ > D2 > ••• > DM, then letting H — min{n,M},L = max{j : 1 < j < (i - l,Dj = 2}, one can solve (2) with the degree bound 3 max deg qj < (-)'£>i •••£>„ (3) (In this estimate and those that follow, all the constants, implicit or not, can be estimated explicitly.) Once one has obtained the a priori estimate (3), the equation (2) can be transformed into a large system of linear equations in the unknown coefficients of the qj. By itself, this estimate only gives a coarse bound on the complexity of the problem. The reason is that this system of linear equations is still too big, and very undetermined. In order to do significantly better one needs to impose additional restrictions on the coefficients, for instance, if one has a notion of "size" for the elements in A one can reduce the complexity of solving (2) into size bounds for the coefficients of one of the solutions {qj}jL1 of (2). For the precise definition of size we refer to the work of Phillipon 25 . In the case where A = Z, the ring of natural integers, then a natural notion of size of a € Z* is given by its logarithmic height h(a) = log 2 \a\. For F a field, A = F[m,... ,um], then one can use h(a) = deg u a. For p(x) — £ ) a a x a e A[x] we define h(p) = max.,- h(aj). In our early work on this subject 9 , Alain Yger and I proved the following effective estimate: For A = Z, if there are solutions to (1) then there is a solution ro,qj of (2) with maxj degfc < (|) 7 n(2n + l)Dv... £>„ maxj h(qj) < K^DfiDt... DJ8(h + logM + Dx log A ) ,

K

where h = maxh(j>j). This estimate is essentially optimal and the constant K(ri) can be made explicit. The proof in above mentioned manuscript with Yger9 was not only heavily dependent on complex analytic tools, but specially the analytic theory of multidimensional residues (and residual currents) played a fundamental role. Moreover, we used the fact that residues could also be computed by analytic continuation of associated zeta functions. I refer to our short monograph on

>

38

residues4 for the development of these ideas, here we have to be satisfied with ar just a few points: Let us assume that fi,..-,fn>9 e holomorphic functions n near the origin of C , and assume that {/j = - • • = / „ = 0} = {0}, then, the residue of the meromorphic function g/fi • • • fn at z = 0, as defined by Poincare, is given by Res(

y

,0) = lim( e

/ 1 • • • fn

->°

9(z) J

2m

J /i (*)•"/»(*) l/l=*

dz,

(5)

dz — dz\ A . . . A dzn, where | / | = e, for e = ( e i . . . e n ) , denotes the cycle {|/i| = ei. • • • > |/n| = £n}- When the Jacobian J of the /_,• does not vanish at 0 we have

-,o) = 9(0) 7 i •••/»'"' J(o)

Res (-

(6)

as expected. This definition of the residue was introduced by Jacobi (at least for polynomials) 19 . The definition (5) can be extended to define a residue current, namely, we replace gdz by a smooth compactly supported (n, 0) differential form
(

^"/^x=<4">'

(7)

l/l=*

where dj is a (0,n) current. What one does next is to relate this current dj to the current-valued holomorphic map in C A ^ |/i • • • fnYXdh

2A

A • • • A df„

= | / r a / , Re A »

0

Using the Bernstein-Sato functional equation one sees that the holomorphic function A K» A f | / | 2 ( A _ 1 a / A
0

c» has an analytic continuation to A = 0 and one obtains

j\f?(X~l

•a 1

dfA
= c„ < J A=0

o-,
,

(8)

39 where cn ^ 0 is an absolute constant. The reason we needed to use residues and several interpretations of them was because the polynomials qj were found by "explicit" interpolation formulas. These interpolation formulas are based, one way or another, on the Abel-Jacobi vanishing theorem. It states the following: let P i , . . . , P n be polynomials in C n without any common zeros at infinity and let Q be another polynomial satisfying the inequality d e g Q < d e g P i + --- + d e g P n - n - l

(9)

then =0,

(10)

in other words, the sum of all the residues of the meromorphic function Q/P\ • • • Pn vanishes. When all the common zeroes of the Pj are simple, we can use (6) to derive the original statement of Jacobi 1 9 ^

J(a)

'

where J denotes the Jacobian of Pj. There are many interesting geometric applications of this theorem, to be found, for instance, in the work of Griffiths17, Kunz 23 , and the more recent survey 15 . In general, and specially in the application to the Nullstellensatz, one cannot find convenient polynomials Pi,...,Pn in the ideal generated by the original polynomials pi,... ,PM which satisfy the additional condition of having no common zeroes at infinity. What one can do is to find P\,..., P n so that the map z^P(z)

=

{P1{z),...,Pn(z))

n

is a proper self map of C . Note that the properness of the map P is equivalent to the Lojasiewicz type inequality: there are constants K > 0,7 > 0, and S > 0 such that if \z\ > K then \P{.z)\>l\z\s, Such a 6 is called a Lojasiewicz exponent for P .

(11)

40

Thus, to find the desired polynomials qj with good degree and size estimates, one needs a generalization of the vanishing theorem of AbelJacobi. The key ingredient is the following: there is a proper affine function 6 : Z" -> R such that for any polynomial Q and any m e ( Z + ) n such that degQ<0(m)

(12)

=0,

(13)

one has

where Pm = (P™,... ,P™)- Note that a proper map usually has zeroes at oo, this is the point that makes this statement a strong generalization of the Abel-Jacobi theorem. The proofs given in our early work 9 ' 4 depend very heavily on identities like (8) for the computation of residues. In order to extend the effective Nullstellensatz to arbitrary integral domains A with a size (irrespective of the characteristic of the field K), Yger and I arrived at the conclusion that the scheme of proof given in the case A = Z should work since, at least, a purely algebraic theory of residues already existed and the corresponding Abel-Jacobi theorem was known 24 . Originally we had many false proofs of the generalized Jacobi vanishing theorem. The main difficulty is that analytic constructs like (7) or (8) seemed very hard to reproduce for the case of cftar(K) > 0. Finally, we managed to prove, by purely algebraic means, the following slightly weaker version of the generalized Jacobi vanishing theorem 11 - 12 : Theorem Let P i , . . . , P„ be polynomials in K[x], assume that deg P, = D for 1 < j < n, satisfying (11) with an integral Lojasiewicz exponent S. Assume further that, for e„ = l/n(n + 1), (l-en)D<6.

(14)

n

Then for any m 6 ( Z + ) such that degQ < n(n + l)\m\(8 - (l-en)D)

-n-1

(15)

we have =0.

(16)

41

Here the algebraic residue identity (16) has to be understood in the sense of Lipman 24 . The proof is rather complicated, as one does not have available any of the analytic tools which can be applied in the complex case thanks to the identity (8). As a consequence of this theorem, one can prove an effective Nullstellensatz estimate of the type (4) for arbitrary integral domains A with a size. We refer to our work n ' 1 2 for the exact statement and its proof. We would like now to mention some other applications of the identity (8), and of the techniques that led to it, both in algebraic geometry and number theory. (See, e.g., our work mentioned in the references below 10 ' 13 , and also the preprint 8 .) They are related to the work of Arakelov x on intersection theory, which lead to the concept of Arakelov currents, and further developments and applications to Number Theory by Faltings 16 , Bost, Gillet, and Soule 3 , and others. Let Z be a pure dimensional effective cycle in P n ( C ) , whose decomposition into s irreducible cycles Zi, all of codimension d, is s

Z = ^miZurrii

6 N*.

A Green current Gz associated to Z is a (d — 1, d — 1) current such that ddcGz + (degZ)wd = 6Z =

9

^mideg(I(Zi))6Zi. *=i

In this identity, I(Z) = Yfi=i -^(^»)mi is t n e ideal sheaf of Z, decomposed in terms of the ideal sheaves of the components Z<. The (1,1) form CJ = dd c log(|xo| 2 + ••• + \xn\2) defines the Kahler metric on P " ( C ) , Sz is the integration current on the cycle Z (counting multiplicities), while Sz{ denotes the integration current (without multiplicities) on the reduced algebraic variety V{I{Zi)). Given global sections P i , . . . , Ft that generate I(Z), that is, k homogeneous polynomials in n + 1 variables, then, Yger and I constructed explicitly the corresponding Arakelov Green currents using the analytic continuation method (8). (See our J. Analyse Math, manuscript 10 for the complete intersection case and that in the Crelle journal 13 for the general case.) There are many important applications of these currents to Algebraic Geometry, Physics, etc., but an extremely interesting one appears in an unpublished (as far as I know) manuscript of Curtis McMullen, which provides a sketch of a simple proof of Fermat's Last Theorem. The ideas used, as far as I understand them, are a combination of Faltings's insights and the use of dynamical systems.

42

Other applications of the analytic continuation method to define residues appear in older joint work with Yger to study ideals generated by exponential polynomials 2 , and in a recent preprint 8 . Overall, these examples show that, without any doubts, residues live in the world in between Algebra and Complex Analysis.

Acknowledgments The research of the author on this subject was partly supported by NSA and NSF. References 1. S. J. Arakelov, Intersection theory of divisors on an arithmetic surface, Math. USSR Izv. 8 (1974), 1167-1180. 2. C. A. Berenstein, D-modules and exponential polynomials, in Structure of solutions of differential equations, T. Kawai and M. Morimoto (eds.), World Scientific Publ. Co., 1996, 81-88. 3. J.-B. Bost, H. Gillet, and C. Soule, Height of projective varieties and positive Green forms, J. Amer. Soc. 7 (1994), 903-1027. 4. C. A. Berenstein, R. Gay, A. Vidras, and A. Yger, Residue currents and Bezout identities, Prog. Math. 114, Birkhauser, 1993. 5. J. E. Bjork, Analytic D-modules and applications, Kluwer Sci. Publ., 1993. 6. D. W. Brownawell, Bounds for degrees in the Nullstellensatz, Annals of Math. 126 (1987), 577-591. 7. C. A. Berenstein and D. C. Struppa, On explicit solutions of the Bezout equation, Systems & Control Letters 4 (1984), 33-39. 8. C. A. Berenstein, A. Vidras, and A. Yger, Analytic residues along algebraic cycles, preprint 138 (2001), Math. Dept. Univ. Bordeaux I. 9. C. A. Berenstein and A. Yger, Effective Bezout identities in Q[zi,..., zn], Acta Math. 166 (1991), 69-120. 10. C. A. Berenstein and A. Yger, Green currents and analytic continuation, J. Analyse Math. 75 (1998), 1-50. 11. C. A. Berenstein and A. Yger, Electr. Research Announcements of the Amer. Math. Soc. (1996), 82-91. 12. C. A. Berenstein and A. Yger, Residue calculus and effective Nullstellensatz, Amer. J. Math. 121 (1999), 723-796.

43

13. C. A. Berenstein and A. Yger, Residue theory in the non-complete intersection case, J. Reine Angew. Math. 527 (2000), 203-235. 14. L. Caniglia, A. Galligo, and J. Heintz, Borne simple exponentielle pour les degres dans le theoreme de zeros le Hilbert, C.R. Acad. Sci. Paris, Ser I Math, 307 (1988), 255-258. 15. D. Eisenbud, M. Green, and J. Harris, Cayley-Bacharach theorems and conjectures, Bull. Amer. Math. Soc. 33 (1996), 295-329. 16. G. Faltings, Diophantine approximation on Abelian varieties, Ann. Math. 133 (1991), 549-576. 17. P. Griffiths, Variations on a theorem of Abel, Invent. Math. 35 (1976), 321-390. 18. P. Griffiths and J. Harris, Principles of algebraic geometry, Wiley, New York, 1978. 19. C.G. Jacobi, Theoremata nova algebraica circa systema duarum aequationum inter duas variabiles propositarum, Gesammelte Werke, Band III, 285-294. 20. J. Kollar, Sharp effective Nullstellensatz, J. Amer. Math. Soc. 1 (1988), 963-975. 21. T. Krick and L. Pardo, A computational method for diophantine approximation, Proc. Effective Methods in Algebraic Geometry, MEGA94, Birkhauser (1996). 22. J. J. Kelleher and B. A. Taylor, Finitely generated ideals in rings of analytic functions, Math. Ann. 193 (1971), 225-237. 23. E. Kunz, Uber de n-dimensionalen Residuensatz, Jahresber. Deutsch. Math. Verein. 94 (1992), 170-188. 24. J. Lipman, Residues and traces of differential forms via Hochschild homology, Contemp.Math. 61, Amer. Math. Soc. (1987). 25. P. Philippon, Sur les hauteurs alternatives III, J. Math. Pures Appl. 74 (1995), 345-365. 26. M. Shub and S. Smale, On the intractability of Hilbert's Nullstellensatz and an algebraic version of NP ^ P, Duke Math. J. 81 (1995), 47-54. 27. S. Smale, Mathematical problems for the next century, in Mathematics, Frontiers and Perspectives 2000, Amer. Math. Soc. 2000. 28. A. Vidras and A. Yger, On some generalizations of Jacobi's residue formula, Ann. Scient. Ecole Normale Sup. 34 (2001), 131-157.

M O M E N T CONDITIONS FOR P O M P E I U P R O B L E M E X T E N D E D TO GENERAL RADIAL SURFACES DER-CHEN CHANG Department

of Mathematics, Georgetown University Washington D. C, 20057, USA E-mail: [email protected] WAYNE E B Y

Department

of Mathematics, University of College Park, 20742, USA E-mail: [email protected]

Maryland

We extend the results of [4] to include integrals with a more general measure. Our goal is to utilize a distribution based on an arbitrary radial measure and including a moment factor, i.e.,

=

Izmf(z)dn(z)

= J" y(r) (f

*r • • • C" /(•)** (•)) r^ 2r

with da area measure on the sphere and w(r) an arbitrary radial function. This is achieved easily in C n by extending the two dimensional result of Zalcman for C 1 . Further some specific cases are illustrated to depict Theorems from [4] as extended to disks. We also prove several results for L p ( H n ) , 1 < p < 2, on certain specific cases, including spheres, disks, and annuli.

1

Radial Moment Conditions for Pompeiu Problem on C n

This paper is written to describe how some results regarding the Morera Problem with moments [4] may be extended to include integrals utilizing a more general radial measure. The results appear in two separate settings. In the first setting we consider arbitrary continuous functions on the space C™, while in the second we are working in the Heisenberg group, H " , and restrict our attention to functions in the space L P (H"), 1 < p < 2. The Heisenberg setting is, in general, a more difficult problem, and the setting of C™ was introduced as an intermediate between the known moment results for C 1 and the unknown case of H n . As we can see that L ^ H " ) , 1 < p < 2, is an intermediate result on the way to investigating what happens for arbitrary functions in L°°(Wl). It is when considering functions in L°°(H n ) that we can produce (counter)examples

44

45

which show that one radius or one moment is not enough. Theorems for multiple radii were established in [1], [3], and the present line of research is directed to discovering what manner of result exists in L°°(H n ) utilizing multiple moments. Note that it is also at the level of L°°(H n ) that the moment conditions begin to introduce conclusions which involve a differential applied to the given function / , rather than conclusions about / itself. As mentioned earlier, the present cases of C(C n ) and -L 2 (H") are easier than the case of L°°(Hn), and they are necessary stopping points along the way before we can arrive at the more difficult results. Results for L°°(H.n) are expected to incorporate combinations of ideas and methods from both of C(Cn) and L 2 ( H n ) , the cases in the current paper. To push the analysis to L°°(H.n) would certainly also require, as well, new tools and ideas to handle some of the new complications that arise in that case (not covered herein). The results and methods for C(C") generalize from C 1 to C n those used by Zalcman in [10], [11] where he develops the first Pompeiu/Morera type theorems which incorporate moment conditions. The results and methods for L 2 (H") generalize those developed in [2] to the case where a moment factor is included in the integral. Such results were collected in the paper [4], This paper generalizes the results of [4] from integrals over spheres to integrals over more general radial surfaces. The motivation to look to these more general measures may be found in the theorem of Zalcman. Let us first intorduce some notations. Denote S(m, fi) is the set of quotients of (nonzero) zeros of F M , a function representing the Fourier transform of the measure and given by Ffi{z)

~^QkKk

+

\m\yAk{2}

The Ak are determined by Ak = /0°° t2k+2^dp,(t) for k — 0 , 1 , 2 , . . . , and s is the smallest value k for which Ak ^ 0. Now we may recall a theorem in [10] as follows: T h e o r e m 1.1 Let f G Ljoc(R?) numbers r\, r
and suppose there exist distinct positive real

I l f(z + peie)pWeim6d0dfir(p)=0 Jo Jo for almost all z e C and r = ri,r%- Then if p- ^ S(m,fi), f agrees almost everywhere with a function g satisfying

vk)

A 5=0 (m 0);

*

-

\jk) A*9=0

(m<0)

46

Remark. This theorem has been slightly altered from its statement in [10] by the addition of the factor plml in the integral and the corresponding change in the determination of Af.. This change is made to include consideration of the disk at this point, but does not change the character of the theorem. This theorem compromises between the issues of generality and tractability on the subject of considering arbitrary measures. Maintaining a radial surface makes considerable difference in terms of being able to evaluate the Fourier transform of the associated measure. However this theorem is extended to consider arbitrary radial measures as a base for the distribution and further extended to include any moment factor. Thus the distribution can be of any fixed homogeneity (not just radial). The other aspect of great interest in this theorem is the connection between the moments in the integral conditions and the corresponding differentials which turn up in the conclusions of the theorem. For these reasons we direct our efforts herein toward this same manner of measure. This theorem is easily extended to R 2 n = C n merely by using the Fourier transform results of [9] in conjunction with the series expansion of the Bessel function Jm{z)

-2^k](k

+ my}2>

to arrive at the following theorem. It will be useful in stating the theorem to first define some notation for multi-indices and derivatives in multiple variables. For the moment condition we use m _

mi

_ _

m„

and the differential condition corresponding to this moment

where |m| = m i +m,2 -I \-mn. Some (or all) of the variables may be shifted to their conjugate in the moment term m _ mi I, — Z1

m.j -mj+i _m„ Zj Zj+l • • • Zn

yielding the differential condition Q\m\

a(z, z)

Q\m\

m =

1

dz? • • • dzpdz™^1 • • • a c - '

47

Furthermore there is a need to introduce the Laplacian A

d

A= Y

.

A r dzjdzj j—i

J

J

Denote S(n, |m|,/x) is the set of quotients of (nonzero) zeros of 00

<—l)k

f—z\ 2k+d

where d = n + |m| — 1 and Ak = Jt2h+2d+1dfi(t). theorem: Theorem 1.2 Let f e L\oc{Cn) numbers r\, r^ such that [ JO

[

Now we may state the

and suppose there exist distinct positive real

zmf(w

+ Z)da(z)d»r{p)

=0

(1)

J\z\=p

for almost every w 6 C n and r = r\,T2- Here da represents surface measure on the sphere. Then if p- ^ S(n, |m|,/i), then f agrees almost everywhere with a function g satisfying

a( z ,z) m

Jy

Remark. In special cases, which we will consider shortly, the function .FM>Tl||m| evaluates to the well known Bessel function, of some index, whose zero set is thoroughly understood. In these cases it is possible to discuss Theorems which involve only one radius and multiple moments, such as those discussed above. Proof: We investigate the zeros of the Fourier transform .F(z m /i r (|z|))(£), which may be evaluated using (3.10) of [9] and series expansion of Bessel functions. We let |£| = R and d = n+ |m| — 1.

JVV(M))(0 = / = r

e-27ri«'zzm^r(|z|) Um-WR-t'+W-1'*

J™

Jn+lml_1(27rRp)pn+^d^r(p)\

48 2k+d

-oo

= (iOm(-l)|m|2vr-S5-^(27rJRr). Note, unless A 0 = 0, we have that F^JRr) ^ 0 for R = 0, i.e., £ = 0. If A0 = 0 it is necessary to consider a nonzero s such that A s ^ 0 but Aj = 0 for j < s. Next step is to eliminate common zeros of a "trivial" nature. We note that S m m T[{ d(z z) r -A )Tr^j (0 = ( ^ ) ( - l ) l l 2 7 r ^ i ^ ( 2 7 r i ? r )

= ^(zm^(|z|))(£) where rd+l

Then we claim the only zeros of !F(Tr)(£) are those £ such that R = |£| is a nontrivial zero for F^^ftnR). The assumptions on r\,r2 are designed to insure there cannot be such a zero for both r = r\ and r = r^ simultaneously. We can then apply the one variable spectral synthesis result of Schwartz [8] to arrive at the desired conclusion. The integral conditions of the theorem give the following sequence of results for both r = r\ and r = r%: f * (z m /i r (| Z |)) = 0 /

0lml

,<9(z,z)r

-A s )f*Tr

\

=0

Thus this theorem is proven as claimed. Note that this theorem requires two radii satisfying certain conditions, thus matching with the line of Pompeiu theorems which require multiple radii. In contrast it is possible to use multiple moments on a single surface rather than several surfaces of different radii. One of the remarkable qualities of the use of multiple moments in the Pompeiu/Morera Problem is the disappearance of the exceptional set of radii which don't work in the theorem. Zalcman first demonstrated this manner of two moment theorem (without exceptional set) for C 1 in [11], and it is generalized to C " in [4] where n + 1 different moments are required. These theorems of multiple moments have heretofore been

49

stated only for spheres but may be stated for disks as well. We record these observations here as they fit within the theme of extending from moments on spheres to moments on other radial surfaces. We begin with a result describing multiple moments on a solid ball in Cn, generalizing a result from [4], where it was done for a sphere. Theorem 1.3 Given a function f £ L}oc(Cn)

we claim that

0M

d(*,*y 7 = o, notation as defined above, if f satisfies the following integral conditions (with moments) for all w £ C " : /(w + z)z"W(z) = 0 / .B(0,r) and, for j — 1 , . . . , n /

/(w + z)zm+a'eW(z) = 0

JB(0,r)

where sign(aj) = sign{m,j) and ej represents ( 0 , . . . , 1 , . . . ,0), 1 in the j t h position. Here B(0,r) = {z G C : |z| < r} is the open ball centered at the origin with radius r. Proof: The proof for disks is identical to that for spheres in [4], save that the Fourier transforms have Bessel functions which are increased by one index. The essence of that proof is unchanged by the shift in indices, and we may still rely on the fact that Bessel functions of different indices have no common zeros. We recall the Fourier transform of a™, the measure associated to the moment on the sphere.

n<m

= (^ m (2^> +|m| (-i) |m| r- 2 " +2|m| - l ^g=^5?

where |£| = R and £ m is denned in the same manner as our moments z m . Jm{x) is the Bessel function of order m. Now observe the shift in index when moving to the area measure for the moment on the disk, /it™: •WXO = A

/

e2^wwmrfa(w))p2"-1dp

JO

' | wJM=P |=p

L

e2^wX[o,r)(|w|)wm^(w),

50 and

Jo p2-rrRr

= ^ m 27rr | m | i?- ( n + | m | - 1 )(27ri?)-("+l m l+ 1 ) / Jo

J„+|m|-i(p)p"+|m|dp

We have demonstrated the shift in the index of these Bessel functions, and the proof follows as in [4]. Other results of [4] may be extended similarly to their analogous results on a disk, and we leave these to the reader. However we include a few important corollaries of this theorem. First consider integral conditions sufficient to conclude that a function in C ra is holomorphic. Corollary 1.4 Let f £ C(Cn) integral conditions f

and let r > 0 be fixed. Consider the following

/ ( w + z)Zjd»(z)

JB{0,r)

= f

/ ( z + w)z?dfi{z)

=0

JB(0,r)

for j = 1 , . . . , n and every w £ C", where each aj is an integer > 2. We assert that f is a holomorphic function if it satisfies these given integral conditions. Finally we want to consider the one dimensional case, generalized from Zalcman's aforementioned two moment theorem [11], to now include moments on disks. The corollary follows by simply letting the dimension n=\. Corollary 1.5 Let f G L11oc(R2) and let r > 0 be fixed. Suppose there exist integers I, m such that for almost all z £ C /

f(z + w)wed/i(w) = 0

J\w\
f

f(z + w)wmdn(w)

=0

J\w\
Then (a) if 0 < £ < m, f agrees almost everywhere with a solution of (-§=Yf = 0; (b) if 0 > t > m, f agrees almost everywhere with a solution of ( J j ) ' f ' / = 0; (c) if £ > 0 > m, m ^ —£, f agrees almost everywhere with a solution of

51

the pair of equations (•§=)* f = 0, ( | j ) | m | / = 0. (essentially) a polynomial.

Thus in this case, f is

Note that the results here are no different when integrating over the disk as compared to when integrating over its boundary, the sphere. This result is implicit in the work of Zalcman [10], [11], though he never brought it out explicitly. Observe that what makes these theorems work is what we know about the zeros of the Bessel functions arising in the Fourier transforms. In particular, Bessel functions of different indices have no common zeros. As we cannot make the conclusions, in general, for the zero sets of Fn^(z) and Fmtll(x) where m ^ n, we therefore cannot extend this kind of two moment theorem to the same level of generality as the previous theorem. We would like to remark that there might be other measures which yield functions without common zeros. (Spheres and disks leading to Bessel functions of different indices are ones that have already been shown to work.) Any such measures would also produce two moment theorems without an exceptional set. However such have not, as yet, been developed or classified. Further we point out that this main theorem can yield results about harmonic functions and mean value relations, of the same nature as those proven by Delsarte [6] in his early work in this area of research. In Delsarte's work for harmonic functions and mean values in R™, he has shown that the exceptional set vanishes when n = 3 so that any two radii work. He has further conjectured the same happens for all n > 2.

2

Certain Moment Conditions on Disks for LP functions on W1

We now turn our attention to the Heisenberg group H™. For n > 1, the Heisenberg group is the set H " = C x R with the group law (z,t)-(w,s) = (z+w,t+s+2Im(z-w)) = ( z x + w i , . . .

,zn+wn,t+s+2lm(z--w)),

where n

.7=1

Whereas up to now we have integrated over Heisenberg translations of the set {(z,0) G H n : |z| = r}, a complex sphere embedded in Heisenberg, we now shift to a complex disk in place of the sphere {(z, 0) € H n : |z| < r}. Along with this shift comes a change from the differential forms Wj(z) for

52

j = 1 , . . . , n to the area measure on the disk, cfyir(z), or on the sphere, dar(z). As a consequence of these two adjustments we now plan to demonstrate a given set of integral conditions are met if and only if our function / is uniformly 0, i.e., f — 0. We list the following four sets of integral conditions and consider what conclusions could be made from each one: For all g e H " zmLsf{z,0)da{z)=0.

I

(2)

J\z\=r

Here L g / ( z , 0) = / ( g _ 1 • (z>0)) is the left-translation of the function / by the element g. The condition (2) is very similar to the sphere condition with which we have worked to this point. However the measure has been changed from the set of differential forms Wfc(z) = dz\ A • • • A dzn A dz\ A • • • A dSk A • • • A dzn,

k=

l,...,n

to the area measure da(z), which is radial. Interestingly enough, the conclusions we reach from these integral conditions are somewhat different from those we have seen to this point in the paper. Next consider an area measure on the solid ball rather than the sphere which is its boundary. When the moment is zero, this one is more closely associated with the Pompeiu problem as compared to the Morera problem. For all g € H " zmLgf{z,0)dfi(z)

/

=0

(3)

J\z\
which may also be written as fr f J0

zmLgf{z,

0)da{z)s2n-1ds

=0

J\z\=s

Finally we consider mean value type integral conditions -^~= "

z m L 6 /(z,0)<Mz) = / ( g ) .

r / \Z->2n-l\

(4)

J\x\=r

where |S2n-i| is the surface area of the unit sphere in Cn. In Euclidean space the analogous integral conditions to (4) have been used to describe harmonic functions, as in [6], [10]. In the Heisenberg group the concept of a harmonic function is more complicated. Nevertheless it is curious to investigate what manner of results we might discover.

53

One further region or interest is an annular region. The integral conditions would then be as follows: /

z m L g /(z,0)d M (z) = 0

(5)

ri<|z|
also written as I ' / Jr\

zmLgf(z,0)da{2)s2n-1ds

t/|z|— s

Note there are many other possibilities which could be considered, including the disk and annulus with radial polynomial weight considered in the next section, but for now we limit ourselves to these. In the case of the Euclidean space C 1 , Zalcman [10, Theorem 3] covered all of these possibilites, and others, at one time by considering a more general radial measure. In the previous section we saw this result and observed how it extended to C™. We approach these conditions (2), (3), (4), and (5) in a manner comparable to what was done in the earlier papers [2], [4] for integral conditons with moments. It is still essential to make some sort of homogeneity assumptions in order to reach our conclusions. However the nature of these assumptions must be adjusted to mesh with the radial property of these measures. In these cases we will want to reduce to the case where we are working with a function that is radial. This reduction can be made again using the radialization operator = [^... / ' 2 , r e i ( r o i * 1 + - + m - * - ) / ( e i * 1 z i , . . . , e i * - « n , t ) - d ^ 1 ' " 'n ( ^ n (27T) Jo Jo However in this case working just with a radial function is not enough to arrive at the conclusions we would like. In order to reach the goal, we are forced in addition to consider a slightly different manner of radial function. We introduce the concept of a function / that is radial centered at w as follows:

nmf

/(M) = /(e*(z-w)+w,i) for all 0 < 8 < 2n. The normal concept of a radial function corresponds to when the center is the origin, i.e., w = 0. In order to make the conclusions we'd like from these radial measures, we reduce to considering functions which are radial centered at w for arbitrary w G C™. Similar to the radialization operators used in the first part of this paper, which are centered at 0, we also introduce radialization operators centered at w.

v/(«. *)=/o27r • • • jf* fvn* - w)+w, * ) ^ f "

54

These will be used in making the reduction to working only with functions which are radial centered at some arbitrary w G C™. In the following we consider how this reduction works. But we first state the following result concerning these integral conditions. Theorem 2.1 Suppose we consider f G i 7 ( H n ) , 1 < p < 2. Then f = 0 if and only if f satisfies integral conditions (2). The same is true when integral conditions (2) are replaced with any of (3), (4), or (5) Let us now introduce the Laguerre series expansions for functions in L 2 (H"). This is a basic tool to study harmonic analysis on the group H n . For (3 > — 1 and i / £ Z + l let L\, be the generalized Laguerre polynomials defined by w

"

v\ dxv

v

;

Let A G R* = R \ {0}. For fi, u G Z + consider the function W*v denned on Cby W^l/(z) = e-2"xlz\2z»-"Li»-^{4Tr\\z\2) 2

v l

if

2

WttV(z) = e- "Wz -' L%-ri(4K\\z\ )

z

if n
w

z

if A<0

KA ) = »,Z( ) For n,i/€

M>!/,

n

n

(Z+) , let W^v be the function on C

A > 0, A > 0,

-

defined by

n

where C*

is a positive constant chosen so that ||W^^||i 2 (C") = 1> i-e-> x

"•"

TT f 7T (ma3c{/ij-,i/j})l)~g / i I (47r|A|)lw-"il+i (min{Mil!/,-})! J '

The readers may consult the book [5] for background of Laguerre calculus and its applications on the Heisenberg group. Here we mention just one of the fundamental properties of Laguerre polynomials. For pi, v, n', v' G ( Z + ) n and A G R*, we have

55

where C^y K,y

= C^,/{C^C*.y)

*X K'yW

= I

(see [7]). Here KA*

- w)W£ > ,(w)e- 4 ' r i A l m <--*)dm(w)

is the "twisted convolution" of W^v and W*, ,. In this paper, we follow the

same notations as in [1], [2], [3], [4] and [5] for Laguerre calculus on the group Hn. Proof of Theorem 2.1: One direction of the proof is trivial. Clearly if a function / = 0 it will satisfy each of the integral conditions (2), (3), (4), and (5). We now concentrate on the other direction, assuming a function satisfies the integral conditons and proving it is zero. Here we outline the strategy we will use for this proof, including a reduction to working only with functions of the aforementioned homogeneity. First we demonstrate that given any funciton / G L 2 (H n ) satisfying the integral conditions (2) [later also (3), (4), or (5)], we can form the functions lZ0f and 7£o,w/ for any w £ C n . (Once we obtain the result for L 2 (H"), we may use standard approximation arguemtn to show the same result holds for L p ( H n ) , 1 < p < 2.) Furthermore these are both in L 2 (H") as well and both satisfy the same integral conditions (2) [later also (3), (4), or (5)]. Then we observe that if 1Zof = 0 and 7£o,w/ = 0 for all w G C™ then it follows that / = 0. These steps complete the reduction step. Finally we will demonstrate that if / G L 2 (H"), / satisfies integral conditions (2) [later also (3), (4), or (5)], and / is radial centered at w for any point w G C", then / must be zero. In particular, beginning with an arbitrary / G L 2 (H") before the reduction step, then the lZ0f and 1Z0,wf must all be zero. It will therefore follow that / is itself zero, as we desire to prove. Here we observe both that 7£o,w/ G L 2 (H n ) and that TZQtWf satisfies integral condition (2). (The integral conditions (3), (4), and (5) work in exactly the same way, so we demonstrate only (2).) Here we write the expression for Lg72.o,w/(z,0) for g = {C,s). '*" #" ^ ..d<j>n L g fto, w /(z,0) = yo •••J /(e i *(z + C - w ) + w , S + 2Im(C-z))-^- 2 7 r ) r i and this may be simplified to /•J7T

/*Z7T

11

.•d(pn

r) n for h = (e^C - e^w + w,s-

2Im(C • z) - 2Im(w • e"^z) + 2Im(w • z)). So

56

now we investigate the integral conditions of (2) for 7£o,w/-

i

z m Z g (ft o , w /)(z,0)d<7(z)

|z|=r

/»27T

/ Jo

/»2?r

/

••• / JO

f

/

\

zmLh/(e^z,0)da(z)

\J\z\=r

J

d(j>i...

d(j)n

{2ir)n

Now we note that the inner integral is zero by the assumption of / satisfying the integral conditions (2). Therefore the entire integral is zero, and we have verified that 7£o,w/ satisfies conditons (2) as desired. To complete the reduction step we assume that TZof = 0 and 7^o,w/ = 0. So first observe that for all t s R ,

#

o=wo/(o,t)= r ••• r/(o,t) ;9-;f"=/(o,*) o=*o,w/(o,t)= r •• r/(w,*)%^=/(w,i).

Jo Jo and also that for all t G R and w € C",

./o Under our assumption that TZof conclude / = 0.

an

\™)

Jo \^l d 72.0,w/ are all zero for w G C", we then

From here on we focus on verifying two points: (i) if / is radial (centered at 0) and satisfies integral conditions (2) then / = 0, and (ii) if / is radial centered at w and satisfies integral conditions (2) then / = 0. We will later do the same with the other integral conditions (3), (4), and (5) but for now concentrate on (2). Let us first work with / radial, as this case is very similar to what has been done in the earlier section. The integral conditions may also be written as either of the convolution equations /*<7™=0

or

a™*/ = 0

We take the same steps of using the Fourier transform in the real t variable and then expanding in Laguerre series. It turns out that we useCT™* / — 0 when m > 0 and / * 0

o? * f = 0 is more relevant. As a consequence we have the following series:

i>e(z+)»

W(z+)n

57 where

and the coefficients

«W(*) = I Km(z)**W^)(Z)W,\m,,,(Z)cfc. And as before, the evaluation of these coefficients forms the detailed part of the proof which we now undertake. We leave out some of the steps as they are similar to what went before. This integral is reduced by A-convolution to the following integral zmW^+m(z)da(z)

c<W /

and then we may rewrite this integral as the following cS,,ve-^2

[

n(|zJf)^L^)(47rA|zj|2)da(z).

This integral has been evaluated in [4] and comes out to ~

/

n

|m|+n

Il(\zj\2r>Lfr\4ir\\zj\2)d*{z)

J\z\=rj=1

= E

cfe£H+fe(47rAr2)

(6)

fe=0

= ^,m,n(47TAr 2 ).

So we finally obtain t
/M(A)cP^ m ,„(47rAr 2 )H£ +m)/1 (z)

We conclude by the orthogonality of the Laguerre functions that for each \x /^(A)cP^, m ,„(47rAr 2 ) = 0.

Then since each of the polynomials P^m.n is zero for only finitely many values of A we know that /M(A) = 0 for every ^ and almost every A. As a consequence / = 0, as we intended to show. We now intend to show the same thing happens when we assume that / is radial centered at w. We go through the steps to verify that the different

58 homogeneity still works with the procedure. Note that since fx is a transform in the real, t, variable only, it does not affect the variable z and therefore does not have any effect on the homogeneity. Since / is radial centered at w, fx must be also. We also write a Laguerre series for this function, now allowing the Laguerre series to be centered at w as well. /A(z-w) =

Y,

/MWW*M(Z-W).

Once again from the integral conditions (2) we have that m A A

0=

CTr

* / (z-w)=

£

/^(A)Km*AVO(z-w).

M6(Z+) n

And now we note for each /i G (Z+) n we have that (a™ *A W* M )(z — w) is again radial centered at w. (^*AVO(e"(z-w))

( e ^ m e 4 7 H A l m e i e ( z - w ) ' e ~ i e ^ Jeie(z

= I

- w - Z))d*(£)

•/|CI=r = eim9

[

^me4«Alm(.-w).«y»A

(z _

w

_ £)d<7(0

J|£|=r = eim0(arm*AW^)(z-w) Therefore we observe that
so that we have the larger series equal to zero:

E WA)( E ^(A)W A +m >-w))=0.

M£Z+"

\ve(z+)n

/

This is familiar territory, and we now want to evaluate the coefficient tuMi„(A) and ensure that it turns out as expected. We have UV,„(A)

= /

(
59

= /

(/

= /

rW^(z-w-Oe47raIm(z-w>^a(0)W^+m,J/(z-w)dm(Z)

^ ( [ KAZ

" (z - w ) ) e - 4 - ^ I m « ( ^ ) W ^ + m ( z - w)dm(z)) da(t)

= [ r w ^ *A wVtV+m(Od*(z) J\t\=r J\Z\=r J\t\=r Note that at this point the integral has become identical to that which we were evaluating when considering / radial (centered at 0). So from here the computations are the same as in that previous case and lead to the same conclusion. We now conclude that when / £ L2(Hn) is radial centered at any w G C n and / satisfies integral conditions (2), we have that / — 0. We have completed this part of the proof, and the same now follows for any / £ L 2 (H n ) with no assumptions on homogeneity. We have therefore completed the proof when considering integral conditions (2). We now proceed to consider integral conditions (3), (4), and (5). All of these cases work much the same way as what has just been completed. In particular the homogeneity assumptions still apply. So then our goal is to show that given / G L 2 (H n ) that is radial centered at any w € Cn and / satisfies integral conditions (3), (4), or (5), we have that / = 0. We use the same method of Laguerre series expansion, and our first change comes when evaluating the integral as the region of integration has been changed. In the case of (3) we now have

Jm
„r

= c5^J <W#M( A )

/|m|+n

e-^'V"-

1

J2

\

2

ckLH+k(4n\p )\dp

60

Then we may rewrite our equation 0 =
o= E

as

/M(A)^(A)n£+m>)

M€(Z+)"

and using the linear independence of the W^ +mjM conclude that /^(A)(?M(A) = 0 for all p,. So then if g^ (A) = 0 has a zero set of zero measure, we may conclude that / = 0 by the same arguements used previously. Now let's get an expression for g^X). We already know the value of the inner integral, just as in the computation for the sphere of integral conditions (2), above. So we may write «>M,„(A) - c<W / r e- 2 ' r A '' 2 (P, i m , n (47rAr 2 ))p 2 "- 1 c Jp. Jo This integral may be evaluated using integration by parts and gives the following result: |m|+ra

r

9fi(X)

= J

e-^'V"-1

ckLM+k(4nXp2)dp

£

/o

fc=0

|m|+n -n\p fc=0

M+k(AirXp

jm|+n

=E

(2TTA)™

£

at,k{2-K\r2Y

t=o

|m|+n

1

£

~ (27rA)n 1

)dp

n+\v\+k-l -2it\r2

fc=0

_

2

2n—1 L j

Jo

ckbk + e- 27rAr2 P n+ |,| +fc _ 1 (2 7 rAr 2 )

k=0

(_b + e -2ir\r2 ;

(2TTA)" V

2n+|i/| + | m | - l (27rAr

2

)) .

So we may now observe that for <7^(A) = 0, we must have P2„+|„|+| m |_i(27rAr 2 ) = —b^lrXr . Since there can be only finitely many solutions to this kind of an equation, we are done. The arguement is that /M(A) = 0 for every p and almost every A, giving the consequence / = 0. When considering conditions (4), it is our overall Laguerre series equation that changes. Instead of °=

£ ( £ /M(AK,*(A))>V A + m >) n n i/e(z+) fie(z+)

61 we have £

W

A

> ) = Z2^ITy

1 £ ( £

/,(%,(A))<

r a

,»

The evaluation of wMi„(A) is identical to the situation of the sphere. We get an integral, as in (9), reducing to a polynomial n+|m|

E ^[JU^Ar 2 ),

fc=0

here abbreviated as P(A). Therefore wIJ,il/(X) = c<5M,„P(A). We then reduce to the equation £/M(A)W^(Z) =

__l__^c/M(A)p(A)>VA+m)/i(z).

We now break into cases, first considering when m = 0. This is the easy case, where we may write

Using the property of the Laguerre basis we then conclude for all /i 6 ( Z + ) n that

The expression cr 1 _ 2 n |S2n-i| _ 1 -P(A) — 1 is a polynomial as well and can have only finitely many zeros. Therefore we conclude /M(A) = 0 for every fi and almost every A. This proves / = 0 as desired. Next consider the more challenging case of nonzero m. Once again we rewrite the equation:

o= £

/**(*) (ra»-i[L , I

„6(Z+)»

V r

|2J2

"-X|

f

w ^ - K,) >

So we need to describe why the set

{cP(A)W* + m ,» - VOz)} M 6 ( z + ) n is a linearly independent set. The result then follows in this case just as in the other. The linear independence of this set of polynomials is a consequence of their degrees in each of the variables. The polynomial

62

(cP(A)VV^ +mM — Wptn) ( z ) is of degree pj + m,j in the variable Zj and Pj in the variable Zj for each j = 1 , . . . , n. Thus { c P ( A ) W * + m » - W^ i/1 (z)} M€ (z +) all have different degrees in (z\,..., zn,2\,... ,zn) and for this reason must be linearly independent. Finally we move to the case of integral conditions (5) wherein the region of integration is an annulus. This situation is comparable to that of the disk (3) with some added complications. The procedure is identical up to the integration used to evaluate pM(A) in the series expansion

/A(z)= J2

u(\)9ll(x)w^mA^

^<E(Z+)"

In the case of the disk (3), we had |m|+n

r

9ll(X)

= /

e

-Wp2„-i

ckLwl+k(4n\p2)dp

J2

which gave the result 9 {x) =

"

( d \ ^ ( 6+e " 27rAr2p ^+iH+i m i-i( 2 ^ 2 )) •

Here we must instead consider the following integral r2

/

|m|+n

e-2^"2p2n-1 1

£

ckLH+k(4nXp2)dp.

fe=0

Using the same techniques we get a slightly more complicated result: r2

\m\+n

/ i

k=0

1 (2TTA)"

2

b

+ e- ^^P2n+H+lml_1(27rXr21)

+ e-27r^P2„+H+|mM(27rAri)

Now it is not as easy to describe the solutions of the equation g^X) = 0. We have e-2^r'P2n+M+lml_1(2nXr2)+e-2^^P2n+luMml_1(2nXr2)

= -b.

The solution set is not as clear as the previous case, but we still claim there are only finitely many A satisfying this equality. This result follows the fact that

63

the set of A satisfying this equality are all isolated (no accumulation point). Just as in case (3) we then conclude that /^(A) = 0 for every p and almost every A. So / = 0 as desired. 3

Moments with Weighted Radial Measures on Disks and Annuli

Finally we have a motivation to ask what happens in the case of either the disk or annulus when a radial polynomial weight is added. Ideally we'd like to consider any arbitrary radial weight. We now observe how a radial weight in the form of a polynomial may be handled on either the disk or annulus. First consider the disk and the polynomial weight w(r2) = a m r 2 m + a m _ i r 2 ( m _ 1 ) + • • • + ao- The integral condition is then written as zmLsf(z,0)w(\z\2)dfi(z)

/

=0

(7)

J\z\
or equivalently f

z m L 6 / ( z , 0)do-(z)) w(p2)p2n-Hp

( /

JO \J\z\=p

= 0.

J

There is a smiliar integral condition for the annular: // r i < | z | < r

z m L g / ( z , 0)w{\z\2)dp(z)

=0

(8)

2

also written as f2 Jri

(zmLJ{z,

f

0)da(z)) w(p)p2n-ldp

= 0.

J|z|=s

We have already proven that these intgeral conditions imply / = 0 when w(p2) = 1 or equivalently when w(p2) = a, a constant. We now claim that the same is true for any polynomial weight w(p2), as described above. We have the following theorem. Theorem 3.1 Suppose we consider f G Lp(Hn), 1 < p < 2. Then f = 0 if and only if f satisfies integral conditions (7) or (8). Proof: The situation is identical to that of the disk (without polynomial weight) covered above until we begin to evaluate the integral.

«v,„(A) = cv„ f [! Jo \J\Z\=P

r^

B

K M ) | w{p2)P2n-Hp J

64 ,r

= cS^vJ

/|m|+n

e-^'VpV"

- 1

\ 2

E

cfcL|1/|+fc(47rAp ) U p

As before to complete the proof for the case (7), we need only to show that the set of A satisfying the equation g^(A) = 0 has measure zero. We know, as shown above, that for wo{p2) = «o we get -2jrAr 2

5M(A)O

k0 + e-J"*r / W n + l m l - i ^ A r 2 )

(2TTA)™

and next consider when wi(p2) = aefpe is only a monomial term. 9li(X)e = aej\-2*x'>2p2n+2e-1 \m\+n

dp

r

= a, £

Cfe

fe=0

/ r^^-'ii^tW)^

^°

1 =

ckLM+k(4ir\p2)\

I £

r

(2TTA)"+^

2

[^ + e _ 2 7 r A r p2n+|Ml+|m|+M-i(27rAr2)^

All in all for w(p2) = amp2m + am-ip2{-m~1"> H

h a 0 , we get

f=o -27rAr' :

=E

(2TTA)»+'

+

(2 7 r A)™+^ 2 "+l' i l + l m l + 2 f - l ( 2 7 r A r

}

i=0 fc

^ re

(27rA)

l

,

+e (2TTA)<

-27rAr2 V ^ -P2n+|M| + | m | + 2 ^ - l ( 2 7 r A 7 - 2 )

^(2^ ^

Li:

£,

(2TTA)'

We can conclude ^ ( A ) = 0 if and only if

Ee=o

k

t

(2TTA)^

K

,

-27TA7-2 V ^ - P 2n+| < x| + | m | + 2 < - l ( 2 7 T A r 2 ) (2TTA)*

_

' e=o By reasoning similar to the other cases, there can be only finitely many A satisfying this equation. Therefore we conclude that in the Laguerre series expansion

/ » = E/"(A)WM»

65

we must have /M(A) = 0 for every fi and almost every A. It then follows that / = 0 as claimed. The same kind of computation can be done for the annulus. We omit the detail here.

Acknowledgments This article is based on the lecture presented by the first author during the RIMS conference on "Prospects of Generalized Functions" which was held from 27th to 30th November, 2001 at the Research Institute for Mathematical Sciences, Kyoto University. This conference was in honor of Professor Mitsuo Morimoto on the occasion of his sixtieth birthday. The first author would like to thank Professor Takahiro Kawai and Professor Keiko Fujita for organizing the conference and their invitation. He would also like to thank their warm hospitality during his visit to Japan. This Research project is partially supported by a grant from National Science Foundation and by a William Fulbright Research Grant.

References 1. M. Agranovsky, C. Berenstein, and D.C. Chang, Morera theorem for holomorphic Hp spaces in the Heisenberg group, J. Reine Agnew. Math. 443, 49-89 (1993). 2. M. Agranovsky, C. Berenstein, D.C. Chang, and D. Pascuas, A Morera type theorem for L2 functions in the Heisenberg group, J. Analyse Math. 57, 281-296 (1992). 3. M. Agranovsky, C. Berenstein, D.C. Chang, and D. Pascuas, Injectivity of the Pompeiu transform in the Heisenberg group, J. Analyse Math. 63, 131-173 (1994). 4. C. Berenstein, D.C. Chang, W. Eby, L. Zalcman Moment versions of the Morera problem in C™ and H n , to appear in Advanced in Applied Math., (2002). 5. C. Berenstein, D.C. Chang, and Tie, Laguerre Calculus and its Applications in the Heisenberg Group, AMS/IP series in Advanced Mathematics # 2 2 , International Press, Cambridge, Massachusetts, (2001). 6. J. Delsarte, Lectures on Topics in Mean Periodic Functions and the TwoRadius Theorem, Notes by K.B. Vedak, Tata Institute of Fundamental Research, (1961).

66

7. P. Greiner, On the Laguerre calculus of left-invariant convolution (pseudo-differential) operators on Heisenberg group, Seminaire Goulaouic-Meyer-Schwartz XI, 1-39 (1980-1981). 8. L. Schwartz, Theorie generale des fonctions moyenne-periodiques, Ann. of Math. 48, 857-928 (1947). 9. E.M. Stein and G. Weiss, Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, New Jersey, (1971). 10. L. Zalcman, Analyticity and the Pompeiu problem, Arch. Rat. Mech. Anal. 47, 237-254 (1972). 11. L. Zalcman, Mean values and differential equations, Israel J. Math. 14, 339-352 (1973).

HEAT EQUATION VIA GENERALIZED F U N C T I O N S SOON-YEONG CHUNG Department of Mathematics, Sogang University, Seoul 121-742, Korea e-mail: [email protected]

Introduction It is well known that solutions of the heat equation in the upper half space is not uniquely determined by its initial value. But many mathematicians, such as Tychonoff[13], Hayne[6], and so on, have constructed a uniqueness class of solutions with some constraint on the growth of solutions. Here, we give a much larger uniqueness class which improves the results ever known so far. In addition, a reflection principle for the solutions of heat equation will be improved and used to get a uniqueness theorem for the temperatures in semi-infinite rod. Throughout this talk, it will be seen how theory of generalized functions, such as distributions, ultradistributions, hvperfunctions, work properly in this area.

1

Notations and Preliminaries

First of all, we introduce briefly the Gevrey class and ultradistributions of Gevrey type which will be very useful later. See [7] or [8] for more details. Let fi be an open subset of Rn, s > 1 and
0 there exists a constant C = C(K, h) > 0 such that s u p l d X z ) !
aeN£

x<EK

where No is the set of nonnegative integers. Here, we use the multi-index notations \a\ = ax + • • • + a„ and da = dpd^ •••0£», dj = •£-, j = 1,2, ••• ,n, for a = {a1,a2,--- ,an) G NJ. For h > 0 we denote by V]^h the set of all functions in £^(Q) with

67

68

support in K. Then V^

forms a Banach space under the norms denned by \da4>(x)\

..... \\4>\\.,K,h =

SUP

'

and is naturally imbedded into X>^ for k > h. We denote by T>$ the set of all functions in Vs^h for every h > 0 and we denote by 2?W (fi) the set of all functions in V^' for a compact subset K of

n. In fact, the fact that s > 1 makes it possible to construct cutoff functions and partitions of unity if necessary. The topologies of above spaces are defined as follows: (i) {(j>j{x)} S £^(fl) converges to zero in £^(fl), if for any compact subset K of fl and for every h > 0, sup

\daj(x)\ , i i , -> 0

as j -> oo.

(ii) {0j(x)} G X>W(n) converges to zero in V^S\ if there is a compact set K of fl such that supp^- C K, j = 1,2, • • •, and <£,• -> 0 in £W(fi). As usual, we denote by 2?W (fl) (by
£'(fl) C

£{s)'(fl),

where V(fl) and E'(fl) are the space of Schwartz distributions and the space of Schwartz distributions with compact support, respectively. It is well known that £W (fl) consists of the ultradistributions in Z)M (fl) with compact support in fl, like the space £'(fi) in V'(fl). In fact, since there exist cutoff functions and partitions of unity in T>^ (fl), the properties of the ultradistributions in P ' s ' (fl) are very similar to those of Schwartz distributions. In particular, the concepts of the support, convolution, and so on are denned in a very similar and natural manner. A partial differential operator p(d) of infinite order is called a partial differential operator of Gevrey order (s) if p(9) = ^ a

Q

3a,

\aa\ < CL^/als,

aGNJ

69

for some L > 0 and C > 0. Then it is easy to see that p{d)(j> e T>(a\Q,) for every e X>M(f2). Moreover, it is well known that the mappings

p{d) : D w ( f i ) -»• D (s) (fl), p(fl): p ( s ) '(n) -»• p (s) '(fi),

£{s)(Q) -> £ (s) (fl) ew'(fi)

-»• f W ( n )

are continuous.

2

Uniqueness Theorems

From now on we denote by E(x, t) the fundamental solution of the heat equation: ts

F(r filI,t,

_ J v - r t ) _ n / 2 exp(-|a;| 2 /4t),

••«>-{r

"l"

t> 0 t<0.

Then the following can be obtained by some tedious calculation. Proposition 2.1 Let g(x) be a continuous function on Rn satisfying that for some constants a > 0 and C > 0 \g(x)\
xeRn.

Then G(x, t) = g(x) * E(x, t) is a well defined C°° function in M.n x (0, l/4a) and satisfies (i) {dt - &)G(x, t) = 0,

0
(ii) \G(x,t)\ < Cexp(2a|z| 2 ),

l/4a 0
l/8a,

fnij G(x,t) -¥ g(x) uniformly on each compact subset o / E n as t -¥ 0+. iJere, * denotes the convolution with respect to x variable. The following is a key to prove the main theorems: Lemma 2.2 For any L > 0, s > 0 and for a small e > 0 there exist functions v(t), w(t) £ CQ°(M) and a differential operator p(d/dt) of infinite order such that p(d/dt)v(t)=S(t)+w(t); suppv C [0,e],

suppw C [e/2,e];

(1) (2)

70

p(d/dt) = Y^ak{d/dt)k,

\ak\ < Chk/k\s;

(3)

k=0

for some positive constants C and h and \v{t)\ < Cexp [-(cL/i) 1 ^ 5 - 1 )] .

(4)

where 6 is the Dirac measure and c is a constant depending only on s. Proof. We set 00

'<*> = £ /

eitxdt , P(it)

l[(l+L(/qs).

where p(C) = (1+C)V(C), Pi(C) 9=1

The function u(x) is the inverse Fourier transform of l/p(it). 1/(1 + iXt) is the Fourier transform of the function \e~xlx,

ux(x)

Notice that

x >0 x <0

to,

Hence u(z) is the convolution of a sequence of functions u\(x) with A= l,l,i,i/2s,i/3s,--- . This implies the properties u(x) — 0 for x < 0, w(ar) > 0 for x > 0, u(x)dx = 1,

and

p(d/dx)u(x)

— S(x).

J —:

Further, for x > 0 and g e N ,

(!)'»H^I/ < max :

-

t

{it)qeitx dt\ p(it) tq

j

,00

77T7 ' T— /

L

\\is+iLt\

7= 1 "

-2

Li

Thus if we use this and (x-y) 9 - 1 (9) u (y)dy Joo (?-l)!

t{x) = f

d*

2 27ry_00|l • iil

|Pl(«)|

1 -max * • ! ! 2 *

:—

'

71

then we have for each q € N and x > 0 xq \u{x)\ < — max|u ( 9 ) (t)| q\

< -

g!

t€R

* ~ V < 28-2e-£ii(*)1/a-1 2L« ~

Jfc! since inf -^ < 2e~'' 2 for t > 0. This implies that u(z) satisfies the inequality (4). In order to estimate a*, we write °°

lnPl «) = ^

/.OO

ln

(l + * W ) « / -

q=i

ln(l + LC/qs)dq.

71

for real C > 0. By substitution q = (L£) 1/,s £, we get y*ao

lnpi(C) < C ( i C )

1/s

/

ln(l + t - s ) d i = d ( L C ) 1 / s

JO

The above estimate obviously holds also for p(Q, with a different constant C\. For complex number £,

IpCOI^PdCD^expCCilLCI1/'). From the Cauchy equality

{ICI=fl} s

with i? = k /L, it follows

N
~

fcfcs

_

k\s

Therefore, the estimate (3) holds. By multiplying u(x) with a function in £( S )(R) which is equal to 1 in (—oo,e/2] and equal to 0 in [e, oo) the function v(x) can be obtained. By the definition of p(Q we can easily see that it is an entire function of order 1/s such that oo

P(C) = 2>*C\ This completes the proof.

\ak\
72

Now we deal with uniqueness of solutions of the Cauchy problem for the heat equation (dt - A)u(x,t) u(x,0)=
=f

in R n x ( 0 , T ) for i £ l " .

(T > 0),

For the uniqueness problem, it suffices, by linearity, to consider only the homogeneous case / = if = 0. It is well known that the temperature of the infinite rod is not uniquely determined by its initial temperature (see [3,5,9,11]). In fact, a very sharp counterexample will be given in the last part of this paper. The following is the famous uniqueness theorem which was originally given by Tychonoff[13]. Theorem A ([11,12,13,15]) Let u(x,t) Rn x [0, T] satisfying that (dt - A)u(x, t) = 0

in

be a continuous function on Rn x (0, T)

and for some constants a > 0 and C > 0 \u(x,t)\ < Cexpa\x\2

on

R" x [0,T\.

n

Then u{x, 0) = 0 implies u(x, t) = 0 on R x [0, T]. In above theorem we require that the solution u(x,t) must be uniformly bounded with respect to t variable. However, sometimes we need a uniqueness theorem with a milder condition in direction of t variable. Some authors have had interests in this direction and have relaxed the growth condition on t. For example, Shapiro [11] has showed following: Theorem B ([11]) Let u(x,t) be a solution of the heat equation in the strip 0 < t < c and bounded in every substrip of the form 0 < to < t < c. Suppose that (i) I K M J U O O = o(t~l)

as

t -»• 0 ;

(ii) lim u(x, t) = 0 except possibly for a countable set E ; (Hi) liminf tl^2u(x,t)

= 0 for every x in E.

73 Then u(x, t) = 0 in the strip 0 < t < c. On the other hand, Chung and Kim [4] showed the following: Theorem C ([4]) Let u(x,t) (0,T) satisfying:

be a solution of the heat equation in M.n x

(i) There exist constants k > 0 and C > 0 such that \u{x,t)\
+ -),

0
(it) lim / u(x, t)(j)(x)dx = 0 for every C°° function <j)(x) such that for every t—*0-r

h>0, \da(j>(x)\exp2k\x\ sup •!— ,,' , ,—[-± < co. a

Then u(x, t) = 0 on W1 x [0, T\. In the above two theorems they relaxed the growth condition on time so that the uniqueness classes determined by them are larger than that of Theorem A. In fact, Theorem C gives a much larger uniqueness class than others. But, nevertheless, the hypotheses are not so natural that one can apply them effectively. The hypothesis (ii) is stronger than u(x, 0) = 0, since there exists a nonzero temperature function satisfying (i) and u(x,0) = 0 (seen later). We give here a much better uniqueness theorem than those in [4,5,6,11]. Theorem 2.3 Let u(x,t) be a continuous function on Rn x [0,T] satisfying (i) (dt - A)u(x,t)

=0

in

Rn x (0,T]

(ii) \u(x,t)\ < Cexp [(a/t)a + a\x\2] in E n x (0,T] for some constants a > 0,0 < a < 1, andC >0. (Hi) u(x,0) = 0

on

M".

Then u(x,t) is identically zero on Rn x [0,T]. Here, T may be oo. Sketch of proof. In view of Theorem A given by Tychonoff we have only to show that u = 0 on K x [0, T0] for sufficiently small T0 > 0.

74

Let s = | ( l + ^ ) > l . Using the functions seen in Lemma 2.2 we define two functions G(x,t) and H(x,t) by

fT G(x,t) = /

u(x, t + T)V(T)(1T,

Jo and H(x,t)

= -

/

u(x, t + T)W(T)CIT.

Jo Then the integrals converge and define continuous functions on K" x [0,To]. Moreover, they satisfy the heat equation and the growth condition \G(x,t)\ < Cexpo|x||2 \H(x,t)\

and G(x, t) = g(x) * E(x, t),

H(x, t) = h(x) * E(x, t)

on E n x [0,T0] where g(x) = G(x,0) and h(x) = H(x,0). If we define u = p(—A)g(x) + h{x), then u belongs to V^ p(-d/dt)G(x,

t) = p(-A)G(x,

Then it is not hard to see that u= lim u(x,t) t-*o+

(K n ) and

t) = u(x, t) - H(x, t). in

V(s)''(Rn)

and u(ip) = lim / u(x,t)ip(x)dx t—*o j

= 0,

which implies that u = 0 as an element of T>^ (Rn). Then it follows that u(x, t) = = = =

p(-A)G(x, t) + H(x, t) p(-A)g*E + h*E \p(-A)g + h]*E u*E = 0,

which completes the proof. In the above proof, the continuity of u(x, t) at t = 0 can be weakened as follows:

75 satisfies the conditions in K n x (0,T)

Theorem 2.4 Ifu(x,t) (i) ( f l t - A ) u ( x , t ) = 0,

(ii) \u(x,t)\ < Cexp [(a/t)a + a\x\2] for some constants a > 0, 0 < a < 1 and C > 0 (Hi) lim /u{x,t)ip(x)dx

= 0,

^ePWfl")

toftere

s = ±(l + i ) ,

tften u(x,t) = 0 in E n x [0,T).

3

Example of Nonuniqueness

Now we show that the uniqueness class given in the previous section is an optimal one. In particular, it will be shown that the growth condition on the time variable is optimal. In the direction of the space variable it is well known (see [5]) that for every s > 0 there exists a C°° function u(x, t) ^ 0 satisfying the followings: (i) (ft - &)u(x,t)

= 0 in

R n x (0,T).

(ii) u is continuous on E n x [0,T). (iii) \u(x,t)\ < C£ exp \x\2+e (iv) u(x,0) = 0 on

on

W1 x (0,T).

Rn.

This shows that Theorem 2.3 is no longer true if we replace exp(a|a;| 2 ) by exp(a|x| 2 + e ),e > 0 in the condition (ii). Now we will show that Theorem 2.3 is also no longer true if we replace the condition 0 < a < 1 by a = 1. To see this let DN be a domain in the complex plane C given by DN = {zeC\z

= x + yi,

x>N,

-n
N>0

and CV be the boundary of ZV- Define a function u(x, t) on K x (0. oo) by «(*, *) = ir~- I E(x~ C, t) exp(e«)dC, 2m JcN where the integral is taken counterclockwise.

(5)

76

Since the function exp(e^) decreases very rapidly as Re£ —¥ oo on the curve CN the integral converges and u{x,t) satisfies (dt - A)u(x, t) = 0 in

E x (0, oo).

Also, Cauchy's integral theorem implies that u(x, i) is independent of TV > 0. Since the integral

± [

|exp(e^)||dC!

is finite it follows that \u{x,t)\
sup CeDN

\E{x-Q,t)\.

Writing C = £ + if] we obtain sup \E(x-C,t)\ CeDn

= -?

(x-tf-V2

sup. exp

\f4nt

4t

N<(

exp

ii ©r
xp

(x-02 4t

since £ = £ + "? implies N < £ and |7j| < 7r by its definition. Then it follows that for some constant a > 0 K M ) I < C(iV)exp (j)

d(x,DN) 4t

exp

21

(6)

where d is the Euclidean distance. Thus we have K M ) I < C(N)exp

(j)

in

W1 x (0,oo

(7)

Let r > 0 and x < r. Since the integral (5) is independent of N we may choose a sufficiently large N > 0 so that 4a < (N - r ) 2 . Then by (6) we obtain sup | u ( M ) I < x
C(N)exp

~4a-(N-r)2' 4t

t>0.

(8)

The right hand side of (8) converges to 0 as t ->• 0+ and u(x, t) converges uniformly to 0 as t —>• 0+ in every half-line (—00, r], r > 0. Therefore, we can conclude that u(x, t) is continuous on E n x [0,00) and u(x, 0) = 0. Now it remains to show that u(x, t) ^ 0.

77

To do this we suppose that u(x, t) = 0 in R x [0, oo). Then we obtain from (5) that

L

exp

(

*

C)2

At

1 • exp(c<)dC = 0

in R n x [0, oo). Applying the Lebesgue dominated convergence theorem we can see that I exp(e<)dC = 0. (9) JcN Since the integral (5) does not depend on N > 0 we may choose N = 0. Then (9) can be written as /•OO

/»7T

exp(-e c )d< - i /

0= - / JO

exp(eJ/i)d2/

J-1C

/»oo

c

+ /

exp(-e )dC

•/o

= -2i

[ ecosy Jo

cos(siny)dy.

But the integral eCOS2/cos(sin2/) > 0 on [0, ir], which leads a contradiction. Thus we can conclude that u(x, t) ^ 0. Remark. In [3], an example with the same estimate as above was seen. But it was more complicated than the above one.

4

Reflection Principles

In order to get a uniqueness theorem on a semi-infinite rod we need some reflection principle of the solutions of the heat equation. The reflection principles were originally given for the question about the analytic continuation of holomorphic functions across a hyperplane(see [10]). A similar consideration has been given also for the harmonic functions(see [10, 14]). It is well known that temperature functions, the C°° solution of the heat equation, behave very much alike harmonic functions or holomorphic functions. Thus it would be interesting to consider a reflection principle for temperature functions too. D. V. Widder [15] has considered this question probably for the first time as follows:

78

Theorem (Continuous version[15]) Let ft be an open subset of the plane R2 given by ft = {(x, t) € R 2 | \x\ < R, 0 < t < T}, R > 0, T > 0, and

ft+ = {(x,t) e f t | z > 0 } , ft~ = {{x,t) 6 f t | x < 0 } , E={(x,t) € ft|z = 0}. If u(x,t) is a temperature function in ft+ which is continuous up to E and vanishes on E then u(x, t) can be continued to the whole ofCl as a temperature function by the relation u{x,t) = —u(—x,t) on ft~. We are now in a position to state a theorem which improves the continuous version of the reflection principle for the temperature functions as follows:. Theorem 4.1 Let ft, fi^, and E be as above. If u(x,i) is a temperature function in ft+ satisfying that there exists s > 1 such that lim u(x,tU(t)dt = 0 x—>0+ J for every 6 V^ (0, T) then u(x, t) be extended to the whole of ft as a temperature function by the relation u(x,t) = —u(—x,t) on ft-. For the proofs of classical reflection principles for holomorphic functions or harmonic functions they usually depended on a very sophisticated functional analysis(for example, see [10]). Here for the proof of the above theorem we apply quite a different method from them. A basic idea of the proof is to reduce the problem to the case of the continuous version by expressing u(x, t) as

u(x,t) = P (-)

g(x,t) + h(x,t)

for a differential operator -P(^) of infinite order and temperature functions g(x, t) and h(x, t) in ft+ which are continuous up to the boundary E. In general, if we combine a uniqueness theorem and a reflection principle we can derive a new uniqueness theorem for temperature functions on a semiinfinite rod. As an application of Theorem 4.1 we will give here a uniqueness theorem for temperature functions on a semi-infinite rod.

79 Theorem 4.2 Let u(x,t) be a continuous function on (0, oo) x [0, T] satisfying the heat equation in (0, oo) x (0, T) and the followings: (i) There exist constants M > 0, 0 < a < 1, and C > 0 such that \u(x,t)\ < Cexp

t

,

+L\x\

(z,t)€(0,oc)x(0,T),

(ii) u(x,0) = 0 on (0,oo), (Hi) For every lim

/ u(x, t)4>(t)dt = 0.

Then u{x,t) = 0 on [0, oo) x [0,T]. Proof. At first, in view of (iii) we can apply Theorem 4.1 to obtain a temperature function u(x, 4 ) o n M x [0, T] which extends u(x, t) given on (0, oo) x [0, T]. Then it is easy to see that u(x,t) satisfies the conditions in Theorem 2.3. Therefore, it follows that u(x, t ) E 0 o n l x [0, T\. This implies u{x, t) = 0 on [0, oo) x [0, T], which completes the proof.

References 1. S.-Y. Chung, Uniqueness in the Cauchy problem for the heat equation, Proc. Edinbugh Math. Soc. 42 (1999), 455-468. 2. S.-Y. Chung, A stronger reflection principle for temeprature functions, J. London Math. Soc. 61 (2000), 543-554. 3. S.-Y. Chung, D. Kim, An example of nonuniqueness of the Cauchy problem for the heat equation, Comm. Partial Differential Equations 19 (1994), 1257-1261. 4. S.-Y. Chung, D. Kim, Uniqueness for the Cauchy problem of the heat equation without uniform condition on time, J. Korean Math. Soc. 31 (1994), 245-254. 5. A. Friedman, Partial differential equations of parabolic type, Englewood Cliffs, N. J.: Prentice Hall, Inc. 1964. 6. R. M. Hayne, Uniqueness in the Cauchy problem for the parabolic equations, Trans. Amer. Math. Soc. 241 (1978), 373-399. 7. H. Komatsu, Introduction to the theory of hyperfunctions, Tokyo: Iwanami 1978 (In Japanese).

80

8. H. Komatsu, Ultradistributions I; Structure theorems and a characterization, 20 (1973), J. Fac. Sci. Univ. Tokyo, Sect IA, 25-105. 9. P. C. Rosenbloom, D. V. Widder, A temperature function which vanishes initially, Amer. Math. Monthly 65 (1958), 607-609. 10. W. Rudin, Lectures on the edge of wedge theorem, CBMS regional conference series in math. Vol.6. 1971 11. V. L. Shapiro, The uniqueness of solutions of the heat equation in an infinite strip, Trans. Amer. Math. Soc. 125 (1966), 326-361. 12. S. Tacklind, Sur les classes quasianalytiques des solutions des equations aux derivees partielles du type parabolique, Nova Acta Soc. Sci. Uppsalla 10 (1936), 1-57. 13. A. N. Tychonoff, Uniqueness theorem for the heat equation, Mat. Sb. 42 (1935), 199-216. 14. N. A. Watson, Class of subtemperatures and uniqueness in Cauchy problem for the heat equation, J. London Math. Soc. 32 (1985), 107-115. 15. D. V. Widder, The heat equation, New York and London: Academic Press 1975.

B E R G M A N T R A N S F O R M A T I O N FOR ANALYTIC F U N C T I O N A L S O N SOME BALLS KEIKO FUJITA Faculty of Culture and Education, Saga University, Saga, 840-8502, Japan E-mail: [email protected] We consider the Bergman kernel for the Ap-ball which is defined by the iVp-norm and is related to the Lie ball. Then we study the Bergman transformation for analytic functionals on the iVp-ball and study a relation with the Fourier-Borel transformation. Introduction Let E = C n + 1 , n > 2. The cross norm on E corresponding to the real Euclidean norm ||a;|| is the Lie norm L(z) denned by

i W = {NI 2 + (INI 4 -k 2 l 2 ) 1 / 2 } 1 / 2 , where z • w = zxwx + z2w2 H p > 1 the function

h zn+iwn+i,

z2 = z • z and ||z|| 2 = z • z. For '•IV

Np(z)=(^{L(zy

+

(\z*\/L(z)y)

is a norm on E (see Baran * or Morimoto-Fujita u . Professor Jozef Siciak kindly communicated us that M. Baran had studied more general cases in 1). Note that Np(z) is monotone increasing in p and l i m p . ^ Np(z) — L(z). We define the open JVp-ball Bp(r) by Bp(r) = {z£-E;Np(z)
p > 1,

B(r) = Bx(r)

= {z e E;L(z) < r},

B[r] = B^r]

= {z e E; L(z) < r}.

and the closed A^p-ball Bp[r] by Bp[r] = {zeE;Np(z)
p>l,

We call B(r) (resp., B[r]) the open Lie ball (resp., the closed Lie ball). When p = 2, AT2(z) = ||z|| is the complex Euclidean norm and B2{r) is the complex Euclidean ball of radius r in E. In our recent research, we studied holomorphic functions and analytic functionals on the JVp-balls through their double series expansion (see our

81

82

papers 2 , 3 , 4 and 5 , for example). We are interested in the integral representation of holomorphic functions on the TVp-balls. In general, it is difficult to find the reproducing kernel in a concrete form, although the Bergman kernel for.the complex Euclidean ball (A^-ball) and that for the Lie ball are known explicitly. On the other hand, by noting that the Shilov boundary of the Lie ball is the Lie sphere, Morimoto 10 considered the Cauchy-Hua integral representation for holomorphic functions on the Lie ball and the Cauchy-Hua transformation for analytic functionals on the Lie ball. In this paper, we review some of our results on the iVp-balls and consider the Bergman kernel for the iVp-balls. Similar to Morimoto 10 , we construct a transformation for analytic functionals on the ./Vp-balls by means of the Bergman kernel, which we call the Bergman transformation. We study the cases of the complex Euclidean ball and the Lie ball in more detail. Then, we review some results on the Fourier-Borel transformation and study a relation between the Bergman transformation and the Fourier-Borel transformation. 1

Double series expansion

We denote by 0{G) the space of holomorphic functions on a domain G of E equipped with the topology of uniform convergence on compact sets, and by O(K) = ind \im{0(W); W : open, W D K} the space of germs of holomorphic functions on a compact set K of E equipped with the inductive limit locally convex topology. We put ||/||C(A-) = sup i 6 A : |/(z)|. It is well-known that a holomorphic function / on a neighborhood of 0 in E can be expanded locally into homogeneous polynomials fk: f — Y^T=ofkWe denote by Vk{E) the space of homogeneous polynomials of degree k in E. It is also known that the homogeneous polynomial fk £ Vk(E) can be expanded into homogeneous harmonic polynomials fk,k-2i' fk(z) = E\=o](z2)lfk,k-2i(z), where z2 = z\ + z\ + • • • + z2n+l and / M _ 2 , is a homogeneous harmonic polynomial of degree k — 21. We denote by "P£(E) the space of homogeneous harmonic polynomials of degree k in E. The dimension N(k, n) of V\ (E) is known as N(k,n)

= {2k + n-

l)(k + n- 2)!/(*!(n - 1)!).

Note that lim N(k,n)^k

= l.

(1)

k—t-oo

The harmonic extended Legendre polynomial Pk
k

ft,»(*,uo = (^nv^) pk,n(4= vz2

is defined by

• -?=),

Vw2

83

where Pk,n(t) is the Legendre polynomial of degree k and of dimension n + 1 . For / e C({0}) and sufficiently small p > 0, define

Note that it does not depend on p. Then we have f{z) = Yl'kLo A( z )- Further, define fk,k-2i by / fk(uj)Pk-2i,n{z^)diJ> Js where dw is the normalized invariant measure on the real unit sphere S. Then we have fk(z) = Ej!^ 2 l (* 2 )7 M -2j(*)- Note that fk u;) = ( 5 2 / ^ 2 + 52/c>z22 + • • • + d2/dz2n+1)Pk,n(z,w) = 0. Thus / € 0({O}) can be expanded locally by means of homogeneous harmonic polynomials as follows: fk,k-2i{z) = N{k-2l,n)

oo [fc/2]

2

/( ) = EE( z 2 )'/M-2iW,

(2)

fc=0 1=0

where

/*,*-«(*) = # ( * - 21, n) | |

_ ^^ft_2J)n(z,a;)dL;.

We call the right-hand side in (2) the double series expansion of / and fk,k-2i the (fc,fc— 2/)-harmonic component of / . We proved the following theorem in Fujita-Morimoto 5 : 5 THEOREM 1.1 (Theorem 2.2 in ) Let f G O({0}) be expanded into (2) and 1 < p < oo. If f G C(S p (r)), i/ien tfie sequence {fk,k-2i} of (k, k- 21)-harmonic components of f satisfies j

limsup f ( 2 f c " ( * , " ° ! ) '

1

||/M-2I||C(S))

< 1/r.

(3)

Conversely, if a sequence {fk,k-2i} of homogeneous harmonic polynomials satisfies (3), tften ifte double series SfcLoSLo (z2Yfk,k-2i(z) converges to a holomorphic function f in 0{Bp(r)). Similarly we have / e C(B p [r]) iff limsup ( ^ ^ Z ^ j

' ||/M_2«[|C(S) J

< 1/r.

84 2

Bilinear form of holomorphic functions on t h e A r p -balls

For z = x + iy, let Vp,r =

dxiAdyiA---

A dxn+1 A dyn+1

JBp[r]

be the volume of i? p [r]. We denote by dVPtr(z) the normalized invariant measure on Bp[r]; that is, „ dxi A dyi A • • • A dxn+1 A dyn+i dVp,r(z) = , 2.1 Let f e 0(Bp(r)) the double series:

and g 6 0{Bp[r\).

PROPOSITION

oo

oo

/(*) = £/*(*) = E k-0 oo

. z = x + iy. Expand f and g into

[*/2]

/* G P*(E), /M_2I e vkA~2l(E),

E ^ V J I W ,

k=Q 1=0 oo [k/2]

fc=0

fc=0

i=0

ITien we Aawe /

f(pz)g(z/p)dVp,r(z)

JBp[r] oo [fe/2]

-

(4)

l*2f'/M-2l(*)ffM-2l(zW,r(*),

=E E / fc=0 1=0

J

BAr]

where p < 1 is sufficiently close to 1. Put F(z) = f(pz)g(z/p).

PROOF.

Since dVp,r(z) = dVp,r{tz) for t e C

with |i| = 1, f JBP[T}

F(z)dVp,r(z)

= f

F(tz)dVp,r(z)

JBp[r]

= 5^ f = f.

JBM

f (

J\t\=i

F(tz)dVp>r(z)dt F(tz)^dV p,r(z). l m

85

Therefore, we have f

f(pz)g(z/p)dVPAz)

JBP[T} r

r

°°

= [ f(tpz)g(E/p)dVPtr(z) JBp[r] °°

JBP[r] J\t\=l t^O

Ai

2WI

J^O

oo

On the other hand, since dVp>r(z) = dVPi7.(T2:) for T £ SO(n + 1), we have /

F(z)dVPir(z)=

JBp[r]

f

F{Tz)dVPtr(z)=

JBp[r]

= / JBP[T]

! JSO(n+l)

f

I

F(Tz)dVpr(z)dS

J B„[r]

F(Tz)dSdVPtr(z),

JSO(n+l)

where dS is the normalized invariant measure on SO(TI + 1). Thus we have /

fk{z)9k{z)dVp,r{z) = f

JBp[r]

= I JBp[r] .

= /

fk(Tz)gk(Ti)dVp,r(z)

JBp[r]

I

fk{Tz)gk{T~z)dSdVp,r{z)

JSO(n+l) lk/2]

f

[fc/2] Tz

/

T,^yf^-^ ^12^y'9k,k-2v(Tz)dSdVPAz)

JBp[r} JsO(n+l) [fc/2] .

l=Q

[l=0

q.e.d. Since the right-hand side (4) does not depend on p, the bilinear form B.[ri= / JBP[T]

f(pz)g(z/p)dVp,r(z)

is well-defined for / e 0(Bp(r)) and g 6 0{Bp[r]).

86

3

Bergman kernel for the iVp-ball

Let HO(Bp(r)) be the Hilbert space of square integrable holomorphic functions on Bp(r); that is, HO{Bp{r))

= if

E 0{Bv{r));

\f{w)\2dVp,T{w)

[

< oo 1 .

Since \ f e 0(Bp(r)); f \f(w)\2dVp,r(w) [ J6p[r]

sup / \f(tw)\2dVp>r(w) 0/Bp[r]

= J / € 0(Bp(r)); [ we may call HO(Bp(r)) Put

< oo 1 J

the Hardy space on

< oo 1 , J

Bp(r).

H M - 2 i ( E ) = {(2 2 )7fc-2/W : fk-21 e P * - 2 ' ( E ) } . Then by Theorem 1.1 we have oo

oo

[fc/2]

k

fT0(Bp(r)) = 0P (B P H) = 0 0 Wfc,fc_2I(BpM), fc=0

where we put Vk{Bp[r]) = Vk{V)\§p[r]

fc=0

(5)

/=0

a,ndHk,k-2l(Bp[r})

=

Hk,k-2i(E)\Sp[r].

We denote by B?(z, w) the Bergman kernel for the JVp-ball with respect to the normalized invariant measure dVp
f

f(w)B?(z,w)dVp,r(w),

z € Bp(r).

(6)

By Proposition 2.1, for / € 0(Bp(r)), we have the following integral representation: T H E O R E M 3.1 Let f e 0(Bp(r)). For p with 0 (z,to/p)dVp,r(w),

zeBp(pr).

(7)

87 LEMMA 3.2

Pk-2l,n(*,U>) = (/?M,r) _1 /

\C\2lPk-2lAC,w)Pk-2lA^OdVp,r(0,

JBP[V]

where fi,l,r=

$C2)'Pk-2l,n(Cu)\2dVpA0,

I.

UJtS,

JBP[r]

which does not depend on u £ S. PROOF.

Let u £ S. Put

H

»(*)

= (/3fc,«,r) _1 / . IC2|2'A-2I,n(C,w)Pfc-2l,n(«,C)dVp,r(C)JBp[r]

Then Hu(u) = 1 and, as a function in z, Hw(z) £ V^~2l(E). SO(n + 1) such that Tu = u. Then HU(TZ)

= {Pi^T1

= Ki,r)~l

Take T £

\e\2lPk-2iAu,Qh-2iATz~QdvPAQ

f

\{T-H)TPk-vAT-lu,T-$Pk„2lAz,T-lQdVPAQ

/ J

BT[r]

1

= (Pll,r)~ f.

\(C)2\2lPk-2lA",(')Pk-2l,n(z,C)dVpAC)

JBp[r]

=

Hu(z),

where we put (' = T _1 C- Therefore since Hu(z) £ p£~ 2 i (E), as a function in z, is a zonal function and satisfies Hu(u) = 1, we have Hu(z) = A_2i,„(*,w) 9

(8) 21

(see Morimoto , for example). Further, as a function in u, Hu (z) £ ~p^~ (E). Thus (8) is valid for w £ E. Putting z = UJ £ S, we have 1 = A-2l,n(l) = ( ^ . J - 1 / .

\(C2)lPk-2l,n(C,u)\2dVpAO,

JBP[r]

which means /3£; r does not depend on u £ 5. LEMMA

3.3 Let
I

Then we have IC2|2Vfc-2l(0A-2l)n(«,C)^p,r(0-

q.e.d.

88

PROOF.

Because there are constants aj € C and UJJ € S such that N(k-2l,n)
=

53

aj-Pfc-2J,n(z,Wj),

the statement is clear by Lemma 3.2.

q.e.d.

The following proposition follows from Lemma 3.3. 3.4 Bpklr{z,w) = (0*, r )- 1 (^ 2 )'(^ 2 ) / Pfc-2i,n^,w) is the reproducing kernel on ~Hk,k-2i(E); that is, for
(z2)lVk-2l(z)

= /

(C2)W2j(0££,l>,0d^,r(C)-

By (5) and Proposition 3.4, we have the following theorem: T H E O R E M 3.5 The Bergman kernel for HO{Bp(r)) is expanded as follows: oo [fc/2] k=0 1=0

where Pil,r=

\((2)lPk-2lA(,")\2dVpAC),

l

COGS.

JBp[r]

4

Bergman transformation

We denote by 0'{Bp{r)) and 0'{Bp[r]) 0(Bp[r]), respectively. For T eO'{Bp[r}) define BpT(w) =

the dual spaces of 0(Bp{r))

(Tz,Bp(z,w)).

Then B£T G C(B p (r)) and we call the mapping Bp : T K> B £ 2 » = the Bergman transformation. More precisely we have

(Tz,BP(z,w))

and

89

4.1 The Bergman transformation establishes the following topological linear isomorphisms:

THEOREM

Wr:0'{Bp[r])^0{Bp{r)), B* : 0'{Bp{r)) We have (T,f)

^

0(Bp[r]).

= ( / . B y D ^ for T e 0'(Bp(r))

TeO'(Bp[r})

and f e 0(Bp(r)),

or for

andf£0(Bp[r}).

PROOF. We prove only the second isomorphism. Let T e 0'(Bp(r)). By the Hahn-Banach theorem, there is a Radon measure \i with supp/x c Bp(r'),r' < r, such that (T, f) = fg , ,, f(z)dp(z) for / e 0(B(r)).

Especially, we have WrT(w) = f

B*{z,w)dn{z),

JBp[r'\

and B£T can be extended holomorphically to Bp(r"),r" = r2/r'; that is, BPT€0(Bp[r}). Let / e 0(Bp(r)). Take p < 1 with r < pr". Then by (7), {T,f)=[

f

f(pw)BP(z,w/p)dVp,r(w)dn(z)

JBP[T'} JBP[V]

= [

f(pw) [

JBp[r]

= /

BP(z,w/p)dn(z)dVp,r(w)

JBp[r'}

f(pw)BerT(W/p)dVpAw)

= (f,B?rT)6

[p].

Thus, B£ is injective. The continuity of B£ is clear. Conversely, let F e 0{Bp[r]). Then there is r" with 0 < r < r" such that F £ 0(Bp(r")). Take p < 1 with r < pr" and define TF e 0'{Bp{r)) by (TF,f)=f

f(pw)F(W/p)dVPAu>)

= (f,F)S[r],

JBp{r]

Pi J

feO(Bp(r)).

Then WrTF{w) =

({TF)z,B^{z,w))

I,

BP(pz,W)F(z/p)dVptr(z)

= F(w), w 6 Bp[r]

lBp[r]

Thus B£ is surjective. The continuity of ( B £ ) - 1 : F \-¥ TF is clear.

q.e.d.

90 5

Bergman kernel for the complex Euclidean ball

In general, it is difficult to represent the Bergman kernel in an explicit form. In case of the TVp-ball, it is known for p = 2 and p-oo (Hua 6 ) . When p — 2, it is the case of the Euclidean ball and it is well-known that r2n+4

B2r(z,w)

2

w)n+2'

r —z•

Since

1 "(fc + n + pi 2 kl (1 " *i)"+ " to and k X

_[k^T((n ~ hs

+

l)/2)k\N(k-2l,n)

2fcr(fc - / + (n + l)/2)/! ^ - 2 i ' " W ,

Theorem 3.5 for p — 2 will be reduced to the following: THEOREM 5.1 Let z,w 6 E with \z-w\ < 1. The Bergman kernel can be expanded into the double series as follows:

B2(z,w)

B2r{z,w) ^[!^}r(*±±)(k =

g

g

+ n + iy.N(k-2l,n)

(z2\l

fw2\l

2 * r ( * - / + ( n + l)/2)«

[?)

[WJ

P

^n(z/r,w/r).

COROLLARY 5.2

r

/.

2

1kr2kv(k

2

Js2[r} 6

' \ \{ \

|(C )'A-2i,„(C^)| ^2„(C) =

r{-f-){k

— / -i- ™±Iyi

+n+

I ? >\ ly.N(k-2l,n)

Bergman kernel for the Lie ball

In this section, we consider the Bergman kernel in the case of p — oo; that is, the case of the Lie ball. The Bergman kernel Br(z,w) = B™(z,w) is known as r4n+4

B

r(z,w)

= (r4 _

2 2r

z •w+

z2w2)n+l'

Note that 7rn+lr2n+2

Vo r

°'

=

2"(n+l)!'

(9)

91

(see Hua 6 ) ; that is, putting dVr = dVoo^, (6) for / £ HO{B(r)) C

f( l

JB{r]{ -^{zlr)-{wlr)

+

is written as

\

{zlr)2{wlrY)n+1

Now we consider the double series expansion of the Bergman kernel for HO{B(r)). The Bergman kernel is related to the generating function of the Gegenbauer polynomials. For A > - 1 / 2 let C^{x) be the Gegenbauer polynomial in x of degree k and of order A, which is determined by the generating function: 1

{l_2xt

°°

+ t^=Ef^^

-1<*<1, I*I<1.

(10)

It is well-known that (10) holds on the interior E(x, t)° of the ellipse E(x,t) = Ux,t)

e C xB(l);max{\x±y/x2

-l\\t\}

= l} ,

where B(l) = {z € C; \z\ < 1}. (For the theory on the orthogonal polynomials, see Szego 12 , for example.) Namely, for a fixed t e E with \t\ = 1/r < 1, the series in the right-hand side of (10), as a function in x, converges for x € E if x belongs in the interior E(r)° of the Ellipse E{r) = < z e C; \z - 1| + \z + 1| = r + and the denominator of the function in the left-side hand of (10) does not vanish for x G E(r)°. Note that E(r) = E(x,t)\t=1/r. 2 Therefore, for z, w E E put * = Vz^Vw and x = -j== • -]£= in (10). Then we have

on the domain I (z, w) e E x E; |\/z2\/u^| < 1, max < z • w ± y (z • w)2 - vz^Vw2

\ = 1^

Now we define the homogeneous extended Gegenbauer polynomials C%(z,w) on E x E as the harmonic extended Legendre polynomials:

Ct(z,w) = (V?)*(v^)*C£(-£= • -^=). vr

v r

92 Note that C£(z, -) = C£(-,z) and C£(z, •) e Vk(E). The Gegenbauer polynomials and the Legendre polynomials are related by +1 c

*

_ r(fc + 2n + 2) ^ - ^ n + ^r^ + i)^

2

^

3

^-

On the other hand, [k/2] +1

CT (Z) = E

a

k,k-2lPk-1l,n(x),

(11)

1=0

where r(g±l)iV(fc-2Z,n) /-1 ak,k-2i = /=r/n\ / Ck (*) p *-2J,n(*)(l-« ) _ 2r(/ + ^ ) r ( f e + n - I + 1)JV(fc - 21, n)

^ 2

ds U

^

(n + l ) ! « ! r ( f c + 2 f i - 0 6

(see Hua ) . By (1) and (12), we have l i m s u p | a M _ 2 ( | 1 / ' : = 1.

(13)

fc—*-oo

By (11),

^(^)*(>^)*cri(4r-?f) Vz2

fc=0 f-i

Vw2

[k/2]

= E E aM_2,(V^)*(V^)*Pfc-2i,n(-^= fc=0 i=0 oo [k/2]

= E

E

W

^ 2

" M - ^ * ) V ) ' A-21,n(«, W).

(14)

fc=0 (=0

Because (13) and \Pk,n{z,w)\ < L(z)kL(w)k, (14) converges for z and w with L(z)L(w) < 1. Thus the Bergman kernel for the Lie ball is expanded into the double series as follows: 6 2 PROPOSITION 6.1 (Hua ) For z,w € E with L(z)L(w) < r , r4n+4

B

r(*,w)

2

= (r4_2r z-w OO

fc=o

+

z2w2)n+1

93 oo [fc/2]

ak,k-2l((z/r)2)l((w/r)2)lPk^n(z/r,w/r),

= E E fc=0 1=0

where ak,k-ii is given by (12). Prom this Proposition, (Pk,i,r)~l = ( Z ^ , , . ) - 1 ball can be calculated: ,_!

(n [Pk l r)

o>,t-M r2k

=

''

COROLLARY 6.2

(Hua

8l,VPk-2l,n(z,w)

6

2 r q + ^)T(k

=

m

Theorem 3.5 for the Lie

+ n-l + l)N(k - 21, n) r2k{n+l)W.T(k+^-l)

)

= ^ 2 1 r

\C\2lPk-2l,n(C^)Pk-2l-,n(z,0dVr(Q.

f JB[r]

We prove for r = 1 because we have

PROOF.

|w 2 | 2! P fc -2i,n(w,0-Pfc-2i',n(z,w)dVi(u;)

ak,k-2i / lB[l] JB[l] 1k,k-2l T2k

\w2\2lPk-2l,n{w,0Pk-2l',n(z,W)dVr(W).

• [ JB\T\ JBW\

When / = V, it is clear by Proposition 6.1 and Lemma 3.2. Let I ^ V. As functions in w, (w2)lPk-2i,n{w,C) S ^ ( E ) and 2 (W yPk-2i',n(z,w) e P f e - 2 C'-')(E). Since I ^ V, by Proposition 2.1, we have /

\W2\2lPk_2iA™,Oh-2rAz,™Wx{w)

= Q, for

IjtV.

JS[i]

q.e.d. Since we know / f(pw)(W2/p2)!Pk-2i,n(^/P^)dVi(w) y*[i] = Ofc,fc-2J / 'B[l]

=

/fc,fc-2i(^)|w2|2'A-2i,n(w/P^)d^lM

fk,k-2l(z)

= N{k~2l,n)

/ /fe,fc-2iM-Pfc-2!,n(«,w)dw,

we have ll/fc,fe-2/|lc(BFllJ

=

l i m s u

P

(ll/*,fc-2j||c(S))

(15)

94 If / 6 0(B(r)), then (15) holds for any p < r. Thus noting that \Pk,n(z, w)\ < L(z)kL(w)k and (13), we have /

xl/fc

limsup f ll/fc,fc-2/ltc(B[i]))

^Vr-

Therefore, in case of the Lie ball, Theorem 1.1 (Theorem 3.2 in Morimoto 8 ) may be restated as follows: THEOREM 6.3 Let f £ 0(B(r)). For p < r, define fk,k-2i{z) = ak,k-2i /

JB[X]

f(pw)(W2/p2)lPk^2i,n(w/p,z)dV1(w).

Then ( \1/k limsup (||/*,fc-2i|lc(Sril)) ^llr

( 16 )

an

dYX=aYdlo\z2)lfk,k-2i{z) converges to f in the sense ofO(B(r)). Conversely, if a sequence {fk,k-2i} of homogeneous harmonic polynomials satisfies (16), then the double series 5Z^=0 ]Cl=o (z2Y fk,k-2i(z) converges to a holomorphic function in 0(B(r)). 7

iVp-Fourier transformation

For an entire function / and a norm N(z) on E, we put ||/IU(r,Ar) = sup{\f(z)\exp(-rN(z));z

€ E},

X(r ) JV) = { / e O ( E ) ; | | / | | x ( r , J V ) < o o } . Then X(r,N) define

is a Banach space with respect to the norm ||/||x(r,JV)- We

Exp(E;(r,iV)) = {/ £ 0(E);Vr' > r, \\f\\X(r>,N) < oo} = f )

X(r',N),

r'~>r

Exp (E; [r, N}) = {/ 6 0 ( E ) ; 3r' < r, \\f\\x(r',N) < oo} = \J

X(r',N),

r' < r

and equip Exp (E; (r, N)) with the projective limit locally convex topology and Exp(E; [r, TV]) with the inductive limit locally convex topology. We denote by Exp'(E; (r,N)) and Exp'(E; [r,N]) the dual spaces of Exp(E; (r,N)) and

95 Exp(E; [r,N]), respectively. For a norm N(z), the dual norm N*(z) of N(z) is defined by j n * ) = sup{|z-C|;iV(C)
\z2\)/2}1/2.

The Fourier-Borel transform TT of T G Exp'(E; (0)) is defined by ^ T ( 0 = (ri)exp(z-0>, where ( is sufficiently small. Then we know the following theorem as a special case of a A.Martineau's theorem: 7 T H E O R E M 7.1 (Martineau ) Letp > l,q > 1 be the conjugate numbers; that is, 1/p + l/q = 1. Then the Fourier-Borel transformation !F establishes the following topological linear isomorphisms: T : 0\Bp[r\)

-^

Exp (E; (r, Nq)),

0 < r < oo,

T : 0'(Bp(r))

-^> Exp (E; [r, Nq]),

0 < r < oo,

JF : Exp'(E; (r,iVp)) ^ > 0(B,[r]), 0 < r < oo, T : Exp'(E;[r,iV p ]) -^ For / G 0(Bp(r)),

0(Bq(r)),

0 < r < oo.

we define the N p -Fourier transform Tp,Tf of / by

FP,rf(0=

/

/(/9w)exp(C-uJ/p)dVp,,-H

where p < 1 is sufficiently close to 1. We call the mapping TP)T '• f >->• -7>,r-/ the -/Vp-Fourier transformation. By the proof of Theorem 4.1, ((BP)-1/W,exp(z,W))= f

f(pw)exV(C-W/p)dVp,r(w).

JBp{r]

Therefore as a corollary to Theorems 4.1 and 7.1 we have COROLLARY 7.2 Letp > l,q > 1 satisfy 1/p+l/q = 1. Then the Nv-Fourier transformation Tp%r establishes the following topological linear isomorphisms: Tp,r : 0{Bp{r))

^

Exp (E; (r, Nq)),

0 < r < oo,

Tp,r : 0{Bp[r]) - ^ Exp (E; [r, Nq]),

0 < r < oo.

96 For T € E x p ' ( E ; (r, Nq)),

we have

1

{To ( B ? ) - oFT) (C) = TP,r

((Tz,exp(z,w)))

= ( Tz, / exp(C, / H exp(z,u7/p)dVp ir .(u;) ) \ JBP[r] I

= (Tz, \

exp(C,w)exp(z,W)dVp,r(w)

J

BP[r]

) . I

Put Ep>r(z, w)=

exp(z • C) exp(C •

w)dVp,r(()

JBP[T]

and consider the mapping £Pir : Tz M- £p,rT(w)

=

(Tz,Ep
for T G Exp '(E; (r, Nq)). Then the mapping £ p , r relates Exp '(E; (r, iV,)) with Exp (E; [r, JV9]) as follows: 0'(Bp(r))^

Exp(E;[r,7V g ])

4- B?

0{Bp(r))

t £p,r

AExp'(E;(r,iVg)).

Since the mappings T and TV,T = T o (B£) _ 1 in the above diagram give topological linear isomorphisms, we have P R O P O S I T I O N 7.3 The transformation cal linear isomorphisms:

Ep
the following

E x p (E; [r, TV,]),

0 < r < oo,

£„,,, : E x p ' ( E ; [ r , J V , ] ) -^> E x p ( E ; (r, JV,)),

0 < r < oo.

£Pyr : E x p ' ( E ; (r, Nq)) ^

Remark on the Lie ball Since ex; ^

^[^]r(^)iV(fc-2/,n)/2^2Wk

for / e 0(B(r)) we have frf(Q

= ^oo,r/(C) = / JBW]

/ ( / ™ ) exp(C •

W/P)dVr(w)

topologi-

97

by (12) and (9). Therefore, 00 [fe/2]

Fr(HB(r))

= \ F(z) = E

j>2)'.F*,*-2i(*) e 0(E) :

00 [fc/2]

E

E

k=o 1=0

2feZ!(fc — Z)! 1 T2fc H-F*,*-2/|lc(B[l]) < °° > • r

Moreover, the double series expansion of Er(z,w) as follow: Er(z, w)=

= E00t7.(z,w)

is given

exp(z • 0 exp(C • w)dVr(Q JB[r] [k/2]

00 [fc/2] [fc/2]

= EE

T(^)N(k-2l,n)\z

r

2 iT(2±i) V 2 iV(fc - 2J,n)(n +

l)lr2k(z2)l(w2)lPk-2l,n(z,w) 22k+lV.T{k - I + 2 f 1 ) r ( i + 2±a)r(fc - J + n + 1)

References 1. M. Baran, Conjugate norms in Cn and related geometrical problems, Dissertationes Mathematicae CCCLXXVII (1998), 1-67. 2. K.Fujita, On the double series expansion with harmonic components, Finite or Infinite Dimensional Complex Analysis: Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, 2000, 77-86. 3. K.Fujita, Double series expansions and their convergence domains, Proceedings of the third Asian Mathematical Conference, World Scientific, 2002, 175-180. 4. K.Fujita and M.Morimoto, Holomorphic functions on the dual Lie ball, Proceedings of the Second Congress ISAAC, Kluwer Academic Publishers, 2000, 771-780.

98

5. K.Fujita and M.Morimoto, On the double series expansion of holomorphic functions, to appear in J. Math. Anal. Appl. . 6. L.K.Hua, Harmonic Analysis of Functions of Several Complex Variables in Classical Domain, Moskow 1959, (in Russian); Translations of Math. Monographs vol. 6, Amer. Math. Soc, Providence, Rhode Island, 1979. 7. A.Martineau, Sur les fonctionnelles analytiques et la transformation de Fourier-Borel, J. d'Analyse Math, de Jerusalem 11, 1963, 1-164. 8. M.Morimoto, Analytic functional on the Lie sphere, Tokyo J. Math., 3(1980), 1-35 9. M.Morimoto, Analytic Funtionals on the Sphere, Translation of Mathematical Monographs 178, AMS, 1998. 10. M.Morimoto, A generalization of the Cauchy-Hua integral formula on the Lie ball, Tokyo J. Math., 22(1999), 177-192. 11. M.Morimoto and K.Fujita, Between Lie norm and dual Lie norm, Tokyo J. Math. 24(2001), 499-507. 12. G.Szego, Orthogonal Polynomials, Amer. Math. Soc, 1939.

E X P L I C I T C O N S T R U C T I O N OF F O U R I E R H Y P E R F U N C T I O N S S U P P O R T E D AT INFINITY

AKIRA KANEKO* Department

of Information

Sciences,

Ochanomizu

University

We generalize in two ways the method of construction of Fourier hyperfunctions supported at infinity proposed by Morimoto-Yoshino. Namely, on one hand, we generalize the growth order to most general admissible ones. On the other hand, we construct the example in general quasi-analytic ultradistributions.

1

Introduction

In [8] Morimoto-Yoshino invented a method to construct a Fourier hyperfunction of one variable supported only at a point at infinity, employing a kind of integral transform. The existence of such Fourier hyperfunctions was theoretically known as early as the introduction of the theory of Fourier hyperfunctions by M. Sato in which the flabbiness of t h a t sheaf was established. Their example, however, was the most concrete one ever known. It is as follows: T h e function exp(e* ) is entire holomorphic and increases just like exp(e x ) on the real axis. This does not serve as a defining function of a Fourier hyperfunction as it is, because it is not of infra-exponential growth on a strip neighborhood of the real axis off the real axis. But it decreases rapidly on the region 7r/4 < \xy\ < 3ir/4. Hence if we choose the path 7 as in Fig.l of §3, the integral 1

/• e x £ ( e £ )

converges and defines a holomorphic function which is bounded for z outside the path 7, hence on | I m z | > e for Ve > 0 via the deformation of the p a t h by the Cauchy integral theorem. Therefore the boudary value f(x) := F(x + iO) — F(x — iO) defines a Fourier hyperfunction which is obviously zero on the finite points. On the other hand, by the Cauchy integral formula we have for x on the real axis F(,)=exP(^)

+

-L/^dC. 2 « J1 (,-x

•Partially supported by GRANT-IN-AID FOR SCIENTIFIC RESEARCH No. 12640158.

99

(C)(2)

100 T h u s F(z) has the growth of exp(ez ) on the real axis, hence its boundary value f(x) becomes a non-trivial Fourier hyperfunction which should be supported by only one point + 0 0 at infinity. In this note we generalize their method of construction to produce Fourier hyperfunctions supported at + 0 0 with as general as possible growth by replacing the very concrete function exp(e 2 ) by a similar holomorphic function of arbitrary, but admitted growth. 2

Holomorphic functions b e y o n d t h e three-line theorem

We first recall the minimum necessary growth for the functions holomorhic on a strip neighborhood of the real axis with which the construction of MorimotoYoshino works. This is given by the classical three-line theorem: L e m m a 2 . 1 . Let F(z) be a function of one variable holomorphic in the half strip | I m z | < e/2, Re z > 0 and continuous up to the boundary. Assume that \F(z)\ < M on the boundary lines \lmz\ = e/2 and Rez = 0. If F(z) = 0(eAe

) for some A > 0 and e' > e, then \F(z)\ < M on the whole

strip.

Proof. Consider in fact G(z) = F(z)/

exp(6 cosh(7rz/e)),

where coshz =

ez + t~z

.

Then G(z) is bounded on the strip. It is obviously bounded by M on the boundary. Hence by the three-line theorem we have \F(z)\ < M|exp(<5cosh(7rz/e))| on the whole strip. Since 6 > 0 is arbitrary, we conclude t h a t |.F(z)| < M.

•

From this lemma we have to pose the following condition on our candidate of Fourier hyperfunction maker: Ve > 0 Va > 0 limsup

00

In the following we estimate several quantities related to a monotone increasing positive function x(t)- F ° r such a function notice the inequality

~xC-)< joX(r)dr
(2.1)

which is obvious from the observation of the corresponding areas. T h u s we will not distinguish tx(t) and the primitive of x(^) in the sequel. The following is our key tool:

101 T h e o r e m 2 . 1 . Letx{t) be any positive function oft > 0 which is monotone increasing to oo as t —> oo. Then we can find a function G(z) such that, setting z — x + iy, we have 1) G(z) is holomorphic in the region D:=

{z = x + iy; x > 0, \y\ <

—-7},

and continuous up to the boundary. 2) There exists constants A, B > 0 such that eAxX(Ax)

<

G

^

< eBxX(Bx)

(22)

on the real axis. 3) There exists a constant C > 0 such that on the curves y = ±—-—- we X(x) have ReG{z)

< -CeCx^Cx\

(2.3)

Proof. Let h(z) be the (unique) conformal mapping from D to the rectangular strip E:={z

= x + iy; x > 0, \y\ < | }

such that it maps the three boundary points ±i/x(0), +00 to ± « ' / 2 , + 0 0 in the same order. By the Ahlfors distortion theorem') (see [1] or an expository reference [7]) estimating the breadth of the image of D n {ari < R e z < #2} in terms of the length of the vertical cut of D, h(z) satisfies on D I>X

PX

7T /

Jo

X(t)dt

- 4TT < Re h(z) < IT /

Jo

x{t)dt{\

+ o(l))

for x large enough. (Usually the distortion theorem is stated for the mapping from a generalized strip region to a standard strip | I m z | < a / 2 with two endpoints ± 0 0 . But we can apply it to our mapping h for a half strip, because in view of the reflection principle we can extend it as —hi^—t) to a full strip obtained by the symmetry with respect to the imaginary axis.) Now set G(z) = exp(h(z)). This clearly satisfies all the conditions 1) -

3).

•

t)l owe this kind of knowledge to the late Professor Kotaro Oikawa, who was my teacher and then colleague.

102 For x(t) — ta, a > 0 we can more concretely find G(z) as above by choosing h(z) — z1+a. The example of Morimoto-Yoshino is the special case a = 1 here. Also, for x ( 0 = exp(exp(- • • exp(i) •••))> w e c a n choose h(z) — z exp(exp(- • -exp(z) • • •)). The most interesting examples are those of the least possible growth which are non-trivial. For such growth we have a more explicit construction: T h e o r e m 2.2. Let
f /o

f(s)2

(2.4)

decreasing to zero. Then the infinite oo

G

monotone

ds < oo

Jo

and that
or equivalently,

of s > 0 which is

product

2

w-n(i+^)

the integral

ex P {£°log(l + - ^ ) d S } gives a function

which satisfies all the conditions

X(t) = t I

of the above theorem

with

(2.5)

~^ds.

Before proof we remark t h a t in case • oo. We have

[°°

I

\z\2

\

Since we are expecting a function of super-linear growth, the second term in the leftmost side is asymptotically negligible as \z\ —• oo. Thus to derive the estimate, it suffices to consider the integral.

103 We have

f MI+^)NfM1+ /o

*>

Jo

V

¥>( s ) 2 y

ds

V

J
¥>(s) '

Here we have, setting
/

izl2 \

/

r1

i

i\

iz

io g (i + _ L ) _ ^l(W)-dt

iog(n.J£L)dfi= /

^ V 1 ^ ) ^ ^ ^ ^ ^ ^ ^ - (2-6) Here we used the monotone decreasing property of f'(s). by the same change of variable we have

r

On the other hand,

log (1+-%)ds = r log (1+1) _j£L__dt \V\i)) p

\z (M*)) r°°

1

dt

I

i M tV(?-H*))

rf< (2.7)

and also f°°

/

/

\z\2

f°°

\

log(l + - ^ W S > ( l o g 2 ) | z | 2 /

1

-p-rfs

(2.8)

We can see that the quantity in (2.6) is dominated by the one in (2.7). This is established by Lemma 2.2 below. T h u s we have seen t h a t the infinite product converges locally uniformly and defines an entire function G(z), which satisfies with x(<) as in (2.5) and with some B > 0 \G{z)\ 2 we have 1*1 > l z l / 2 , hence

l0g 1 +

(

^)l

=1 g 1+

° | ^l^l0g|1

+

Ms)2

>log

1 + 4ip( y s

104 By the same calculation as above and in view of (2.8), the integral of the last quantity can be estimated from below by A | z | x ( A | z | ) for another constant A > 0. Namely, we have \G(z)\ > e ^ ' l x U M )

for |y| < 1, x > 2.

The decreasing estimate 3) is more delicate: We calculate the argument of the infinite product: oo

]C

ar

s( 1 +

x2 — y2 + 2xyis

ar

J2

ip(n)2

J

x2 — y2 + 2xyi ip(ny

« (X +

)+

ar

« i1 +

Y,

n>ip-1(2x)

x2 — y2 + 2xyi
)

•

When x > 2 a n d |y| < 1 each term in the sum for n < ip 1(2x) can be estimated as arg ( l +

x2 — y2 + 2xyi\ 2x\y\ 2x\y\ = Arctan ip(n)2
Hence we obtain

ar

1

| J2 s i + l

x2 - y2 + 2xyi
n<tp- (2x)

<

f1

2\y\

)\<-l

f1

2

J0 4 ^ + |^(^-i(2^)) *-./„ ^ A r c t a n ^ ^ L

¥>_10)

2x\y\

V{s)2 + f x2

1

ds

\y\

fiTT^^-ipx))

4 . 4 ^ "T^Arctan —;= |i/|x(2x).

Here in the last inequality we used Lemma 2.2. On the other hand, each term in the sum for n > ip~l(2x) can be estimated as Arctan

/ 44 2x\y\ \ 2x\y\ -L-^2 2 \5
< arg 1 + < Arctan

x2 — y2 + 2xyi^
= Arctan

2

f(n)

2x\y\ + x2 — y2

2x\y\ ip(n)2

hence, as 3 2ir|?y| 5 iyj(n)

2

arg ( l +

x2 - y2 + 2xyi>
t

<

2x\y\
105 provided x is large enough. Thus the sum for n > ip~ (2x) can be estimated as V^

, x2 -y2

(,

n>v?- 1 (2rc)

+ 2xyi\ V

^

y

2x\y\

n>¥>~ 1 (2x)

V

;

Thus if x is large enough the total argument is majorated by ( l + - ^ A r c t a n - ^ ) \y\X(2x)

~ 3.683 • • •

\y\X(2x).

For the estimate from below, we can assume t h a t y has a fixed sign. Let, e.g. y > 0. Then we have, •^

/

arg 1 +

£

x2 -y2

l

Since the difference of / I1+ V

+2xyi\

^

3

|2/| (2a;)

)>5 *

-

the constants satisfies 4 4 \ 3 — A r c t a n —= 1 = 3.083 • • • < x, v^3 V ^ 5

we can find c > 0 such t h a t when a; is large enough the total argument of G(z) on the line y — ±c/x(2x) remains in a compact interval inside (x/2,3ir/2). Therefore with a constant C> 0 we obtain \G(z)\ < -CeCx^Cx) thereon. D L e m m a 2.2. Assume that a positive monotonously to oo as t —+ oo ara
TTten ii

satisfies

function

r

\

as < oo.

t

I

s2

J*

tf>(t) of t

>

0

grows

The proof is elementary. In fact, we have

r*t*>r*?*The left-hand side is in general of small order with respect to the right-hand side. But case ip(t) — tq they become the same.

106

Figure 1. Path of integration.

3

C o n s t r u c t i o n of Fourier h y p e r f u n c t i o n s s u p p o r t e d at infinity

Now we follow the construction of Morimoto-Yoshino employing the holomorphic function prepared in the preceding section. Let G(z) be a function given in Theorem 2.1. P u t

i i^maidc

(3I)

where 7 is a p a t h as described in Fig.l, which passes through the region where G(z) has negative real part. In view of the estimate (2.3) for G, we see that the above integral converges and the result is a bounded holomorphic function of z outside 7. Since by Cauchy's integral theorem we can deform 7 as close to the real axis as possible, F(z) is holomorphic and bounded on | I m z | > e for every e > 0. Since we can deform 7 so t h a t it is as farther as possible to the right, we see also that F(z) is entire holomorphic. But it is not bounded on the real axis. The growth can be seen by the following deformation of the integral by means of Cauchy's integral theorem: for real x inside 7 we have

^)=exp(G(,)) + -L/^P(/C. 27H 7 7 C- X Since the second term on the right-hand side is bounded, F(x) shows just the same growth as exp(G(a;)). Now f(x) — F(x + iO) — F(x — iO) gives a well-defined Fourier hyperfunction which is zero at the finite points and also at —00. It is nevertheless non-trivial because if F(x + 2O) — F(x — iO) = 0 as Fourier hyperfunction, F(z) would have t o become of infra-exponential growth on a strip neighborhood of the real axis. T h u s s u p p / must be equal to one point + 0 0 .

107 We can construct examples in several variables by making the product:

g(x)=f(Xl)S(x'). n

Since the base space D has no product structure, the above (tensor) product is not trivially meaningful in the corresponding space of Fourier hyperfunctions of n variables. It can be interpreted individually as follows: {g,
:=

be any rotation matrix. Then the Fourier

9A(X)

:=

g{A~lx)

is defined by {9A,
gA{x)ii(A)dA,

JSO(n)

where dA is the Haar measure of SO(n). We can differentiate these further by local operators of infinite order in the variables x. It is an interesting theme to study how general is such a construction to obtain Fourier hyperfunctions supported by the sphere at infinity. 4

Fourier u l t r a d i s t r i b u t i o n s s u p p o r t e d at infinity

First we recall the space of ultradistributions a la K o m a t s u [ 6 ] . Since we are concerned with quasianalytic type, however, the results cited below mainly come from de Roever [ 2 ] . Let Mp denote a monotone increasing sequence of positive numbers satisfying the following conditions. (M.0) extending real analyticity: for some B,C Mp>CBpp\,

p =

> 0 0,1,2,...;

( M . l ) logarithmic convexity: Mp2<Mp_!Mp+1,

p=0,l,2,...;

108 (M.2) stability under convolution: for some A, B > 0 Mp
min MqMp_q,

p=

0,1,2,...;

0
The condition ( M . l ) is equivalent t o the monotone decreasing property of Mp/Mp+i. From this follows the inequality Mp_qMq

< MpM0

for 0 < q < p,

(4.1)

hence

X> ? M P _,M ? < 2»MpM0,

(PC, = ( j ) = ^ ^ ) .

T h e condition (M.2) is grosso modo reciprocal to this. We further pose the following condition: (M.3) quasianalyticity:

,

M

P

Also we require the following in order to utilize the characterization by boundary values: (M.4) concavity: p—rz—

is monotone decreasing.

From the sequence Mp we derive the following function of t > 0: ^(t) = s u p l o g — A . p

(4.2)

M

P

This is a non-negative function satisfying the following properties: (/i.0) sublinearity: for some B, C > 0 H(t) < Bt + C; (n-1) fi(t) is a convex function of logi, monotone increasing t o infinity; (fi.2) n and 2/z are equivalent in the sense t h a t for some B,C > 0 H(t/B)

-C<

2n(t) < fi(Bt) + C;

(4.3)

The first inequality of (4.3) is rather trivial in view of the monotonousness of//. T h e second one corresponds t o (M.2). Now, (M.3) corresponds to the following:

109 (/i.3) quasianalyticity:

Two sequences {Af p }^i 0 and {Np}pxL0 exist B, C > 0 such that C-lB~vMp
are said to be equivalent if there p = 0,1,2,....

(4.4)

Equivalent sequences obviously define the same space of ultradifferentiable functions or ultradistributions. The equivalence is translated to fi as follows: fi(t) and v(t) are said to be equivalent if there exist B,C > 0 such that C~1v(t/B) Now (MA)

< n(t) < Cv(Bt).

(4.5)

corresponds to the following

(fiA) concavity: T h e least concave majorant of /i(t) is equivalent to

n(t).

Given a function n(t) as above, we deduce conversely the sequence JVp = Ar 0 sup< p e-"W,

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

This again satisfies (M.0) to (M.2) if// satisfies (fi.O) to (p. 2), and also (M.3) resp. (MA) if fi does (/i.3) resp. (fiA). If/i comes from Mp via (4.2), this JVp is equivalent to the original sequence Mp in the above sense. Now we define the Fourier type variant of Komatsu's spaces which was introduced in [ 5 ] . Recall t h a t the function /i(t) characterizes the decay order of the Fourier transform for the differentiability measured by Mp through the Paley-Wiener type theorem. T h u s , denoting N = { 0 , 1 , 2 , . . . } , we can introduce the following space of Fourier type test functions: S { M ' } = {v{x)

e S; 3B,C>

0, \Da
M]a\e~ ^

x

^

B

for Va G

Nn}.

This space is invariant under the Fourier transform. Hence we can define the Fourier transform on its dual space <S'' p' of "M p -tempered ultradistributions". Similar spaces are treated by Ehrenpreis and Gelfand-Shilov already long ago. W h a t we newly remark here is t h a t this space is localizable as explained in the sequel to the directional compactification Dn U S " - 1 of R" and gives a sheaf 5 ' ' pi of "M p -tempered ultradistributions", whose restriction to Rn agrees with the usual sheaf 2)'^ "' of ultradistributions. Thus, more generally, for a compact subset K C Dn, we define the space of ultradifferentiable test functions as follows: S^Mp}(K)

= {
n Rn);

3C > 0, 3B > 0, Vx G K, Va G Nn

\Da
110 The growth condition is meaningful only if K contains points at infinity. Notice t h a t compact subsets of Dn, when restricted to Rn, are not necessarily bounded in usual sense. If K is thin or not enough regular, we define the corresponding space by the inductive limit of similar spaces defined on a compact neighborhood of A'. Then S'-Mp'(K) becomes a D F S ( = d u a l FrechetSchwartz)-type space. The dual of S^M^(K) is an FS(=Frechet-Schwartz)p type space S (K) of the corresponding class of ultradistributions supported by K. Taking locally finite sums of these for sections on an open set, we can legitimately define the sheaf S'* p' of M p -tempered ultradistributions as mentioned above. The relation between this and the usual sheaf of ultra-distributions of class {Mp} is similar to the one between the sheaf Q of Fourier hyperfunctions and the sheaf B of hyperfunctions. For the quasianalytic type ultradistributions as we are considering now, the sheaf <S'' p' becomes flabby and the restriction 5 ' { M p } ( D n ) - • V'{Mr]{Rn) is surjective just as Q(Dn) — Q(Rn) = B{Rn) is. Hence we can easily derive the theoretical existence of elements of S " (Dn) supported at one point at infinity. We are going to present below a more concrete example using the construction of the preceding sections. The last preparation is the characterization of ultradistributions by boundary values. Let j(t) denote the function defined by t(t)

, tPplMo = sup log — — — .

, x (4.6)

Mp

P

The difference from the definition for fi(t) is t h a t this one contains the new factor p\ in the numerator. Its relation to n(t) is 7 « = sup{/i(p)-£}, P>O

/*(*)= i n f { 7 ( - ) + - } .

t

P

P

(4.7)

P

The condition of quasianalyticity in terms of 7 is

where j'(t) is understood as the derivative of any regular function which is equivalent to the original one. Typical example of these functions in quasianalytic class is as follows, which corresponds to each other as above modulo equivalence: Mv = f(logPr We have

(0 < a < 1),

,(t)

= ^ l p

7(t)

= i!^.

(4.8)

111 T h e o r e m 4 . 1 . (Komatsu [ 6 ] , de Roever [2]) A hyperfunction f(x) £ B{Q) pi belongs to V if and only if it can be represented as a finite sum of boundary values of holomorphic functions Fr(z) from the wedges J? + iT which satisfy for some A > 0 \Fr{z)\
12/1

J

locally uniformly in x £ Q and y/\y\ £ T C\ S n _ 1 . We need a Fourier variant of this characterization. In this article we skip to present the detailed theory for this and prepare only the following global version: T h e o r e m 4 . 2 . Let F{z) be a function satisfies for some A > 0, B > 0,

\F(z)\ < C e x p {A7(A±-) 1

Then the boundary value f(x) ,{M ]

in a wedge Dn + ir

holomorphic

+

Bn(B\x\)\. J

\y\

— F(x+ir0)

which

as a Fourier hyperfunction

belongs

n

toS ' {D ). Proof. The Fourier transform of f(x) as a Fourier hyperfunction is calculated as follows: Let Xj( 2 )i 3 — l,.--,N be a set of cut-off functions, i.e. each Xj(z) i s holomorphic on a strip neighborhood of the real axis and decreasing exponentially and locally uniformly with respect to the direction outside the closed convex cone A'j with vertex at the origin, such t h a t $3,- = 1 Xj — 1- For an elementary example of such family see [4]. For our application to the case of one variable, it suffices to choose X\{z) — e V ( e Z + 1)> X2{z) = l / ( e Z + 1) and A\ =R+,A°2 = R~. Now JV

where ^•(0=

/ JRn+iey

e-"<Xj(z)F(z)dz,

with a fixed y £ F. Then each Gj has the estimate, for Ve > 0 | G j ( C ) | 0 sufficiently small sup {XT] + e\x\\r)\ + Bfi(B\x\)}

there exists c > 0

0

As is shown in [2], formula (1.26), this is equivalent with 7 ( l / M ) . On the other hand, by [ 2 ] , formula (1.21), the quantity

inf{eyt+Ay(A^)}
I

E\y\

i

is equivalent with /i(|£|) provided (MA) the new constants A > 0, B > 0,

*

I-

A

e\y\

J

is assumed. Thus we obtain, with

|G,-«)| < Cexp {A»(A\Z\) +

Bj(B^)}

locally uniformly in TJ/\T]\ G —AJ n S n _ 1 . Now let J(Q be an entire function of infra-exponential type which further satisfies, with some A' > 2A, Ie2AM(A|f|) < | J ( C ) | <

CeA'KA-\i\)

on the strip neighborhood of the real axis. Such J(£) can be produced by an infinite product of the type (2.5), where 2, C,2 should be understood as £i -\ hCn-) For * n e details of the proof of the growth estimate see [3] or [ 4 ] . Then the inverse Fourier image of each Gj(£ — iAjO)/J(£) becomes a continuous function. We have N

1

f(x) = J(D)^- [g(0/J(0]

= J2J(D)Fj(x

+ ir°)'

j= l

where each (27T) n jRn_illU)

J(Q

is continuous up to the real axis. Notice t h a t the local operator J(D) preserves the class <S'* v . For the global sections this can be easily seen through the Fourier transform and multiplication by 3(C). Since J(D) works as a local operator (i.e. sheaf homomorphism) to Q, and <S'^ T' is a subsheaf of Q, both flabby, we can conclude t h a t J(D) acts also locally to 5 ' * . T h u s we finally obtain a global section of 5 ' * "' as the boundary value. • Now we present the main result of this section.

113 T h e o r e m 4 . 3 . Let G(z) be a holomorphic function as given in Theorem 2.1 in relation with the growth-indicating function x(t) there. Then the function F(z) = eG^ satisfies for some B > 0 \F(x + iy)\ < C e x p [exp {flX'1

( A ) }] •

(4.9)

Hence F(x + z'O) — F(x — iO) becomes an Mp-tempered ultradistribution supported by one point +oo at infinity, where Mp is a sequence which corresponds to j(t) := exp(5<x _ 1 (-B<)) via (4.6). Proof. Let \y\ > e and deform the p a t h 7 of the integral (3.1) up to x > R so t h a t the path is included in t h e region \y\ < c, i.e. R = x _ 1 ( l / e ) - Since the integrand is bounded except for the vertical segment connecting (R, ± e ) , we can estimate F(z) in (3.1) as follows: \F(z)\

< Cexp{eRx^)

< Cexp{ex'l^'c^s)

< Cexp(ex_1(1/l!/|)/l!'l).

By Theorem 4.2, we see t h a t the boundary values of such a holomorphic function belongs t o a class of quasianalytic ultradistributions as above. • For example, if we choose x(t) = ta (a > 0), the above boundary value belongs to a class with /i(<) = / l o t ia/(i + 0 ) and j(t) =

jl + l / o

e

fl+1/a

• T h e marginal

case /i(i) = ^-^ cannot be achieved because X - 1 ( 0 m (4.9) increases in any case. For general \ , the condition of quasi-analyticity for the corresponding function y(t) reads as

Jf

^-Z-dt = 0 0 .

Since X _ 1 ( 0 i s increasing t o 00, this condition seems to be satisfied more than necessary. Therefore our estimate (4.9) might be a little improved. For our way of construction, it is obvious t h a t there exists a relation between the growth of F(z) on the real axis and the regularity of the boundary value defined by it. Moreover, the relation is inverse t o t h e common sense: the faster x(^), accordingly F(z), increases on the real axis, the slower x _ 1 ( 0 increases, hence the milder the boundary value F(x ± iO) becomes. It is an interesting problem t o examine if this is the case independently of the method of construction. (Notice t h a t apart from the restriction of support, a function of any growth can be extended u p t o the infinity in any class of quasianalytic type ultradistributions.)

114

References [1] Ahlfors L. : Untersuchungen zur Theorie der konformen Abbildungen und ganzen Funktionen, Acta Soc. Sci. Fenn. 1, N o . 9 (1930), 1-40. [2] de Roever J.W. : Hyperfunctional singular support of ultradistributions. J. Fac. Sci. Univ. Tokyo Sect.lA 3 1 (1985), 585-631. [3] Kaneko A. : On the structure of Fourier hyperfunctions Proc. Japan Acad. 4 8 (1972), 651-653. [4] : Introduction to Hyperfunctions (in Japanese), Univ. of Tokyo Press, I: 1980, II: 1982 (English translation from Kluwer, 1988). [5] : Liouville type theorem for solutions of linear partial differential equations with constant coefficients Annates Polonici Math. 74(2000),143-149. [6] Komatsu H. : Ultradistributions I J. Fac. Sci. Univ. Tokyo Sect. 1A 2 0 (1973), 25-105. [7] Komatu Y. : Theory of Conformal Mappings (in Japanese), Kyoritsu, I: 1944, II: 1949. [8] Morimoto M. and Yoshino K. : Some examples of analytic functionals with carrier at the infinity Proc. Japan Acad. 56 (1980), 357-361. Department of Information Sciences, Ochanomizu University, 2-1-1, Otsuka, Bunkyo-ku, Tokyo 112-8610 J a p a n e-mail: [email protected]

On infra-red singularities associated with QC photons Takahiro KAWAI * Research Institute for Mathematical Sciences Kyoto University Kyoto, 606-8502 Japan and Henry P. S T A P P t Lawrence Berkeley National Laboratory University of California Berkeley, CA 94720 U.S.A.

1

Introduction

Block and Nordsieck [1] showed in 1937 how to remove the infra-red divergences from quantum electrodynamics: compute the low frequency classical part to all orders in the fine structure constant, compute a probability, exploit a cancellation in the classical part when a probability is computed, and use perturbation theory only on the surviving remainder. Many applications were made, but in the period from 1965 to 1975 workers such as Kibble [2], Chung [3], Storrow [4], and Zwanziger [5] showed that the momentum-space applications of this method gave incorrect behaviors at large distances, or equivalently in the singularity structure. In particular, the pole-factorization •Supported in part by JSPS Grant-in-Aid 11440042 ^Supported in part by the Director, Office of Science, Office of High Energy and Nuclear Physics, Division of High Energy Physics, of the U.S. Department of Energy under Contract DE-AC03-76SF00098

115

116

property associated with the large-distance fall-off of the propagator of a charged stable particle was disrupted. This creates difficulties with the reduction formulas and with the interpretation of the theory, because the defining characteristics of a physical particle are disrupted. In a 1983 paper [6] the second-named author showed how this problem could be overcome by going to a coordinate-space treatment. In coordinate space one can center the emissions of the classically describable soft-photon bremsstrahlung radiation associated with a deflection of a charged particle on the coordinate-space vertex at which the deflection occurs. This improves the accuracy of the classical part of the calculation. It also allows the photon interaction to be separated into two parts, a classical part and a quantum part, and allows all the contributions of "classical" photons, which are photons that interact only via the classical part of the interaction, to be gathered into a single exactly definable unitary operator that effectively drops out of computations of probabilities. The remainder is expected, by the BlockNordsieck argument, to be nicely convergent in the infra-red regime, because the soft-photon classical part has been treated exactly. One unexpected feature of this treatment pertains to the propagators of the QC photons, which are the photons that connect onto the chargedparticle lines by a quantum coupling at one end and a classical coupling at the other: these photons are propagated by a retarded propagator, rather than the usual Feynman propagator. In 1995 we published a set of three papers [7] that examined in depth the remainder terms involving the soft quantum couplings, and concluded that they were all finite, and gave no contribution to the singularity structure that was as strong as the leading pole contributions. However, in this treatment we used, without explanation, the Feynman propagators for all lines. The reason was that the second-named author had argued informally that we could go back to an earlier form of the result in which only Feynman propagators appeared. However, this author has recently re-examined that argument and has concluded that the final form given in the 1983 paper [6] must be used. The present paper addresses the resulting problem of incorporating the retarded propagators. A key causality property that was available before, namely that positive energy flows always forward in time, is no longer available for QC photons, and the earlier arguments now fail. We have not encountered any reason to believe that the desired analyticity properties will actually fail, but have not so far been able to construct, as we did before, a general proof that these

117

good properties will always hold. In the present paper we examine some simple cases and find that the expected properties do hold. We also find, and will report, some interesting mathematical properties of these functions that emerge from the microlocal analysis. The fact that retarded propagators should appear in an expression that is the result of just summing and rearranging the usual Feynman series may seem strange. The proof was given in the 1983 paper [6], and will not be reconstructed here. But the reason is easily described. Consider a Feynman diagram D consisting of one or more charged-particle closed loops, plus a set of hard-photon external lines connected to the diagram at a set of coordinate points x = (xi,X2,... ,xn), plus a set of soft-photon lines. [Diagrams with external charged-particle lines are treated by exploiting the proved pole-factorization properties associated with the internal chargedparticle lines.] Let F£,(x) be the scattering operator in photon space that corresponds to this coordinate-space diagram, but with no classical photons. Then the result of adding to it the contribution associated with all numbers of classical photons, both internal and external, is F°{x) = exp(a* • J(L(x)))F% exp(-(a • J(L(x)))) x exp(i${L(x)) - (J*{L{x) • J{L(x)))) = U(L{x))F^(x), where U(L(x)) is the unitary operator formed by bringing together the three exponential factors from the preceding lines, and the subscript "opr" signifies that the Feynman propagators for QC particles have been changed to retarded propagators. The change in the propagator type is caused by the fact that the creation and annihilation operators for photons are no longer normal ordered; the factor exp(—a • J(L(x))) that annihilates classical (Ctype) photons has been moved to the left of the operator JFj£ that creates quantum (Q-type) photons. This change introduces an extra mass-shell delta function for negative-energy photons flowing from a C-type vertex to a Qtype vertex. This changes the Feynman propagator to a retarded propagator, for QC-type photons (cf. (2.3) below). We need the unitary operator U{L(x)) on the left so that it will effectively drop out when a probability is computed, as explained in detail in the 1983 paper [6]. The problem is then to show that the remaining factor has for the dominant analytic structure the same structure found for theories of massive particles. In our 1995 papers [7] we studied the simplest case with six external lines

118

and six internal lines, with the external momenta arranged so as to put three of the internal particles far away from the mass shell, and the other three close to the mass shell. We then need to verify that the dominant singularity structure along the usual triangle diagram singularity surface is exactly the one associated with the triangle diagram surface when all relevant particles are massive. That singularity type is logarithmic. Our purpose here is to show that this result continues to hold, at least for the simplest cases, when retarded propagators, rather than Feynman propagators, are used for the QC photons.

2

Characteristic features of the problem

In this section we explain some characteristic features of the problem that we encounter by using the retarded propagator l

(2.1)

{k0 + iO)2 - k2

instead of the Feynman propagator 1 k + tO '

(2.2)

2

where k is the momentum from the C-vertex to the Q-vertex, flowing always from right to left. Note that they are related in the following manner: (2-3) V

- = j ^ - — + (2m)6-(k2), (Jbo + iO) - k2 k2 + i0 y ' K h 2

where 6~(k2) stands for 0(-A;o)<5(A;2) (cf. e.g. [8]). Here, and throughout this paper, we use the same symbols and notations used in our 1995 papers [7]. In our formalism each right-hand end-point C lies on one of the "hard" vertices vi,v2 and v3. Hence our conclusion in [7, p.2510 ff.] concerning the rightmost vertex VR is unaffected by the inclusion of C-vertices; each VR must coincides with some Vi, i G {1,2,3}. However, the arguments pertaining to the left-most vertex Vj, are disrupted. A typical example is shown in Figure 1.

119

Figure 1. A diagram with VR = Vi and VL = Q. A starred line stands for a pole factor in the sense of [7]. Diagrams of this sort raise the following question: what is the effect of the singularities associated with the inside triangles formed by Vi, v2 and Q? An important point to note is that points where kx + k2 = 0 with k\ = k2 = 0 may be relevant to the resulting singularities; such points are irrelevant if all photon propagators are Feynman propagators, i.e., if only QQ photons are considered. As we will show in Section 3, the study of such singularities is an interesting new issue in microlocal analysis. The vanishing of ki + k2 leads to a singularity in the residue factor l/p2{ki + fa)- To supply appropriate dbzO's to the denominators in the residue factors we need to decompose the domain of integration according to the relative magnitudes of \ki\'s so that |fa| = |fa| does n o t touch the boundary of each integral (considered in polar coordinates). (Cf. [7, p.2497].) Since such a decomposition is not unique, we have to consider all relevant terms simultaneously to assert that the net contribution is not stronger than the logarithmic singularities along the Landau surface associated with the triangle diagram. That is, we have to simultaneously consider diagrams in Figure 2 if we want to discuss the effect of the diagram in Figure 1.

(b)

(c)

Figure 2. Diagrams accompanying the diagram in Fig. 1. We note that we have to analyze each diagram in Figures 1 and 2 separately making use of different techniques. (Cf. Section 3.)

120

3

Study of some basic examples

In this section we study some basic examples of QC couplings and confirm that the resulting singularities are not stronger than the ordinary triangle diagram singularities, i.e., the logarithmic singularity. Actually we confirm they are strictly weaker than the logarithmic singularity. Let us first study the diagrams in Figures 1 and 2. For the sake of uniformity the diagram in Figure 1 is labeled as (3.1.a) and diagrams (b) and (c) in Figure 2 are respectively labeled as (3.1.b) and (3.1.c). In what follows we often omit the dotted external lines for the sake of simplicity. Using the power-counting result ([7, p.2496 ff.]) and assigning -HO uniformly to each residue factor ([7, p.2507]) the integral Fc associated with the diagram (3.1.c), which is the simplest to analyze, is given in the polar coordinate system for ki's (cf. [7]) by (3.2) below. Here and in what follows we use the following labeling and orientation of lines of triangle diagram:

(3.2)

Fc(gi,q2)=

fd4p3-2

^—r-2

^y—-

with (3.3)

G = / dri / dr2 ( d ^ K ^ A - 1) Jo Jo J 1

x I dAQ,25{9,2fl2 - \) (P3-n(fii + r 2 0 2 ) ) 2 - m 2 + i O 1 1 1 2 2p2{Sl1 +r2Sh) +iO(Qlfi -HO) - fi? (fi2,0 + i0)2 - H22 where Qj = (Qjfi,-Uj) for Q, = (QJi0, Oj), K > 0 and c > 1. Here K denotes a cut-off parameter, c designates the domain of integration, and we

121

have dropped small and unimportant terms in the denominators and ignored numerator factors. By ignoring physically unimportant contribution from r\ = K, we find (3.4)

G = (log(pf - m2 + tO)) x H,

where (3,5)

H= f dr2

[dtQiSiSltd!

I) fd4Q2S(Q.2fi n22-l)

1 2p2{Sll + r29,2) + iO 2p3{£li + r2Q2) - tO 1 1 2 (fi1>0 + iO) - T?; (fi2i0 + iO)2 - ?f2 ' We now verify that H is non-singular if we ignore the non-physical contribution from r2 = c. If H is non-singular, (3.4) immediately entails that Fc has the form (3.6)

Atp\og(ip + iO)+B

with ^4 and B being analytic near the Landau surface {ip = 0}. As we show below, microlocal analysis summarized in the form of Landau table is effective in confirming the analyticity of H. The Landau table for the integral H is as follows, if we ignore the contribution from r2 = c. dp2 dp3 1 Q.I + r2Cl2 0 -(fix + r2Q2) 2 0 0 0 3 0 0 4 0 5 0

dQi p2 -p3

~r2P3

CTxfix

0

0 ±ftx

(72^2

dfi2 7"2P2

0

Here Oj = signfij)0. Note that the sixth row corresponding to S(Q2Q2 — 1) has been omitted in the above table as in [7]; the closed loop condition for the dr2-column, which is also omitted in the above table, guarantees that the row is irrelevant to the singularity of (3.5) (cf. [7, Appendix A]). Needless to say, the closed loop condition for the dr2-column originates from the r2integration in (3.5). On the other hand, the integral (3.5) does not contain

122

reintegration. Hence we have to include the fifth row in the above table. Fortunately, however, there is no net contribution from the inclusion of the fifth row. In fact, the closed loop conditions are: (3.7)

aijD2 - a2P3 + a 3 ^ i ^ i + A A = 0 ,

(3.8)

oeir2p2 - <*2?"2P3 + a4<72ft2 = 0

with otj non-negative and 05 real. Multiplying (3.7) by fix and (3.8) by f22 respectively, and then summing them up we find (3.9)

/3 5 fii^i = 0,

i.e.,

p5 = 0

by noting that ot\ (resp., a 2 ) vanishes unless p 2 (fii + ?*2fi2) (resp., Pz{Q,\ + r 2 fi 2 )) vanishes. The relation (3.9) means that we may ignore the fifth row. By the mass-shell constraint on p 2 and — p3, Qip2 — a 2 p 3 cannot be a light cone vector unless ( a i , a 2 ) = 0. Thus (3.7) and (3.8) imply a;- = 0 (j = 1,2,3,4). This means that H is well-defined and non-singular. Therefore Fc has the form (3.6). Next we study the integral associated with the diagram (3.1.b). In this case the residue factors are non-singular when flj = fi| — 0- Hence nothing peculiar to QC-couplings can occur; the analysis of the integral is basically the same as that for non-separable diagrams in QQ couplings ([7, p.2514p.2515]). For the convenience of the reader we briefly discuss how the computation is done. Let us concentrate our attention to a point near r-^ = 0 and r 2 = 1. Since the closed loop condition for d^-column (j = 1,2) in the Landau table forces the Landau constant associated with Q^ (j = 1,2, respectively) to vanish, it suffices for us to consider the following integral:

(3.10)

Fb(q2,q3)=

fdn 0

x

/

d4fii

f dr V2 1-K'

/

d 4 f i 2 l o g ( ^ ( § 2 - r i r - 2 f i 2 , 9 3 - n f i i ) + «0),

where K and K' are sufficiently small positive constants. Here we have used the fact that the singularity of the integral associated with the diagram in

123

Figure 3 is a logarithmic one near the triangle diagram singularity surface {

dip - r - = -Oipi, oq2

dip ^— = a3p3 , aq3

where a-j is the Landau constant associated with pj, we find (3.12)

d
If we consider the problem in a neighborhood of {rj = 0, QJ = Q% — 0, fii/f^}, we can readily confirm that the right-hand side of (3.12) does not vanish. (Cf. the remark at the end of this paragraph.) With this nonvanishing property of dip/dri we can easily compute the integral (3.10) to find it has again the singularity of the form (3.6). Thus the singularity structure of Fb is again described by (3.6). (The non-vanishing property ofdip/dri can be verified as follows: if £l\ = —Q2 and r2 = 1, the right-hand side of (3.12) is equal to (3.13)

(aipi + a3p3)fi2 •

Using the closed loop condition for the ordinary triangle diagram we find it is equal to (3.14)

-a2p2Q2 •

Since we consider the problem at a point where a2 ^ 0, this is different from 0. If Ui = Q2 and r2 = 1, then, as aipi — a3p3 is a massive vector, (aipi - 0:3^3)^1 cannot vanish either. Thus the right-hand side of (3.12) is shown to be different from 0 at the point in question.) Let us now study the integral Fa associated with the diagram (3.1.a). By the same computation done for Fc, we findjthat the singularity of Fa is again of the form (3.6) if the following integral H is well-defined and analytic

124

ignoring the non-physical contributions from r2 = 1 ± K. l+K

(3.15)

I d4Qx

H= I dr2 1-K

x

|n,|=i

f d4£l2 |n 2 |=i

1 2p2(tll + r2Q,2) + iO 2pi(fii + r29.2) + iO 1 1 =*

=y>

(fii,o + iO)2 - fi2 (122,0 + *0)2 - ^ 2

The trouble is, however, that the usual reasoning based on microlocal analysis cannot guarantee the well-definedness of H; in the Landau table for the integral H all columns may sum up to 0 with some non-zero Landau constants. (The so-called u — 0 problem.) This means the product of several factors in the integrand of (3.15) is not guaranteed to be well-defined by the general theory of microlocal analysis [9]. If such a point "naturally" appears, we usually find that the singularities of relevant factors are rather tame and that the tameness (such as continuity) makes their product welldefined. Fortunately we can find some "tameness" in this case, although the "tameness" we encounter below is a quite novel one. To find the tameness we have to compute the integral explicitly. To perform the explicit computation we replace the propagator by 5(fi2) and use the frame where p2 = (m, 0,0,0). Letting Xj denote the component of Clj in the "j^i direction, and normalizing ?n = 1 for simplicity, we can then rewrite the integral (3.15) on a neighborhood of {fij = -£22} in the following form up to a constant factor: l+K

(3.16)

I = \-K

dr2

dxi / dx2 -1

1 1 — r2 + iO 1 — axi — r2(l — ax2) — iO' where a is a non-zero and small positive analytic function of p\. Here the —iO in the second factor is due to the fact that PitoP2,o < 0. The Landau

125

table for (3.16) is as follows:

dr2 -1

dx\ 0

1 — ox 2 a 0 ±l(end) 0 0

do 0

dx2 0 -ar2 0

X\ -

±l(end)

r2x2 0 0

Here l(en<j) indicates that the component survives only at the end point x, = + 1 or - 1 . It is evident that u = 0 points appear when x\ = x2 = ± 1 and r2 = 1. However a straightforward computation shows l

(3.17)

l

/ dx1 / dx2r J J 1 - axi - r 2 ( l - ax2) -3

-

IO

-1

- J z- [ ( l + o)(l - r2) log{(l + o)(l - r 2 ) - iO} a r2 - (1 - o)(l - r 2 ) log{(l - a)(l - r 2 ) - iO} + 9], where (3.18) g =(2or 2 + (1 + o)(l - r 2 )) log{2ar 2 + (1 + a)(l - r 2 ) - iO} + (-2or 2 + (1 - o)(l - r 2 )) log{-2or 2 + (1 - a)(l - r 2 ) - iO} is non-singular near r2 = 1 (as a is small and non-zero). Thus we have clearly found the origin of the u = 0 problem, and at the same time, understood why it is not a real problem. First the first two terms in the right-hand side of (3.17) respectively come from X\ = x2 = —1 and Xi = x2 = 1, and they are boundary values taken from the domain {Im(l - r2) < 0}, while the integrand of / contains a factor (1 — r 2 + iO) - 1 . Fortunately the singular part of (3.17) contains a factor that kills this singularity (1 - r 2 + iO) - 1 . Thus the integrand of J is well-defined in spite of the existence of a u — 0 problem. Then it is clear that the resulting function is analytic in a. Therefore H is well-defined and analytic; hence the singularity of Fa associated with the diagram (3.1.a) is again of the form (3.6). Remark. The argument given above shows that, if we assign — iO instead of -HO to each residue factor in diagrams (3.1.a), (3.1.b) and (3.1.c), then a u = 0 problem appears in the diagram (3.1.c) and the computation of Fa

126 becomes simple. It is also worth mentioning that the resulting singularities of Fa,Ft and Fc are all of the form (3.6), which is much weaker than the ordinary triangle diagram singularity. To see what occurs in more complicated diagrams let us examine the following diagrams given in Figure 4; each of them contains two QC photon lines and one QQ photon line. As we will see below, we can analyze the functions associated with these diagrams by the same method as that used to analyze integrals associated with diagrams (3.1.a), (3.1.b) or (3.1.c); the difference is just a combinatorial complexity. (a)

CL

(b)

^

^

zm~» /Hi-* +i0 Q Q

+i0

+t0 +i0

-NO

-NO

+i0

(c)

^ ^

/I%

+i0

-NO

'

+i0

^ . ,"zm.-."zM

-NO -NO -NO Figure 4. Diagrams with two QC photon lines and one QQ photon line.

The reason we treat these 9 diagrams in Figure 4 at the same time is that the uniform assignment of ±i0 to the residue factors on the bottom segment (i.e. +i0) and those on the right slope (i.e., ~i0) forces us to use several different techniques to analyze each diagram. For the sake of simplicity we discuss the problem in the region where \kj\ (j = 1,2,3) are of the same magnitude, i.e. r 2 ^ 0, r 3 ^ 0 with rj > 0 so that both (pi(^2 -^3^3) - « 0 ) - 1 and (^2(^1 + ^2^2) + *0) _1 may become singular. Let us now study the singularity structure of the function associated with each diagram in Figure 4. In what follows we freely use the power-counting result obtained in [7, p. 2496 ff.] in rewriting the integration over A:-variables to that over the (r, Q)-variables.

127

Case(a): Let us study the integral associated with the diagram given in Figure 4(a). The Landau table is: dfii 1 2 3 4 5 6 7 8

ri{pi+Ti{ni+r2T3n3))

0 ri(p2+ri(fli+r2llg))

P2 0 crfii 0 0

dQ2 0

dp

dQ3 rir2r3(pi+rj(na+r 2 r3n3))

pi+ri(ni+r2r 3 fJ 3 )

~P\

r3px

-n2+r3n3

rir2{p2+Tl(ni+T2U2))

0

P2+n(nj+r 2 fi2)

0 0 0 0 fi8

fii+r2n2

T2P2 0 0 0,2 0

P3 0 0 0

Here p denotes a loop momentum of the triangle diagram and we omitted the rows corresponding to non-singular residue factors such as (pi£l2 — iO)-1 etc. The symbol a indicates the line 1 is with a retarded propagator, as usual. The symbol cr3Q-3 may be used, but for simplicity we omitted a3. Since a2pi + o/.$r2p2 (<*2, <*4 > 0) cannot be a light-cone vector unless it vanishes, the closed loop condition for the dfi2 column implies a2 = a^ = 0. Hence we may detour the singularity at pi(fl2 — r3Q3) = 0 and that at p2(£l\ + r2Q,2) = 0. Hence by integrating over the triangle loop momentum p first, we are to calculate the following integral: fff d^d^d4^ f drx f dr2 f dr3 J J J Jo Jr2&\ Jr3«l |ni|=|n2|=|nal=i x /A(92 - rir 2 r 3 n 3 +r 1 r 2 n 2 ,93 - fi^i - rxr2Q2), where /A denotes an analytic function multiple of the triangle singularity, i.e., \og(ip(q2, q3) + iO). Here we have used the power-counting result for Qcouplings to rewrite d4k to drdQ, without extra divergent factor like rf 1 . We now calculate (3.19)

(3.20)

Fa=

— (q2 - nr2r30,3 + r1r2Q2,q3 - r^

- r 1 r 2 n 2 )| ri= o

= Tr~{-r2r3n3 + r2Q2) - A ^ i + ^ 2 ) dq2 oq3 =aip1{r2r3Q3 - r2Q2) - a3p3(Qi + r 2 Q 2 ), where a,- denotes the Landau constant associated with (p?-m 2 -H0) _ 1 in the triangle singularity. By choosing the detours so that Impi(r2fi2 — r2r3£l3)

128

may become much bigger than Imp3(S7i + r2Q,2) we find J^-|n=o does not vanish. Hence the integral (3.19) can be readily computed to produce the singularity of the form (3.6). Case(b): In studying the integral associated with the diagram in Figure 4(b), it is easier to analyze the associated integral by choosing a new loop momentum p3 + ki = p as the integration variable. Then the Landau table is as follows: dCli

1 2 3 4 5 6 7

0 0

dSl2 Tiri(pi+rir2(fl2+r3flt))

rir2(P2+nr2f!2)

P2

^2P2

-np3

0 0

CTQI

0 0

n2 0

dn3 rir2r3{pj+riT2({l2+r3n3))

dp Pl+nr 2 (fi2+r3n 3 )

0 0 0 0 0

P2+nr 2 n2

ns

fil+r2ft2 ps-nfli

0 0 0

Here pj denotes pj +fcx.Since a3y>2 — aATip3 cannot be a light cone vector unless it vanishes, the closed loop condition for the dfij-column implies a 3 = 0. Hence we can detour the singularity p2{£l\ + ^2^2) = 0. This time the integral Ft that corresponds to Fa is obtained by replacing the integrand of Fa by (3.21)

U(q2 - rir2r3n3,q3 - r ^ + r2fi2)).

On the other hand we find (3.22)

^-{g2 - rir 2 r 3 fi 3 ,«3 - MQi + r 2 fi 2 ))| n = 0 =aipir2r3Cl3 - a3p3(D.i + r2Q,2).

Hence by choosing the detour so that Imp3(Qi + r2fi2) is much greater than Impifi 3 we find the same result as for the diagram in Fig. 4(a). Case (c): To study the integral associated with the diagram in Fig. 4(c), we use a method different from that used for analyzing the integral associated with the diagram in Fig. 4(b) (although the same method may be employed). Choosing p3 + ki +fc2= p3 as a new variable, we integrate (p3 - m2 + iO 2ri(p3(fii + r2f22) - i0))~l over dri. The contribution from rx = 0 is then

(3.23)

l f c * t f i — ' + ")l( W n , + Uj- a )-

129

Then the "Landau table" for the product (3.24) (pj(n2 - r3n3) - »"0)-1(p2(n1 + r2fl2) + 2'0)-1(p3(fii + r2fl2) - iO)-1 x ((fii,o + iO)2 - Qly'iQl + tO)-1((fis,o + iO)2 - j ^ ) " 1 is given as follows: 1 2 3 4 5 6

dfi! 0 Pi

-Pz CTjfii

0 0

dtt2

dQ.3

dpi

-Pi r2P2 -f 2 P3 0

rzp\

-ft 2 + r3ft3

^2

0 0 0 0

0

CT3Q.3

0 0 0 0 0

dpi dp3 0 0 Qi + r2VL2 0 0 -(Oi+r2Q2) 0 0 0 0 0 0

Since -a\p\ + a2r2p3 - a3r2p3 cannot be a light-cone vector, the closed loop condition for the dfi2-column implies ot\ = a2 = a3 = 0 (as r 2 / 0); then we also find a 4 = a 5 = a 6 = 0. This means that the product (3.24) is well-defined and its integration over diQ,\d4Q,2d4Q.3 gives an analytic function. (Strictly speaking we have to take into account of <5(f2if2i — 1) as reintegration has been done to get the factor (2p3(Qi + r2fi2) — iO)-1. But, the argument is exactly the same as that for the integral H given by (3.5) and we do not give the detailed argument here. This remark applies also to the discussion in Case (g) and Case (h) below.) Thus, by combining (3.23) with other two poles and integrating them over pi, we find the function associated with Fig. 4(c) is again of the form (3.6). Case(d): The Landau table for the integral associated with the diagram in Fig. 4(d) is as follows: dfti 1 ri(pi+ri(ni+r2rsn3)) 0 2 ri{p2 + riCli) 3 4 0 ai€l\ 5 0 6 0 7

dQ2 0 -Pi 0 0 0 fi2 0

dn3 rir2T3(pi+ri{Qi+T2T3fl3))

r3Pi 0 0 0 0 03^3

dp Pi+ri(ni+r2r3n3)

-Cl2 + r30,3 J>2 + fifil P3

0 0 0

We then obtain a2 = 0 by the closed loop condition for the dfi2-column. Hence we can find a distortion avoiding pi(fl2 — r3Q3) = 0. The integral

130

corresponding to (3.19) is obtained by replacing the integrand by fA(q2 r\r2r3Q3,q3-r1Q,i). Since

-

-^—(Q2 ~ rxr2r3Q,3,q3 - rifii)| r i = 0

(3.25)

HalPl)(r2r^)

- (°3P3)Ol-

Thus by choosing a detour so that lmpiQ.3 is much bigger than lmp3£li, we find d(p/dri\ri=o does not vanish, and hence the integral in question has the form (3.6). Case (e): The residue factors in the integrand of the integral associated with the diagram in Fig. 4(e) are all non-singular. Hence it suffices to confirm J^"(?2 _ ^1^2(^2 + ^3^3)) 93 - '*i^i)ln=o is different from 0. In fact dtp

(3.26)

{q2-rir2(£l2 + r3fl3),q3 - r i f i ^ l n=o

=a1r2r3p1(Q2 + r3Cl3) - 03^3^1 • Since we are considering the problem near {Qi = ~Q2 = —fi3},this is close to (01P1 + a3p3)w - diPiU) = a3p3oj (w = Cl2)- Thus, reflecting the fact that no residue factor is singular in this case, the condition dip/dri\ri=0 ^ 0 is automatically satisfied. Hence the resulting singularity is again of the form (3.6). Case (f): The Landau table for the integral associated with the diagram in Fig. 4(f) is as follows:

nr 2 (pi+ri(f!i+r2fi2))

dtt3 0

0

-Pi

rsPi

n(P2+nfh)

0 0 0

fi2

0 0 0 0

0

^3^3

dfii 1 n(pi+n(ni+r2fl2))

2 3 4 5 6 7

0 (Tlfii

0 0

<m2

dp J>i+ri(f!i+r2n2) -n2+r3fl3 P2+nfti

P3

0 0 0

The closed loop condition for the oKVcolumn then implies a 2 = 0. (Note that, if r3 = 0, the residue factor corresponding to the second row is nonsingular and hence a2 — 0.) Hence we can detour Pi(Q,2 - r3Cl3) — 0. In this

131

case the integral that corresponds to (3.19) has /A(ft - »"ir2f22!93_ r i ^ i ) as its integrand. Since (3.27)

^-(g2 - rir2Cl2,g3 - riQi)\ri=0 =air2pi£l2 - asp&i,

at the point in question this is close to a-[P\ui + a$pzu) = —a2p2u (w = Cl2). In this case again d

(pi(ft! + r2r3Q3) + iO)-1(pi(Q2 - r3Q3) - ^O)"1 x(p2(Cl1 + r2Q2) + tO) _1 ((ni l0 + iO)2 - Q 2 )" 1

x ( ^ + io)-1((nSlo + tO) 2 -J^)- 1 . The "Landau table" for this product is: dQx

1 Pi 2 0 3 Vi 4 aSlx 5 0 6 0

dQ2

dn3

dpi

0

f2f 3 Pl

-Pi

npi 0 0 0

^i + r 2 r 3 fi 3 -Q,2 + r3fi3 0 0 0 0

?"2P2

0

n2 0

03^3

dp2 0 0 S7a + r2fi2 0 0 0

It is now clear that the product is well-defined and the final integral is again of the form (3.6). Case (h): To analyze the integral associated with the diagram in Fig. 4(h), we use the same technique as that used in Case (g). This time the product to be considered after performing the integration / drx{p\ - m2 + iO + 2ri(pi(n x + r2S72 + r2r3Q3) + i0)) _1 Jo is (3.29)

(pi (fii + r2Q2 + r2r3Q3) + i0)~l x(p2(fii + r2n2) + tO)-1((fii,o + «0)2 x(tf2 + iO)-\{Q3fi + iO)2 - fi2)"1,

til)'1

132

and the associated "Landau table" is 1 2 3 4 5

dft3 WzPi

dfii

dQ.2

Pi

T2Pl

P2 CTlfii

T2P2

0 0

n2

0 0 0

0

a3n3

0

dpi dp2 £li + r2Cl2 + r2r3Cl3 0 0 fii + r2Q2 0 0 0 0 0 0

Since we are considering the problem in a region where r2, r3 » 1, it is clear that no u = 0 problem arises, and thus the resulting integral is of the form (3.6). If we allow r 3 = 0, then a u = 0 problem arises. Inclusion of the point r 3 = 0 would result in y(log(y> + iO))2-singularity, instead of (3.6). Case (i): To analyze the integral associated with the diagram in Fig. 4(i), again we perform the integration K

L

driipl - m2 + iO + 2r1(p1(fi1 + r2fi2) + iO))"1

first and pick up the contribution from ra = 0. Then the product to be considered is (p 1 (f2 1 +r 2 Q 2 ) + tO)-1 x(pi(fi2 ~ r3fi3) - i0)" 1 (p 2 (^i + r2Cl2) + i0)~l

(3.30)

x ((fii,o + iO)2 - fi?)_1(^2 + iO)-\{Q3fi + idf - Ql)-1. Writing down the Landau table for the product (3.30), one can readily see that it admits a u = 0 point just as in the case of the integral associated with the diagram (3.1.a). To analyze this troublesome product we first consider the (r3, fi3)-dependent factor and integrate it over dr3d4Q3, i.e., we first consider the following integral: (3.31)

/

JI-K

dr3 /

^|n 3 |=i

d^sfa^-rsnsJ-tOr^fiiwi + iO)2-^)-1.

The associated "Landau table" is as follows. 1 2

eK73

dpi

d£l2

r3pi

- f i 2 + ^3^3

~P\

CT3f23

0

0

133

Since the closed loop condition for the dQ3-column implies ai = a.^ = 0 near r 3 = 1, (3.31) is well-defined and analytic in ( p i , ^ ) . The part of (3.30) which is irrelevant to (r 3 ,fi 3 ) is, i.e., (3.32)

(p^fij + r2Q2) + t O ) " 1 ^ ^ ! + r2Q2) + iO)"1 x((fi],o + i O ) 2 - ^ ) - 1 ( ^ + tO)-1,

is the integrand we encounted in (3.14). (The difference between (fi| + *0) -1 and ((^2,0 + zO)2 — fi2)_1 ' s n o t important.) Since the u = 0 problem for the integral (3.14) has been resolved, we can perform the integration of (3.30) over (r 2 ,r 3 ,f2i, $"22.^3) near r2 = r3 = 1, and we then obtain an analytic function of {jp\,P2)- Thus the integral associated with the diagram in Fig. 4(i) is again of the form (3.6). The study done in this section indicates that the effect of the QC problem discussed in Section 2 should be strictly weaker than the ordinary triangle diagram singularity, although we have not yet proved the fact in general; since the infra-red finiteness has been confirmed in general ([7, p.2496 ff. ]), the problem of confirming the weakness of the resulting integrals should be an interesting mathematical problem.

References [1] F. Block and A. Nordsieck, Phys. Rev. 52, 54 (1937). [2] T. Kibble, J. Math. Phys. 9, 315 (1968); Phys. Rev. 173, 1527 (1968); 174, 1883 (1968); 175, 1624 (1968). [3] V. Chung, Phys. Rev. 140, 1110 (1965). [4] J. K. Storrow, Nuovo Cimento 54, 15 (1968). [5] D. Zwanziger, Phys. Rev. D 7, 1082 (1973); 11, 3504 (1975). [6] H. P. Stapp, Phys. Rev. D 28, 1386 (1983). [7] T. Kawai and H. P. Stapp, Phys. Rev. D 52, 2484 (1995); 52, 2505 (1995), 52, 2517 (1995).

134

[8] T. Kawai and H. P. Stapp, Algebraic Analysis (Academic Press), 1, 309 (1988). [9] M. Kashiwara, T. Kawai and T. Kimura, Foundations of Algebraic Analysis, Princeton Univ. Press, 1986.

O N THE LINEAR HULL OF E X P O N E N T I A L S I N Cn A N D APPLICATIONS TO CONVOLUTION EQUATIONS LE HAI KHOI Hanoi Institute of Information Technology 18 Hoang Quoc Viet Rd., Can Giay Dist., 10000 Hanoi, E-mail: Lhkhoi@ ioit.ncst.ac.vn

a

Vietnam,

In this paper we concerned with properties of the closed linear hull of exponentials in the space of entire functions in C n . Some applications to convolution equations are considered 1

Introduction

The closure of the linear hull of exponentials in spaces of holomorphic and entire functions plays an important role in many problems of complex analysis. In a general setting this question reads as follows. Let X = (xfc)fcli be a sequence of nonzero elements of a Frechet space H. Denote by [X; H] the closed linear hull of (xk) in H, that is [X; H] = span {xk}^=1

in H.

We suppose that X is not a complete system in H (otherwise, [X; H] would coincide with H). It is required to find conditions for a system X to be either a basis (an absolute basis) or a representing system (an absolutely representing system) in [X;H]. Here we recall that X is said to be a representing system (an absolutely representing system) in H if any element x £ H can be represented in the form of a series oo

that converges (respectively, converges absolutely) in the topology of H. Uniqueness of the representation is not required. As an example of possible applications, let £ be a continuous linear operator from one Frechet space i?i into another Frechet space H^- Consider a homogeneous equation Cu = 0, uGHx.

(1.2)

"Supported in part by the National Basic Research Program in Natural Sciences (Vietnam) and Research Institute for Mathematical Sciences, Kyoto University (Japan).

135

136 Denote by Ec the set of all linear combinations of specific (in some sense) solutions of (1.2) and, the question is when any solution of (1.2) can be approximated by elements from Ec, i.e., when the relation Ec = £ - 1 ( 0 ) holds. Suppose that the equation (1.2) has a sequence of solutions X = (zfc)fcLi which we call elementary. It is clear that [X;if] C £ - 1 ( 0 ) . Suppose that [X; H] = £ _ 1 ( 0 ) . Then there arises a natural question of when an arbitrary element u € £ - 1 ( 0 ) can be represented (in a unique way or not, with respect to Xk) in the form of a series (1.1) that converges (respectively, converges absolutely) in Hi. It is clear that the latter question is solved if and only if X forms a basis (an absolute basis) (respectively, a representing system (an absolutely representing system)) in [X; H], which means that we can use the results on linear hull said above. There is a standard functional analysis approach based on duality theory for the above mentioned problem (see, e.g., Korobeinik (1975) [5, 6]). As in the concrete situation of exponential systems considered in this note, a convergence of a multiple Dirichlet series is equivalent to its absolute convergence, we limit ourselves only to absolute bases and absolutely representing systems in Frechet spaces. Now let A = (Afc), be a sequence of complex vectors in C n with limjfc-xx, |A* | = oo. Denote by Ax the space of entire funtions in C n with the usual topology of uniform convergence on compacta. Then we know the following: a) the exponential system £A =

C , is never a

basis in AQOJ b) depending on the choice of (A^) it may happen that S\ forms a representing system (an absolutely representing system) in A^; c) in the case when b) holds, £A is a complete system in A,*,, i.e., span{e}£! = Aoc. Note that in a topological vector space any basis is a fortiori representing system and in turn, any representing system is a complete system. Moreover, the inverse implications are, in general, not true. Denote by [5A; AM] the closed linear hull of the exponential system £\ in the space A^. If / £ [£\; A<x], then there exists a sequence of polynomials that converges to f(z). We are interested in a question under what conditions the system £& is either an absolute basis or an absolutely representing system in the space

137 2

Preliminaries

Consider a multiple Dirichlet series oo

^ c / ^ , z e C " .

(2.1)

The following characterization for coefficients of this series when it converges in all of C n is given by Le Hai Khoi (1999) [9]. Theorem 2 . 1 . If the Dirichlet series (2.1) converges for all z £ C™ and |Afc| -> oc as k -» oo, then lim sup ° S ^ |

= -oo.

(2.2)

fc->oc |A I Conversely, if the coefficients of (2.1) satisfy condition (2.2) and if, in addition, lim sup -rfrv(2.3) log A; < +oo, A then the series (2.1) converges absolutely for the topology of the space A^. Theorem 2.1 shows that with the sequence of coefficients satisfying condition (2.2) and the sequence of frequencies satisfying condition (2.3), the series (2.1) represents an entire function in C n . In connection with Theorem 2.1, we can associate to the given sequence (Afc)^_1 of complex vectors in C n the following two sequence spaces Ei = {c= (ck);3M VA; \ck\ < e M l A *l} E0 = {c = (cfc); \ck\1/lx"1

-> 0,fc -> oo} .

We can define these spaces in a uniform way by requiring log |cc | r < +oc, lim sup . ',,fc < fc-KxT |Afc| I = - 0 0 . The space E0 is a proper subspace of Ei, as the element (ck) with Ck = eM\\ | ; jy e R belongs to Ei, but does not belong to EQ. For a sequence space E denote by Ea its Kothe dual, i.e., Ea = < d = (dk); Y^ Ckdk converges absolutely for all c= (ck) £ E>.

138

Also we put into consideration the following sequence space Ep = < d = (dk);^2Ckdk

converges for all c = (ck) G E

It is clear that Ea C E0 and also, E C Eaa. (see Le Hai Khoi (1999) [9]).

We have the following result

Theorem 2.2. If condition (2.3) is satisfied, then the following relations hold rpP

jpa

ipaa £J0

77 .

pa

T-I/3

77 . TTiaa — h/o, H/l —

17 .

17 hi\.

Everywhere in what follows condition (2.3) for the sequence of frequencies (Afc) is assumed to hold. Thus if the Dirichlet series (2.1) converges at each point of C n , then it converges absolutely for the topology of the space A^. Since A^ is a Frechet space, the space EQ, endowed with the topology denned by a family of pseudonorms

l\c\P =

Jt\ck\p(e^);P6v\,

where V is a family of pseudonorms defining the topology of A^, is also a Frechet space. The dual space EQ of E0, in this case, can be identified with Z7/3 -^0 -

z?a 17 - ^ 0 — •C'l •

Let L be a linear operator acting from EQ into A^ by the following rule oo

fc=i

Then it is clear that L is a continuous linear operator from E0 into Ax. In this case we can describe the dual operator V of L. It can be verified that the dual operator V: A'^ -¥ E'0 acts by the rule

L'p= (fa, z*e<x" >'>))" V

'

£ 4

K= l

which precisely means that it puts the sequence (fi(Xk)\

from Ey into corre-

spondence with the analytic functional /x G A'^, where /t G Ax is the Laplace

139 transformation of p, and A\ is the space of entire functions of exponential type inCn. Before going on we note that the closed linear hull [£A;^4OO] = span {e^"' 2 )} of the system £\ in A^ is also a Frechet space. 3

A n absolute basis of exponentials

We can easily see that £\ is an absolute basis in [5A; Ago] if and only if L is a bijection of Eo onto [£\; Aoo], i.e., if and only if L is an isomorphism into i ^ (onto [ £ A ; A » ] ) - By the duality theory (see, e.g., Edwards (1965) [2]), the last assertion holds if and only if L'lA'^) = E'Q. We get the following result. Theorem 3.1. Let A = (Xk)k_1

be a sequence of complex vectors in Cn

satisfying the condition (2.3). Then the exponential system £\ = (e^ 'ZM V

/ fc=l

is a basis (or the same, an absolute basis) in the space [£A: A^] if and only if a system of equations ji(Xk)=uk,

A = 1,2,...,

has a solution in the space A\ for any u — (uk) € -Ei • Thus we arrive at the question of when the interpolation problem g(Xk) =uk,

k = 1,2,...,

is solvable in the space Ai of entire functions of exponential type for any u — (uk) € Ei, provided A = (Xk) C C n satisfying condition (2.3) is given. Note that u = (uk) e E\ if and only if Vfc > 1 |u*| < AeBlA<:l for some constants A, B independent of k. This means that the statement in Theorem 3.1 is the same as the requirement that the restriction map Ai -t Ei denned by a rule

is onto. This is precisely the notion of the so-called universal interpolating sequences in Ai. Thus Theorem 3.1 can be reformulated as follows. Theorem 3.2. Let A = (Afc), = 1 be a sequence of complex vectors in C satisfying the condition (2.3). Then the exponential system £\ = I e^x '^ ) is a basis (or the same, an absolute basis) in the space [£&; A^] if and only if the sequence (Xh) is a universal interpolating sequence for the space Ai.

140

The universal interpolation problem have been studied in many articles and moreover, in a more general context. There are different approaches to this problem. The necessary and sufficient condition for universal interpolating sequences was obtained by C.A. Berenstein, B.A. Taylor, A. Squires, B.Q. Li,... (see, e.g., Berenstein & Li (1994) [1]). Some sufficient conditions concerning the so-called sparse sequences or lacunary sequences were obtained by C O . Kiselman (1984) [4], T. Kawai (1987) [3], ... It is well known that (see, e.g., Berenstein & Li (1994) [1]) a sequence (Afc) of complex vectors in C™ is a universal interpolating sequence for A\ if and only if there exist n functions / i , . . . , /„ £ A\ such that = {£ e C n ; A ( 0 = ••• = /„(£) = 0}

A C Z(h,...,fn) and for some s, C > 0

\detJfu...,u(Xk)\>ee-c\xhK

k 6 N,

where Jf1,...j„ is the Jacobian matrix of / i , . . . , / „ . The following example of a universal interpolating sequence is well known. Let hj (j = 1 , . . . , n) be an entire function of exponential type in the plane of a complex variable

Zj

and Z(hj) =

(A^)~_

with ^ ( A ^ J I > ee" 0 !**"! for

some constants e, C independent of k. In this case (see, e.g., Leont'ev (1972) [10]) (AJ, J

is a universal interpolating sequence for the space Ax,Zj

of

entire functions of exponential type in C z . , (j = 1 , . . . , n). Then the combined sequence of complex vectors in C n defined as follows

A:=(\k =

(\Z\...,\l%\ii)eZ(hj))~=i,

is a universal interpolating sequence for Ai. 4

An absolutely representing system of exponentials

As in the previous section, we see, on the one hand, that S\ is an absolutely representing system in [£A; ^4OO] if and only if L(E0) = [£\; Aoo]. On the other hand, as the following inclusions always hold span{e} C L(E0) C [ £ A ; 4 » ] , we have [S\\ Aoo] = span{c} C L(E0) C [£A; 4 M ] , which implies that

L(EQ)

=

[£A;-AOO]-

141

Combining these facts gives that £A is an absolutely representing system in [£A;AOO] if and only if L(EQ) = L(E0). By the duality theory (see, e.g., Edwards (1965) [2]), L(E0) is closed in Aoo if and only if L'iA'^) is closed (weakly or strongly) in E'Q. Thus we get the following result. Theorem 4.1. Let A = ( A * ) ^ be a sequence of complex vectors in Cn

(

,

\ OO

/fe=l

is a representing system (or the same, an absolutely representing system) in the space [£\; Ax] if and only if a set A4 = { ( A ( A f c ) ) ~ 1 ; / i 6 > 4 i } is closed (weakly or strongly) in the space Ex. The question in Theorem 4.1 seems rather tough. Below will be given a "temporary" solution of this problem. The method presented here is in the spirit of Leont'ev (1951) [11] using a technique of differential operators of infinite order. For A = ( A i , . . . , An) e C™, we write A = Ai H (- An. Suppose that the sequence of frequencies (A*) satisfies the following condition fA^A-(fc^m), |limsupfc_foo|J^
^• >

in other words, (Afe) are supposed to be pairwise different and have a finite upper density. Obviously, this condition implies condition (2.3). As already noted in the Introduction, if / £ [£&; Aoo], then there exists a sequence of polynomials rrij

Pj(z) = Y,cjke<xk'*\j

= l,2,...,

(4.2)

*=i

such that it converges to f(z) in Aoo. We have the following property of the coefficients of (4.2). Proposition 4.2. Suppose that (Afc) satisfies condition (4.1). If the sequence {Pj{z))^Ll in (4.2) converges for the topology of the space A^, then 3

lim Cjk = Cfc, k = 1 , 2 , . . .

i-foo

Moreover, these limits define in a unique way the limit function.

142

In the proof of this proposition we use the following differential operators of infinite order (see, e.g., Morzhakov (1971) [13]): k

with the symbol

iM=n

= X>t*.

1

k=0

fc=i L

and

() i A £(P)(£)[/] = X >kP l dzx + '"

+

dZ>

f(z),p=

n

'"'Jf

1,2,...,

k=0

with the symbol ap(t)

a(t) J

k=p+l

= E4"«' fc=0

For these operators there are relations

L(D) [e] = ea(A) and L ( p ) ( 0 ) [ e M = eop(A), p = l , 2 , . . . . We see that £(!?)[/] = L(D)

lim P,(z) = lim J—>00

J-i-OO

yV f c a(A>< A *- 2 > 1 = 0, *Jfe = l

as A* are all simple roots of a(t). Thus functions / € [£A; A»] are solutions of the equation L(D)[f] = 0. Note that not every function satisfies this equation, for example a function e^x^ with a(A) ^ 0 is not a solution of such an equation. ^From Proposition 4.2 it follows that we can associate in a unique way to each function / 6 [£&; Ax,] the series c*e <**.*> *=i

and if this series converges, then its sum is f(z).

(4.3)

143

Thus the question now is to find conditions for series (4.3) to converge in We have the following result. Theorem 4.3. Let (\k)™=1 be a sequence of complex vectors in Cn satisfying condition (4.1). The exponential system £A is a representing system (or the same, an absolutely representing system) in the space [£\; Aoo] if and only if the following condition hold log

limsupT-p• • + A*

5

1 a'(A} + -.- + A*)

< 0

°'

Convolution equations

Let fj, E A(Cn)' be a nonzero analytic functional carried by a convex compact K C C". This functional defines a continuous linear convolution operator My,: A(ft + K) -¥ A(to) which is given by Mlt[u](Q = (ji,z>->u(<; + z)),

(€Q,z€K,

where ft is any convex domain in C n . Here A(M) denotes a space of holomorphic functions in a domain M c C " with the usual topology of uniform convergence on compact subsets of M. Convolution operators in spaces of entire functions and holomorphic functions in convex domains of C " have been studied by many mathematicians. There are two problems that merit attention. P I . Surjectivity of the convolution operator The first results on surjectivity of the convolution operator MM were obtained by Ehrenpreis and Malgrange for the case when ft = C™. Later, Martineau considered a particular case, when K = {0}, i.e., a differential operator of infinite order, and showed that for any convex domain ft in C™ the operator MM is surjective. For different cases of ft and K some sufficient and necessary conditions were found by Morzhakov, Napalkov, Lelong & Gruman, Sigurdsson, ... (see, e.g., Sigurdsson (1991) [14]). Finally, the answer to this problem was given by Krivosheev (1991) [7]. P 2 . Approximation of solutions of the homogeneous equation Malgrange was the first who proved (for the case ft = C n , without any restriction on /x), that the linear combinations of all polynomial-exponential solutions of the homogeneous equation

Ma[u](0 = 0,

144

i.e., solutions of the form P{z)el~z'<'\ where P(z) are polynomials, are dense in the set M~x(0) of all solutions of this equation. Martineau also gave the answer for the case K = {0} (with an arbitrary convex domain U). For a general case of Q and K there were some results obtained by Morzhakov, also by Berenstein, Taylor, Kawai, Meril & Struppa (for a system of equations), etc. (see, e.g., Meril k Struppa (1987) [12]). Concerning the problems P I and P2, we should make some remarks. R l . The results obtained on surjectivity of MM were nonconstructive ("existence theorems") and did not permit the effective construction of a particular solution of the equation M M M(C) = /(C), C e n, in A(£l + K) for a given right-hand side / in A(Q). R 2 . The results on the structure of the kernel M~l(Q) of MM were for a set of all exponential-polynomial solutions and that set was defined again not effectively. There arise the following, more sophisticated, questions. Q l . Is it possible to construct effectively a particular solution of a (surjective) convolution equation? Q2. On what conditions can solutions of a homogeneous convolution equation be approximated merely by exponential solutions? Consider a Dirichlet series in a convex domain H C C n oo

£ Cfc e< A ^>, c e n .

(o.i)

jt=i

Denote by Jffn(C) — sup2gQ 3?(z, C) the supporting function of Q,. Similarly to Theorem 2.1, there is a characterization for coefficients of series (5.1) for this holomorphic case given by Le Hai Khoi (1985) [8]. Theorem 5.1. If Dirichlet series (5.1) converges at each point of a bounded convex domain fl C Cn, then hmsup

!

—pr^i—^

< 0.

(5.2)

Conversely, if the coefficients of (5.1) satisfy condition (5.2) and if losk

^M=0' then the series (5.1) converges absolutely for the topology of A(Q).

(5 3)

"

145

In the sequel the condition (5.3) is always supposed to hold. This means that the convergence and the absolute convergence of Dirichlet series are the same. 6

A particular solution

Consider a convolution operator

MM:.4(n + / 0 - » A ( n ) , where Q is a bounded convex domain in C™. Suppose that £\ forms an absolutely representing system in A(Cl). Let f(Q e .4(f)) be a given right-hand side of the equation

(6.1)

MM = /. Then /(£) can be represented in A(Q) in the form oo

/(C) = $ > e < A * , C > > C 6 f i ,

(6,2)

Jb=l

with coefficients (cj,) satisfying, due to Theorem 5.1, the condition

,. 1

P

TS

loglcfcl + //Q(Afc) ^ n a

\*\

-

We would like to find a function u(( + z) E A(Cl + if) of the form

fc=i

that satisfies equation (6.1). Note that

M^[«](C)=f;dfcA(A*)e <**.0 fc=i

where /i(A) = (/x, z *+ e^x'z^) is the Laplace transform of the analytic functional fi. Suppose that the following conditions hold: A(A f c )^0, V* = l , 2 , . . . ;

liminf^'^l^l-^^^^O.

146

We form a series oo

u

(C + ^) = E ^ ) e < A f c ' C + 2 > ' C€il,zeK,

(6.3)

which is well defined and its sum /i(C + z) belongs to A(Q + K). It is clear that CO

M„[«](C) = X>e < A * , c > = / ( 0 , ^ e n fc=l

Thus, the operator MM is surjective and moreover, is effectively surjective, from A{Q + K) into A{Q). We obtained the following result. Theorem 6.1. Suppose that the following conditions hold: (i) The exponential system £A forms an absolutely representing system in A(Q); (ii) fi(Xk) ^ 0, Vfc = l , 2 , . . . ; Tien the convolution operator MM is effectively surjective from A(Q + K) onto A(Q,). Moreover, for any right-hand side /(£) G A(fl) of the form (6.2) we can find explicitly a particular solution of the equation (6.1) and of the form (6.3). 7

A homogeneous equation

Denote W» := M-HO) = {ue A(il + K); M > ] ( ( ) = 0, VC € 0 } , and .EM = span {all exponential-polynomial solutions of M^u] — 0} . We wish to have more than the fact that E^ = W^, namely we study a question whether any function from W^ can be written in the form of series with frequencies (A*). Since M„[e A ](0=/i(A)e< A -<>, where eA(C + z) = e,C eQ,ze K,XeCn, the exponential e<+*> is a solution of the homogeneous equation if and only if jj,(\) = 0 .

147

So the problem now becomes as follows: suppose that A = (Afc) is a discrete subset of the zero variety of (l. When does the exponential system £A form either an absolute basis or an absolutely representing system in WM? Let A = (Afc) be a sequence of complex vectors in C " and fi{\k) = 0, Vfc > 1. Denote by [£A;A(n + K)} = s p a n f e ^ . ^ * ) } ^ the closure of the linear hull of the exponential system £\ in the space A(Cl + K). It is clear that [£A; A(£l + K)] C WM. We say that W^ admits A—spectral synthesis in A(Cl + K),iiWfl = [£A; A(Q + K)}. Now suppose that W^ admits A—spectral synthesis in A(Cl + K). Then the question (we are interested in) becomes to find when the system £A forms either an absolute basis or an absolutely representing system in the closed linear hull [£A; A(Q + K)]. In the sequel we consider the case when Cl = C™. As already said above, Malgrange proved, in this case, that any solution of the homogeneous equation MM[u] = 0 can be approximated by exponentialpolynomial solutions, i.e., E^ = W^. Since fi + K = C n , the question now is to find when £\ forms either an absolute basis or an absolutely representing system in [£\, Ace]. We get the following result. Theorem 7.1. Let A = ( A * ) ^ be a sequence of complex vectors in C " satisfying the condition (5.3). Let further, W^ admits A-spectral synthesis in A(Cn). In order that any solution from Wy. = M~ 1 (0) of the homogeneous equation M^u] = 0 can be represented in the form of Dirichlet series with frequencies (Afc) it is necessary and sufficient that the exponential system £A = (e(x '*M

is either an absolute basis or an absolutely

representing

system in the space [fAi'^x]Thus the problem considered in this section reduces to that of §§3-4 and we can use the results obtained there. Acknowledgements. This note was written during author's stay at Research Institute for Mathematical Sciences (RIMS), Kyoto University, Japan. The author would like

to express his gratitude to RIMS for the hospitality. He is grateful to Prof. Takahiro Kawai for valuable discussions and comments. A special note of gratitude goes to Prof. Christer Kiselman for continuous encouragements and attention to this work. The author thanks the referee for useful remarks and comments. References 1. Berenstein C.A., Li B.Q., Interpolating varieties for weighted spaces of entire functions in C", Publ. Mat. 38 (1994), 157-173. 2. Edwards R.E., Functional Analysis: Theory and Applications, Holt, Rinehart & Winston, New York, 1965. 3. Kawai T., The Febry-Ehrenpreis gap theorem and linear differential equations of infinite order, Amer. J. Math. 109 (1987), 57-64. 4. Kiselman C O . , Croissance des fonctions plurisousharmoniques en dimension infinie, Ann. Inst. Fourier (Grenoble) 34 (1984), 155-183. 5. Korobeinik Yu.F., On a dual problem. I. General results. Applications to Frechet spaces, Mat. USSR Sbornik 26 (1975), 181-212. 6. Korobeinik Yu.F., On a dual problem. II. Applications to LN*-spaces and other questions, Mat. USSR Sbornik 27 (1975), 1-22. 7. Krivosheev, A. S., A criterion for the solvability of nonhomogeneous convolution equations in convex domains of the space C n , Math. USSR-Izv. 36 (1991), 497-517. 8. Le Hai Khoi, Holomorphic Dirichlet series in several variables, Math. Scand. 77 (1995), 85-107. 9. Le Hai Khoi, Coefficient multipliers for some classes of Dirichlet series in several complex variables, Acta Math. Vietnam. 24 (1999), 169-182. 10. Leont'ev A.F., Representation of functions by generalized Dirichlet series, Math. USSR Izv. 6 (1972), 1265-1277. 11. Leont'ev A.F., Series of Dirichlet polynomials and their generalizations (Russian), Trudy Mat. Inst. Steklov 39 (1951), 1-215. 12. Meril, A. & Struppa, D. C , Convolutors of holomorphic functions, in: Complex Analysis II, Lect. Notes in Math. 1276 (1987), 253-275. 13. Morzhakov V.V., On the theory of the applicability of differential operators of infinite order in spaces of functions of several complex variables (Russian), Litovsk. Mat. Sb. 11 (1971), 843-859. 14. Sigurdsson, R., Convolution equations in domains of C n , Ark. for Math. 29 (1991), 285-305.

H Y P E R F U N C T I O N S A N D KERNEL M E T H O D DOHAN KIM Department of Mathematics, Seoul National University, Seoul 151-742, Korea E-mail: [email protected]

1

Introduction

Sato 22 introduced the hyperfunctions as the difference of boundary values of an analytic function in the upper half plane and one in the lower half plane, especially in one dimensional case. In several dimensional case Hormander and Komatsu introduced the hyperfunctions as the boundary values of harmonic functions to avoid the heavy cohomological machinery. We refer to 2,17,19,20,22,23 for historical details. In this further direction Matsuzawa introduced the hyperfunctions and distributions as the initial values of the solution of the heat equation satisfying certain growth condition. Refining this approach further we can define many analytic and global properties on hyperfunctions, for example, positive definite hyperfunctions, periodic hyperfunctions and almost periodic hyperfunctions. For this, representing the hyperfunctions as the initial values of solutions of the heat equation we define the positive definiteness and periodicity of hyperfunctions in terms of their defining functions. In this survey article we briefly introduce the above hyperfunctions and their characterization theorems, for example, Bochner-Schwartz theorem for positive definite hyperfunctions and characterization of periodic hyperfunctions and try to compare our results in hyperfunctions and their analogs in distributions. As these natural definitions are obtained we can easily prove the following results in hyperfunctions: (i) Every positive definite hyperfunction is a positive definite Fourier hyperfunction, which is the parallel to the well known fact in the theory of distributions that every positive definite distribution is a positive definite tempered distribution. (ii) Every periodic hyperfunction can be represented as an infinite sum of derivatives of bounded continuous periodic functions. (iii) Fourier coefficients ca of periodic hyperfunctions are of infra-exponential growth in Rn, i.e., \ca\ < CeelQl for every e > 0. Our result contains the

149

150

result of Sato and Helgason which deals with the case of R 1 . See 11,13 for related results.

Also, generalizing this method further we introduce the Mehler kernel approach in representing the tempered distributions and generalized functions of Gelfand-Shilov type as initial values of the solution of the Hermite heat equation satisfying certain growth conditions. These results are natural generalization of the well known results in distributions and ultradistributions. In the proof of these results in distributions and ultradistributions, we use the heat kernel method and introduce the space BL? of Sato hyperfunctions of LP growth which generalizes the space V'LP of Schwartz distributions of LP growth. This heat kernel method, which represent the above generalized functions as the initial values of smooth solutions of heat equations as in Matsuzawa 21 , Kim-Chung-Kim 18 and Chung-Kim 7 , can overcome difficulties due to the sheaf theoretical definition of the hyperfunctions while dealing with the analytical and global properties in the theory of hyperfunctions. Applying this idea we relate the periodic hyperfunctions to the periodicity of its defining function. Note that also in the case of distributions periodicity in terms of the denning function for the distributions and the original definition for periodic distributions coincide naturally. See (2.1) for the definition of the defining function. In Section 2 using this heat kernel method we state a structure theorem for distributions and Fourier hyperfunctions in 21 and 16 respectively. In Section 3 we also introduce the space BLP of Sato hyperfunctions of LP growth and give a structure theorem for this space. In Section 4 we define periodicity for hyperfunctions u in terms of its collection of restrictions Uj to Clj by following A. Martineau's approach in Theorem 2.3 and its following remark. We relate the definition of the periodic hyperfunctions and periodicity of its defining function and show that every periodic hyperfunction is a Fourier hyperfunction and furthermore a bounded hyperfunction, which is parallel to the fact that every periodic distribution is a tempered distribution and furthermore, a bounded distribution. Finally, we show that Fourier coefficients cQ of periodic hyperfunctions are of infra-exponential growth, i.e., for every e > 0 \ca\ < Ce e l a l, which is a natural generalization of polynomial growth of the Fourier coefficients of distributions and of Gorbacuk's result for the case of ultradistributions. In Section 3, We briefly describe the Mehler kernel method to the space of tempered distributions and the generalized functions of Gelfand-Shilov type and state an extension of Strichartz's result on the characterization of eigenfunctions of the Laplacian.

151

2

Generalized functions as initial values of solutions of the heat equation

We first briefly introduce analytic functionals, hyperfunctions and Fourier hyperfunctions. See 12 ' 14 - 16 for more details. Definition 2.0.1. Let K c l n be a compact set. Then A(K) is the space of all real analytic functions in some neighborhood of K. In other words, ip € A(K) if ip is a C°° function in a neighborhood of K and there are positive constants C and h such that

sup ev
h\<*\a\

where we use the multi-index notations: \a\ = a\ + ... + an for a = ( a i , . . . .Q-n) £ NQ where N0 is the set of non-negative integers and da =

d?*...dz~,dj

= d/dxj.

We denote by A'(K) the strong dual space of A(K) and call its element an analytic functional carried by K. We set ^ i ' ( l n ) = UKA'(K) and the support of u G A'(Rn) is the smallest compact set K CRn such that u £ A'(K). We now define the space B of hyperfunctions following A. Martineau as in12. Definition 2.0.2. Let Q be a bounded open set in l n . Then the space B(fl) of hyperfunctions is denned by B(Cl) = A'(Cl)/A'(dn). We now state the localization theorem to define hyperfunctions in every open set in R™. Theorem 2.0.3. Let Clj, j = 1,2,..., be bounded open subsets of M." such that fi = li^Clj. If Uj £ B(Qj) and for all i, j we have u» = u, in fii n fij (that is, supp (UJ — itj) n fii n fij = 0) then there is a unique u £ 5 ( 0 ) such that the restriction of u to fij is equal to u. By virtue of the above localization theorem we can define hyperfunctions u G B(M.n) as a collection of Uj £ A(Clj) such that Uj = u*, in fij n Q^, 1 < j , k < oo. We now introduce a real version of the Fourier hyperfunctions. Definition 2.0.4. 16 (i) We denote by T the set of all infinitely differentiable functions ip in E n such that . .

for some /i, k > 0.

\dap(x)\expk\x\

152 (ii) We say that ipj -¥ 0 as j -¥ oo if \yj\h,k -* 0 as j -»• oo for some A, Jb > 0. (iii) We denote by P the strong dual of T and call its elements Fourier hyperfunctions. We denote by E(x, t) the n-dimensional heat kernel E(x 1

t) =

' '

{(4^)-"/2exp(-|o;|2/4i), \0,

t> 0 t<0.

Note that E(x, t) belongs to the space T for each t > 0. Thus 17(3:, t) =uy{E{x-y,t))

(2.0.1)

is well defined in R" + 1 = {{x,t)\x eW,t> 0} for all u E T' and called the defining function of u. We now represent some generalized functions as the initial values of smooth solutions of heat equation. Theorem 2.0.5. 21 (i) Let u € V'iW1). Then there exists U(x,t) € C°°(ffi™+1) and satisfies the following conditions: {d/dt-A)U(x,t)

= 0 in

iR^+1.

For any compact set K C Mn there exist positive constants N = and CK such that l^(*. *) I < CKt'N,

t>0,

x£K

(2.0.2) N(K)

(2.0.3)

+

and U(x. t) -* u as t ->• 0 in the sense that for every v> £ C^°

U(
= lim

U{x,t)tp(x)dx

(2.0.4)

Conversely, let U(x,t) £ C°°(E" + 1 ) satisfy (2.0.2) and (2.0.3) . Then there exists a unique u G V(Rn) satisfying (2.0.4). (ii) Let u € B(Rn). Then we can canonically define U(x,t) e C ° ° ( l " + 1 ) which satisfies the heat equation (2.0.2), and the following conditions: For every compact subset K C M" and for every e > 0 there exists a constant CC,K > 0 such that \U(x, t)\ < Ce,K exp(e/i),

O O . i e K

(2.0.5)

153

and U(x, t) -> u as t -» 0+

(2.0.6)

in the sense that U(x, t) - Uj(x, t) -¥ 0 as t -> 0 + in fij, j = 1,2,..., where Uj is the defining function of Uj G A'(Qj) as in (2.0.1) and u = («,-) € £ ( R n ) , and 1 " = Ujfij. The converse is also true as in (i) (hi) Let u G S'(Rn). Then tf(x,i) = uy{E{x-y,t)) belongs to C°°(R^ + 1 ) and satisfies (2.0.2) and the following condition: There exist positive constants C, M such that \U(x,t)\
+ \x\)N

in

RI+1.

(2.0.7)

+

and U(x, t) —>• u as £ —>• 0 in the following sense: for every tp £ S u(if) = lim / Z7(x, i)y(a;) cte. Conversely, every C°°-function defined in E™+1 satisfying (2.0.2) and (2.0.7) can be expressed in the form U(x,t) = uy(E(x — y,t)) for some u G <S'. Theorem 2.0.6. 16 Let u G F'(Rn). Then the defining function U(x,t) satisfies (2.0.2), (2.0.4) for every
0 there exists a constant Ce > 0 such that \U(x, t)\
exp [e(\z\ + 1/t)]

(2.0.8)

for t > 0, x G I P . Conversely, let [/(a;,*) G C°°(]R" +1 ) satisfy (2.0.2) and (2.0.8). Then there exists a unique u G ^"'(M™) such that U(x,t) = uy(E(x — y,t)). 3 3.1

Heat kernel method and hyperfunctions Positive definite hyperfunctions

S. Bochner proved the following theorem in 3 . Theorem 3.1.1. (Bochner) If / is a continuous function in W1 then the following conditions are equivalent: (i) / is positive definite, that is. for any x*L.... ,xm G Kn and for anycomplex numbers £ i , . . . , £m 771

Y, f(xj - XkKjCk > 0.

(3.1.1)

154

(ii) / is the Fourier transform of a positive finite measure /z, i.e.,

/(a:) = Je-iXxdn(\).

(3.1.2)

(iii) For any C°° function ip with compact support

/ /

f{x - y)ip(x) 0.

(3.1.3)

where y>(x)* = 0 for every nonnegative test function (p> and is said to be positive definite (or of positive type in Schwartz 24 if u(
0 for any test function 0. Also, note that every positive definite distribution is a positive definite tempered distribution, that is, the class of positive definite distributions and the class of positive definite tempered distributions are the same. Also, the above theorems are generalized to the more general case of hyperfunctions and Fourier hyperfunctions. For this we first need the following definition. Definition 3.1.3. A hyperfunction on an open set H is positive if its restriction on each bounded open subset V of fi is represented by a positive analytic functional carried by V. Theorem 3.1.4. 7 (i) Every positive hyperfunction is a measure. (ii) Every positive Fourier hyperfunction is an infra-exponentially tempered measure. (iii) Every positive Aronszajn trace is a measure.

155

Here, a positive measure /i is said to be infra-exponentially tempered if for every e > 0 / e ~ e W dp < oo. We now introduce the Bochner-Schwartz theorem for the hyperfunctions. In other words, every positive definite hyperfunction is the Fourier transform of an infra-exponentially tempered measure, consequently the class of positive definite Fourier hyperfunctions and the class of positive definite hyperfunctions are the same, which is the parallel result of Theorem 3.1.2 for the theory of hyperfunctions. In order to prove this main result we note that a hyperfunction is defined locally as analytic functionals, in other words, that the space B of hyperfunctions is defined locally as the space of analytic functionals which is the dual space of analytic functions, but not globally. Hence it is difficult to define the global concept of positive defmiteness for the hyperfunctions. To overcome this difficulty we apply the heat kernel method of T. Matsuzawa as in 21>16>7. We first make use of the representations of the generalized functions including distributions, hyperfunctions and Fourier hyperfunctions as the initial values of the solutions of the heat equation (see Theorems 2.0.5 and 2.0.6) and then we define the positive definite generalized functions in terms of the defining function. As a consequence of these results we define the positive definite hyperfunctions (see Definition 3.1.5). As this natural definition of positive definiteness for the hyperfunctions is given, we can easily prove the Bochner-Schwartz theorem for the hyperfunctions. We are now in a position to define the positive definite hyperfunction in terms of the defining function and the growth condition. Definition 3.1.5. A hyperfunction u is positive definite if the defining function U{x, t) of u is a positive definite function for each t > 0, that is, n

J2

U(Xj-xk,t)CjCh>0

j,fc=i

for every xi,... ,xn £ E n , & , . . . , Cn £ C and for each t > 0. For another definition of positive definite hyperfunctions different from ours and related Bochner-Schwartz theorem we refer to Ouchi's paper, some applications of hyperfunctions to the abstract Cauchy problem and stationary random process, that appeared in RIMS Surikaiseki Kennkyusho Kokyuroku around 1970. We now state and prove the main theorem.

156

Theorem 3.1.6. The following conditions are equivalent: (i) u is a positive definite hyperfunction. (ii) u is a positive definite Fourier hyperfunction. Combining the Theorem 2.0.5 Theorem 2.0.6 with the Bochner-Schwartz theorem we have the following result: Theorem 3.1.7. The following conditions are equivalent: (i) u is a positive definite distribution. (ii) u is a positive definite tempered distribution. (iii) u is the Fourier transform of a positive tempered measure. (iv) The defining function U(-,t) of u £ V is a positive definite function for each t > 0. (v) The defining function U(-,t) of u 6 S' is a positive definite function for each t > 0. As a parallel result of the Bochner-Schwartz theorem we have the following: Theorem 3.1.8. The following conditions are equivalent: (i) u is a positive definite hyperfunction. (ii) u is a positive definite Fourier hyperfunction. (iii) u is the Fourier transform of a positive infra-exponentially tempered measure. (iv) The defining function U(-,t) of u € T1 is a positive definite function for each t > 0. 3.2

Hyperfunctions of V growth

We first introduce the new space BL? of hyperfunctions of Lp growth which is a natural generalization of V'LP for hyperfunctions and will be used to characterize the periodic hyperfunctions. We also represent these hyperfunctions as the initial values of solutions of the heat equation. Definition 3.2.1. We denote by ALI (1 < <7 < oc) the space of all functions p e C°°(Rn) satisfying „ „

IP'VIIL.

157

for some constant h > 0. We say that tpj -» 0 in AL* as j -> oo if there is a positive constant h such that *

h\a\a\

J

We denote by BLP (1 < p < oo) the dual of ALI, where 1/p + 1/^ = 1. In particular, every element in BL oo is called a bounded hyperfunction. Note that every element in V'Loo is called a bounded distribution. It is easy to see the following topological inclusions: F<->AL- (1 < 9 < o c ) , BLP ^ T' {K P < oo). We will prove the structure theorem for BLp (1 0 and e > 0 there exist functions v(t) in C£°([0, e]) and an ultradifferential operator P(d/dt) such that \vlk)(t)\
fc

= 0,l,---, 0
OC

P(d/dt) = Y,a*(d/dVk> P(d/dt)v(t)

l°*l < C i A i / * ! 2 ,

0
(3.2.1)

=S + w(t),

where «;(*) G Cc°°([e/2,e]). Lemma 3.2.3. For every y> € „4.£<, (1 < g < +oo), let
t > 0.

(3.2.2)

Then tpt £ At? and ¥> in -4L<J as t -> 0^. Let u e B^P . Then the defining function

U(x,t)=uy(E(x-y,t)),

(i,t)e8f

is a well defined C°° function, since E(x- •, t) belongs to ALI {1/p+l/q = 1) for each (x,t) e i " + 1 . Theorem 3.2.4. A hyperfunction u belongs to BLP (1 < p < oo) if and only if u can be written as u = P(-A)g(x)

+ h{x)

where P(—A) is given in Lemma 3.2.2 and g and h are bounded continuous functions belonging to Lp.

158

Theorem 3.2.5. Let u G BLP- Then the defining function U(x,t) of u belongs to C°°(R™+1) and satisfies the following: (dt-A)U(x,t) = 0 in l^ +1 :

(3.2.3)

For every e > 0 there exists a constant C > 0 such that \\U{x,t)\\Lf^i)
(3.2.4)

+

and U( •, t) —> u as t -> 0 in the sense that u((f>) = lim / U{x,t)4>{x)dx, (t>eALi. (3.2.5) *-»-o+ ./R» Conversely, every C°° function defined in M" +1 satisfying condition (3.2.3) and (3.2.4) can be written as U(x,t)=uy(E(x-y,t))

in Ri+1

with a unique element u € BLP . Remark 3.2.6. In Theorem 3.2.4 we can obtain that u *

Periodic hyperfunctions

In this section we give a definition of periodic hyperfunctions u in terms of the collection of analytic functional with compact support Uj G A'(flj). Also, we relate this definition of periodic hyperfunctions and the periodicity of the defining function U of u. Definition 3.3.1. A hyperfunction u — (UJ)J £ Z G B(Rn), where Uj G A'(Clj) with Rn = Ujftj is periodic if Tau = {raUj) — u for all a G Z n . Here, rQUj := Uj(x — a) G ^ ' ( r a f i j ) with r a fij = {x + a\x G fij}. From this definition we can easily obtain the following Theorem 3.3.2. If a hyperfunction u is periodic then the defining function U(x, t) of u is also periodic. Making use of the periodicity of the defining function U of a periodic hyperfunction u we now prove the following Theorem 3.3.3. Every periodic hyperfunction is a bounded hyperfunction. Remark 3.3.4. Every periodic hyperfunction u is a periodic Fourier hyperfunction, i.e., u is periodic in T'. Therefore, we obtain that Tau(ip) = u(
159

T h e o r e m 3.3.5. The trigonometric series u = X^ aeZ „ cae2niax is the Fourier series of a periodic hyperfunction, i.e., rau — u for all a e Z n if and only if for every e > 0 there exists a constant C > 0 such that \ca\ < Ce^a\ for every a 6 Z n , where \a\ = \cti\ + • • • + \an\. R e m a r k 3.3.6. One dimensional version of the above Theorem 3.3.5 was given by Sato by using quite different method from our heat kernel method. We finally estimate the partial sum of Fourier series. T h e o r e m 3.3.7. Let u be a periodic hyperfunction. Then for every e > 0 there exists a constant C > 0 such that |* m (u,a:)| = J2

\c<*e2*ia-x\
\a\<m

3.4

Almost periodic hyperfunctions

Let f(x) be a complex valued continuous function defined on K. A number r is called an e-almost period of f(x) if sup_ 0 0 < a ; < 0 0 | f(x + T) - f(x) | < e. If for any e > 0 there exists a number 1(e) such that every intervals of length /(e) contains an e-almost period of / , then f(x) is said to be almost periodic. It is well known that the following three statements are equivalent: (i) / is an almost periodic function. (ii) The set of translations fh for h £ R forms a relatively compact set with respect to the uniform topology. (iii) f(x) is the uniform limit of a sequence of (generalized) trigonometric polynomials Pm(x) = ]T)n=i anexp(i\nx), Xn e E. Schwartz (Stepanoff) used (ii) to define almost periodic distributions with the appropriate topology, and proved that the following are equivalent for any bounded distribution T: 1. T is almost periodic. 2. T is the finite sum of derivatives of functions in Cap. 3. T*(p£Cap

for all tp £ V.

Cioranescu 4 used (iii) to define almost periodic ultradistributions with the appropriate topology, and proved the following are equivalent for any bounded ultradistribution T:

160

1. T is almost periodic. 2. T *tp € Cap for every ip £ V^M^ which is the space of ultradifferentiable functions of class (M p ) of Komatsu in 18 . 3. There are two functions f,g€ Cap and an ultradifferential operator P of class (Mp) such that T = P(D2)f + g. We give below a result generalizing those of Schwartz and Cioranescu to the case of hyperfunctions. We also obtain the similar result for quasianalytic ultradistributions T, which, in fact, includes all the above results. We can replace the condition (M.3) imposed by Cioranescu by weakening the condition (M.3) to (C) to prove our characterization theorem for some class of quasi-analytic ultradistributions. For the proof we apply the characterization of bounded hyperfunctions in Chung-Kim-Lee 8 and the heat kernel method. Let D = -id/dt. Then an operator of the form P(D) = ^ anDn is called an ultradifferential operator (of class p\) if for every L there exists C such that |o„| < CLn/Mn,

n € N0.

(3.4.1)

The following result establishes the existence of a parametrix of an ultradifferential operator, which will be very useful later. Lemma 3.4.1. 21 For any h > 0 and e > 0 there exist functions v(t) € Cc°([0>el)> w(t) e C£°([e/2,e]) and an ultradifferential operator P(d/dt) such that \v(k\t)\
k =

\v(t)\
QA,---, 0
OO

P(d/dt) = Y,ak(d/dt)k,

\ak\ < dh't/kl2,

0 < hi < h,

k=0

P(d/dt)v(t)~w(t)

= 6.

We now recall the characterization of the bounded hyperfunctions as in 8 . E(x, t) belongs to the Sato space T for each t > 0 and E(x — -,t) belongs to Ai\ for each (x,t) e R x M + . So, for each T £ BL°° its Gauss transform u(x, t) = Ty(E(x - y, t) is a C°° function in R x Mr. Theorem 3.4.2. 8 The following statements are equivalent: (i) T

G

BL

161

(ii) T *
€ T. (iii) There exist two functions / and g belonging to Cj and an ultradifferential operator P of class {p\2} such that T = P{D2)f + g. (iv) The defining function u(x,t) of T belongs to C°°(R\) following: (dt - A)u(x,t)

and satisfies the

= 0 in R2+;

for every e > 0 there exists a constant C > 0 such that ||t«(a;,t)IU-(R) < C e e / t

in R2.;

and u( •, t) -¥ T as t -> 0 + in the sense that T(
tpeA^-

Here, C& is the space of bounded continuous functions on ]R. we now define almost periodic hyperfunctions. Definition 3.4.3. A hyperfunction T £ BL°° is called almost periodic if T is the limit of a sequence of trigonometric polynomials Pm(x) = X^n=i "n exp(«A„ar) in the space BL"> with respect to the strong topology, where A „ e l and a„ G C depend on m. For the proof of our main result we need the following lemmas. Lemma 3.4.4. 8 For any
let
t > 0.

Then ipt £ Aii for every t > 0 and tpt -> ip in ALI as t -> 0+. We now prove the continuity of the ultradifferential operator. Lemma 3.4.5. Let P(d/dt) be an ultradifferential operator of class {p! 2 }. Then the operator P(A) : A^ —• ^ L 1 is a continuous linear mapping. We are now in a position to state and prove the main result. Theorem 3.4.6. For T £ BL«. the following statements are equivalent: (i) T is almost periodic. (ii) T * ip 6 Cap for every ip £ T. (iii) There exist two functions / and g belonging to Cap and an ultradifferential operator P of class {p!2} such that T = P(D2)f + g. (iv) The defining function u(x, t) of T is almost periodic.

162

As an application to the Dirichlet problem for the half plane for the case of hyperfunctions we state the following theorem without proof, which generalizes the result in 4 . Theorem 3.4.7. Let T G BL<*> be almost periodic. Then there exists a harmonic function u(x, y) in the right half-plane such that (i) for every x > 0, the function y —»• u(x,y) is almost periodic; (ii) u(x, y) - * T in BL°° as x -> 0.

3.5

Mehler Kernel Approach

For x,£ € R™, k G NQ and |w| < 1, the Mehler formula is represented by

5>*(*)M0 ^*! = 7r- ? (i-™ 2 r f e~* s£i—€ia-4fs*-«.

(351)

Using (3.5.1), we introduce the Mehler kernel in the following form : k

7rf(l-e-4*) = f o r t e (0,oo) a n d z , £ G l n . Using the Mehler kernel E(x,£,t), as analogs of the Theorems 2.0.5 and 2.0.6 we now state the following theorems that represent the tempered distribution and the generalized function of Gelfand-Shilov type (i.e. the elements in (<S£)' for a > 1/2) as initial values of the solution U(x,t) of the Hermite heat equation (dt - Ax + \x\2)U(x, t) = 0. Theorem 3.5.1. Let T > 0 and E(x,£,t) be the Mehler kernel. Let u E S'(Rn). Then the defining function U(x,t) := (u(£),E(x,£,t)) is a C°° function in E n x (0, T) satisfying the following :

(i) (dt-Ax

+ \x\2)U(x,t) = 0.

(ii) For some positive constants C and N, sup r e R „ \U(x,t)\ f->0+.

< C t~N as

(hi) limt_>0+ U(x,t) = u(x) in «S'(Mn). Conversely, every C°° function U(x, t) in E™ x (0, T) satisfying (i) and (ii) can be expressed in the form U(x,t) — (u(£),E(x,£,t)) for unique u in S ( R n ) . Theorem 3.5.2. Let T > 0 and E(x,£,t) be the Mehler kernel. Let u G (<S£)'(Kn). Then the defining function U(x,t) := {u(£),E(x,Z,t)} is a C°° function in Rn x (0, T) satisfying the following :

163

(i) (dt-£x

+ \x\2)U(x,t)

= 0.

(ii) For all e > 0 there exist a constant C > 0 such that sup.,.gH„ \U(x,t)\ < Cexp

(et^=i\

(iii) lim t ^ 0 + U(x,t) = u{x) in («S£)'(Rn). Conversely, every Cx function U(x, t) in E n x (0, T) satisfying (i) and (ii) can be expressed in the form U(x, t) = («(£), E{x, £, t)) for unique u in (££)' (ffi"). As an application of the above theorems, we can give an extension of the following theorem in the spaces S (E™). Theorem 3.5.3. (Theorem 3.2, 25 ) If / is a function on 1 " satisfying \\(-A+\x\2)jf\U<MnJ for some positive constant M and all j £ No, then f(x) = C e~^~. Theorem 3.5.4. Let u £ S (E n ). Suppose that there exist some 8,p £ NQ and a constant L — L{0,p) > 0 such that - A + \x\2)j u(x), 4>(x)} | < L nj ||0|| 9|/) for all j £ N0 and all (j> £

(3.5.2)

S(Rn).

|x|2

Then u(x) = C e~ 2 for some constant C. The proofs of Theorem 3.5.4, Theorems 3.5.1 and 3.5.2 will be published elsewhere. Acknowledgments This work is partially supported by KOSEF R01-1999-00001. References 1. N. Aronszajn, Traces of analytic solutions of the heat equations, Asterisque 2 et 3, (1973), 5-68. 2. G. Bengel, Das Weylsche Lemma in der Theorie der Hyperfunktionen, Math. Z. 96 (1967), 373-392. 3. S. Bochner, Lectures on Fourier integral, Princeton University Press, Princeton, 1959. 4. I. Cioranescu, The characterization of the almost periodic ultradistributions of Beurling type, Proc. Amer. math. Soc. 116 (1992), 127-134.

164

5. J. Chung, S. Y. Chung and D. Kim, line caracterisation de I'espace de Schwartz, C. R. Acad. Sci. Paris Ser. I Math., 316 1993, 23-25. 6. J. Chung, S. Y. Chung and D. Kim, A characterization for Fourier hyperfunctions, Publ. RIMS, Kyoto Univ., 30 (1994), 203-208. 7. S.-Y. Chung and D. Kim, Distibutions with exponential growth and Bochner-Schwartz theorem for Fourier hyperfunctions , Publ. RIMS, Kyoto Univ., 31 (1995), 829-845. 8. S. - Y. Chung, D. Kim and E. G. Lee, Periodic hyperfunctions and Fourier Series, Proc. Amer. Math. Soc. 128 (2000), 2421-2430. 9. V. I. Gorbacuk, On fourier series on periodic ultradistributions, Ark. Mat. 34 (1982), 144-150. 10. I. M. Gelfand and G.E. Shilov, Generalized functions IV, Academic Press, New York, 1968. 11. M. Hashizume, K. Minemura and K. Okamoto, Harmonic functions on Hermitian hyperbolic spaces, Publ. RIMS, Kyoto Univ., Hiroshima Math. J. 3 (1973), 81-108. 12. L. Hormander, The analysis of linear partial differential operators I, Springer-Verlag, Berlin- New York, 1983. 13. A. Kaneko, Representation of hyperfunctions by measures and some of its applications, J. Fac. Sci. Univ. Tokyo Sec. IA, 19 (1972), 321-352. 14. A. Kaneko, Introduction to hyperfunctions, KTK Sci. Publ., Tokyo, 1988. 15. T. Kawai and T. Matsuzawa, On the boundary value of a solution of the heat equation, Publ. RIMS, Kyoto Univ., 25 (1989), 491-498. 16. K. H. Kim, S.-Y. Chung and D. Kim, Fourier hyperfunctions as the boundary values of smooth solutions of heat equations, Publ. RIMS, Kyoto Univ., 29 (1993), 289-300. 17. H. Komatsu, Boundary values for solutions of elliptic equations, Proc. Int e r n a l Conf. on Functional Analysis and Related Topics (Tokyo, 1969), Univ. of Tokyo Press, Tokyo, 1970, pp. 107-121. 18. H. Komatsu, Ultradistributions I, J. Fac. Sci. Univ. Tokyo Sec. IA, 20 (1973), 25-105. 19. H. Komatsu, Microlocal analysis in Gevrey classes and in complex domains, Microlocal analysis and applications (Montecatini Terme, 1989), pp. 161-236, Lecture Notes in Math., 1495 (1991), Springer, Berlin. 20. H. Komatsu and T. Kawai, Boundary values of hyperfunction solutions of linear partial differential equations, Publ. Res. Inst. Math. Sci. 7 (1971/72), 95-104. 21. T. Matsuzawa, A calculus approach to hyperfunctions II, Trans. Amer. Math. Soc, 313 (1990), 619-654. 22. M. Sato, Theory of hyperfunctions, I, J. Fac. Sci., Univ. Tokyo, Sec. I,

165

8 (1959), 139-193. 23. P. Schapira, Probleme de Dirichlet et solutions hyperfonctions des equations elliptiques, Boll. Un. Mat. Ital. (4) 2 (1969), 367-372. 24. L. Schwartz, Theorie des distributions, Hermann, Paris, 1966. 25. R. S. Strichartz, Characterization of Eigenfunctions of the Laplacian by boundedness conditions, Trans. Amer. Math. Soc. 338 (1993), 971-979. 26. D. V. Widder, The heat equation, Academic Press, New York, 1975.

GENERALIZED FOURIER T R A N S F O R M A T I O N S : T H E W O R K OF B O C H N E R A N D C A R L E M A N V I E W E D IN T H E LIGHT OF T H E THEORIES OF SCHWARTZ A N D SATO CHRISTER 0 . KISELMAN Uppsala University, P. 0. Box 480, SE-751 06 Uppsala, Sweden E-mail: [email protected] Salomon Bochner (1899-1982) and Torsten Carleman (1892-1949) presented generalizations of the Fourier transform of functions defined on the real axis. While Bochner's idea was to define the Fourier transform as a (formal) derivative of high order of a function, Carleman, in his lectures in 1935, defined his Fourier transform as a pair of holomorphic functions and thus foreshadowed the definition of hyperfunctions. Jesper Liitzen, in his book on the prehistory of the theory of distributions, stated two problems in connection with Carleman's generalization of the Fourier transform. In the article these problems are discussed and solved.

Contents:

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

1

Introduction Bochner Streamlining Bochner's definition Carleman Schwartz Sato On Carleman's Fourier transformation Liitzen's first question Liitzen's second question Conclusion References

Introduction

In order to define in an elementary way the Fourier transform of a function we need to assume that it decays at infinity at a certain rate. Already long ago mathematicians felt a need to extend the definition to more general functions. In this paper I shall review some of the attempts in that direction: I shall explain the generalizations presented by Salomon Bochner (1899-1982) and Torsten Carleman (1892-1949) and try to put their ideas into the framework of the later theories developed by Laurent Schwartz and Mikio Sato. In his book on the prehistory of the theory of distributions, Jesper Liitzen [1982] gives an account of various methods to extend the definition of the Fourier transformation. This paper has its origin in a conversation with

166

167

Anders Oberg, who pointed out to me that Liitzen had left open two questions. I shall try to answer these here. Thus this paper is in a way an historical survey, but not exclusively. 2

Bochner

In his book Vorlesungen iiber Fouriersche Integrale [1932], translated as Lectures on Fourier Integrals [1959], Salomon Bochner extended the definition of the Fourier transform to functions such that f(x)/(l + \x\)k is integrable for some number k. The usual Fourier transform of / is defined as

Hf){(i) = /(£) = / f(x)e-^dx,

£ € R,

JR

provided the integral has a sense; e.g., if / is integrable in the sense of Lebesgue. If / is sufficiently small near the origin we may form

and this integral now has a sense if f(x)/xk is integrable. If both / and f/xk are integrable, then the fcth derivative of gk is equal to / . So gk is a kth primitive function of / in the classical sense. This is the starting point of Bochner's investigation. To overcome the somewhat arbitrary assumption that / is small near the origin, Bochner [1932:112, 1959:140] adjusted the integrand by using the Taylor expansion of the exponential function and defined

E(a,k) X j - / f{x)e~la\"~L"{ka'X)dx,

aeR,

k 6 N,

{-lX)k

27T./R

where ( k-i

,

s,-

.

£ ^ ,

Lk(a,x)

= < .7=0

>0,

|*KI,

J

'

|*| > 1 .

k

The symbol X means that the difference between the two sides is a polynomial of degree less than k. Thus E is undetermined, but its kth derivative is not influenced by this ambiguity. Bochner's Fourier transform is this kth derivative, a formal object. Calculations are done on E, not on its derivative

168

In his review of Schwartz [1950,1951], Bochner [1952:79-80] remarks that any distribution in Z>'(R") agrees in a given bounded domain with the kth derivative of a continuous function for some sufficiently large k. Thus the Fourier transforms that Bochner constructs are locally not less general than distributions. Bochner's review portrays the theory of distributions as not going much beyond what he himself has presented in his book [1932]; "it would not be easy to decide what the general innovations in the present work are, analytical or even conceptual" [1952:85]. Later generations of mathematicians have been more appreciative. In several papers, starting in 1954, Sebastio e Silva developed the idea of defining distributions as derivatives of functions. He used an axiomatic approach; see, e.g., [1964]. 3

Streamlining Bochner's definition

In particular, if / vanishes for \x\ ^ 1, then the definition of E(a, k) simplifes to k 1 f e-iax We may therefore split any function / into two, f — fo + fi, where fo{x) = 0 for |x| > 1, fi(x) — 0 for | i | ^ 1. For / i we then define the function E(a,k) as above without the need to use Taylor expansions, while /o, a function of compact support, has a Fourier transform in the classical sense; the latter is an entire function of exponential type. Another way to avoid the division by {—ix)k is to divide instead by some power of 1 + x2. This can easily be done in any number of variables, defining x2 as an inner product, x2 = x • x = ^2 x2. Since the function 1 + x2 has no zeros, the mapping / i-> (1 + x2)f is a bijection. We may define F.Af)(0=

[

f(x)e-ixS(l

+ e2x2)-sdx,

(£R",

s,eeR.

Then the usual Fourier transform is obtained when s or s vanishes:

11 f(x)(l + x2)~t G L 1 (R), then Ts,e{f) xs a bounded continuous function for all s ^ t and all e ^ O . By applying a differential operator of order 2m, m 6 N, we can lower the index s by m units:

(l-e2A)mFsM)

= Fs-mAf),

169

where A is the Laplacian, A = ^d2/dx2. In particular, if (1 + x2)~mf is integrable, then !Fm,e{f) has a sense, and (1 - e2A.)mJrm<e(f) is a generalized Fourier transform of / . This is a somewhat streamlined version of Bochner's idea of defining a primitive function of the Fourier transform of / . The word primitive must now be understood in terms of the differential operator l—e2 A. We shall come back to this idea in section 9. The transform !Fs,eU) depends continuously on (s,e): for (p e S ( R n ) , TsAv) -> F*o,eo(£) This is easy to prove using norms which define the norms defined in (5.4) below or those given we get the same statement for u € <S'(Rn). In is zero, then for all

as

~> {so,e0) e R 2 . the topology of <S(R")) either in Proposition 9.3. By duality particular, if one of so and SQ

(s,e) ->• (s0,e0)-

Carleman

In 1935, Torsten Carleman lectured on a generalization of the Fourier transformation at the Mittag-Leffler Institute near Stockholm, Sweden. His notes, however, were not published until nine years later. In his book [1944] he quotes Bochner [1932] and the work of Norbert Wiener. In June, 1947, Carleman participated in a CNRS meeting in Nancy organized by Szolem Mandelbrojt and presented his theory there; see Carleman [1949]. Carleman's approach is quite different from Bochner's and foreshadows the definition of hyperfunctions. In fact, in modern terminology, he defines the Fourier transform for a large class of hyperfunctions of one variable. He remarks in the beginning that he will cover the case of functions which are integrable in Lebesgue's sense on each bounded interval and which satisfy the condition (4.1)

f \f(x)\dx = 0(\x\"), £->±oo, Jo for some positive number K. This condition is equivalent to the one imposed by Bochner, i.e., that / ( x ) / ( l + \x\)k be integrable for some k. He then remarks that the usual Fourier transform of an integrable function can be written g(z) = 4 =

f ° e-izyf{y)dy

= 9l(z) -

g2(z),

where 9l(z)

= J =

/

y ZTT J_oo

e-"yf{y)dy

and

g2(z) = - - L

\Jlit Jo

[°°V""f(y)dy;

170

the function g\ is well-defined and continuous for Im z ^ 0 and is holomorphic in the open upper half plane; similarly with g2 in the lower half plane. So the Fourier transform of / appears as the difference between the boundary values of two holomorphic functions g\ and g2, each defined and holomorphic in a half plane. In the case we are now considering, i.e., when / is integrable, the holomorphic function gj has boundary values in a very elementary sense: it admits a continuous extension to the closed half plane, given by the same integral. We shall write B{g\,g2){x) for the difference limy_+o+ ( 0 and Imz < 0 such that there exist nonnegative numbers a and /? and, for all #0 in the interval ]0,7r/2[, a number A(6Q) such that (4.2)

| / i ( r e * ) | < A(60)(ra+r^),

r > 0,

60 < 6 < TT - 60,

and (4.3)

| / 2 ( r e " ) | < A(90)(ra + r^),

r > 0,

TT + 60 < 6 < -90.

Let us call such a pair (/1, / 2 ) a Carleman pair of class (a, /?). He then defines [1944:48] another pair of holomorphic functions G, H by (4.4)

G(z) = - L / e~izyMv)dy and H{z) = -L= / V 27T JL

e-^f2(y)dy.

V27T JL'

Here £ is a half line in the upper half plane issuing from the origin, and similarly with L' in the lower half plane. Thus, for a particular choice of L, the function G will be defined in a half plane {z;lm(zy) < 0}; by letting L vary in the upper half plane, we will get a function defined in the complement of the positive real half axis; similarly H will be defined in the complement of the negative real half axis. In particular the difference H -G is defined in CxR.

171

The integrals are well-defined if j3 < 1; if not, Carleman has to resort to the kind of trick that Bochner used: he defines the m t h derivatives as (4.5)

G^{z) = -±=^e-»y(-iy)mMy)dy

and (4.6)

H^{z) = J = [

e-izy(-iyrf2(y)dy,

V27T JU

so that G and H are determined only up to a polynomial of degree at most m — 1. (This ambiguity will not affect the definition of the Fourier transform as we shall see.) The factor ym attenuates the singularity at the origin. He chooses m such that 0 ^ / 3 — m < 1; in fact, any m > /? — 1 will do. Next he defines gi(z)

= H(z) - G(z) for lmz > 0 and g2(z) = H(z) - G(z) for Imz < 0,

and remarks that it is easily proved that g± and g2 satisfy inequalities similar to those for / i and f2, \gi(reie)\ < A^ir"'

+ r""'),

60<8
and \g2{reie)\ < Ai(0 o )(r a ' + r ^ ' ) ,

-TT + ^O < 6 < -6»0>

where we may choose a ' = /? — 1 ^ — 1 and ,5' = a + 1 ^ 1 if we assume that /3 7^ 1,2,3,... . If /3 = 1,2,3,..., there appears a logarithmic term in the estimate at infinity, and we may take a' as any number strictly larger than P — 1 while P' = a + 1 as before. The interchange between a and ft means that the growth of the fj near the origin is reflected in the growth of the gj at infinity and conversely. A convenient comparison function is r 7 _ 1 ' 2 + r - 7 - 1 / 2 , i.e., with a = 7 — | , /? = 7 + | . Then we achieve symmetry for 7 7^ | , | , | , . . . . Thus Carleman's Fourier transform C F ( / i , / 2 ) of the pair / = (A,.£2) is the pair (31,^2); let us denote it by g = S(f). He needs to interchange the gj, so he defines a new operation T by T(g) = (/i 1 ,/i 2 ), where hi{z) = g2{z) and h2{z) = gi(z). Carleman's version of Fourier's inversion formula [1944:49] then reads (T o S o T o S)(fi, / 2 ) = (/1 + P, / 2 + P ) , where P is a polynomial; the latter does not influence the difference between the two functions. Since the calculation has to be done on the derivatives, the proof [1944:50-52] is a bit involved.

172

5

Schwartz

To extend the Fourier transformation Laurent Schwartz took the formula (5.1)

/

/(0s(0# = /

f(x)g(x)dx

as his starting point. The formula holds under quite general conditions and for most definitions of the Fourier transformation; no constant is needed. In particular it is true if both / and g are integrable on R™. To be precise, Schwartz [1966:231] defined (5.2)

HfW)=l

me-2i"**dx,

(€R n ,

so that the inversion formula reads

fix) = / Hf){0
x e R".

JR"

Formula (5.1) makes it natural to define Schwartz's Fourier transform SF(u) of a functional u by (5.3)

SF(u)(V)=u(^),

¥>€*,

Schwartz [1966:250]. In this way SF(u) is defined as a functional on a space of test functions $ provided u itself is defined on the space $ of all transforms of functions in $ . Schwartz made this situation completely symmetric by defining $ so that $ = $. Since he wished $ to contain V(Rn), it must also contain P ( R n ) U 2?(R n ), and this is indeed the case for the Schwartz space <S(Rn). It is defined as the space of all smooth functions on R n such that the norms
ifKf)

= I

JR"

f(xMx)dx,

f € Lf o c (R n ),

v> 6

V(Rn).

173

Then, in view of (5.1), Schwartz's Fourier transform of [/], defined by (5.3), is the distribution defined by the function / : SF([/])(¥>) = [/](£) = [/](¥>),

/ 6 i ' ( n

which means that SF extends the classical Fourier transformation to a larger class. For any c\,c2 ^ 0, the mappings
(1 + cix2)(p,

(5.5)

tp i-> (1 - c2A)(p

are topological isomorphisms of the space <S(Rn) of Schwartz test functions. Here, again, we write x2 for the inner product x • x = Y2X] a n d A for the Laplacian ]T d2/dxj. They correspond to each other under the Fourier transformation in the sense that, for c\ = 4n2C2, F{{l + cxx2)tp) = ( l - c a A ) £

^ ( ( 1 - c2A)tp) = (1 + ci£ 2 )£.

and

By duality the mappings (5.5) give rise to isomorphisms of <S'(R n ), u i->- (1 + cix2)u,

(5.6)

u H-> (1 -

c2A)u.

To define not only the Fourier transform u(i~) for all £ € R n but more generally the Fourier-Laplace transform u(Q for all £ £ C " (at least as a functional), it would be desirable to find a space $ such that (5.7)

PC$C5,

and such that (5.8)

for all tp £ $ and all C € C", f

e^*
is well defined.

In 1961 I attempted to define the Fourier-Laplace transform in C™, inspired by Schwartz's definition of <S(Rn). I realized then that it is not possible to require (5.7) and (5.8) and keep the symmetry in the sense that $ = $. Indeed, the function defined as VKO = exp ( - 1/(1 - ||£|| 2 )) for ||£|| < 1 and V>(£) = 0 for 11£|| ^ 1 is in D(R") but its Fourier transform tp = ip does not satisfy JRn \e~^-x(p(x)\dx < oo for any ( e C " \ R n . By abandoning the requirement that $ be equal to $, Gel'fand & Shilov [1953] found other interesting spaces of test functions. In particular they defined the Fourier transform of a distribution as a functional on V. See also Ehrenpreis [1954, 1956]. Hormander [1955] announced a very general theory of this nature.

174

In my work of 1961,1 kept the symmetry $ = $ and instead relaxed the condition (5.7) that $ contain V. In this work I defined a space W of test functions consisting of all entire functions i p o n C " such that the norms |M|ro=

sup

|^)|e™» R e *ll,

m€N,

||Imz||^ro

are all finite; W is equipped with the topology defined by these norms. The Fourier transformation is an isomorphism of W onto itself, and the same is true of the dual space W . I studied the Fourier transformation and convolution in these spaces and developed several of their properties but my work was not published. Kelly McKennon independently discovered the same space and published his results in [1976]; he was kind enough to mention my work (McKennon [1976:178]). Hormander [1998] gives a full account of the ideas he presented in his short note [1955]. 6

Sato

Mikio Sato presented his theory of hyperfunctions in [I958a,b,c, 1959, I960]. Boundary values of holomorphic functions (without any growth condition) are the basic objects of his theory; in particular, all distributions in one variable are represented as the difference of such boundary values from the upper and lower half planes. The Fourier transform in one variable is defined for pairs of functions with infra-exponential growth, generalizing Carleman's conditions. The theory of Fourier hyperfunctions in several variables is a theme outside the scope of this article. Let us only mention that it was developed by Kawai [1970a,b] and further developed by Morimoto [1973, 1978] and Saburi [1985]. 7

On Carleman's Fourier transformation

In this section we shall comment on Carleman's theory and also show how Carleman pairs can be constructed. Carleman's theory does not lend itself easily to calculations. For the pair of functions representing the Dirac measure placed at the origin one has to take P = 1 in (4.2), (4.3) and so has to use m ^ 1 in (4.5), (4.6). It is easy to calculate explicitly the functions G' and H' in (4.5), (4.6), and the jump in H — G is found to be the constant -A= as expected. For the Dirac measure placed at a point a ^ O w e may take ft — 0; it is, however, difficult to calculate G and H from (4.4), although their difference H - G can be easily

175

found. For a > 0, H - G is 0 in the upper half plane and - - 4 = e lza

lower, so that the jump is -k=e~

lza

in the

as we should expect. One even receives

V lit

the impression that Carleman avoids examples and applications of his theory to simple generalized functions. We note that if (/i,/2) is a Carleman pair of class (a,/3), then the pair (2/1,2/2), which is of class (a + 1,/? — 1), has a Carleman transform which is just i times the derivative of the transform of (/1, / 2 ) . Similarly, the derivative of (/1, / 2 ) , which is a pair of class (a — 1, /? + 1), has a transform which is iz times the transform of (/ 1 ; / 2 ) . Thus the usual rules hold. However, Carleman does not mention these simple rules. Bremermann & Durand [1961:241] write that Carleman's work is limited to I? and IP functions. As we have seen, this is not so: the Carleman pairs are much more general. Along the rays through the origin Carleman assumes that the fj have a temperate behavior (see (4.2) and (4.3)), but there is no restriction in the growth of Ao(9o) or Ai(6o) when 6Q tends to zero. If we impose a temperate growth also on A0(6Q), then the condition can be written as \fj(z)\ ^ C|Imz|~ 7 (|2| a + |^| _ / 3 ), which means temperate growth both at infinity and at the real axis, and we get exactly the temperate distributions. Thus Carleman's classes are more general than the temperate distributions. On the other hand, the hyperfunctions are even more general, because for them we do not impose temperate growth at infinity or the origin. To make the last remark clearer we may map the upper half plane onto the unit disk by a Mobius mapping, with the origin going to the point 1 and infinity going to —1, say. Then the temperate distributions correspond to pairs of holomorphic functions of temperate growth at the boundary of the disk, which means that \f(z)\ ^ C(l — | z | ) - a , \z\ < 1, for some constants a and C, while the hyperfunctions impose no restriction on the growth at all. The intermediate Carleman pairs have a temperate behavior along all circles through 1 and —1. To define (91,92) it would actually be enough to assume that /1 and / 2 grow slower than e*'*' for every positive e along every ray (infra-exponential growth). This, however, would allow for a faster growth of (51,32) at the origin, and it would then not be possible to attenuate the singularity simply by multiplying with a power of y as in (4.5), (4.6); another definition of the transform would be needed. Although Carleman does not offer any comment on this problem, I would surmise that this is the reason why he limited the admissible growth to powers of |^j along the rays. Given an integrable function we have seen how its Fourier transform is the difference between the boundary values of two holomorphic functions, each

176

defined in a half plane. But how do we represent the function itself as such a difference? The answer is: by forming its convolution with 1/z. It follows from Plemelj's formulas lim — = vp I - ) =F ni5, y->o x + iy V v xJ I ±v>o " that the difference between the limits from the upper and lower half planes of 1/z is —2m5. So, apart from a factor, 1/z represents the Dirac measure, the most fundamental distribution. Let us define a function E(z) = i/(2nz) for z 6 C \ {0} and convolution with E by (E*f)(z)

= / E(z-t)f(t)dt

= I E(t+iy)f(x-t)dt,

z = x+iy S C \ R ,

JR

JR.

whenever the integral has a sense, e.g., if f(x)/(l + \x\) is integrable. We may also form the convolution E * u for any distribution u with compact support; it is holomorphic in C x s u p p u , where we consider the support of u as a subset of the complex plane. Let us now see when the two holomorphic functions have a limit at the real axis in the classical sense. Proposition 7.1. If f £ C 1 (R) and f(x)/(l + \x\) is integrable, then h{z) = (E* f){z), Ixaz > 0, is the restriction of a continuous function defined in the closed upper half plane. Proof. Any function of class C1 can be written as f(x + t) = f{x)+t

Jo

f'(x + ts)ds = f(x)+tg(x,t),

x,teR,

where g(x,t) = JQ f'(x + ts)ds is a continuous function of (x,t) G R 2 . We shall study the behavior of (E*f)(z) when Rez belongs to a bounded interval [-a, a]. We assume first that / has compact support. We choose a positive number b which is so large that f{x - t) vanishes when x € [-a,a] and t $. [—6,6]. Then for x = Rez £ [—a,a] and y = Imz > 0,

* fb f(x-t) , if(x) fb dt i fb t h(z) = — / ^ — A d t = -^-i/ — / —g(x,yy y ' 2W-6 t + iy 2TT J_bt + iy 2TT J_b t + iy

'

-t)dt. ;

The first integral in the last expression can be evaluated, and it is easily seen that it tends to |/(a;o) as x + iy -» x0 with y > 0. In the second integral we note that t/(t + iy) tends to 1 almost everywhere as y ->• 0 and

177

that \t/{t + iy)\ ^ 1. Lebesgue's theorem on dominated convergence can be applied and we see that the second integral tends to a limit too. The extension to R is therefore given by lim

x+iy-*x iy->z<j 0

1 h(x + iy) = -f(xQ) Z

y>0

i frb - — / g(x0, Z7T J

-t)dt

i

1 i fb f1 -J{%o) — / dt f'(x0 - ts)ds, 2 J V u/ 2 ?"r .J-b Jo

xQ e

[-a,a].

Next we consider the general case and write / = J2jez fj u s m g a partition of unity, where fj has its support in the interval [j — 1, j +1], say. The argument just presented applies to any finite sum of the E * fj. For indices j > a + 1 and points z satisfying Re,z ^ a < j — 1 we have the estimate

$E*fj)(z)\

=

l_iE(z-t)fj(t)dt<Mj_i_a)jj_i\fjmt.

Therefore the sum X)j>o+i ^ * U converges uniformly for Rez ^ a in view of our hypethesis that f(t)/(l + |i|) is integrable. The terms with j < —a — 1 can be estimated in the same way, and we are done. Tillmann [1961a,b] and Martineau [1964] studied systematically the boundary values in the sense of distributions of holomorphic functions. In the framework of Proposition 7.1 we can form the difference of the extensions from the upper and lower half planes. We see that h{x + iy) — h(x — iy) tends to f(xo) as x + iy —» xo while y > 0. However, this conclusion holds even if we assume only that / is continuous as the next result shows. Proposition 7.2. If f E C°(R) and f(x)/(l

+ \x$ is integrable, then

(E * f){x + iy) - (E * f)(x - iy) -» f(x0) locally uniformly as x + iy —¥ xo while y is positive. Proof. We have fix -1) h(z) = (E * f)(z) = j - / Jl*."'dt, 27r JR t + iy so that

z =x +

iyeC\K,

for positive y. This is the Poisson integral of / ; ^ a sjf a is a well-known approximate identity, so the integral tends to f(xo) as x + iy -» xo while y > 0, even locally uniformly.

178

Thus the difference H{z) = h(z) - h(J), a harmonic function, is much easier to work with than each of the terms when it comes to passage to the limit. (However, there are of course other difficulties connected with the harmonic functions: for instance, they do not form an algebra as do the holomorphic functions.) This transform H is mentioned by Arne Beurling in his note [1949:10]; he called it la transformee harmonique and used it to define the spectrum of / . In his book [1983], Hormander chose this as the main definition of hyperfunctions. A systematic development of the theory of hyperfunctions as boundary values of harmonic functions was undertaken by Komatsu [1991, 1992]. 8

Liitzen's first question

In his book Jesper Liitzen wrote [1982:192]: I do not know whether Carleman's function pairs under the conditions (42) always represent distributions. Tillmann's growth condition in [1961b] suggests that this is not the case. The conditions (42) that Liitzen refers to are the conditions (4.2), (4.3) of the present paper. We shall confirm Liitzen's conjecture. In doing so we shall allow ourselves to use freely the language of the later theories of distributions and hyperfunctions. Fix a point o € R, o ^ 0, and define f(z) = exp (

),

z e C \ {a}.

\z — a) Since / is bounded in a neighborhood of 0 as well as in a neighborhood of oo, the pair of functions obtained by taking the restriction of / to the upper and lower half planes is a Carleman pair of class (a,/3) = (0,0), but it does not represent a Schwartz distribution. Indeed, if it did, then this distribution would have its support contained in the singleton set {a}, and so would be a finite linear combination of the Dirac measure at a and its derivatives. Since (a; - a)Sa = 0, and similarly (x - a)mu — 0 when u is a derivative of 5a if only m is large enough, the pair representing u would be entire after multiplication by (z-a)m for some m. Now this is obviously not the case with e x p ( l / ( z - a ) ) ; the singularity at a is essential and cannot be removed just by multiplying with some power of z — a. We use here the fact, well known since the work of Sato, that all distributions, in particular all distributions with compact support, can be represented by pairs of functions holomorphic in the upper and lower half plane, and that

179

this representation is unique up to adding an entire function to both functions in the pair. So the zero distribution is only represented by a pair (/i, / 2 ) where the fj are restrictions of the same entire function. 9

Liitzen's second question

Liitzen writes in his book [1982:192] I have not been able to rigorously prove that Carleman's and Schwartz's Fourier transforms of a tempered distribution are equal; but formal calculations strongly suggest that this is the case. We can confirm Liitzen's suggestion: Theorem 9.1. For any temperate distribution u G <S'(R), Carleman's Fourier transform agrees with Schwartz's Fourier transform; more precisely, u is represented by a Carleman pair (/i,/2) and the difference between the boundary values, taken in the sense of distributions, from the upper and lower half planes of Carleman's Fourier transform CF(/i,/2) is equal to Schwartz's Fourier transform SF(u) of u. Writing as before E{z) = i/(2irz), defined for z £ C \ {0}, we know that a distribution u G £'(R) is the difference between the boundary values of the holomorphic function (E * u){z), z 6 C \ R. In fact, it is not necessary that u have compact support; it is enough that u is so small at infinity that the convolution has a good sense. In particular we may assume that u is a continuous function which satisfies \u{x)\ ^ C(l + |:E|)~ Q , X G R, for some positive a. If / is a function in C X (R) DL 1 (R), we may form by convolution a Carleman pair {fi,h) to represent it (Proposition 7.1). We then know (Proposition 7.2) that the Carleman Fourier transform of this pair is a Carleman pair representing the classical Fourier transform / of / . We also know that the Schwartz Fourier transform SF(/) of a function / G i 1 ( R ) agrees with the classical Fourier transform. When comparing the definitions, we must agree on a definition of the classical Fourier transform. Let us use in the sequel Carleman's definition

HfW

= f(0 = -4= / f(x)e-^dx,

£ G R.

V27T JR

Modifying Schwartz's definition accordingly, we can say that B ( C F ( / i , fi)) = f and SF(/) = [/] for / G C J (R) n L : ( R ) . We express this fact by saying that CF and SF agree on these functions.

180

To go from these special functions to distributions we shall use the rules F{xf) = iF(f)' and Ftf') = i{F(f), which hold for both Carleman's and Schwartz's definitions. By applying them twice we see that ^ ( ( 1 - A ) / ) = (1 + e)Hf)

and ^ ( ( 1 + x2)f) = (1 - A ) ^ ( / ) .

This yields, for functions / e C ^ R ) n

L^R),

SF((1 + x2)f) = (1 - A)SF(/) = (1 - A)CF(/) = CF((1 + x2)f) and SF((1 - A ) / ) = (l + ^ 2 )SF(/) = (1 + ^ 2 )CF(/) = CF((1 - A ) / ) . Repeated use of these rules proves that the Schwartz and Carleman transformations agree on all generalized functions of the form (Ps o P s _i o • • • o P\)f for some / £ C 1 (R) n L 1 (R), where each Pj is one of the operators 1 + x2, 1 — A. But this class of generalized functions is equal to all of <5'(R) as shown by Theorem 9.4 below. (We will actually need that result only for k = 1 and m = 0.) The mapping 1 - A has an inverse, which in one variable is convolution with the function w(x) = | e - l z l , x € R . This is an integrable function, and its derivative in the sense of distributions is w'(x) = w{x) for x < 0 and w'{x) = -w(x) for x > 0, which is also an integrable function, and ||w||i = \\w'\\i = 1. Its second derivative in the sense of distributions is the measure w" = w — S, whose total mass is 2. We thus have three well-defined convolution operators
w*
||(i + x2y(u * vOiU < Cpll(i + *2)Vlloo,

f e 5(R).

Proof. Writing ip(x) = (1 + x2)ptp(x) we see that we have to prove that u*

(1 + X2)P

£

a X )P V

[1 +

2

when HV'lloo ^ 1. For u = 5 this is clear; for u = w,w' it suffices by symmetry to prove that

f Jo

o-y (l + (x-y)2)P

-,d*V ^^

c

P

M (1+X , ?2)P' w

x

eR.

181

This is easy when x ^ 0, for then 1/(1 + (a: - y)2)p ^ 1/(1 + x2y. x > 0 we consider two integrals. First 2 p A/a E/2 0U + -I- (x^ - y) < / r)r

When

(i + x2Y

J^x/2 x/2

Over the interval [0, x/2] we may estimate as follows: f*/2

Jo

p

-y

/-x/2

dv

(l + (z-v)V *J0

e

-j,

(1 + (a;/2)2)P

dy

< (l^ +?a;w/ /) "io« " ' * <^ ( l +C'x )P' 2 4 p

y

2

Proposition 9.3. T/ie topology of the space <S(R) is i/ie weakest topology such that all norms

- A)VI,

P, Q € N ,

are continuous. More explicitly, ll^'flVlloo ^ C||(l + x 2 ) p (l - A ) V I U ,

V G <S(R),

w/tere p = j / 2 and q = /s/2 w/ien fc S 2N; g = (fc + l ) / 2 when k 6 2N + 1. T/ie norms are essentially increasing in their indices: \\ 0. ^ Q1 • (Hence it suffices to use the norms \\
\u(tp)\ ^ C|M|„,„

ip G S ( R n ) .

Proof. When k is even we have to prove that ||(1 + X 2 ) ^ 2 V l l o o ^ C | | ( l + Z 2 H 1 -

A)Vlloo,

which may be written as

||(i + x2y{w"y* * viloo ^ c||(i + x2Fviloo. To prove this we use the lemma q times with u = w". When k is odd we have to prove that ||(1 + x2YD2«-Vlloo

^ C||(l + x2Y(l

- A)Vlloo,

which may be written as ||(1 + x2y(w")««-V

* w> * V;||oo ^ C||(l + x2)PV||oo-

182

Here we use the lemma q — 1 times with u = w" and once with u = w'. Finally, the inequality |M| P , 9 ^ C|Mlp,g'> where q ^ q', follows from q' — q applications of the lemma with u = w. Theorem 9.4. Given any temperate distribution u G S'(R) and any numbers k,m G N , there exist a number s G N and a function f £ Ck(R) satisfying (1 + x2)mf G £ X (R) such that u = (Ps o Ps_x o • • • o Px)f, where each Pj is equal either to 1 - A or to multiplication by 1 + x2. This theorem is similar to that of Schwartz [1966:239]; however, it is adapted to the operator 1 - A and its proof is more direct. Proof. If u is a temperate distribution, we know that H

for some constants C, p, and q; see (9.1). There is a distribution v such that ( l - A ) « ( l + a;2)pu = u, for the mappings (1 - A) 9 and (1 + x2)p are isomorphisms. We see that |u(V>)| ^ CIIV'llo.o, if ip is of the form (1 + x2)p{l - A ) V for some y G 5 ( R ) . But this means that the estimate holds for all ip G «S(R), and thus v is a measure of finite total mass. The convolution product g = (1 — A ) - 1 v = w * v is a bounded continuous function. We can then form h = (1 — A)~rg, which is a bounded function of class Ck if 2r ^ k. Indeed, (1 - A)""1 maps CJ' D I/°° into Cj+2 n L°°, j G N , so (1 - A ) " r maps C° n L 0 0 into C 2 r n L°°. Finally / = (1 + a;2)"™-1/* is such that (1 + a; 2 )" 1 / G L^R-)Collecting what we have done we see that u = (1 - A)«(l + x2)p(l 10

- A)r+1(l +

x2)m+1f.

Conclusion

In his lectures in 1935, Torsten Carleman represented the Fourier transform of a function of temperate growth as a pair of functions defined in the upper and lower half planes, respectively. He also extended the Fourier transformation to be defined on such pairs. In modern parlance, he defined the Fourier transformation of a class of hyperfunctions, but only in one variable. Although very different in nature, his definition agrees with the one given later by Laurent Schwartz for temperate distributions. His calculus, however, is valid for a class of hyperfunctions strictly larger than the temperate distributions.

183

Carleman's monograph [1944] was probably not well-known; the same goes for the proceedings article [1949]. However, a pirate edition of Carleman's book [1944] was published in Japan after the war (Professor Hikosaburo Komatsu, personal communication, November 28, 2001). Nevertheless, it seems that the work of Carleman did not play a role in the early development of the theory of hyperfunctions in Japan. Liitzen [1982:191] writes that the connection was pointed out only by Bremermann & Durand in [1961]. Indeed they quote both Carleman [1944] and Sato [1958a].

References Beurling, Arne 1949 Sur les spectres des fonctions. Analyse harmonique; Nancy, 15-22 Juin 1947, pp. 9-29. Colloques internationaux du Centre national de la Recherche scientifique, XV. Paris: CNRS. 133 pp. Bochner, Salomon 1932 Vorlesungen uber Fouriersche Integrate. 229 pp. (Reprinted 1948 by Chelsea Publishing Company, New York.) 1952 Review of Theorie des distributions by Laurent Schwartz [1950, 1951]. Bull. Amer. Math. Soc. 58, 78-85. 1959 Lectures on Fourier Integrals. Annals of Mathematics Studies, Number 42. Princeton: Princeton University Press, x + 333 pp. Translated by Morris Tenenbaum and Harry Pollard. Bremermann, H. J.; Durand, L., Ill 1961 On analytic continuation, multiplication, and Fourier transformations of Schwartz distributions. J. Mathematical Physics 2, 240258. Carleman, Tforsten] 1944 L'integrale de Fourier et questions qui s'y rattachent. Publications Scientifiques de l'lnstitut Mittag-Leffler, 1. 119 pp. (Reprinted 1967 with an additional note on page 107.) 1949 Sur l'application de la theorie des fonctions analytiques dans la theorie des transformers de Fourier. Analyse harmonique; Nancy, 15-22 Juin 1947, pp. 45-53. Colloques internationaux du Centre national de la Recherche scientifique, XV. Paris: CNRS. 133 pp. Ehrenpreis, Leon 1954 Solution of some problems of division. I. Division by a polynomial of derivation. Amer. J. Math. 76, 883-903.

184

1956

Analytic functions and the Fourier transform of distributions. I. Ann. of Math. (2) 63, 129-159.

Gel'fand, I. M.; Silov, G. E. 1953 Fourier transforms of rapidly increasing functions and the questions of uniqueness of the solution of Cauchy's problem. [Russian.] Uspehi Matem. Nauk (N.S.) 8, no. 6(58), 3-54. Hormander, Lars 1955 La transformation de Legendre et le theoreme de Paley-Wiener. C. R. Acad. Sci. Paris 240, 392-395. 1983 The Analysis of Linear Partial Differential Operators I. Distribution Theory and Fourier Analysis. Berlin: Springer-Verlag. ix + 391 pp. 1998 On the Legendre and Laplace transformations. Ann. Scuola Norm. Sup. Pisa CI. Sci. (4) 25, 517-568. Kawai, T. 1970a The theory of Fourier transforms in the theory of hyperfunctions and its applications. [Japanese.] Master's Thesis, University of Tokyo. 1970b On the theory of Fourier hyperfunctions and its applications to partial differential equations with constant coefficients. J. Fac. Sci. Univ. Tokyo Sect. IA, 17, 467-517. Komatsu, Hikosaburo 1991 Microlocal analysis in Gevrey classes and in complex domains. Microlocal analysis and applications (Montecatini Terme, 1989), 161-236. Lecture Notes in Mathematics, 1495. Berlin: SpringerVerlag. 1992 An elementary theory of hyperfunctions and microfunctions. Partial differential equations, Part 1, 2 (Warsaw, 1990), 233-256. Banach Center Publ., 27, Part 1, 2. Warsaw: Polish Academy of Sciences. Liizten, Jesper 1982 The Prehistory of the Theory of Distributions. New York: SpringerVerlag. viii + 232 pp. McKennon, Kelly 1976 Analytic distributions. J. reine angew. Math. 281, 164-178. Martineau, Andre 1964 Distributions et valeurs au bord des fonctions holomorphes. Theory of distributions. Proceedings of an International Summer Institute held in Lisbon, September 1964, pp. 195-326. Lisbon: Instituto Gulbenkian de Cincia.

185 Morimoto, Mitsuo 1973 On the Fourier ultra-hyperfunctions I. [Japanese.] Res. Inst. Math. Sci. Kokyuroku 192, 10-34. 1978 Analytic functionals with noncompact carrier. Tokyo J. Math. 1, 72-103. Saburi, Yutaka 1985 Fundamental properties of modified Fourier hyperfunctions. Tokyo J. Math. 8, 231-273. Sato, Mikio 1958a On a generalization of the concept of functions. Proc. Japan Acad. 34, 126-130. 1958b On a generalization of the concept of functions. II. Proc. Japan Acad. 34, 604-608. 1958c Theory of hyperfunctions. [Japanese.] Sukagu 10, 1-27. 1959

Theory of hyperfunctions I. J. Fac. Sci. Tokyo Sect. I, 8, 139-193.

1960

Theory of hyperfunctions II. J. Fac. Sci. Tokyo Sect. I, 8, 387437. Schwartz, Laurent 1949 Theorie des distributions et transformation de Fourier. Analyse harmonique; Nancy, 15-22 Juin 1947, pp. 1-8. Colloques internationaux du Centre national de la Recherche scientifique, XV. Paris: CNRS. 133 pp. 1950/51 Theorie des distributions. Vol. I, II. Publications de I'lnstitut de Mathematique de l'Universite de Strasbourg, nos. 9 and 10; Actualites scientifiques et industrielles, nos. 1091 and 1122. 1966 Theorie des distributions. Paris: Hermann, x m + 420 pp. Sebastio e Silva, J. 1964 Integrals and orders of growth of distributions. Theory of distributions. Proceedings of an International Summer Institute held in Lisbon, September 1964, pp. 327-390. Lisbon: Instituto Gulbenkian de Cincia. Tillmann, Heinz Giinther 1961a Distributionen als Randverteilungen analytischer Funktionen II. Math. Z. 76, 5-21. 1961b Darstellungen der Schwartzen Distributionen durch analytische Funktionen. Math. Z. 77, 106-124.

T H E EFFECT OF N E W STOKES CURVES IN T H E E X A C T STEEPEST D E S C E N T M E T H O D TATSUYA KOIKE Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto, 606-8502 Japan E-mail: [email protected] YOSHITSUGU TAKEI Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502 Japan E-mail: [email protected]

1

Introduction — Brief review of exact steepest descent method and several problems of it

The exact steepest descent method was born in [4] by combining the ordinary steepest descent method with the exact WKB analysis. (See, e.g., [2] for the notion and notations of the exact WKB analysis used in this report.) It is a straightforward generalization of the ordinary steepest descent method and provides us with a new powerful tool for the description of Stokes curves as well as for connection problems of ordinary differential equations. Still in [4] some restrictions were imposed for its applicability. In this report, in order that we may remove such restrictions and apply it to more general equations in the future, we discuss the effects of several kinds of new Stokes curves in the exact steepest descent method. Let us here review the exact steepest descent method briefly. An equation to be discussed is an ordinary differential equation with polynomial coefficients of the following form:

P^= £

a^V-^

= 0,

(1)

0<j<m 0
where a ^ is a complex constant and 77 > 0 is a large parameter. By the Laplace transformation ip = J exp(r]x£,)ij)(£)dt, with respect to an independent variable x with a large parameter 77 (1) is transformed into

P^ = Y,a^m-k(-^)

186

^ = 0.

(2)

187

In the exact steepest descent method, following the idea of Berk et al. ([7]), we take a WKB solution ^fc (more precisely, the Borel sum of V^) and consider its inverse Laplace transform (j)

exp(vxOi>kdt = I , ( exp U f x£ - / xk(£)d(, j + • • • I d(,

(3)

to discuss a solution of the original equation (1). Here Xk(0 is a root (with respect to x) of the characteristic equation

P(*,0=f J2 a ^ V = 0

(4)

0<j<m 0
and Of!' is a steepest descent path of Refk(x, £) passing through a saddle point of fk{x,Oi where fk(x,Q — %(,— J Xk(Qd(; denotes the phase function of (3). Note that, since the integrand of (3) is the Borel sum of a WKB solution ipk, the so-called Stokes phenomenon occurs and tpk becomes a linear combination of ipk and V>£' (as was first observed by Voros [11]) when the steepest descent path a),3' crosses a Stokes curve of type (k > k') for P. Hence, taking this Stokes phenomenon into account, we find that we should globally consider a linear combination of integrals of the following form: / J i}

-i

exp(rjx£)''l>kd£ + ck, /

J*y

exp^xQ^v

d£, + ck" / J

exp{r]x{,)ipkl,d{, + •

w

(5) where akv is a steepest descent path of Refv emanating from a crossing point of ak3' and a Stokes curve of type (A: > k') {ak,, , uk,,,, • • .are also steepest descent paths obtained by similarly repeated bifurcation procedures), and cy (ch" . . . as well) is a constant determined by the connection formula which describes the Stokes phenomenon at the crossing point. The configuration of these steepest descent paths (the whole of which is called an "exact steepest descent path") is closely related to asymptotic behaviors (including exponentially small terms) of a WKB solution of (1). For example, a Stokes curve of (1) is characterized as a point where an exact steepest descent path passing through a saddle point hits another saddle point ("Exact Steepest Descent Path Ansatz" or "ESDP Ansatz" for short, cf. [4]). The exact steepest descent method is, in a word, a method of investigating global asymptotic behaviors of solutions of (1) by tracing the configuration of exact steepest descent paths. However, if we try to apply this method to general equations, we may encounter several difficulties mainly because the Laplace transformed equation

188

(2) often has new Stokes curves and/or some singular points. The purpose of this report is to discuss how these difficulties can be overcome by studying a few examples with the aid of a computer. To be more concrete, we investigate the following situations: In [4], to avoid these difficulties, we imposed the restriction that the Laplace transformed operator P is of at most second order (i.e., the degree of the coefficients of P is at most two). If we try to remove this restriction, the effect of a new Stokes curve for P becomes a problem. In Section 2 we first investigate the effect of a new Stokes curve for P. Next in Section 3 we consider the case where P has a singular point. In particular, when P has a singular point of "simple pole type", there appears a Stokes curve emanating from such a singular point ([8], [9]). In Section 3 we investigate the effect of a Stokes curve emanating from a singular point of simple pole type. Furthermore, in Section 4 we deal with the case where the characteristic polynomial p(x,^) defined by (4) is factorized as p(x,0

= (£ - a)(£ - po(x))(Z ~ PiO»0)

(6)

with a being a constant. In a generic situation a root £j(x) of (4) with respect to £ gives a saddle point of the phase function fk{%,0 of (3) and the integral (5) along an exact steepest descent path passing through £j(x) corresponds to (the Borel sum of) a WKB solution ipj of (1) with the phase factor T] j x £j(x)dx. However, in the situation where p(x,£) is factorized as (6) only po(x) and P\(x) give saddle points of (3) and hence the number of saddle points is strictly smaller than that of WKB solutions of (1) (that is, a WKB solution with the phase factor rj Jx adx = r\ax cannot be expressed in the form of (5)). In section 4, taking up an example which has its origin in the problem of non-adiabatic transition probabilities in quantum mechanics, we consider the case where (4) has a root (with respect to £) independent of x. Finally in Section 5 we give a summary and concluding remarks.

2

The effect of a new Stokes curve

In this section we study the following equation: P ^ = -(12 + 1 4 i > ^ + ((6 + 3i)x2 + 2 - l i t - (24 + 28? + 4ci)?T 1 ) yfir dx - (a;3 - —(1 + 2i)x - (12 + 6i - 2ic0)r]-1x) r)2i/j = 0,

(7)

189

where en and c\ are arbitrary complex constants. This equation is an example of Carroll-Hioe type equations discussed in [3]. As a matter of fact, by setting

n=2-i,

r2

= _ ( - + 2i),

r3=0,

fti2 = - 3 + 4i, fi23 = 1 - 3i

in Equation (CH) of [3, p. 629], we obtain (7). As (7) is of second order, its Stokes geometry can be completed without introducing any new Stokes curve. The result is shown in Figure 1, where and in subsequent Figures 2, 5 and 8 a Stokes curve (with its type being specified by a symbol "+ < —" etc.) is designated by a solid line, while a wiggly line designates a cut which is placed to define a characteristic root of (7) as single-valued analytic function. Note

Figure 1. Stokes curves of (7).

that the origin x = 0 in Figure 1 is a regular singular point of "double pole type" discussed in [1, Section 3]. In what follows we will observe that a new Stokes curve of the Laplace transform of (7) plays an important role when we try to detect a Stokes curve of (7) by using the exact steepest descent method.

190

The Laplace transform of (7) is d3ip

,„

„.^

d2ip

+ ( (12 + 14z)£ 2

+ ((2-llt)-4cuT 1 )fr 3 ^

(9)

+ 21C07J M r y 2 ^ = 0

and the configuration of its ordinary and new Stokes curves is drawn in Figure 2. In Figure 2 (and in Figures 5 and 8 as well) a small dot designates

Figure 2. Stokes curves of the Laplace transformed equation (9).

an ordinary or virtual turning point and a broken line means that no Stokes phenomenon occurs on that portion of the curve. Note that P itself is a Carroll-Hioe type operator. Hence we readily find that it is transformed into a Laplace type operator by a change of independent variables z = £2 and consequently it possesses an integral representation of solutions. Figure 2 can be confirmed to be the correct Stokes geometry of (9) by using the integral representation. As is observed in Figure 2, (9) has several new Stokes curves. These new Stokes curves are necessary to detect a Stokes curve of (7). For example, we find that, in order to detect a Stokes curve 7 in Figure 1, a new Stokes curve 7 (of type (0 < 2)) passing through an ordered crossing point A in Figure 2

191

is necessary in the following way: Let us take two points XQ and Xi near 7 as is shown in Figure 1 and describe the configuration of exact steepest descent paths at these two points XQ and x\. We then obtain Figure 3. (In Figure 3 (a)

(b)

Figure 3. Exact steepest descent paths at x — XQ (a) and x = x\ (b). (a)

(b)

Figure 4. Exact steepest descent paths at x = xo (a) and x = x\ (b).

and subsequent figures describing exact steepest descent paths as well a solid line designates a steepest descent path and a dotted line designates a Stokes curve of the Laplace transformed equation.) Figure 3 shows that between XQ and x\ a steepest descent path a, which a steepest descent path passing

192 through a saddle point £ + bifurcates at its crossing point with 7, hits another saddle point £_. This clearly visualizes the necessity of the new Stokes curve 7Furthermore, we next let xo and xi be closer to a turning point 00 (cf. Figure 1). Then the steepest descent path passing through £ + crosses an ordered crossing point A and, for example, at XQ and x\ the configuration of exact steepest descent paths becomes that described in Figure 4. In Figure 4 a bifurcated steepest descent path a obtained by repeated bifurcation from the steepest descent path passing through £ + (that is, the steepest descent path passing through £ + bifurcates another steepest descent path at its crossing point with a Stokes curve of type (1 < 2), and further it bifurcates a at its crossing point with a Stokes curve of type (0 < 1)) hits a saddle point £_. In this manner a new Stokes curve is built in the exact steepest descent method very exquisitely to the effect that it explains the change of configuration occurring when a steepest descent path crosses an ordered crossing point very well. 3

The effect of a singular point of simple pole type

We next consider the following example in this section:

PiP = x^

- x^pL _ tf** _ ^

ax6 ax1 ax where 6 — exp(2i7r/5). Its Laplace transform is given by P^-f]

n£-i)-77T ( r «- 4"£ )-- ^ $ + + (< <

(0f + i0 + (-6f

+ 2)IJ-2)

rj2^

= 0

(1Q)

(ii)

= 0.

The Laplace transformed equation (11) has a singularity of "double pole type" at f = 0 and of "simple pole type" at £ = 1. In particular, there appears a Stokes curve emanating from the singular point £ = 1 of "simple pole type". In what follows we investigate the effect of such a Stokes curve emanating from a simple pole type singularity. The configuration of Stokes curves of (10) is shown in Figure 5. We first take two points XQ and x\ near a Stokes curve 7 (cf. Figure 6) and draw the configuration of exact steepest descent paths at these points. The resulting figures are Figures 7(a) and 7(b). A change of the configuration can be readily observed: A steepest descent path a passing through a saddle point

193

Figure 5. Stokes curves of (10).

Figure 6. Magnification of Figure 5 near the Stokes curves 7.

£0 bifurcates another steepest descent path at, at a crossing point of a and a Stokes curve 7 emanating from the singular point £ = 1 of simple pole type, and a and at, simultaneously hit a saddle point £1. This shows the relevance of a Stokes curve emanating from a simple pole type singularity in the exact steepest descent method. Remark 3.1 In Figure 7(a) a steepest descent path at, bifurcated at a crossing point of a and 7 intersects again with 7. We can verify that this second intersection point of at, and 7 is passed also by the original steepest descent path a. As a matter of fact, letting £, denote the first intersection point of a and 7 (i.e., the bifurcation point of at,), we find that 7, a and at,

194 (a)

(b)

6

Figure 7. Exact steepest descent paths at x = xo (a) and x = x\ (b).

can be described respectively by 7 : Im / (xk, - xk)d£ = 0,

(12)

a : Im

(13)

ab : Im

z(£-f*)

/

xk'd£\ = 0

(14)

with some indices k and k'. Then the second intersection point £** of er;, and 7 should satisfy Im /

A.

(a;*/ - a;fc)d£ = 0

(15)

and I m I £(£»* - £*)

_

/

a;*'dC

= 0-

(16)

= 0,

(17)

Summing up these two equations, we obtain I m I £(£»» - £ » )

/

which implies that a also passes through £,

xkd£J

195 If we further let x0 and x\ approach closer to a crossing point B of Stokes curves (cf. Figure 6), we obtain Figure 8. It appears that only a hits a saddle (a)

(b)

Figure 8. Exact steepest descent paths at x = xo (a) and x = x\ (b).

point £1 and a Stokes curve 7 emanating from £ = 1 is irrelevant in Figure 8. However, the degeneracy in Figure 8 is "multiple-ply"; other steepest descent paths are bifurcated at crossing points £* and £** of a and 7 and these bifurcated steepest descent paths simultaneously hit £1 with overlying a (cf. Remark 3.2 below). Thus a Stokes curve emanating from a simple pole type singularity is relevant also in Figure 8. Remark 3.2 The Stokes curve 7 emanating from a singular point £ = 1 of simple pole type can be described by Im / {xk> - xk)d(, = 0, •/

(18)

Im / (xk' - xk)d£ = 0,

(19)

or equivalently by

where £t denotes a point on 1Z, the Riemann surface of xk and xk> ramified at £ = 1, which has the same projection with £ but is different itself from £. Since 7 passes through £* and £»», we have Im / (xk> - xk)d£, = Im / (xk< - xk)d^ = 0. JV. J it.

(20)

196 Taking these relations into account, we can verify that the steepest descent paths bifurcated at £» and £** overlie a by the same reasoning as in Remark 3.1. 4

The effect of a constant characteristic root

In this section we discuss an example of the form dx3

^ S 2 + (-2 + (i + N 2 + |c 1 | 2 )r,- 1 )r ? ^(21 ) dx

2

- (ix + ( - 1 + i\c0\2x + I|CI| 2 )T7 _ 1 ) rfij, = 0 with Co and C\ being arbitrary constants, which is equivalent to the following 3 x 3 non-adiabatic level crossing problem in quantum mechanics: .d t d^V>

-1 0 0 \ = ri

0 x 0

0 0 -x)

/ 0 c0 0 +TT1/2

CO 0 ci

(22)

UcTO

The configuration of Stokes curves of (21) is shown in Figure 9. There are three 1<2

0<2 0<2

0<1 ()<1

2<1

K0

2
0<1

0<2

2<1

2<0 2<0

K0K0

1<2

Figure 9. Stokes curves of (21). ordinary turning points (all of them are double) and two virtual turning points for (21). (The correctness of Figure 9 can be confirmed by the same reasoning as that employed in [5, Remark 2.1].) Since the characteristic polynomial of (21) can be factorized as (£ — «)(£ 4- ix)(£ — ix), the difficulty explained in

197 Section 1 appears for (21) due to the existence of a constant characteristic root £o = i- In what follows we investigate the effect of this constant characteristic root £o = iFigures 10(a), . . . , 10(f) respectively describe the configuration of exact steepest descent paths at points XQ, ..., x$ near a crossing point A of Stokes curves in Figure 9. In Figure 10, besides exact steepest descent paths passing (a)

(b)

(<0

(d)

(e)

(f)

Figure 10. Exact steepest descent paths at x — XQ (a), x — x\ (b), x = x5 (f). through saddle points £i and £2, we have added steepest descent paths of Re/± = Re ( a * - / x±dA

(«"/

= Re ( a * - f (±i$dn

(23)

emanating from the constant characteristic root £0 = * a n d steepest descent paths bifurcated from them also ("exact steepest descent path emanating from £0 = «")• Note that, since £0 = i is a "new" turning point for the Laplace transformed equation in the sense of [10], a Stokes curve passing through £0 = i is also included in Figure 10. As is clear from Figure 10 (for example,

198

from comparison between Figures 10(c) and 10(d)), a Stokes phenomenon for Borel resummed WKB solutions of (21) occurs at a point where such an exact steepest descent path emanating from £o = i hits a saddle point. This example strongly suggests that a constant characteristic root like £0 = i of (21) should be dealt with in the same manner as a saddle point. Remark 4.1 From the above considerations it may appear that only one of the exact steepest descent paths of Re/± emanating from £o = i should be relevant. However, generically speaking, both exact steepest descent paths (n exact steepest descent paths in case the Laplace transformed equation is of n-th order) must be taken into account, as is shown by an example in [6, Section 4] (cf. Figures 18 and 19 of [6]). Both exact steepest descent paths being relevant might be related to the fact that £0 — i is a "new" turning point in the sense of [10]. 5

Concluding remarks

As the examples in Section 2 and 3 show, we should deal with a new Stokes curve and a Stokes curve emanating from a singular point of simple pole type as if they were an ordinary Stokes curve in defining exact steepest descent paths. These new Stokes curves and Stokes curves emanating from simple pole type singularities are built in the exact steepest descent method very exquisitely. Furthermore, when the characteristic equation has a root (with respect to £) being independent of x, such a constant characteristic root plays the same role with a saddle point. In order to establish the exact steepest descent method for generic equations, we are required to take into account (and to prove rigorously) these effects. In particular, when there exists a constant characteristic root £o = OL, it is an interesting and important problem to find out an exact description of (the Borel sum of) a WKB solution of Pip = 0 with the phase factor r\ax in terms of the inverse Laplace integrals. (The fact that ^ = a is a "new" turning point in the sense of [10] might be a key to attack this problem.) Acknowledgments The authors would like to thank Prof. T. Kawai and Prof. T. Aoki for valuable advice and discussions with them. This work is supported in part by JSPS Grant-in-Aid (No. 13740096 for T.K. and No. 11440042 and No. 13640167 for Y.T.).

199

References [I] T. Aoki, T. Kawai and Y. Takei: Algebraic analysis of singular perturbations — On exact WKB analysis. Sugaku, 45(1993), 299-315. (In Japanese. Its English translation is published in Sugaku Expositions, 8(1995), 217-240.) [2] : New turning points in the exact WKB analysis for higher order ordinary differential equations. Analyse algebrique des perturbations singulieres, I; Methodes resurgentes, Hermann, 1994, pp. 69-84. [3] : On the exact WKB analysis for the third order ordinary differential equations with a large parameter. Asian J. Math., 2(1998), 625-640. [4] : On the exact steepest descent method — a new method for the description of Stokes curves. J. Math. Phys., 42(2001), 3691-3713. [5] : Exact WKB analysis of non-adiabatic transition probabilities for three levels. RIMS preprint (No. 1331), to appear in J. Phys. A. [6] T. Aoki, T. Koike and Y. Takei: Vanishing of Stokes curves. In this volume. [7] H. L. Berk, W. M. Nevins and K. V. Roberts: New Stokes' line in WKB theory. J. Math. Phys., 23(1982), 988-1002. [8] T. Koike: On a regular singular point in the exact WKB analysis. Toward the Exact WKB Analysis of Differential Equations, Linear or Non-Linear, Kyoto Univ. Press, 2000, pp. 39-54. [9] : On the exact WKB analysis of second order linear ordinary differential equations with simple poles. Publ. RIMS, Kyoto Univ., 36(2000), 297-319. [10] : On "new" turning points associated with regular singular points in the exact WKB analysis. RIMS Kokyuroku, No. 1159, 2000, pp. 100110. [II] A. Voros: The return of the quartic oscillator — The complex WKB method. Ann. Inst. Henri Poincare, 39(1983), 211-338.

FOURIER'S H Y P E R F U N C T I O N S A N D HEAVISIDE'S P S E U D O D I F F E R E N T I A L OPERATORS HIKOSABURO KOMATSU Department of Mathematics, Science University of Tokyo Wakamiya-cho 26, Shinjuku-ku, Tokyo, 162-0827 Japan E-mail: [email protected] Mathematical folklores are convincing, especially when told by great mathematicians, but they are not necessarily based on the historical facts. "Fourier formulated his expansion theorem but failed to give a proof." "Heaviside invented operational calculus." "The (S-function was introduced by Dirac." "The theory of distributions clarifies all mysteries in operational calculus." Reading Fourier's book and Heaviside's papers, we give negative evidence to the above. A knowledge of hyperfunctions and ultradistributions makes it easy to understand these works.

1

Fourier's J-function and operational calculus

Schwartz' book "Theorie des distributions" 21 starts with citations of Heaviside's paper on operational calculus5 and Dirac's on quantum mechanics2 in which Dirac unified Heisenberg's matrix mechanics and Schrodinger's wave representation by the use of his ^-function. These are supposed to be the first places where distributions were actually employed. It is little known, however, that Fourier introduced the J-function much earlier as an integral kernel, as Dirac, and formulated his integral formula as 1 f00 S(x - a) = — I cos(q(x - a))dq. fl" Jo

(1)

His book "Theorie analytique de la chaleur" 3 claims this repeatedly. In particular, the statement on p. 449 is incontestable. He also gives many proofs of (1) and, in particular, the last one given on pp. 546-551 is essentially the same as that of Dirichlet, Riemann and Jordan and is sufficient for developing his "fonctions quelconques" which are actually piecewise real analytic functions. Fourier did not know the e-S arguments. Therefore, it is easy to find fault with his proof. However, reading Riemann's papers 18 ' 19 etc., we find that our concept of arbitrary functions was given birth to by Fourier's integral representation of arbitrary functions, and that accordingly one had to redefine the integrals for such functions. The J-function might have been easier to understand when the infinitesimal and the infinity existed in reality.

200

201 Moreover, Fourier has developed on pp. 511-546 an operational calculus in order to solve not only the initial value problem of the heat equation but also many other linear partial differential equations. He starts with an expansion of a general solution into a formal power series, and then, writes the result in an operational expression. For example, consider the initial value problem: dv

d2v

^

n

*=&?'

f > M £ R

'

(2)

v(0,x) = (f>(x), x £ R. Then, we have v(t, x) = 4>{x) + t ^

(x) + | / 4 > (x) + | / 6 ) (*) + .••,

which may be written = etD (f)(x), where D =

(3)

d/dx.

The power series (3) converges only for entire functions cf>{x) of order < 2. Applying etD to cos(q(x — a)), we get etD cos(q(x - a)) = e~tq cos(q(x - a)). Fourier obtains the heat kernel as its integral with respect to q: -

J_^ e-t<2 cos(q(x - a))dq = —e~

"«

.

(4)

His proof given on p. 477 makes use of a shift of the integration path in OO

e~q dq /

-oo

as in our textbooks. Fourier attributes the integral formula of solutions v(t,x) = — 7 = / e~JL*?~(f>(a.)da=-j= e~q2(j>(x + 2qVi)dq (5) n 2V7r£ J-OO V J-oo to Laplace in the latter form but he never forgets to mention also that Laplace's paper 13 appeared later than the year 1807 in which he submitted the first part of his book to the Academy (and its committee with Laplace as a member rejected to have it published). In the same way, the solution of the initial value problem of the wave equation in two dimensions is expressed as . r* . u(t,x,y) = (cos(tV-&)(l>)(x,y)+ (cos(sV-A)ip) (x,y)ds. (6) Jo

202

Fourier almost stops here but it is not difficult hence to obtain the explicit formula of solutions due to Poisson and Herglotz (see e.g. van der Pol-Bremmer 17 or Komatsu 11 ). Another case in which Fourier succeeded in obtaining the integral formula of solutions is the initial value problem of the equation

of vibration of a thin elastic rod. Solutions to linear ordinary differential equations with analytic data are analytic except at singular points of equations and data, and solutions for nonanalytic data can be approximated by analytic solutions for analytic data. Fourier seems to have thought that this was also the case with partial differential equations. For example, he claimed that the solution of the Laplace equation was represented as (6) with —A replaced by A with two initial values. The right representation is, of course, the Poisson integral (e~ty/~^<j>)(x,y) with one data. Along with the usual initial value problem (2), Fourier considers also the initial value problem d2v _ dv

0^2 =-of, t,xeR, v(t,0) = (t), i e R , —v(t,0)

= ip(t),

(8)

teR,

in the direction of space as time. In this case, we have, as the formal power series representation of solutions, V(t,x) = 4>{t) + | ^ ' ( t ) + ^ ( t )

+^ W ( 0 + •••

+xm + l^'w + | y 2 ) m + ^ (3) w + • • • = (cos{x^/-d/dt)4>){t)

+ I

(9)

(coa(yy/-d/dt)x/)){t)dy.

Jo

The power series (9) converges if functions 4> and ip are in Gevrey class of index (2), i.e., if their p-th derivatives are bounded by Ceepp\2 for any e > 0. The initial value problem (8) in the direction of x is weakly hyperbolic of constant multiplicity and of irregularity 2 (cf. Komatsu 8 ). Hence it is well posed in Gevrey classes (s) and {s} for 1 < s < 2 and 1 < s < 2, respectively.

203

Thus it follows that if (*) and ip(t) are ultradifferentiable functions (resp. ultradistributions) of one of the above classes, then v(t, x) defined by (9) is an ultradifferentiable function (resp. ultradistribution) of the same class in (t,x) and it is a unique solution of (8). If the data (i) and ip(t) are in an intermediate class of functions or generalized functions, then the solution v(t, x) is no more a function but an ultradistribution of class (2), in particular, in t. On p. 544 Fourier gives an integral formula of the fundamental solution but it does not converge in the sense of a function. Since there are nontrivial ultradifferentiable functions (t) and ip(t) of class (2) and with compact support, we can construct a nontrivial infinitely differentiable solution v(t, x) of the heat equation which vanishes for t < 0. Therefore, we do not have the uniqueness of solutions to the initial value problem (2), although Fourier believed in the uniqueness and even gave a proof. 2

The heat equation and Fourier's boundary condition

Fourier's results mentioned above are all stated in the last Chapter IX on pp. 425-601. Few people reach there because it requires a great patience to read through the first parts of the book, which look like an inedited collection of many papers written at diverse times. The most important contribution of Fourier is, of course, his derivation of the heat equation, and then his solution of it by means of trigonometrical series. Fourier derives the heat equation from the following two physical laws governing homogeneous solids in heat conduction: 1. The conservation law of heat : The total amount of heat is conserved, where heat = specific heat C x density D x volume x temperature v. 2. Newton's law of internal heat conduction : The heat propagates in proportion to the gradient of the temperature. Its ratio K is called the heat conductivity. A modern book like Bergman-Schiffer1 spends only one page or two for this derivation. Let the heat propagate in a homogeneous solid of the shape Q C R 3 . Take an arbitrary volume V d O. Then we have %- f CDvdV at Jy

=K f JQV

gradwndS,

204

where n is the unit outer normal on the smooth boundary dV. follows from the Gauss formula that [ CD^-dV

Jv ly

ot

Hence it

= [ Kdiv grad vdV.

Jv JV

UL

Since V is arbitrary, this is equivalent to the heat equation dv

K

(d2

d2

m = cD{w

+

d2 \

+

d? w)v

(10)

in the domain Q. In Fourier's book 3 equation (10) appears for the first time only on p. 122. His derivation is awkward because he doesn't know the Gauss formula. Moreover, he mentions neither of the two physical laws explicitly. On the other hand, his considerations on the boundary conditions are much more explicit. He distinguishes two cases: 1. In direct contact with a heat sink or source : When the body D is in direct contact with a heat sink of constant temperature c on a part S of its boundary dQ, the temperature v has to satisfy the Dirichlet condition v\s = c

(11)

2. Exposed to a medium: When the body is exposed to a medium of constant temperature, Fourier assumes that the loss of heat per unit surface is proportional to the difference of the temperatures of the surface and the medium. Let h be its ratio called the conductivity relative to the medium. We make use of a scale of temperature in which the temperature of the medium is 0. Then, we have for any part S of the boundary dfi —K / grad v • n dS = h / vdS. IS

JS

Since S is arbitrary, this is equivalent to the boundary condition: dv K— + hv = Q an

on dn.

(12) '

In its full generality condition (12) appears on p. 141. In a special case we find it already on p. 110. Fourier was interested in the thermal history of the earth. In order to estimate the age of the earth from its birth at a burning temperature, he needed the condition. Condition (12) is often called the Robin condition but Gustafson-Abe 4 shows that Gustave Robin (1855-1897) has little to do with the condition

205

and that the textbook 1 of Bergman-Schiffer may be responsible for making this popular although Bergman and Schiffer introduced only Robin's function which is a fundamental solution to the elliptic equation div(K grad v) = 0 under boundary condition (12). Fourier has not considered the adiabatic case in which the Neumann condition dv

- 0

on dQ

(13)

should be satisfied. 3

Fourier's trigonometrical expansions and hyperfunctions

Fourier considers only on pp. 160-210 the first mathematical problem, which is not an initial value problem of the heat equation but the stationary problem of finding a harmonic function (x, y) on the semi-infinite slab ' 0 < x < oo, 7T

7T

2 - " -

2

under the boundary conditions (z,-|)=tf>(z,|)=0, 0(O,y) = l,

x>0,

(14)

-\
(15)

The first Fourier expansion in the book 4 f sign cost/ = - \ cosy

cos3y cosoy — + —;

cos7y

cos9y !

] ^

. „. (16)

was introduced in order to solve this problem as 4 f (x,y) = -ie 4

f

= - Re < e-{x+iy) K

{

e~3x cos 3u

„x

cosy

e-3(x+iy)

e-5(x+iy)

+ 3

e~ 5x cos hy + •>

= 5

e~7x cos 7w 2- + .. A

} = - Rearctan(cc + iy). )

7T

206

By a separation of variables it is easy to see that e~mxcosmy,

m = l,3,5,...,

are harmonic functions satisfying boundary condition (14). In order that a linear combination of these functions satisfy (15), we must have 7T

1 = a cos y + b cos 3y + c cos 5y + d cos 7y +

7T

~2
(17)

with coefficients a,b,c,d,..., etc. independent of x and y. The way Fourier determined them is extravagant. He equates the 2m-th derivatives of both sides of (17) at the origin, and obtains an infinite number of equations 1 = a+

b+ c+ d+

e+

f+

g+

0 =a+326+52c+72d+92e+ll2/+1325+ 0 =a+346+54c+74d+94e+ll4/+134g+ 6

6

6

6

6

(18)

6

0 =a+3 6+5 c+7 (i+9 e+ll /+13 p+ 0 = a+386+58c+78d+98e+ll8/+138g+ Then, he solves the first m equations as equations for the first m unknowns with the other unknowns deleted, and shows that the solutions tend to the coefficients of (16) a s m tends to oo by the Wallis formula and a five pages long calculation. This could not be said a proof of (16) but he gives two independent proofs for this expansion, and more than two proofs for the general expansions. In the second proof given on pp. 177-179 he considers the partial sum t< \

cosZy

cos5y

_ „ _ , cos(2m - l)w

After an easy calculation we have 2-f- sin 2y = ( - l ) m _ 1 (cos(2m - l)y - cos(2m + 1)) = (-I) m 2sin2m2/sin2/ ay and hence y sin 2mx (-1) m fry sir cos a;

(19)

We know that /(0) tends to 7r/4 and that the integral converges to 0 uniformly on the interval [-n/2 + e, n/2 - e] for any e > 0. Fourier appeals to a repeated

207

integration by parts. We also remark that the integrand is essentially the same as the Dirichlet kernel near the singular points ±n/2. Fourier's third proof given on pp. 189-190 is mysterious. He writes: "If the sum of two arcs is equal to a quarter n/2 of the circumference, the product of their tangents is 1 and therefore we have -IT = arctan u + arctan —; (20) 2 u and we know since a long time ago the series which gives the value of the arc; thus we have the following result:

now if we write eXy^

in place of u in equations (20) and (21), we will have -7T = arctan eXyjf~^ + arctan e _ X v / ~ T

(22)

and 1 cos3u cos5w cos7y cos9w , , -7r = c o s 2 / - — ^ - + —^--^JL —-JL tc.; (23) + + e the series of equation (21) is always divergent, and that of equation (23) is always convergent; its value is \it or —\n." A most natural interpretation of this proof would be obtained by the theory of hyp erfunctions (see Sato 20 or Morimoto 14 ). We imbed the unit circle T in the complex plane C or the Riemann sphere P . Then, the space of hyperfunctions on T is defined by the quotient vector space B(T) = 0 ( C \ T ) / C ( C ) 2 C ( P \ T ) / 0 ( P ) ,

(24)

where O stands for the space of holomorphic functions. Since P \ T is the union of the open unit disk D and its reciprocal D _ 1 , we have the representation B(T) = (0(D) +

OCD-^/C

Cn2 n ;limsup " A / W < 1 [ .

= \ J2 U=-°o

|n|->oo

The Laurent expansion X^oc c«z™

ma

y alwavs

(25)

J De

divergent but the series

oo

Y,

cnzn,

\z\ < 1,

F(z) = {

(26) c

- J2 "*"> 1*1 >1> V

n=—oc

208

is convergent on each domain and represents a unique hyperfunction f{eie) G B(T) such that

Cn =

h l f(ei(h>e~in<>de-

(2?)

The hyperfunction f(e'e) is identified with the boundary value of defining holomorphic function F(z): f(eie)

= lim{F(reie)

- F(r-V9)).

(28)

r/l

The distribution 26(9 + f ) - 25 (9 - •§) is represented by

and hence its indefinite integral sign cos 9 by

H K 4 - - H - ' - ^ - ' ) } - »•> Taking its boundary value by (28), we have (16) or (23) for -7r/2 < y < n/2. The difference of the signature for reciprocal powers in (21') from Fourier's in (21) comes from the negation — in (28) of the boundary value of the function F(z) on the outer disk D _ 1 . From p. 210 on Fourier discusses the development of an arbitrary function into a trigonometrical series. First he considers odd entire functions 4>(x) and shows by an essentially same method as above that their restrictions to (—IT, IT) can be developed into the Fourier sine series (x) = bi sin x + &2 sin 2x + 63 sin 3x H . (29) At first sight this proof with a 20 pages long calculation may look absurd to the reader of today but Peetre 16 has shown that the main part of the proof can be saved for entire functions (x) of sufficiently small exponential type. Fourier also remarks that the coefficients are obtained by the integrals 2 fn bm = — (j)(x) sin mxdx. (30) 71" Jo

Fourier's conviction that an arbitrary function can be developed comes from his observation that if we integrate in (30) up to an a < n, then (29) holds for 0 < a; < a and the right-hand side vanishes for a < x < IT. We can prove this fact very easily for a real analytic function (j)(x) defined on a neighborhood of [0, a] by the theory of hyperfunctions with the help of Littlewood's Tauberian theorem 23 .

209 After he formulated his expansion theorem, Fourier writes on p. 260: "The expression \ + J l ^ c o s ^ a ; - a) represents a function of x and a such that if we multiply it by an arbitrary function F(a) and, if, having written da, we integrate it between the limits a = —TT and a = ir, we will change the proposed function F(a) into a similar function of x multiplied by the half circumference n." 4

W. Thomson's theory of submarine cables

Telegraphy was invented by C. Wheatstone in England and by S. F. B. Morse in the United States in 1837 independently. Long distance telecommunication systems were of vital importance for both countries to maintain their huge territories. Already in 1850 they started to construct submarine cables for telegraphy but first cables didn't show their expected performance. Signals were so blurred that they became illegible at the receivers. In 1855 W. Thomson published the first paper 22 analyzing the signal transmission on long cables. The paper is composed of extracts from three correspondences he exchanged with G. G. Stokes. Thomson took into consideration only the electro-static capacity between the wire and the sheath and the electric resistance of the wire, and ignored the leakage and the selfinduction. Let C and R be the capacitance and the resistance per unit length, respectively. Then, the electric potential v(t, x) and the electric current j(t, x) of the wire obey the equations

For the sake of simplicity we consider an infinitely long cable x > 0. Let the cable be quiet for t < 0. Then, we put an electric motive force cf>(t) at a; = 0 and observe the electric current j(t) at x > 0. Eliminating j , we have the following problem: d2v(t,x) _ „rdv{t,x) dx2 dt ' v(t,x) = 0, t<0, x>0,

' ( 32 )

v(t,0) = 4>(t), * > 0 . The solution Thomson and Stokes gave is symbolically v(t,x) = e~V^x^Ti<j>(t),

(33)

210

and actually 1

/"* VRCx

, , wRCX* s)ds

= 2^Lv^w^ ^^

-

(34)

Their derivations are somewhat different but we can compute the kernel by 1

f°°

-xVrTdT = ^ w ^ >

w

where

r>0,

-^Vr, IT = <

! - * ' /—

I y/2

'-T,

T < 0.

Except for the coefficient RC the differential equations are the same in Thomson's problem (32) and in Fourier's (8), but the numbers of the initial values are one for (32) and two for (8). The solutions expressed by (33) represent waves going in the positive direction in x, and there are also waves going in the opposite direction. We cannot exclude the latter type of solutions without the quietness condition in the middle of (32). We remark, however, that Fourier's solutions (9) satisfy the quietness condition if <j>(t) = ip(t) — 0 for t < 0. This shows that physical phenomena cannot be described by differential equations only. For Thomson's problem (32) it is intuitively clear that we have only waves going in the positive direction, so that only one initial condition is allowed. This intuitive fact is not necessarily explained clearly in a mathematical or physical language. Heaviside relied on the causality as shown later. We may also appeal to the dissipation of energy 12 . Thomson did not give the details how he computed integral (35) but we can find in the paper 22 Stokes' complicated proof based on the Fourier sine transformation. Expression (34) of a general solution is also due to Stokes. In 1866 a transatlantic cable was successfully constructed and Thomson was knighted for his contributions. Later he became Lord Kelvin. 5

Heaviside's operational calculus

Oliver Heaviside (1850-1925) is an Englishman born in a poor family and had no formal education of college level. Since his mother was a sister of Wheatstone's wife, he was able to get a position of telegraphist but resigned at the age of 24 and spent the rest of his life in writing mathematical and physical papers for a weekly commercial newspaper "The Electrician."

211

His theory of operational calculus 5,6 was invented first to make the Fourier-Thomson theory more accessible to engineers and then to establish the theory of cables for telephone. He knew from his experience that the leakage and the self-induction of cables improve their performance at high frequencies in spite of all our intuition. He was also one of the very few who understood Maxwell's theory at that time and was able to compute the selfinductance of cables, which was not negligible. The electromagnetic theory we learn today at universities is actually what Heaviside rewrote in terms of vector analysis, which is another invention of his. Let L and G be the inductance and the leakage conductance per unit length. Then, he showed in 1881 that the voltage v and the current j obey

( 36 )

aa - — =Gv + C— dx dt' and hence the differential equation for v is 82v

/„

, d\

(„

„d

This is usually called the telegraphist equation but the Heaviside equation would be a more appropriate name. After we have an operational representation like (33), we have to carry out difficult integration in order to obtain the solution v(t, x) or the integral kernel (35). Heaviside employed instead the fractional power series expansions in operator p = d/dt and computed the integrals as series of distributions with the use of the following correspondence of operators and distributions under the causality condition: ,-c

l

+

pa6(t) = { T(-aY S^(t),

(38)

a = 0,1, 2

For example, Thomson's problem (32) is solved as follows: v{t,x) = e-VW5x^${t)

= {l-qx+— =(coshqx-smb.qx)
(33)

— + ...JW) (39)

212

where q = VRC^p

= VRCy/d/dt.

(40)

To compute the integral kernel (35), let cf>(t) be the delta function d(t). Then, cosh go; being a differential operator (of infinite order), we have (cosh qx)S(t) = 0,

t^O.

On the other hand., we have (

m+l

d\m+*

fdY

{dt) 1

y/*\

m+1 /'d\ .dt)

2

S(t)

1

(-l)m+1

(2m+1)!

Vh

20F

m!4m*™+l

Hence it follows that —(sinhqx)S(t) = 0 for t < 0 and that ,

.21 (RHr2)"1

yfWx ^ 1 / 2Vrt* ^ 0 m\\

RCx2\m At J

E^l"—)

/d\m+i

VRCx 2^¥

XC.'

=~^e-^,

t>0.

This is his method which Schwartz21 commented as 'audacious'. It was more so at the end of the nineteenth century. Heaviside's paper 5 originally consisted of three parts. After two first parts were published, the third one was rejected as nonsense and was never published (cf. Heaviside6 and Nahin 15 ). Today we can righteously interpret most of his results with the help of the theory of hyperfunctions and ultradistributions 10 ' 11 . In the above case, for example, cosh qx and sinh qx are an ultradifferential operator of class (2) and a pseudodifferential operator of class (2), respectively, so that the series converge as ultradistributions of class (2) and coincide with the above sums on {t; t ^ 0}. On the other hand, since e~x^ is integrable as a function in r, the integral (35) is a continuous function in t. Thus, we have had a rigorous proof of (35). A much more fantastic calculation is given almost at the end of Part I of Heaviside5 (cf. Heaviside6 p. 439). Let a > 0 and p > 0. We set

1=

/o ^ w r d x = h + l =^>+«*
Then, we have by integration by parts

213

_ e-ppa fp e~xxa + ~r(a + 1) J0 T(a + l)dX

_ e~ppa ~ T(a + 1)

Wl

_ e-Pp"- 1 f°° e~xxa-1 W2 ~ T(a) +Jp T(a)

_ e-ppa~l ~ T(a)

J dX

p a+1

e-

+

+

p T(a + 2) e~ppa-2 f(^l)

+

+

e~ppa+2 T(a + 3)

+

" ' '

e~ppa-z T(^=T) + ''" '

We note that the above expansion of w\ always converges but that of w2 is a divergent asymptotic expansion except for the case where a is an integer and the series terminates after a finite number of terms. Then, he regards, for a constant h > 0, ehp=

Y

—

^

k= — oo

(41) '

as an expansion of the shift operator ehpf(t)

= f(t + h).

(42)

Applying this to the Heaviside function l(t) = p~1S(t), _.

1N

1{,+h)

sinrra ^

i}

he obtains

j^+k

= —£j- (^m*-

(43)

Then, he claims that when a = 1/2, this reduces to Fourier's (21) with u = y/h/t. This shows his strong interest in Fourier's mysterious formula. In turn, Dirac, whose major was electrical engineering, seems to have obtained his ideas from Heaviside's theory 15 . References 1. S. Bergman and M. Schiffer: Kernel Functions and Elliptic Differential Equations in Mathematical Physics, Academic Press, New York, 1953. 2. P. A. M. Dirac: The physical interpretation of the quantum mechanics, Proc. Royal Soc. London, Sec. A, 113 (1926-27), 621-641. 3. J. Fourier: Theorie Analytique de la Chaleur, Firmin Didot, Paris, 1822; Jacques Gabay, Sceaux, 1988. 4. K. Gustafson and Takehisa Abe: The third boundary condition — was it Robin's?, Math. Intelligencer, 20-1 (1998), 63-71. 5. O. Heaviside: On operators in mathematical physics, Proc. Royal Soc. London, 52 (1893), 504-529 and 54 (1894), 105-143.

214

6. 0 . Heaviside: Electromagnetic Theory, Vol. II, The Electrician, London, 1899; Chelsea, New York, 1972. 7. H. Komatsu, Ultradistributions, I, Structure theorems and a characterization, J. Fac. Sci., Univ. Tokyo, Sec. IA, 20 (1973), 25-105. 8. H. Komatsu, Linear hyperbolic equations with Gevrey coefficients, J. Math. Pures Appl., 59 (1980), 145-185. 9. H. Komatsu, Ultradistributions, III, Vector-valued ultradistributions and the theory of kernels, J. Fac. Sci., Univ. Tokyo, Sec. IA, 29 (1982), 653717. 10. H. Komatsu: Operational calculus and semi-groups of operators, Lecture Notes in Math., 1540 (1993), 213-234. 11. H. Komatsu: Solution of differential equations by means of Laplace hyperfunctions, in Structures of Solutions of Differential Equations, World Scientific, Singapore, 1996, pp. 227-252. 12. H. Komatsu: Fourier's operational calculus and Balakrishnan's square roots, in preparation. 13. P. S. Laplace: Memoire sur divers points d'analyse, J. Ecole Polytech. 8 (1809); (Euvres Completes, vol. 14, 1912, pp. 178-214. 14. M. Morimoto: An Introduction to Sato's Hyperfunctions, in Japanese, Kyoritu, Tokyo, 1976; Amer. Math. Soc, 1993. 15. Paul J. Nahin: Oliver Heaviside: Sage in Solitude, IEEE Press, New York, 1988. 16. J. Peetre: On Fourier's discovery of Fourier series and Fourier integrals, preprint, (2000), 15pp. 17. B. van der Pol and H. Bremmer: Operational Calculus based on the two-sided Laplace Integral, 2nd ed., Cambridge, 1955. 18. B. Riemann: Grundlagen fur eine allgemeine Theorie der Functionen einer veranderlichen complexen Grosse, Inauguraldissertation, Gottingen, 1851; Gesammelte Mathematische Werke, pp. 3-45. 19. B. Riemann: Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe, Abhand. Konig. Gesell. Wiss. Gottingen, 13 (1854); Gesammelte Mathematische Werke, pp. 227-265. 20. M. Sato: Theory of hyperfunctions, I, J. Fac. Sci., Univ. Tokyo, Sec. I, 8 (1959), 139-193. 21. L. Schwartz: Theorie des Distributions, Hermann, Paris, 1950-1. 22. W. Thomson: On the theory of the electric telegraph, Proc. Royal Soc. London, (1855); Mathematical and Physical Papers, vol. 2, pp. 61-78. 23. N. Wiener: The Fourier Integral and Certain of its Applications, Cambridge Univ. Press, 1933; Dover, New York.

GEOMETRIC A S P E C T S OF LARGE DEVIATIONS FOR R A N D O M WALKS O N A CRYSTAL LATTICE MOTOKO KOTANI Mathematical

Institute,

Graduate School of Sciences, Tohoku Sendai 980-77,Japan E-mail: [email protected]

Mathematical

Institute,

Graduate School of Sciences, Tohoku Sendai 980-77,Japan E-mail: [email protected]

University,

Aoba,

University,

Aoba,

TOSHIKAZU SUNADA

This note discusses briefly some geometric aspects of large deviation for random walks on a crystal lattice. The motivation arises from our attempts to apply ideas in the Bloch-Floquet theory, developed originally for periodic Schrodinger operators on the Euclidean space, to the transition operators. Indeed, if L denotes the transition operator associated with a periodic transition probability on a crystal lattice, then I — L is regarded as a discrete analogue of the Laplacian which is equivariant under the action of a lattice group. As was established in our previous work, unitary characters of a lattice group play a significant role in the proof of the central limit theorem. What we shall observe in this note is that real characters show up in the study of large deviations. We also see that the Gromov-Hausdorff limit of metric spaces homothetic to a crystal lattice has something to do with the large deviation property of random walks. The detailed accounts of this on-going research will be published elsewhere.

1

The Gromov-Hausdorff limits of crystal lattices

Let us start with a simple example. Consider the square lattice Z 2 . We take this square lattice as a graph in the usual manner and also as a metric space with the graph-distance d. Namely, d(x, y) is the number of edges on a shortest path joining the vertices x, y. Given a positive constant e, we consider the metric space (Z 2 ,ed) homothetic to (Z 2 ,d). We then ask ourselves what the lime4.0(Z2,ed) is in the sense of Gromov-Hausdorff. The answer is, as we may expect, R 2 with the taxi-cab distance d±, say di({xi,yi),{x2,y2)) Question

= \zi -x2\

+ \yi — a/21-

Let X be the hexagonal lattice. What is lim^n (Jf, ed)?

We can generalize this question to the case of more general crystal lattices (the precise definition will be given latter), and the answer is

215

216

Theorem 1 (1) (a special case of Gromov's result 2 ) Let (X, d) be a crystal lattice with the graph-distance. There exists a normed linear space (L, || • ||) of finite dimension such that lim(X,ed) = (L,di), where di(x,y) = ||x -^y||. (2) The unit ball V = {x £ L | ||x[[ < 1} is a polyhedron. Example For the hexagonal lattice, L = R 2 and V is the hexagon (including the interior) in R 2 . A more explicit description of the norm || • || (and V) is given at the end. The purpose of this note is to relate the theorem above to the large deviation property of random walks on crystal lattices. 2

Homological drifting of random walks on finite graphs

Before going to the large deviation problem, we shall make a simple observation of random walk on finite graphs. Let X0 = (VQ,EO) be a finite connected graph, possibly with multiple edges and loop edges, V0 being the set of vertices and E0 being the set of all oriented edges. The origin and terminus of e £ EQ will be denoted by o(e) and t(e), respectively. The inverse edge of e is denoted by e. We shall consider a random walk on X0 given by a transition probability p; say a positive valued function p : E0 -> R satisfying

£

p(e) = 1 (x € V0),

where Eo,x = {e £ Eo \ o(e) = x}. In the usual manner, the transition probability p gives rise to a probability measure Px (x € Vo) on the space of paths with the origin x: flx = {c= (ei,e 2 ,---) I o(c) = o(ei) = x}. For each c £ Qx, we put xn(c) = t(en), x0(c) = x. The following theorem is classical. Theorem 2 (convergence of empirical measures) There exists a positive valued function monV0 such that -($n(c) H

l-<5* n ( c ))-•m

a.e.c£ttx,

217

Actually m is the (unique) invariant probability measure for the transition operator L defined by

(Lf)(x) = £

p(e)/(t(e)),

that is J](L/)(a;)m(a;) = 5 ; / ( x ) m ( a ; ) a

(*

»

£

p(e)m(t(e)) = m(x)).

e£E0,s

From a geometic view, one may regard the theorem above as stating the convergence of a sequence of 0-chains. To explain this, define, as usual, the 0-chain group by C 0 (X 0 ,R) = {^2axx\axe xev0

R}.

For the later purpose, we also introduce the 1-chain group Ci(X0,R) = { ^ a e e | a e £ R}, e€£o

where the relation e = — e (e 6 EQ) is imposed. If we choose an orientation of X0, meaning a subset E^" C EQ such that £ j " n £ o + = 0 and E$ \J E$ = E0, we may identify C i ( X 0 , R ) with the linear space with the basis EQ. The boundary operator d : Ci(Xo,H) -> Co(X 0 ,R) is defined by d(e) = t(e) o(e), and the 1-homology group i?i(.Xo,R) is just Ker d. (Hi(Xo,Z) and HI(XQ,Q) are also defined by replacing R with Z and Q, respectively.) Now the theorem above can be stated in terms of 0-chains as — (xi(c) + n

h xn(c)) —> y^m(a;)a; ^—'

a.e. c £ fij,.

In view of this, it is natural to consider an analogue of this convergence in the 1-chain group. Namely, by putting ef(c) = e, for c = (ei,e2, • • •), we ask ourselves whether the limit of the 1-chains -(ei(c) + ••• +e„(c)) n exists. The answer is also simple.

218

Theorem 3 (1) There exists a l-chain 7 P e C i ( X 0 , R ) such that lim — (ei(c) H

h e„(c)) = 7 P

a.e. c £ fi^..

(2) 97 P = 0, so that 7 p £ Hi(X0,R). The proof of (1) relies on the ergodic theorem as in the case of 0-chains. We consider the ergodic dynamical system (Q,P,T) defined by

n= U iix, p(i) = ^ p I ( i n o > ( i )

(A eft),

x

T(ei,e2,---)

- (e2,e3,---)-

Putting F(c) = ei(c), we find ^(ei(c) + --- + e n (c)) = i $ > ( T * c ) k=0

—> / FrfP = 53p(e)m(o(e))e = n

7p,

e6£

as n -> oo. The second assertion is a consequence of the invariance of m. Here is a more geometric description of the homology class 7 P . For each n, we join xn{c) and x by a shortest path to obtain a closed path c„, and denote by [cn] the homology class of c„. Then 1, , n We shall ask ourselves what the possible values of 7 P are when p runs over all transition probabilities. To give an answer, write A) = {lp £

HI^XQJR)

I p is a transition probability on I 0 } .

Choose an orientation EQ C EQ, and define the ^-norm on Ci(X0,K)

It is obvious that || • ||i does not depend on the choice of EQ.

by

219 Theorem 4

p 0 = {ae#i(Xo,R.) I IHIi 0 are, and how its faces are characterized. Theorem 5 (1) "Do is "rational" in the sense that all extreme points ofV0 are in Hi(Xo,Q). (2) a e HI(XQ,Q) is a vertex ofT>o if and only if a = c/||c||i for a circuit (simple closed path) c in XQ . (3) ot\ = Ci/||ci||i,...,Q/t — Cfc/||cfe||i are in the same face of T>0 if and only if there exists an orientation EQ such that the orientations of simple closed paths c* are compatible with EQ'. In particular, the convex hull of OJI, . . . , ah is a face if and only if {ai,..., a^} is maximal among all sets of vertices satisfying this condition. The homology class 7 P is a sort of quantity used to measure homological drifting of the random walk. In fact, we obtain Theorem 6 j p = 0 if and only if p gives a symmetric random walk, i. e. p(e)m(o(e)) = p(e)m(t(e)), or equivalently L is symmetric with respect to the measure m. We may also establish another geometric feature of T>0. For a G Hi(X0,Z), denote by 1(a) the minimal length of all closed paths c in XQ with [c] = a. Theorem 7 (A graph analogue of a result due t o Gromov) # { a e JJipfo.Z) I 1(a) < x} ~ vo\(T>0)xk

(x ->• oo),

where k = rank (Hi(X0,Z)) and vol(I>o) is the volume ofV0 with respect to the Lebesgue measure on Hi(Xo,'R) such that volfaiXo,*.)

/ HtiXoZJ)

= 1.

Problem Are there any relationships between vol(2?o) and other graph invariants of XQ ? Note vol(X>o) £ Q3

Crystal lattices

Now we shall proceed to the case of infinite graphs. The class of infinite graphs treated here is rather special, but still has rich important examples.

220

We consider a crystal lattice, a connected infinite graph X on which a free abelian group V acts as an automorphism group. We assume that 1. T acts freely on X so that the quotient X0 = T\X has a graph structure. In other words, X is an abelian covering graph of X0. 2. X0 is finite. Such a group T is said to be a lattice group for X. Here are several classical examples of crystal lattices. (1) the square lattice (2) the triangular lattice (3) the hexagonal lattice (4) the kagome(basket mesh) lattice In those examples, the lattice group T is the group generated by two vectors depicted by arrows. Thus those examples are exhibited with "special" realization. Namely, X is realized in R 2 in such a way that T is a discrete subgroup of R 2 . In this view, we introduce the notion of periodic realizations of an abstract crystal lattice as follows. A piecewise linear map $ of X into Rfc (k — rank T) is said to be a periodic realization if there exists a homomorphism p : T —• Rfc such that $(ax) = $(x) + p(cr), and p(T) is a discrete subgroup of R*. Identifying V R with R* via p, we can say that a periodic realization is a piecewise linear map $ : X —> F ® R satisfying $(crx) = $(x) + a 1. There always exists a periodic realization of a given crystal lattice X with a lattice group T. Using a periodic realization, it is straightforward to establish the estimate c\nk < # { x e X | d(x0,x)

< n) < c2nk

with some positive constants c\ and c2. Thus the positive integer k = rank T does not depend on the choice of a lattice group T. We call k the dimension oiX. Now we consider a random walk on X given by a T-invariant transition probability p, or equivalently the lift of a transition probability p0 on XQ. Given a periodic realization $, we put f n (c) = $(x n (c)) for c e tix(X). We thus obtain a T R-valued process {^„}^L0.

221

To describe the asymptotic behavior of {£„} as n tends to oo, we need to inroduce a surjective homomorphism p : Hi(X0,Z) -> T. Let a £ Hi(X0,Z) and represent a by a closed path c in X0. Take a lift c of c in X. Since o(5) and t(c) project down to the same element in Xo, there exists a £ T such that t(c) = (a) = er, we extend p to a surjective homomorphism /oR:H1(X0,R)-+r®R. As a consequence of Theorem 3, we obtain Theorem 8 lim -f„(c) = pRhpo)

a.e. c G n x ( X ) .

n—>oo 71

We may also prove Theorem 9 The following conditions are equivalent. (i) m(7p 0 ) = o. (2) There exists a periodic realization $ such that {£„} is martingale. (3) TAere exists a periodic harmonic realization $ . (4) Positive harmonic functions on X are constant. Here a (vector-valued) function / on X is said to be harmonic if Lf = f. Now comes a discussion about large deviations for the process {£„}. Theorem 10 A large deviation property holds for {£„}. Generally speaking, the large deviation problem for a process {£„} concerns the asymptotic of Px(^„ € A). On the other hand, the central limit theorem looks at lim„_»00.Px(-4^£„ £ A). To give more details, we let ( , ) : ( r ® R) x Hom(r,R) -> R be the pairing map between T ® R and its dual (T ® R)* = Hom(r, R). Theorem 11 Let x € Hom(r,R). (1) lim -log£(e<*"' x >)=c(x) n—too XI

exists. (2) ec^ is the maximal positive eigenvalue of the "twisted" transition operator Lx : C(EX) -4 C(EX), where C{EX) = {s : X -> R | s(ax) =

exi^s(x)},

and Lx = L\C(BX) (it *s easv to check that L(C(EX)) C C(EX)). (3) c is real analytic, and the hessian of c is strictly positive definite everywhere. Thus the correspondence x *-* (Vc) (x) is a diffeomorphism of Hom(r, R) onto an open subset V in F ® R.

222 (4) V — pn(D0),

and hence is independent of p. Moreover

D = { x e r ® R | Hxlli < l}, where ||x||i = inf{||a||i | a e ifipf 0 ,R-),PR(a) = x}. Therefore V is a convex polyhedron, symmetric around the origin, and rational in the sense that the vertices ofT> are in T ® Q. (5) (Vc)(0) = /) R ( 7 p o ). In view of (1) and (3) in the theorem above, one may employ a general recipe in the theory of large deviations 1 . Define the entropy function / : r ® R - > R U {oo} by 7(z) = s u p ( ( z , x ) - c ( x ) ) . x We easily find that / takes finite value on V and infinite on V . Indeed, I{z) = (z,Xo) - c(xo),

(Vc)(xo) = z,

and hence, / is real analytic on V. The large deviation principle claims -I{miA)

< l i m i n f - logP x {-£ n G intA) n^» n n

< limsup - logPx{-£„

eA)<

-I(A),

where I{A) = inf{/(z) | z € A}. This is what Theorem 10 means. Using V, we establish a precise asymptotic for the number of vertices in a crystal lattice. Theorem 12 # { x € X | d{x0,x) < n} ~ (#X 0 )vol(P)n fc

(n -> oo),

where vo\{V) is the volume ofV with respect to the Lebesgue measure on T(g)R such that vol(r ® R / T ® Z) = 1. Finally, we go back to the theorem mentioned at the beginning. Theorem 13 lim{X,ed) = {T ej.0

where dY(x,y) = | | x - y | | i .

®R,di),

223

Remark It may be worthwhile to point out that, in the central limit theorem for simple random walk, an Euclidean distance shows up instead of d\. To explain this, define

n«ii! = £ w 2

(* = £ ««e e

e6E+

#I(*O,R))

e€E+

and ||x|| 2 = inf{||a|| 2 | a e H1(X0,R),pR(a)

= x],

for x e T R. Then for the simple random walk, one has 1 P ( - ^ £ „ eA)^ V" where a = (#J5o) _ 1 .

1

/"

/ (47ra)fc/2 JA

3

llxll2 exp(-^)dx, 4a

References 1. R. S. Ellis, Large deviations for a general class of random vectors, Ann. Prob. 12(1984), 1-12. 2. M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Birkhauser, 1999. 3. M. Kotani and T. Sunada, Albanese maps and off diagonal long time asymptotics for the heat kernels, Comm.Math.Phys.209(2000), 633-670. 4. M. Kotani and T. Sunada, Standard realizations of crystal lattices via harmonic maps, Trans. Amer. Math. Soc. 353(2000), 1-20 5. S. A. Sawyer, Martin boundaries and random walks, Contemporary Math. 206(1997), 17-44.

B O E H M I A N S ON T H E S P H E R E A N D ZONAL SPHERICAL FUNCTIONS MITSUO MORIMOTO Department of Mathematics, International Christian University, 3-10-2 Osawa, Mitaka, Tokyo, 181-8585, Japan E-mail: [email protected] There are two definitions of Boehmians on the sphere; the first one relied on zonal delta sequences, while the second on central delta sequences. In this note we introduce semi-groups of zonal delta sequences to distinguish several spaces of Boehmians on the sphere and investigate their relations. We also show the second method reduces to the first one.

1

Introduction

The theory of Boehmians was introduced and studied by P. Mikusinski et al. 2 ' 3 ' 4 in an abstract algebraic form. But in the mentioned papers Boehmians were considered mainly on the Euclidean space. D. Nemzer H' 12 - 13 considered Boehmians on the unit circle. In the report 9 we discussed two methods of construction of Boehmians on the sphere; the first construction given in Mikusiriski-Pyle 7 relied on zonal delta sequences, while the second construction given in Mikusinski-Morimoto6 on central delta sequences. The latter has been extended by Mikusinski 5 to define Boehmians on manifold with a compact Lie group action. In this paper we shall introduce a semi-group of zonal delta sequences to specify spaces of Boehmians and compare these spaces. This idea of semigroups can be applied to Boehmians on the Euclidean space but we refrain ourselves only to the spherical case. In the first part we shall treat semi-groups of zonal delta sequences to construct Boehmians on the sphere following Mikusiriski-Pyle 7 and prove a comparison theorem. In the last section we shall discuss semi-groups of central delta sequences to construct Boehmians on the sphere following MikusinskiMorimoto 6 and show that two kinds of delta sequences amount to the same space of Boehmians. 2

Functions on t h e s p h e r e

A general reference of this section is our monograph 8 . Let G = SO(d + 1) and e = (1,0, • • • , 0) € Rd+1. The isotropy subgroup

224

225

of e is denoted by K = {g € G; ge = e} = SO(d). We shall denote by S = S d the d-dimensional unit sphere: S = G/K. Let L2(8) be the Hilbert space of square integrable functions on § equipped with the inner product {f,g)s = f§f{x)9(x)dx, where the integration is done with respect to the normalized invariant measure. A fc-spherical harmonic function s*. on S is, by definition, the restriction to S of a homogeneous harmonic polynomial Sk of degree k: Sk = Sk\§. We shall denote by " ^ ( S ) the space of fc-spherical harmonics and by N(k,d) the dimension of Hk(§): N(k,d) = dim rik(S) = 0{kd~l). Lemma 1. The spaces rik(8) are orthogonal subspaces of L 2 (S): for Sk € nk(§) and st € ft'(§), k # /, we have {sk,si)s = 0. Further, L 2 (S) is the direct sum ofUk(S): L2(S) = © £ L 0 ^ * ( § ) Let C(S) be the space of continuous functions on S equipped with the supremum norm ||/||c(S) = sup{|/(x)|;a: £ §}. A function <j> £ C(§) is called zonal if 4>(gx) = (f>{x) for all g G K and x £ S. We shall denote by Z0 (S) the space of zonal continuous functions on S. The following lemmas are well-known. Lemma 2. For any <j> e Z0(S) there is a function 4> on the closed interval [—1,1] such that 4>{x) = (x • e), where x • e denotes the inner product of x and e. Lemma 3. If 4> & Z0(8)D7ik(S) satisfies the normalizing condition (j>{e) = 1, then the function 4> is equal to the Legendre polynomial of degree k: (x) = Pk(x-e). The following lemmas describe some properties of the Legendre polynomial Pk(t): Lemma 4. 1) Pk(x • e) is a zonal k-spherical harmonic function; 2) P fc (l) = 1, - 1 < Pk(t) < 1 »/ - 1 < * < 1; 3) sup{|Pfc(a: • y)\;x € §} = 1 for any y 6 8. Lemma 5. Pk(x • y) is the reproducing kernel o/H f c (§); that is, for Sk £ nk(S) we have sk(x) = N(k,d)

/ sk(y)Pk(y J%

• x)dy.

Especially, we have N(k, d) Js Pk(y • z)Pk{y • x)dy = Pk(x • z) for x,z eS. The orthogonal projection / H-> sk(f) of L 2 (S) onto Hk(§) is given by

sk(f)(x) = N(k,d) [f(y)Pk(yx)dy. Js

(1)

Sk(f) is called the fc-spherical harmonic component of / and we have the

226 spherical harmonic expansion of / in the sense of L2(S): oo

/(*) = £ a* (/)(*)•

(2)

fc=0

Let C 1 (§) be the space of continuously differentiable functions on S. It is known that the spherical harmonic expansion (2) for / € C 1 (S) is uniformly convergent. For the simplicity, we shall use the following space oo

oo

CM(S) = {fix) = £ > * (/)(*);**(/) € Hk(§),J2\\sk(f)\\C(s) fc=0

< oo}.

k=0

Then we have C 1 ^ ) C C ( 1 ) (§) C C(§) C L 2 (S). Put Z<1}(S) = Z„(S) n L e m m a 6. If <j> € Zo (§), i/ien tue /ia«e oo

0(a:) = Y,

N k

( > d)ck(4>)Pk(x • e),

(3)

fc=0

where the scalars ck(4>) o,re defined by ck(4>)= [ (f>(y)Pk(y • e)dy Js

(4)

and satisfy oo

£iV(M)M<«|
(5)

fc=0

Proof. By Lemma 3 we have sk((f>)(x) = N(k,d)ck((f>)Pk(x-e) with a constant ck{(j>). By (1) we haves* (<£)(z) = N{k,d)ck{cj>)Pk{x-e) = N{k,d) ^{y)Pk{y x)dy. Putting x = e, we get (4). (5) results from Lemma 4. • Let C°° (S) be the space of infinitely continuously differentiable functions on § and C " (S) the space of real-analytic functions on §. The spaces C°° (§) and C""(§) can be characterized by the behavior of the spherical harmonic expansions: C°°(§) = {f£

CW(§); ||*fc(/)||c(S) is rapidly decreasing},

CU(S) = {/ e ^ ( S J j l i m s u p (/||« f c (/)|| c ( s) < !}• fc—yoo

v

(6)

227 3

Convolution on the group

Suppose always G = SO (d + l ) , e = (1,0,-•• ,0) £§,K = {g£G;ge = e}^ SO{d) and § = G/K. A point of § is denoted by x = ge. We denote by C(G) the space of continuous functions on G. For i*\, F2 £ C(G) we define the convolution Fy * F2 by Fi * F2(g) = f F1{gg^)F2{g1)dg1

= f F1{gl)F2(g-1g)dgu

JG

(7)

JG

where the integration is done with respect to the normalized Haar measure on G. Then, equipped with the convolution, C(G) is a non-commutative associative algebra. A function / on S is identified with the right ivT-invariant function F defined by f(x) = f(ge) = F{g): F(gk) = F(g) for all g £ G and k £ K. By (7) i f / e C(§) and 4> £ G(G), then ^ * / G C(S). Put Z0{G) = {£ C(G)\(h1gh2) = 4>(g) for all /u,/i 2 e ^ , j e G}. We identify Z0(G) with the space Z0(§) of continuous zonal functions on S. L e m m a 7. For <j> £ Z0(G),

we have <j>(g) = (g~l), g € G.

Proof. By Lemma 2 we have (g) = 0(pe • e). Because of the G-invariance and the symmetry of the dot product, we have (ge) • e = e • (g~le) = (g~le) • e, and then <j>(g) = 4>(g~l). L e m m a 8. If <j>, ip £ Z0(G),

• then we have <j> * V = i> * -

Proof. By Lemma 7 we have

(gi)^(gT19)d9i= / •Ksi^Ms - 1 si) d si JG

VG

= V' * *(9),

where the last equality results from the fact that Z0(G) is closed under convolution. • Theorem 9. If F £ C(G) and <j> £ Z0(§), and we have F*
/

then F * is right

K-invariant

F{gi)(j)(gie-ge)dgu

JG

where 4>{x) = (x • e). Especially,

f*Hx)=

if f £ C(E>), then we have

f f(y)<j>(yx)dy.

(8)

228 Proof. F*<j>(x) = f F{g1)cj>{g-1ge)dgl = f F{g^{g-lge JG

• e)d9l

JG

= / F{g1)<j>(ge-gi_e)dg1= \ F(gi)<j>(gie • ge)dg1. JG

JG

Because fG f(ge)dg = fs f(x)dx,

we get (8).

•

Corollary 10. Let f e C^(S) and
f*)sk(f)(x).

(9)

k=o

Proof. By (2) we have f(x) = £ 2 L 0 «*(/)(*). 2 X o ^ ( M ) c * ( 4 > ) P f c ( i ) . By (8), we have „

f*cf>(x) =

OO

By (3) we have j>(t) =

OO

r£sk(f)(y)^2N(l,d)cl((p)Pl(y.x)dy.

Therefore, by Lemmas 1 and 5, we get (9).

•

Corollary 11. For <j>, ip £ Z0(B) we have oo

4>*iP(x) = Y/N(k,d)ck(<j>)ck(
• e) = {x).

k=0

4

Boehmians on the sphere, 1

In this section, we follow Mikusiriski-Pyle 7 to construct spaces of Boehmians on § by means of zonal delta sequences. Definition 12. {n} € Z0 '(§) N is called a zonal delta sequence if f*4>n -4 / uniformly as n -» oo for any f £ C^(S). By the definition, we have f*4>n{x) = fsf(y)n(x) = f§n(y)f(x • y)dy. Therefore, if {„} is a zonal delta sequence, then ck(n) = f§n(y)Pk(y • e)dy tends to Pk(e • e) = 1 as n -> oo. Definition 13. A subset A 2 of Zo (S) N is called a (commutative) semi-group of zonal delta sequences if it satisfies the following conditions: (A 2 1) {4>n} € A 2 is a zonal delta sequence; (A 2 2) If {„}, {ipn} € Az, then {<j)n*ip„} 6 A , .

229

For a semi-group A2 of zonal delta sequences, we put A(AZ) = {({/„}, {*„}); {fn} e C(1)(S)N,{<M e A 2 , / n *m = fm*„}) ~ ({gn}, {^n}) by /„ * ^ m = 9m * nforall m, n. Lemma 14. The relation ~ is an equivalence relation on A(AZ). Proof. We have only to prove the transitivity. Suppose {{fn},{n}) ~ ({ffn},{^n}) and ({fln},^}) ~ ({^n},{Xn})- Then by Lemma 8 {fn * Xm) *^l

= {fn * V>0 * Xm = ( # * n) * Xm = {dl * Xm) *
Then by the property (A21), we have /„ * Xm = hm * (j>n; i.e., ({/„}, {<£„}) ~ {{hn},{Xn}). • The space $(A 2 ) of A2-Boehnians on § is defined to be the quotient space of -4(A2) by the equivalence relation ~: B(AZ) = .4(A 2 )/ ~. The equivalence class of ({/„}, {„}) is denoted by [{/„}, {<j>n}] or more simply by [/„/<£„]. The space B{AZ) is a linear space by the following definition: \[fnln] = [A/„/&J, [fn/4>n] + [9n/lpn]

A G C;

= [{fn * 4>n + Sn * n)/{n * Ipn)]-

The condition (A 2 2) is necessary for the definition of addition. Identifying / G C{§) with [/ * <j>n/4>n] G £(A 2 ), we consider C(§) C B(A 2 ). 5

Spherical harmonic component of a Boehmian

Let A 2 be a semi-group of zonal delta sequences. Let ({/„}, {<£„}) G -4(A 2 ). Then, by Corollary 10, we have «&(/„ * m) = ck{(j)m)sk{fn)- The relation fn * (t>m = fm * <}>n implies ck{4>m)sk{fn) = Cfc(<£„)sfc(/m). Because ck{(f>n) tends to 1 as n -> oo, there is n 0 such that ck{<j>n) ^ 0 for any n > n 0 . Therefore, sk{fn)/ck{4>n) are defined for large n and independent of such n. Suppose ({/„}, {<£„}) ~ {{g„}, {ipn})- Then the relation f„*ipm = gm*(j>n implies ck{^m)sk{fn) = sk(fn * i/jm) = sk(gm * 4>n) = ck(<j>n)sk{gm). Therefore, we have sk{fn)Ick{<j)n) = sk{gm)/ck{xjjm). We define the ^-spherical harmonic component of the Boehmian F = [fn/n] to be Sk{fn)lck{n)-- Sk{F) = Sk{fn)/Ck{n).

230

6

Examples

Definition 15. Let m be a positive integer or oo. Let A2(C™) be the set of sequences {4>n} which satisfy the following conditions: (NQ.l) 4>n is a Cm zonal function for any n; (NQ.2) (f)n>0 for any n; (NQ.3) / s n{x)dx = 1 for any n; (NQA) For any e > 0 there is no such that the support of cj>n is contained in Vt — {x 6 §; 1 — e < x • e < 1} for any n>no. L e m m a 16. A Z ( C Q " ) is a (commutative) semi-group of zonal delta sequences. In Mikusiriski-Pyle 7 the space of Boehmians was defined by the semigroup A;(C£°) and it was proved that C°°(§)' C £(A 2 (C£°)), where C°°(§)' is the space of Schwartz distributions on §. Definition 17. Let m be a positive integer, oo or u. Let Az(Cm) be the set of sequences {n} which satisfy the following conditions: (Q.l) n is a Cm zonal function for any n; (Q.2) (j)n>0 for any n; (<2.3) / s 4>n(x)dx = 1 for any n; (Q.4) For any e > 0, the integral L v^ <j)n(x)dx tends to 0 as n tends to oo, where Ve = {x e §; 1 - e < x • e < 1}. L e m m a 18. A z (C r n ) is a (commutative) semi-group of zonal delta sequences. AZ(C™) is a sub-semi-group of A 2 ( C m ) . We propose here a new kind of semi-group of zonal delta sequences. Definition 19. A function M(k) on N is called a scale if (M.l) M(0) = 0, M(k) is increasing; (M.2) M(k) > k for sufficiently large k. For example, M(k) = kp for p > 1 or M(k) = ek - 1 are scales. For a scale M(k) we put oo

M

z0 (S) = { e zP(S), fa) = £>(MM *=0 there are 0 < C\ < C2 and 0 )ck{xl>)\ <

C2C'2e-^+^MW.

The theorem results from Corollary 11.

•

Let A™ be the set of zonal delta sequences {cj)n} such that cf>n £ Z^(S) for all n. Then Af is a semi-group of zonal delta sequences. If M(k) = kp and p > 1, then A f is denoted by A 2 p ) . The quotient space B(Af) = , 4 ( A f ) / ~ is called the space of Boehmians on S with scale M(k). The spaces B(Azp)) will be considered in the following sections. Note that A f n A 2 (C 0 m ) = 0. In fact, if n)\ < Ce""*. By (6), <£„ is real-analytic and cannot satisfy the conditions (NQ.3) and (NQA). Note alsoAip)nAi<')=0ifp^g. We try to compare semi-groups of zonal delta sequences. Definition 21. Let Az and A'z be two semi-groups of zonal delta sequences. We say that A'z is stronger than Az and denote Az < A'z if {„} £ Az and {V„} e A'z imply {cj>n * xpn} G A'z. If Az C A^, then Az < A'z. We cannot prove the transitivity of the strength relation. Lemma 22. Let m and m' be positive integers or oo. Then we have A 2 (C 0 m ) < A 2 (C 0 m '),

A 2 (C 0 m ') < A 2 (C 0 m )

Az(Cm)

Az (Cm>) < A , (Cm).

and

Lemma 23. Ifl
< Az(Cm'),

q, then we have A 2 p) < Azq).

Proof. If p < q, then there is C > 0 such that kp < Ckq. Therefore, we have the inclusion. • 7

Comparison theorems

Theorem 24. / / A 2 < A 2 , then there is a natural linear mapping B(A 2 ) —> B(A'Z). Proof. Let [/„/n}) G ^4(A 2 ). Take a zonal delta sequence {ipn} £ A^. Then we have ({/„ * ipn},{n * ipn}) e -4(A'2). We claim that [fn * ipn/n * ipn] € B(A'Z) is

232

well-defined. Suppose ({/„}, {<j>n}) ~ ({/£}>{#»}) and {ip'n} is another zonal delta sequence in A z . By the definition we have (fn * V>n) * ( 0 m * # J = ( / n * # J * 1>n * ^ r o

Therefore, ({/„ * Vn}, {<£n * V>n}) ~ ({/„ * V4>, { # , * #,})•

•

Theorem 25. / / A 2 < A^ and A'z < A., then there is a natural linear isomorphism B(AZ) £ B(A'Z). By Lemma 22, B(AZ(C?)) = B{AZ{C^')) and B{Az{Cm)) £m S(Az(C ')). To describe a sufficient condition for the natural linear mapping to be injective we need the following Definition 26. A zonal spherical function cf>(x) — 2~ZfcLo N{k,d)ck()Pk{x-e) is called strict if Ck(<j)) ^ 0 for all k. A zonal delta sequence {„} is called strict if there is no such that <j>n are strict for all n > no. A subgroup Az of zonal delta sequences is called strict if it contains a strict zonal delta sequence By the definition, A^ are strict for any scale functions M(k). Theorem 27. If Az < A'z and A'z is strict, then the natural linear mapping B{AZ) -> B{A'Z) is injective. Proof. We employ the same notation as in Proof of Theorem 24. We can assume {ipn} £ A^ is a strict zonal delta sequence. Suppose fn*ipn/n*'ll>n = 0; that is, ({/„ * ipn},{m * ipm = 0- Because {ip'm} is a zonal delta sequence, we have fn*4>n = 0. Because {ipn} is strict, / „ = 0 for large n, which implies [fnln] = 0 . • By Lemma 23 we have Corollary 28. Ifl
Hyperfunctions on the sphere

Theorem 29. We have C°°(§)' C C""(S)' C B(A^]), where C°°(S)' is the space of distributions on S, C""(S)' is the space of hyperfunctions on 8 and £(Ai 1 ] ) is the space of Boehmians with the scale M(k) = k on S. Proof. Let / e C""(§)'. The fc-spherical harmonic component of / is given by Sk(f)(x) - N(k,d)(f(y),Pk{x-y))y€s and ||sfc(/)||c(s) is of infra-exponential growth, i.e., for all e > 0 there is C > 0 such that ||sfc(/)||c(S) < Ceek. Define

233 n(x) = Y.'kLoN(k,d)e~k/nPk(x

• e). Then {„} is a zonal delta sequence in

° Define further fn{x) = ZZo^^^if)^)Then ({/„}, {0„}) e ^(Ai 1 5 ). If we associate / € C""(§)' with the Boehmian [/„/<£„] € ^(Ai 1 '), we have the sought imbedding. Further we have sk(f) = S& ([/„/„]). D 9

Initial values of temperature functions on the sphere

A function U(x,t), x e S, t > 0, is called a temperature function if (d/dt — &$)U(x,t) = 0, where As is the Laplace-Beltrami operator on § (see Morimoto-Suwa 1 0 ) . We put Uk(x, t) = N(k, d) fs U(y, t)Pk(x • y)dy. Then we have (d/dt)Uk(x,t)=N(k,d)

Js = N(k,d) f Js

= N(k,d)

f(d/dt)U(y,t)Pk(x-y)dy A%U{y,t)Pk{x-y)dy

fu(y,t)AsPk(x-y)dy

Js = N(k,d) [U(y,t)(-k(k + d- l))Pk(x • y)dy Js, =-k(k + dl)Uk(x,t). Therefore, there is uk e Hk(S) such that Uk(x,t) = e _ *<* +d - 1 > t u fc (a;). If is i n t > 0, then U{x,t) = J2V=oe~k(k+d~1)tui'(x) C (1) (§)- In particular, for all t > 0, there is C > 0 such that ||ufc||c(s) < Ce*2*.

(11)

h( k+d 1 t

Conversely, if (11) is satisfied, then U(x,t) = YlT=o e~ - ~ ^ uk(x) is a temperature function. The formal sum ^ £ L 0 u * (x) *s c a n e d the formal initial value of the temperature function U(x, t). T h e o r e m 30. The formal initial value of a temperature function is a Boehmian in £(Ai 2 ) ), where B(A?}) is the space of Boehmians with the scale M(k) = k2 on S. Proof. Define n(x) = Y,V=o N(k,d)e-k<-k+d-1'>/nPk(x • e). Then {n} is a zonal delta sequence in A\ '. If a formal sum Y^T=ou* > w n e r e uk e H fc (§), satisfies (11), then fn(x) = IXoe~ f c ( f c + d ~ 1 ) / n Ufc(z) € C^{S). We have / „ * <j>m(x) = ^Zo^k(k+d~m/n+1/m)uk^) = fm * n(x); that is, ({/«}, {»}) e .4(Ai 2) ). Then by the mapping £ ~ 0 uk ^ [fn/<j>n] e S(Al 2 ) ) we can identify the formal initial value of a temperature function J2T=o uk with the Boehmian [/„/<^„]•

234

10

Formal spherical harmonic series

We shall show that any formal spherical harmonic series can be considered as a Boehmian on S with some scale. Theorem 31. For any formal sum £)sfc, Sk € %*(§), there is a scale M(k) and a Boehmian [fn/(fin] € B(A^) such that the k-spherical harmonic component of [fn/n] is the given SkProof. Put M(k) = (sup{log + ||«/||c(s) - l°g+ IWIc(s),£}) 2 , l
where log + 1 = max{log*,0}. Then M(0) = 0, 0 < k2 < M(k), and M(k) is increasing with respect to k. Further, we have ||s*||c(s) < CevM(kh Define ^n(ar) = £ r = o ^ ( M ) e - M « / " P * ( * - e ) and fn(x) = ET=oe~M(k)/n^^)Then { / „ } e C W ( S ) and {m = fm * cfin, ({/„}, {4>n}) £ -4(A^). The Boehmian [fn/4>n] S B(A*f) satisfies the required condition. • 11

Boehmians on the sphere, 2

In this section, we follow Mikusiiiski-Morimoto by means of central delta sequences. A function <j> on G is called central if

6

to define Boehmians on §

0 * / = / * 0 for all fEC(G).

(12)

We denote by Ce{G) the space of continuous central functions on G. The following simple characterization of Ce(G) will be useful (see Loomis l): Ce(G) = {(j> £ C(G);(fi(gh) = {g) = JG n} e Ce(G)N is called a central delta sequence ifn*f -4 / uniformly as n —> oo /or any / £ C ^ ' ( § ) . Definition 34. A subset A c of C e (G) N is a (commutative) semi-group of central delta sequences if it satisfies the following conditions: (A c l) {
235

For a semi-group A c of central delta sequences, we put A(AC) = {({/„}, {MY, {U} e CW(Sf, {cf>n} e A c , fn*<j>m = fm* K for all m, n} In .4(AC) we define the relation ({/„},{<£„}) ~ ({#«}, {>«}) by / „ * V>m = € C(G) we define the transformation A^ : ip £ L2(§) i-» cp * tp e 2 L (S). Then we have A^A^ = A^*^, for any <j>, ip e C(G). This is the left regular representation of G. In the direct sum decomposition L2(G) = 0 %*(§), Hk(S) are subspaces invariant under the representation A^. Thus, A^ is decomposed into direct sum: A^ = 0 £ l o A^ ', where AS ' = A ^ - ^ g ) . If e Ce(G), we have A J ^ A ^ = A ^ A ^ for all V £ C(G). k

(Ay,'H (B))

Because

is an irreducible representation of G, by the Schur's lemma,

there is a scalar Xk{4>) e C such that A^ = Xk()\k(Tp) = \k(
Ce(G).

(13)

Lemma 35. For cp € Ce(G) we have Afc() = /
(14)

Proo/. Because the function {x *->• Pfc(x • e)} belongs to "Hfc(S), we have

L

4>(h)Pk(h-lx • e)dh = \k{
IG

l

Because h~ x • e = x • he and Pk(e • e) = Pk{l) = 1, we have (14)

•

Lemma 36. Let

/*^ = ^*/ = 5;^(^)«*(/). where sk(f)

is the k-spherical harmonic component of f.

Proof. / * 0 = tf * / = A , / = £ ~ o A?>8fc(/) = Er=o W M / ) .

•

236 For a central function , we define z{ge)= I (gh)dh= / <j){hg)dh, JK

JK

where the integration is done with respect to the normalized invariant measure on K. Then <j>z is a zonal function. By the invariance of the Haar measure on G, we have, for h e K, Afc(^) = / (j>{gh)Pk(e • ghe)dg = / JG

cj>(gh)Pk(e-ge)dg.

JG

Integrating with respect to h £ K, we have Afc(<£) = /
J$

z(x)Pk(e-x)dx.

Therefore, we have 4>z{x) = Y,T=o N(k,d)\k((j))Pk(x

• e); that is,

Ck{z) = \k{)-

(15)

Lemma 37. Let , ip 6 Ce{G). Then we have (4>*^)z =(f>z *^zProof. Because of (13) we have oo

{4> * i>)z(x) = Y,N{k,d)\k{
* i/))Pk{x • e)

k=0 oo

= J2N(k,d)\k(<j>)\k(i>)Pk(x k=o

• e) = <j>z * ipz(x).

n

Theorem 38. Let A c be a semi-group of central delta sequences. Then (A c ) 2 = {{(4> n ),};{«MeA c } is a semi-group of zonal delta sequences. Further, we have B(AC)*B((AC)Z).

(16)

Proof. By Lemma 37, (Ac)z is a semi-group of zonal delta sequences. By (15) we have, for / € C^(§) and a central function <j>, oo

/ * = ^ k=0

whence we get (16).

oo

X

k()Sk(f) = Yl ck(4>z)Sk{f) = f*z, k=0

•

237

References 1. L. H. Loomis, An Introduction to Abstract Harmonic Analysis, Van Nostrand, New York, 1953. 2. J. Mikusinski and P. Mikusinski, Quotients de suites et leurs applications dans I'analyse fonctionelle, C. R. Acad. Sci., Ser. I, 293 (1981), 463 464. 3. P. Mikusinski, Convergence of Boehmians, Japan. J. Math., 9 (1983), 159 - 179. 4. P. Mikusinski, Boehmians and Generalized Functions, Acta Math. Hung. 51 (1988), 271 - 281. 5. P. Mikusinski, Boehmians on manifolds, Internat. J. Math. & Math. Sci., 24(2000), 583- 588. 6. P. Mikusinski and M. Morimoto: Boehmians on the sphere and their spherical harmonic expansions, Fractional Calculus and Applied Analysis, 4(2001), 25-35. 7. P. Mikusinski and B. A. Pyle: Boehmians on the sphere, Integral Transforms and Special functions, 10(2000), 93-100. 8. M. Morimoto, Analytic functionals on the Sphere, Translations of Mathematical Monographs, Volume 178, AMS, 1998. 9. M. Morimoto, Two definitions of Boehmians on the sphere, submitted to the Proceedings of ISAAC 2001, Berlin 10. M. Morimoto and M. Suwa, A characterization of analytic functionals on the sphere II, H.G.W.Begehr et al. (eds.), Proceedings of the Second ISAAC Congress, Vol. 2, 799 - 807, 2000, Kluwer Academic Publishers. 11. D. Nemzer, Periodic Boehmians, Internat. J. Math. Math. Sci. 12(1989), 685-692. 12. D. Nemzer, Periodic generalized functions, Rocky Mountain J. Math. 20 (1990), 657-669. 13. D. Nemzer, Periodic Boehmians II, Bull. Austral. Math. Soc. 44 (1991), 271-278.

A N E W LAX PAIR FOR THE SIXTH PAINLEVE EQUATION ASSOCIATED W I T H so(8) MASATOSHI NOUMI AND YASUHIKO YAMADA Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan E-mail: [email protected], [email protected] A new Lax pair for the sixth Painleve equation i-fyj is constructed in the framework of the loop algebra so(8)[z,2 -1 ]. The Backlund transformations for Fyi are interpreted as gauge transformations of the corresponding linear problem.

Introduction In this article, we propose a new representation of Lax type for the sixth Painleve equation. This representation, formulated in the framework of the loop algebra so{8)[z, z~l] of type Df', provides a natural explanation of the affine Weyl group symmetry of P V i- After recalling a standard derivation of iVij we describe in Section 2 fundamental Backlund transformations for PyiIn Section 3, we present our Lax pair for P V i associated with so(8)[^,^ _ 1 ], and explain how the Backlund transformations arise from the linear problem. For the general background on Painleve equations, we refer the reader to the reference.2 The authors would like to thank Professor Kanehisa Takasaki for valuable discussions in the early stage of this work. 1

The sixth Painleve equation

The sixth Painleve" equation is the following nonlinear ordinary differential equation of second order for the unknown function y = y(t): V

2\y

y - \

y-tJ{V)

\ t

+

t - l

+

y-t)y

where ' stands for the derivation with respect to the independent variable t, and a, (3, 7, S are complex parameters. A standard way to derive the sixth Painleve equation is to employ the monodromy preserving deformation of a second order Fuchsian differential equation on P 1 , with four regular singular points and one apparent singularity.

238

239 Consider the system of linear differential equations (dl + a1{x,t)dx+a2(x,t))u

= 0,

dtu = ( M M R + b2(x,t))u,

(2)

for the unknown functions u = u(x,t), where dx = d/dx and dt = d/dt. We assume that the coefficients a,j(x,t) and bj(x,t) are rational functions in x, depending holomorphically on t, and that the first equation is Fuchsian with Riemann scheme x=0x=lx=tx— 0 K0

0 KI

q £ = oo ^ 0 Kt

0 2

p > Koo + p J

(3)

with respect to the variable x. In this scheme, K0, KX, Kt, KX and p are generic complex parameters subject t o the Fuchs relation K0 + Ki + Kt + Koo + 2/9 = 1.

(4)

We also assume that the singularity x = q, which may depend on t, is nonlogarithmic. Under these assumptions, the coefficients ai(x,t), a2(x,t) are expressed in the form , ,,. 1 - «o 1 - «i 1 - «t ai (x, £) = 1 x —— 1 + rc-f

1 x -
respectively, where p = Resx=q(a2(x,t)dx),

H = —Resx=t(o2(x,t)dx).

(6)

Furthermore, the coefficient H is determined as a polynomial in (q,p) with coefficients in C(t); explicitly, it is given by H = ^ y y [ P2q(q -l)(q-t)+KXq(q -t)

p{K0(q - l)(g - t)

+ (Kt - l)q(q - 1)} + p ^ +p)(q-

t)].

The compatibility condition of the linear differential system (2) then turns out to be expressed as the Hamiltonian system tfvi:

dt(q) = -^,

dt(p) = - ~ ,

(8)

240

with polynomial Hamiltonian H in (7); namely, t{t - l)dt(q) = 2pq(q - l)(q - t) - {K0(Q - l)(q - t) t(t-l)dt(p)=

+ Kiq(q-t) -p2(3q2-2(l

+

(Kt-i)q(q-l)} + t)q + t)+p{2(K0+Kl

+ Kt-l)q

'

- /C0(l +t) - Kit - Kt + 1} - p(Koo + p).

This system of nonlinear equations is in fact equivalent to the sixth Painleve equation Fyi (1) for y = q, with parameters

<* = -2-. ^ - ~ y .

T - y

5

(10)

- ^ —

We remark that, in place of (2), one can naturally make use of the Schlesinger system of rank two, with regular singular points x — 0, \,t, oo. 2

Discrete symmetry of H\\

It is known that the sixth Painleve equation admits a group of Backhand transformations which is isomorphic to the (extended) affine Weyl group of type D\ ' (see the fundamental paper of Okamoto, 7 for instance). In describing the Backlund transformations for Hyi, it is convenient to use the parameters a 0 = Kt,

Qi =

KX,

a2 = p,

a 3 = «i,

a4 =

K0

(11)

with linear relation a 0 + <*i + 2Q2 + Q3 + a 4 = 1, SO that t(t - \)H = fab - 1)(« - *) "P{(<*0 - l)q(q - 1) +a3q(q - t) + a4(q - l)(q - t)} + a2(a1 + a2)(q - t). In the following, we identify the parameter space for ifyi with the affine space V = C 4 with canonical coordinates e = (si,s2,£3,£4), and regard aj as linear functions of V such that c*0 = 1 - £1 - £2, 03 = £ 3 - £ 4 ,

Qi=£l~£2, 04 = £ 3 + £ 4 .

0:2 = £ 2 - £ 3 ,

,,„.. >

We identify V with the Cartan subalgebra of the simple Lie algebra so(8) (of type D 4 ); {£1,... ,£4} is then a canonical orthonormal basis of V*, and ao,ai,«2,^3,04 are the simple affine roots. Note that the null root 6 = a0 + ai + 2a2 + a 3 + a 4 is normalized to be the constant function 1.

241

By a Backlund transformation, we mean a transformation of dependent variables and parameters that leaves the system invariant. Let us show an example of Backlund transformation for Hyj. Define new variables q,p by q = q,

:• (14) q—t Then one can verify directly that, if the pair (q,p) satisfies the Hamiltonian system (8), then the pair (q,p) again satisfies the same system with parameters ao, c*2 replaced by — cio, a2 + <*o, respectively; we refer to this Backlund transformation as so- To be more precise, let us consider the field of rational functions JC = C(ai,a2,a3,a4,q,p,t)

P = P

(a 0 = 1 - a i - 2a 2 - a 3 - a 4 ),

(15)

and the Hamiltonian vector field 6 =

§H_d__dH_d_

£_

dp dq

dt

dq dp

acting on K as a derivation. We regard this differential field (JC, S) as representing the Hamiltonian system ffvi- We define the automorphism so : /C —> K, by setting «o(ao) = -a0,

s0(a2) = a2 + a0,

s0(aij) = a,

(j ^ 0,2),

(17)

and so(q)=Q,

—j, s0(t)=t. (18) q t Then one can show that the automorphism so : K. —>• K commutes with the Hamiltonian vector field 6. In this sense, a Backlund transformation can be defined alternatively to be an automorphism of the differential field that commutes with the derivation. Table 1 is the list of fundamental Backlund transformations for H\\. We consider two subgroups W = (so,Si,s2,ss,s4)

«o(p) =P~

CW - (so,Si,S2,S3,S4,ri,r 3 ,r 4 ) C A\its(K.)

(19)

of differential automorphisms of /C, generated by the Backlund transformations in Table 1. Then it turns out that W and W are isomorphic to the affine Weyl group and the extended affine Weyl group of type D\ , respectively. The Backlund transformations Si (i = 0,1,2,3,4) and rj (i = 1,3,4)

242 Table 1. Backlund transformations for flyi

ao

ai

a2

a3

a4

9

P

«0

-a0

ai

a2 + a 0

a3

a4

9

p - -22f q-t

Si

ao

~a\

a2 + a i

a3

a4

a

P

S2

a 0 + a 2 a\ + ct2

— a

a3 + a2 a4 + a2

2

*

S3

ao

ai

a2 + a 3

-a3

a4

Q

S4

ao

ai

a 2 + 04

a3

-a4

q

n

ai

ao

a2

a4

a3

rs

a3

a4

a2

ao

ai

5*4

a4

a3

ai

a2

P

p

PP-

t(q-l) q-t t_ 1

a3

-^r ^

(g-t)((«-t)p+a2) t(t-l)

* (g-l)((q-l)p+a2) t-1

in fact satisfy the fundamental relations (» =•- 0,1,2,3,4),

»? = 1

(w =

i 3 — SjSi SiS2Si

= S 2 SjS 2

0,1,3,4),

(i = 0,1,3,4),

r ^ - = rA

(i = 1,3,4) ({*, i,fc} = {i,3,4})

riSj=satU)n

(i = 1,3,4; j = 0,1,2,3,4),

r? = l

(20)

where o~i (i = 1,3,4) are the permutations defined by «TI = (01)(34),

as = (03)(14),

<x4 = (04) (13).

(21)

We also remark that each element w £ W defines a canonical transformation: w({
MelC),

(22)

where { , } stands for the standard Poisson bracket defined by {,} =

dip dip dp dq

dip dtp dq dp'

(23)

243 Remark 2.1 The fundamental relations for the generators s0, si, s2, S3. S4 of the affine Weyl group of type D± is described as follows in terms of the Cartan matrix A = {aij)i • 0 : s? = l

(1 = 0,1,2,3,4)

SiSj = SjSi SiSjSi

— SjSiSj

if

(a,ij,aji)

II

^^"ij? ^"j*J — \

(24)

= (0,0), •'•J

•*•/}

where - 2 0 -1 0 . 0

A-

0 2 -1 0 0

-1 -1 2 -1 -1

0 0 -1 2 0

0 0 -1 0 2

(25)

The action of sz on the simple affine roots a,- is given by Siiotj) =Oj -ctiCLij

(i,j = 0,1,2,3,4).

(26)

Note also that W is isomorphic to the semidirect product W » Cl of W and fi = { l , r i , r 3 , r 4 } acting on W through the permutations {1,01,0-3,04} of indices for the generators Sj; ft is identified with a group of diagram automorphisms of the Dynkin diagram of type D\ '. If we set
¥>i = 1,

¥>2 = - p ,

¥>4 = q,

(27)

the Backlund transformations s, are expressed as 1

\

,

a i

y-'t

{
(28)

consistently with the birational Weyl group actions discussed previously.5'6 In this particular case, the matrix U = (uy )i ,_ 0 is given by 0 0 U = - 1 0 L0

0 1 0 0 0 0 0 0 - 1 0 1 0 0 1 0

0 0 -1 0 0

(29)

Observe that U12 = «2i = 0; this degeneracy is caused by the normalization that one of the regular singular points of (2) is placed at x = 00.

244

Remark 2.2 The extended affine Weyl group W of type D\ ' is expressed as the semidirect product of the weight lattice P of type D 4 and the Weyl group W{Di) = (81,82,83,84) acting on P:

W £PxW(D4),

P= 0 !

(30)

i=l

where •cji are the fundamental weights of type D4 defined by TJD\ = £\ ,

TJ72 = £ l + £2,

ro3 = | ( e i + £2 + £3 - £ 4 ) ,

-£74 = | ( £ i + £ 2 + £3 + £4)-

(31)

Note also that the weight lattice P is the set of all elements • n2£2 + n3e3 + n 4 e 4 ) ( n i , n 2 , n 3 , n 4 e Z) (32) 2(ni£l such that, either all the rij's are even, or all the n / s are odd. The translations TZJi (i = 1 , . . . , 4) corresponding to Wi are expressed as Tai L

tC3

= riS1S2S3S4«2Sl, ^3S3S2SlS4S2S3,

r r o 2 = S0S2S1S3S4S2S1S3S4S2, T r o 4 = r 4 S 4 S2SlS3S2S4,

(33)

in terms of the generators Sj and r,-. These elements transform the simple affine roots ctj as follows.

a0 T T T -'tS74

ai

Q2

"3

a4

a4 a0 + l ai — 1 a2 «3 ao + 2 Oil a2 - 1 a3 Ct4 a 1 a4 a0 + l <*i «2 3 a0 + 1 ai 0:2 a3 a4 — 1

(34)

Regarded as automorphisms of K. = C(ai,a2,a3,a4,q,p,t), Troi (i = 1,2,3,4) provide a commuting family of Backlund transformations for Hyi, which are called Schlesinger transformations (such Backlund transformations that act as shift operators on the parameter space). A certain part of this discrete symmetry of Hyi can be explained by the monodromy preserving deformation of a second order Fuchsian equation (2). In fact, each st (i = 0,1,3,4), except s2, arises from a simple transformation of the unknown function u = u(x, t), and each r* (i = 1,3,4) from a fractional linear transformation of the coordinate x. This framework does not seem, however, to explain the particular Backlund transformation s 2 in Table 1, which is essential in understanding the whole picture of discrete symmetry of

245

ifvi- In the next section, we propose a new Lax pair for if V i, from which all the Backlund transformations in Table 1 can be understood naturally. 3

Lax pair associated with so (8)

Consider the following system of linear differential equations for the column vector ip = (ij}i,ip2,---,ip8)t of eight unknown functions ipi = ipi(z,t) (i — 1,2,...,8): (zdz + M)\l> - 0,

dtr/> = Brj>,

(35)

with the compatibility condition [zdz + M,dt-B]=

0.

(36)

We assume that the matrices M and B are in the form 1 0 0 0 0 0 0 £i -1 -1 0 0 0 0 £2 -P 0 0 0 0 0 £3 q -1 1 0 0 0 0 1 e4 0 -Q M (37) 0 0 0 0 S4 1 0 -1 0 0 0 0 0 -z -£3 P 0 0 0 0 0 -1 (t -q)z -£2 0 (q - * ) z z 0 0 0 0 -£1and Xi 0 0 0 0 0 • "Wl 2/1 0 u2 X2 - J / 3 - 2 / 4 0 0 0 X4 0 0 u3 0 0 0 x3 Ui 0 0 0 0 —Xi 0 2/4 B (38) 1 0 0 —U4 0 0 0 -x3 2/3 0 0 0 0 0 -u3 -x2 -2/1 —z 0 0 0 0 0 -Xi -U2 z 0 . 0 0 0 0 0 -ux. respectively, where e,- are complex constants, and the variables q, p, Xj, yj and u- are functions in t. As before, we set Q0 = 1 - £l - £2,

Oil = £l - £2i

0!3=£3-£4,

0-4 = £ 3 + £ 4 .

a2 = £2 - £ 3 ,

(39)

Theorem 3.1 Under the compatibility condition (36), the variables x,, t/j and Uj are determined uniquely as elements of K. = C(ai,a2,a3.a4,q,p,t). The compatibility condition is then equivalent to the Hamiltonian system ifvi of the sixth Painleve equation.

246

The variables Xj, yj and Uj appearing in B are determined explicitly as follows: Q-t

Q

(q - t)p + X2

=

9-1

a1+a2

(40)

TiTi)—'

and t(t - 1) ui = -q(q - l)p - a2q +

a

^ ~ l t - so±|4=i )

t(t - 1) u 2 = -
ffld^i?

(41)

t(t - 1) u 3 = (2g - l)(g - % + (ai + 2a 2 )o + sa±s±t + «0+at-i ^ t(t - 1) u 4 = ~(q ~ t)p + ^ ^ t + °n+°4-i. We now explain how this representation is related to the Lie algebra so (8) and its loop algebra. With the notation of matrix units Eij = (SiaSjb )8a 6 _ 1 , we set 8

J = Y,Ei$-i-

(42)

We consider the following realization of the simple Lie algebra so (8): so(8) = { I e Mat(8; C) | JX + X* J = 0},

(43)

where X 4 denotes the transposition of X. Let us define the Chevalley generators Ej,Hj,Fj (j = 0,1,2,3,4) for the loop algebra so(8)[z, z~l] by E0 = z(E$2

—

Ej\),

E3 = -E34 — ^56) 1

E2 = -E23 — E67,

Ei = £35 — £46.

FQ = z~ (E28 — En), F3 = -B43 _ E%5,

Ei = E\2 — £78;

F\ = E21 — EST,

(44)

F2 = E32 — Ere,

Fi = E53 — E6i,

and Hj = [Ej,Fj] (j = 0,1,2,3,4). For a vector a = (01, o 2 , a3, a 4 ) given, we also use the notation 4

H{a) = 'Y^ai{Eu - £79_ij9_i) i=l

(45)

247

for the corresponding element in the Cartan subalgebra of so (8), so that H0 = # ( - 1 , - 1 , 0 , 0 ) ,

# ! = # ( 1 , - 1 , 0 , 0 ) , ff2 = # ( 0 , 1 , - 1 , 0 )

# 3 = #(0,0,1,-1),

HA = # ( 0 , 0 , 1 , 1 ) .

(

j

Notice that the two matrices M, B belong to a Borel subalgebra of the loop algebra so{$)[z, z~1]; in fact, they are expressed in the form M = H(e) + {q- t)E0 + Ex - pE2 + (q - l)E3 + qE4 +[Eo,E2] + [E3,E2] +

[Ei,E2),

B = H(u) + E0 + xxEx + x2E2 + x3E3 + Z4E4

(47)

+2/i [E1!, E2] + 1/3 [E3, E2] + J/4 [E4 ,E2], where e = (ei,£2;£3>£4) and u =

(ui,U2,u3,U4).

Remark 3.2 The affine Lie algebra so(8) is realized as a central extension of the loop algebra so(8)[2,2 - 1 ], together with the derivation d = zdz (see the textbook, 4 for the detail): sb(8) = s o ( 8 ) ® C [ z , z ~ 1 ] © C c e C d ,

(48)

where c denotes the canonical central element. In this context, the simple affine roots atj (j = 0,1,2,3,4) are denned as linear functional on the Cartan subalgebra 4

h = 0C/ii©Cd,

/ i 0 = # o ® l + c,

hi=Hi®l

(1 = 1,2,3,4)

(49)

i=0

such that (hi, otj) = aiyj

(i = 0,1,2,3,4),

{d, aij) = S0J

(50)

for j = 0,1,2,3,4; also they are extended to linear functionals on the whole affine Lie algebra so (8) through the triangular decomposition. Note that our Lax pair mentioned above is formulated in fact in the framework of sb(8)/Cc = so(8)[z,z- 1 ]©Crf: M = d + M = zdz + Me sb(8)/Cc.

(51)

Since (c,Oj) — 0, we can regard a, (j = 0,1,2,3,4) and the null root 6 — c*o + ai + 2a2 + &3 + Q4 as linear functionals on sb(8)/Cc. Then we have aQ{M) = 1 - £ 1 - £ 2 , a3(M) = e3-£4,

ai(.M) = £ 1 - e 2 , a4(M) = £3 + e4,

a2(X)=£2-£3, S(M) = 1.

In this sense, our notation for the parameters (39) is consistent with that of simple roots for so (8).

248 In our framework, the Backlund transformations for Hy\ are obtained as the gauge transformations sk$=Gk$

(* = 0,1,2,3,4),

rk$=Tk$

{k = 1,3,4)

(53)

of the linear problem (35), defined by certain matrices Gk, Tk in the loop group of 50(8) = { I E SL(8;C) | X1 JX = J } .

(54)

The matrices Gk and Tk will be specified below. Theorem 3.3 The Backlund transformations sk andrk for Hyi are recovered from the compatibility conditions sk(M) = GkMG-kx - zdz{Gk)G^\

sk(B) = GkBG^

+ dt(Gk)G^\

(55)

and rk(M) = rkMT^1-zdz(Tk)rj;1,

rk(B) = rkBrk-1+dt(Tk)Tk-1.

(56)

In (53), the matrices Gk are determined as Go = l + ^ - * o , q-t

Gi = l + a i ^ i ,

G3 = l + ^ - F 3 , q-1

G4 = l + — F 4 . q

G2 =

1-^F2, p

(57)

The matrices T i , r 3 and T4 are given explicitly by 0 0 0

0

0

/ " * N -J—\A(t-i) y ^ - i ) 0 - ^ " ^

0

0

0

Ti =

o 0

o zy/t{t - 1 )

o 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

—7=t==

0

0

0

- v^- 1 *

o

o

0

0

o

0

o 0

o 0

o 0

o

—7^= \Z*(*-i)

o 0

o

o

.1 yt(t-i)

0

V^-v 0

Zy/t(t-l)

0

o 0 (58)

0 0

r3 =

0

0

0

v^v* 0

0

0

0

0

-<7 y/-ty/z

0

0

1 y[=iyfz~

0 0 0

0

0

0

0

0

0

y/=i qy/z

sTiyfz

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

V^Vz

0

0

0

u

0 0

y/l^ly/z u

V=i z 0 - V-W 1

0

0

0 y/T^t

0

0

/T^ty/z

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

y/T=ty/z~

0

y/l^t 1-9

0

-i=a_

/l^ty/z

1

yf^ty/z

y/l-ty/z

n=t

(l-«)v^

We remark that the matrices Tj. (k — 1,3,4) are expressed as r1 = D ( a 1 ) e x p ( ^ 2 ) 0 — l C i , T3 =

D(a2)exp{1-E2)z-^C3,

T4 = £>(a 3 )exp

(^E2)z-^C4,

where we have used the notation D(a) = d i a g ( a i , a 2 , a 3 , a 4 , a 4 1 , a 3 1 , o 2 1 , a 1 x)

fl^ty/z

0

250

for a vector a = (01,02,03504), a,j ^ 0. In (61), zWk denote the following diagonal matrices associated with the fundamental weights of so(8): zt»i z

=^(1.0,0,0)

w2

= I >

(

Z j l j l ) 1

)

|

=zH(l,l,0,0)=sD^Z)Z)1)1^

2ro3 = 2 ^ ( 5 . 2 . 2 . - 2 ) =

D(Z5,Z2,Z2,Z~2),

2™" = ^ " ^ 2 . 2 . 2 . 2 ^ = D(Z

2, Z 2 , 2 : 2 , « 2 ) .

The matrices C* are essentially permutation matrices; with the notation of permutation matrices S„ = (^
in (61) are denned by y/tjt-i)

y/tit^ij'^/i^'

^^ a3 = n — - Il—L1

i-t

2_ ^E* _j=i\ 9-1

(64)

VT^i

-t

'^/^ZT^ '

(65)

-1 \

We also remark that, for each k = 1,3,4, the adjoint action of the matrix z~ZUkCk induces the automorphism of the loop algebra so(8)[z,z - 1 ] corresponding to the diagram automorphism ax = (01)(34), a3 = (03)(14) or 04 = (04) (13), respectively. The remaining part of Fk concerns the normalization of the matrices M and B. Note that the system of differential equations (35) has a regular singularity at z = 0 with exponents ±Sj (j = 1,2,3,4), and an irregular singularity at z = 00. Assuming that ej are generic, let us take a fundamental system of solutions * = $(z,t) of (35) with normalization such that 00

$(z,t) = Y,yn(t)z-H^+n

(66)

71=0

around z = 0, and that *o(*) is upper triangular; such a * is determined up to the multiplication of constant diagonal matrices. Then, for each k = 0,1,2,3,4, the Backhand transformation s* is interpreted as the transformation of the fundamental system of solutions * i - > $ = G t *5fc,

(67)

251 where Sk = exp(-Ek) exp(Fk) exp(-Ek) denote a lift of sk to the loop group of 5 0 ( 8 ) . Similarly, for each k = 1,3,4, the Backlund transformation rk is interpreted as the transformation * H. $ = Tk*Rk, where Rk =

(68)

z~^"Ck.

Remark 3.4 The system of differential equations (35) can be equivalently rewritten into a chain of systems of rank 2. We first extend the indexing set for ipi and Si to Z by imposing the periodicity condition ipi+s = zipi,

ei+8 = Si - 1,

e 9 _i = -et

(i G Z).

(69)

Then (35) is equivalent to a system for the 2-vectors V>i = W>2i> V^i+i)* (i G Z) in the following form: (zdz + Mi)& + Ni$i+l

= 0,

{-dt + Ai% + Bi^i+1

= 0,

(70)

where Mi, Ni, Ai, Bi are 2 x 2 matrices whose entries depend only on t. This system is formally transformed into (A + Mi)0i + Nnpi+l = 0,

(-% + Ai)$i + Bi$i+1

= 0,

(71)

w

where (pi = A, w «-» -d\. Noting that Nt are invertible, one can rewrite (71) as dm = Um,

0i+1 = Wifr

(i £ Z),

(72)

where Ui = Ai-BiNr1(\

+ Mi),

Wt =-Nr1(\

+M{)

(t G Z).

(73)

1

This type of 2 x 2 nonlinear chains is investigated by V.E. Adler in relation to Painleve equations. It is not clear yet, however, how our system (72) can be related to the one employed there for obtaining FyiIn this paper, we have presented a new Lax pair for the sixth Painleve equation in the framework of the loop algebra so(8)[z, z~l] of type D\'. We also explained how the affine Weyl group symmetry of Pyi can be obtained from the linear problem. We expect that the Lax pair discussed in this paper could be applied as well to other problems concerning Painleve equations. Also, it would be an important problem to understand properly the relationship of our representation with various approaches to the sixth Painleve equation. 2

252

References 1. V.E. Adler: Nonlinear chains and Painleve equations, Physica D 73 (1994), 335-351. 2. R. Conte (Ed): The Painleve Property: One Century Later, CRM Series in Mathematical Physics, Springer-Verlag, New York, 1999. 3. K. Iwasaki, K. Kimura, S. Shimomura and M. Yoshida: From Gauss to Painleve — A Modern Theory of Special Functions, Aspects of Mathematics E16, Vieweg, 1991. 4. V.G. Kac: Infinite dimensional Lie algebra, Third Edition, Cambridge University Press, 1990. 5. M. Noumi and Y. Yamada: Affine Weyl groups, discrete dynamical systems and Painleve equations, Commun. Math. Phys. 199(1998), 281295. 6. M. Noumi and Y. Yamada: Birational Weyl group action arising from a nilpotent Poisson algebra, in Physics and Combinatorics 1999 (Eds. A.N. Kirillov, A. Tsuchiya and H. Umemura), pp. 287-319, World Scientific, 2001. 7. K. Okamoto: Study of the Painleve equations I, Ann. Math. Pura Appl. 146(1987), 337-381.

O N A GENERALIZATION OF T H E L A U R E N T E X P A N S I O N YUTAKA SABURI Chiba Jr. College, 1-3-1 Kounodai, Ichikawa, 272-8512 Japan Email: [email protected]

Abstract A generalization of the Laurent expansion of holomorphic functions of one variable to the case of linear reductive complex Lie groups is discussed.

Introduction In the preceding work [5], we discussed on characterizations of Fourier series of real analytic functions and hyperfunctions on compact Lie groups by means of complex analysis. In the present paper we show again that the same analysis leads us to a generalization of the Laurent expansion of holomorphic functions of one variable to that of holomorphic functions on open neighborhoods of maximal compact subgroups of linear reductive complex Lie groups. We give its brief illustration in the following. Let us review the Laurent expansion of holomorphic functions on neiborhoods of the unit circle T in the complex plane C. For r > 0, put Vr = {z € C ; e _ r < \z\ < er}. Let / be a holomorphic function on Vr, and assume that / extends continuously to the closure Cl(Vr) of Vr. Then / has its Laurent expansion f(z) = J2ieZ fiz1, and the coefficients /; have their integral representations: (1)

27TV-1 J\z\=l

Furthermore the Cauchy integral theorem assures us to shift the cycle of the integrals without changing their values, and we have the following estimates for /,: l-K\f— 1 ,/| z | = esgn(D>-

<sup,eCi(v,)l/(*)|e-p|" 1991 Math, subject Classification: 22E30, 30B99

253

(*€Z).

(2)

254

Thus we have the characterization for the Laurent coefficients of holomorphic functions on annular neighborhoods of T. Namely, they are of exponential decay with respect the integral parameter I. Note that the domain of holomorphy of the function / is sharply reflected in the type of the exponential decay of its Laurent coefficients. We note that the unit circle T in C can be regarded as the 1-dimensional torus and its complexification is the 1-dimensional complex multiplication group C x = {z € C; z ^ 0}. Prom this view point, the functions $i(z) = zl ( z £ C x , i £ Z ) are matrix elements of all holomorphic irreducible representations of C x , and the restrictions of these functions to T are matrix elements $;( e%/=Tt ) = e v/ ~ T " (t € R/2-KZ) of all irreducible unitary representations of T. Furthermore the Laurent coefficients for a holomorphic function / on an annular neighborhood of T coincide with the Fourier coefficients of the restriction of / to T. Generally, a connected linear reductive complex Lie group G has the real reductive structure as well, and it forms a complexification of its maximal compact subgroup K. The Borel-Weil theorem assures that any (equivalent class of) irreducible unitary representation of K can be realized as the restriction of a finite dimensional irreducible holomorophic representation of G to K. The collection of matrix elements from all the non-equivalent irreducible unitary representations forms an orthonormal basis of L2(K). The expansion of a function on K by these matrix elements is called its Fourier expansion. Hence the restriction of any holomorphic function / on an open neighborhood of K in G has its Fourier expansion, and the extension of the expansion coincide with / on some open neighborhood of K by virtue of the uniqueness of analytic continuation as long as the series converges. We are going to call the expansion of holomorphic functions on an open set of G by the matrix elements of finite dimensional irreducible holomorphic representations of G as the Laurent expansion. In [5], we actually, but partly, discussed on this matter. But our interest was concentrated on the characterization of Fourier series of real analytic functions (and hyperfunctions) on compact Lie groups. The method used there was in the line along that illustrated above in the case of 1-dimensional torus T. The aim of this paper is to characterise exactly the Laurent series of holomorphic functions on open tubular neighborhoods of maximal compact subgroups of connected linear reductive complex Lie groups owing to the results in [5]. The main result is Theorem in Section 2. There are forerunning works of our study in [3] and [4], where Morimoto characterized the expansion of holomorphic functions on complex neighborhoods of the sphere and the Lie sphere by the matrix elements of fi-

255

nite dimensional irreducible holomorphic representations of SO(n; C) and SO(n; C) xCx, respectively. 1 Preliminaries from representation theory In this section we summarize the theory of irreducible unitary representations of maximal compact subgroups of linear reductive complex Lie groups. We refer to the reader Chapters IV-V in [1], and Chapter 2 in [6] for this subject. Let G be a connected linear reductive complex Lie group. Then G has real reductive structure as well. Let K be a maximal compact subgroup of G. Then K is connected, and G forms its complexification. We denote the Lie algebras of G and K by g and t, respectively. Put p = y/^lt, then the direct sum g = t © p gives the Cartan decomposition of g for a Cartan involution OX = -X* (X £ g) of g, where X* denotes the conjugate transpose of X. Let T be a maximal torus of K, and t its Lie algebra. Put o = v/-Tt and c = tc = t © o. Then o is a maximal abelian subalgebra of p and c is a Cartan subalgebra of g. Let A be the connected Lie subgroup of G corresponding to o. Then we have the following Cartan decompositions of G: G = K exp p = KAK,

(3)

As to the former decomposition, we note that for each g £ G there exist unique elements k(g) £ K and X(g) £ p such that g = k(g) expX(g). Put B{X, Y) = Tr(Xy) for X, Y e g. Then B gives an invariant nondegenerate symmetric bilinear form on g, and its restriction to p is positive definite. We introduce an inner product on p: (X, Y) = B(X, Y) for X, Y £ p. Put ||X|| = (X, X)1/2 for X e p. Furthermore put \\g\\ = \\X(g)\\ for g £ G. Then the followings hold: \\kgk'\\ =||0|| 1

Hff- !! =11*11

for

g £ G and *, k> £ K,

toT9£G.

(4)

(5)

Let a' denote the dual sapace of o. Since the restriction of the bilinear form B to o is positive definite, for each fi £ a' there exits unique H^ £ a such that fi(H) = B(H, Hp) holds for any H £ a. We can then define an inner product on a' by putting (fj,, v) = B(H^, Hv) for ]x, v £ a'. Put ||/x|| = (/z, /J) 1 / 2 for ix £ a'.

Since c = oc, each element in a! has its unique natural extension as an element in c'. So we regard the elements in o' as elements in the dual space c' of c which are real valued on o. Let S(g, o) denote the set of all nonzero restricted roots of g with respect to a, and ga the root space for a £ S(fl, a). Note that H(g, a) coincides with the set of all nonzero roots

256

of g with respect to c in the above sense and that each of the root spaces Q° is a 1-dimensional complex vector space. Let $ be the positive system of E(g, o) with respect to some lexicographic order on a'. An element fx 6 a' is said to be dominant, if it satisfies the condition (fi, a) > 0 for all a € $ . Put L = {H £ t; exp H = 1G}. An element \i in c' is said to be analytically integral or T-integral if it satisfies the condition /x(L) C 27T\/—TZ. Irreducible representations of the compact Lie group K are finite dimensional highest weight modules. Tow irreducible representations of K are equivalent if and only if their highest weights coincide. Put A = {A; A is dominant and T-integral}.

(6)

For A £ A there exits an irreducible unitary representation $ A of K with the highest weight A, and {$A}ASA gives a complete representative system of equivalent classes of irreducible unitary representations of K. The Borel-Weil theorem assures that for each A e A there exits a holomorphic representation $(A) of G whose restriction to K is equivalent to $\. Namely, <£(A) is the left regular representation of G on a finite dimensional vector space T(A) of relatively invariant holomorphic functions on G, and the inner product on T(A) is given by that on L2(K). Put d(X) = dim c T(A). We identify $(A) with its matrix representation [$ij(A; g)]i
(7)

gives an orthonormal basis of L2(K), and we have the following formulation on the Fourier series of this Hilbert space. For k € Z > 0 , let M(k; C) denote the set of all k — by — k complex matrices, and we put 5(A) = U\eA M(dW', C). For s £ 5(A) and A € A, let us denote the A-th component of s by s(A) £ M(d(X); C). Then the space 5 2 (A) = {se 5(A); || S || 2 = £

d(A)Tr(«(A)*(A)') < oo}

(8)

AeA

is a Hilbert space with the Hermitian inner product («i, *a) = £

d(A)Tr(Sl(A)s2(A)*)

for

su s2 e 5 2 (A).

(9)

AEA

For / e L2(K) and A e A put FfiiW

= (/, *ii(A))^(jfj

- T O ) = [FfiiW]i
(1 < i, 3 < d(X)), e M(d(X); C),

(10) (11)

257 and put J7

= P 7 ( A ) ) A 6 A e 5(A). 2

(12)

Then T defines a unitary operator of L (K) onto 5 (A), and its inverse T~l is given by T-ls{k)

= £

2

(s 6 5 2 (A), * 6 K).

d(A)Tr(s(A)**(A; A))

(13)

AeA

Thus the Fourier expansion of / G L2(K) can be written as follows:

f(k) = T~xTj = J2 dW Tr(^/(A)'$(A; *))

(k e

tf).

(14)

A£A

2 The Laurent expansion of holomorphic functions on neighborhoods of K The discussion in this section is based on the fundamental results in our preceding work [5]. We note that we denoted by U a maximal compact subgroup of G in [5]. For an open subset V of G, let 0(V) denote the space of holomorphic functions on V, and we put Ob{V) - {/ G C(Cl(V));

f\v

e G(V)},

(15)

where Cl{V) denotes the closure of V in G, and C(Cl(V)) the space of continuous functions on Cl(V). The space Ob(V) becomes a Banach space with the usual sup-norm over Cl(V). Let Vi and V2 be open subsets of G with Vi C V2, then the natural restriction mapping Ob(V2) «-> £>i(Vi) is compact. For r > 0, put Vr = {g G G; \\g\\ < r}. The set Vr is an open neighborhood of K in G. Then the spaces 0(Vr) and 0(G) are FS spaces with the following projective limit topologies (cf. [2]): 0(V P ) = lim p r o j j e z > 0 Ob(V{1_1/j)r),

(16)

O(G) = limproj i G Z > 0 O 6 (V}).

(17)

We can regard the spaces 0(Vr) to be subspaces of the space A(K) of real analytic functions on K by virtue of the uniqueness of analytic continuation. In this sense, we had the following estimate for the Fourier coefficients of the elements in Ob(Vr) in [5]: Lemma 1 (Lemma 3 in [5]). Let r > 0 and f € Ob(Vr). following inequality holds for all A G A: |:F/y(A)|<||/||re-p»A"

(1 < *, 3 < d(A)),

Then the (18)

258

where \\f\\r denotes the sup-norm of f over Cl(Vr). We note that the corresponding positive number to above r is denoted by e in Lemma 3 in [5]. For the proof of this lemma, we used the integral representation of the matrix elements of irreducible unitary representations of K (U in [5]) by virtue of the Borel-Weil theorem. Then the proof went in the line along the complex analytic method to estimate the Laurent coefficients of a holomorphic function on a neighborhood of the unit circle in the complex plane illustrated in Section 0. As announced in Section 0, we call the expansion of holomorphic functions on open subsets in G by the matrix elements of finite dimensional irreducible holomorphic representations of G as the Laurent expansion. Now we are going to describe the characterization of the Laurent series of elements in 0{Vr) and

0(G). Indicated by Lemma 1, we prepare the space of Laurent coefficients of elements in 0{Vr) and 0(G). For n 6 E, we put S„(A) - {s g 5(A); ||«||, = sup A € A e -"« A H{Tr( S (A) S (A)*)} 1 / 2 < 00}.

(19)

The space S1(A) is a Banach space with the norm || • ||^. If 77' > 7?, the natural inclusion mapping 5,,(A) c-4 S^'(A) is compact. Hence the following spaces are FS spaces: S0(vr)W

= lim p r o j j e z > 0 S_ ( 1 _i/j) r (A),

So(G) (A) = Hm p r o j i € Z > 0 S-j (A).

(20) (21)

Now we can describe the characterization of the Laurent series of elements in 0(VT) and 0(G): Theorem, (i) Let r > 0. The linear operator T gives topological linear isomorphisms of 0(Vr) and 0(G) onto So(vr)W and SO(G)W, respectively. The inverse T~l of F is given by T~ls(g)

= Y,

d(A)Tr(S(A)4$(A; g))

(g e Vr or G)

(22)

ASA

for s in S'o(vr)(A) or So(G)(^)> respectively. (ii) Let r > 0. For s in S©(yr)(A) or SO(G)(&), the Laurent series (22) converges in the topology of 0(Vr) or 0(G), respectively. (iii) Consequently, we have the Laurent expansion f(g) =

T-1Tf(g)

= ] T d(X)Ti(^f(X)t^(X;g)) A£A

(9 6 Vr or G)

(23)

259

of f in 0(Vr)

or 0(G),

respectively.

For the proof of this theorem we need one more Lemma 2 ((2.10) in [5]). For all A £ A, the following inequality holds: l*«(A; g)\ < el|A|IIMI

(geG,i<

i, j < d(X)).

(24)

Proof of Theorem. We only prove (i) and (ii). Then (iii) is a consequence of them. Since the proofs of (i) and (ii) in the cases of 0(Vr) and 0{G) go similarly, we only show them in the case of 0(G). First we show (ii). For R > 0, put n(R) = *j{A G A; ||A]| < R}. Then n(R) is bounded from above by a polynomial in R. Put p = J^azv (dimR ga) a. The Weyl dimension formula d(X) = ELe# (X + p/2, a)/(p/2, a) shows that d(X) is a polynomial in A of degree (!*• Then, combining these with Lemma 2 we see that the series (22) converges in 0(G) for s E SQ(G)WThus we have shown (ii). Furthermore we see that the series (22) deifines an entire holomorphic function on G for s £ So(G)WNext we show (i). Note that the operator T of L2(K) is a linear isomorphism onto S2(A) and that the natural restriction mapping of 0(G) into A(K) is injective by virtue of the uniqueness of analytic continuation. Hence, to see that the operator T gives a topological linear isomorphism of 0(G) onto 5 0 ( G ) ( A ) , it is sufficient to show that both TO(G) C 5 0 (G)(A) and T~lSo(G)(h) C 0(G) hold, and both T and T~l are continuous with respect to the topologies of 0(G) and SQ(G)(A), respectively. The fact TO(G) C So(G)(A) and the continuity of T follow from Lemma 1. From (ii), we see that T~XSQ(G)(^) C 0(G) holds. Similarly the continuity of J7"1 follows from the fact that n(R) is bounded above by a polynomial in R, Weyl dimension formula, and Lemma 2. Thus we have shown (i) and hence Theorem. • Acknowledgement The author would like to express his sincere gratitude to Prof. T. Oshima. He suggested the author that (2.10) and Lemma 3 in [5] lead to a generalization of Laurent expansion described in this paper. References 1. Knapp, A. W., Representation Theory of Semisimple Lie Groups, Princeton Univ. Press, Princeton, 1986.

260

2. Komatsu, H., Projective and injective limits of weakly compact sequenses of locally convex spaces, J. Math. Soc. Japan 19 (1967), 366-383. 3. Morimoto, M., Analytic functionals on the Lie sphere, Tokyo J. Math. 3 (1980), 1-35. 4. , Analytic functionals on the sphere and their Fourier-Borel transformations, Complex Analysis, Banach Center Publ., 11, 223-250, PWN-Polish Scientific Publishers, Warsaw, 1983. 5. Saburi, Y., A complex Analytic Study on the Theory of Fourier Series on Compact Lie Groups, Publ. RIMS. Kyoto Univ., 33, No. 6 (1997), 953-966. 6. Wallach, N. R., Real Reductive Groups I, Academic Press, San Diego, 1988.

D O M A I N S OF C O N V E R G E N C E OF LAPLACE SERIES JOZEF SICIAK Jagiellonian University, Institute of Mathematics, id. Reymonta 4, 30-059 Krakow, Poland E-mail: [email protected] An open subset D C Kra is called a Laplace domain (resp. complex Laplace domain), if there exists a series 5Zo° Qj where Qj is a homogeneous harmonic polynomial of n variables of degree j such that D = {a G M71; the series is convergent in a neighborhood of a in M n } (resp. D = {a G C n ; the series is convergent in a neighborhood of a in C™}). On the plane R 2 the only Laplace domains are disks with centre 0. For all n > 2 Laplace domains exhibit grater variety, even nonconvex ones are among them. So far we do not know a geometrical characterization of Laplace domains for n > 2. We show that a balanced domain D in C " is a complex Laplace domain iff it is a domain of harmonic holomorphicity. A convex balanced domain is a complex Laplace domain iff whenever the normal vector v exists at a point a of dD, it is a null vector, i.e. v^ 4- . . . + v% = 0.

1

Harmonic functions and Laplace series

Let 'H(D) denote the Frechet space (with the topology of locally uniform convergence) of complex valued harmonic functions of n real variables in a domain D e l " (n > 2). Definition 1.1 A series of homogeneous harmonic polynomials of n real variables with complex coefficients CO

^2Qv(x),

QV{X) = J 3 cvax",

cvaEC,

AQ„=0,

(1)

is called a Laplace series. We know that every complex valued function harmonic in a Euclidean ball centered at 0 is a sum of a Laplace series. In particular for every / e *H{D) CO

f(a + x) = Y^Qu(x),

\\x\\

aeD,

(2)

Mathematics Subject Classification: 31B05, 32A05, 32A07, 32A37, 32D05, 32D10, 32D15. Key words and phrases: Laplace series, Laplace domain, harmonic function, harmonically holomorphic function, domain of harmonicity, domain of harmonic holomorphy, Lie norm, Lie ball. Research supported by KBN grant 2 P 0 3 A 04514.

261

262 where

Qv{x) = Qv(a;x)= £

^p^"-

\a\=v

R e m a r k 1.2 Every domain D C R™ (n > 2) is a domain of harmonicity, i.e. there exists a function / G Ti-(D) which cannot be continued harmonically beyond D (in t h e sense t h a t for each point a G D its Taylor series (2) is divergent a t some points of a ball centered at 0 with radius r > dist(a, dD)). T h e claim follows from t h e fact t h a t 7~C(D) is a Prechet space, and for every point 6 G dD the function fb(x) := l/\\x — b\\n~2 (n > 2) (resp. fb(x) := log ||a; — 6||, n = 2) is harmonic in D and has a singularity at the point b. 2

L a p l a c e d o m a i n s i n Mn ( r e s p . i n C")

D e f i n i t i o n 2.1 We say t h a t a domain G C R n (resp. G c C C n ) is a Laplace domain (resp. complex Laplace domain), if there exists a Laplace series ^2QV such t h a t is convergent in a neighborhood of the point a in M n } is a domain of convergence of this series in R n , respectively Gc = {a G
G = GcnMn,

where V'(z) := l i m s u p Zj\Qv{z)\,

ip*(z) := limsup-0(C)-

One may check t h a t either tp* = oo, or tp* < oo for all 2 G C n . Hence G (resp. G c ) is either an empty set, or a balanced neighborhood of 0 in l n (resp. in C n ) , i.e. x G G = * {i:r;i G R, |i| < 1} C G (resp. z G Gc^{A2;AGC,|A|
263

ri, \zi + izi\ < r 2 } ) {TJ > 0), where r = m i n ( r i , r 2 ) . Namely, in R 2 a Laplace series can be written as oo

a0 + 2_J[aj{xi

— i^y

+ bj(xj + ix%y\,

a,j,bj £ C,

which implies t h a t tp(zii z2) = max(|zi — ^ l l i m s u p y\a,j\, j->oo

V

\zi + iz?\ l i m s u p

y\bj\),

i-^oo

which implies t h a t G (resp. Gc) has t h e special form. So far we do not know any geometrical characterization of Laplace domains for n > 3. However it is known t h a t 8 : 1°. If D C M.n is a Laplace domain then the tube D x K

mRn

is a Laplace

domain

xRk.

2°. If D C R n is a Laplace domain and T : W1 —» M.n is either a homothety or a linear orthogonal mapping then the image T(D) is a Laplace domain. 3°. The sum D = l L > o ®i °f an increasing sequence of Laplace domains and the interior D := intf]-eJ Dj of a family of Laplace domains is a Laplace domain. 4°. Product Property: If D C Rm,G C R " are Laplace domains, G=(D xM.m)n (W1 xG) is a Laplace domain m K m + n .

then D x

2

5°. In M.3 the ellipsoid £r := {x\ + x\ + ^f- < 1} is a Laplace domain if and only if r > 1. 6°. In R " + 1 (n > 3) the ellipsoid £r := {(x,t) e l " x l ; ||a;||2 + ^ (resp. is not) a Laplace domain, if r > 1 (resp. 0 < r < n~2/3). not know what about remaining r.

< 1} is We do

7°. The parallelepiped {\xi\ < 1, |ac21 < 1, I23J < r } (resp. cylinder {x\-\-x\ 1, |rK31 < r}) is not a Laplace domain, if r < 1.

<

We do not know what about if r > 1. 8°. For every n > 3 there exist non-convex

Laplace domains

in R n .

T h e properties 4°, 5°, 6° follow from 3° and from t h e following Lemma.

264 L e m m a 2 . 2 ( 8 ) If a Laplace series ^2^ Qi/(x,t), x £ M n , t 6 R, is convergent in a neighborhood of the ball {\\x\\ < l , t = 0 } , then it is convergent in the set {||a;|| + c\t\ < 1}, where c = n 3 / 2 / o r n > 3, and c—1 for n = 2. Sketch of the proof of 8°. If 0 < p < 1, then t h e domain Dp := {x 6 K"; |:EI| P + . .. + \xn\p < 1} is not convex. It is known (see e.g. 8 , Proposition 1.2.) t h a t there exists a sequence of homogeneous polynomials {
= l i m s u p l i m s u p y/\qv(y)\,

x e ln.

Also it is known t h a t for every v there exists a harmonic homogeneous polynomial Qv(x, t) of n + 1 variables {x, t) of degree v such t h a t Qv{x, 0) = qv{x) a n d •j^Qu(x, 0) — 0 in M". T h e domain of convergence G of the series ^ Qu{x, t) is n o t convex, because t h e intersection of G with t h e hyperplane {t = 0} is identical with t h e non-convex set Dp. • 3

A Remark o n Analytic Continuation

Let V(Af) denote t h e set of all subsets A of the set J\f = {1, 2 , . . . } of n a t u r a l numbers. Let XA denote t h e characteristic function of A G V{Af). Endow V(Af) with t h e topology determined by t h e metric 00

II

P(AB):=V2-' " y f^[

l

—XB||i/ II

+ \\XA

.. ,

A,BeV{M),

-XB\\V

where | | X A - X B | U :=ma,x{|x^(i)-Xs(i)|;i =

1,2,...,^}.

One can easily check t h a t V{J\f) is a complete metric space, and therefore it has Baire property. Moreover, in this topology a sequence {Aj } of elements of V(Af) converges t o A S V(J\f) if and only if for every integer kr, there exists an integer jo such t h a t Aj n [0, k0] = A n [0, k0] for all j > j 0 . P r o p o s i t i o n 3 . 1 Let {fk} be a sequence of holomorphic functions in an open subset fl of Cn. Then the following two conditions are equivalent (1) the series ^ll fk converges locally normally in £1, i.e. for every point a of tt there exists a neighborhood U of a such that the series ^ ^ ° ||/fc||u is convergent; (2) the series Y^

\fk\ converges locally uniformly

in f2.

265 Proof. It is clear t h a t (1) => (2). Suppose now (2) is true, and let E(a,r) :— {z e C n ; \ZJ — o,-| < r(j = 1 , . . . , n ) } be a polydisk whose closure is contained in £2. Then there is a positive constant M such t h a t Y^T \fk(z)\ < M for all z in t h e closure of E(a, r). By the Cauchy integral formula | / k ( * ) | < Wk = = ( — ) " / T1"7-

Jo

•••/

|/fc(o1+rcitl)...,an + re"-)|dt1...dtn,

JO

for all z e E(a, §) a n d A; > 1. By Lebesgue monotonous convergence theorem t h e series J21 /x^ is convergent and so is t h e series Y^T \\fk\\u with U = E(a, | ) . D L e m m a 3 . 2 ( s e e L e m m a 4 of 6 ) Let {fk} be a sequence of holomorphic functions in an open neighborhood J7 of the closure of a ball B = B(w,r) such that the series J ^ i ° fk is absolutely convergent at each point z of B, and its sum f := Y21 fk is holomorphic in B. Let a be a boundary point of B. Then either the series is normally convergent on a neighborhood of a, or there exists a subset M of 'P(N) of the first category such that for every A 6 'P(N) \ M the holomorphic function / A ( ^ ) : = 12k£A fk(z)> z E B, has a singular point at a (in other words, f^ cannot be continued analytically through a). Proof. Given a natural number m, let TLm denote t h e set of all A € V(N) such t h a t there exists a holomorphic function h on t h e polydisk Em := E(a, —) such t h a t |/i(^)| < m on t h e polydisk a n d h(z) = fA{z) for all z G B D Em. We claim t h a t t h e set TLm is closed in t h e space P ( N ) . Indeed, let {Aj} be a sequence of elements of 7Lm convergent t o A € 'P(N). Let {hj} b e a sequence of holomorphic functions on Em such t h a t |/xj(z)| < m on Em and hj(z) — / A (^) o n t h e intersection BC\Em. Observe t h a t for every k0 there exists jo such t h a t | / A , - ( Z ) - / A ( * ) | < Y,k>k0 \fa(z)\ f o r all z e B and for all j > jo- It follows t h a t t h e sequence {hj} is convergent at each point of BnEm. By Vitali theorem t h e sequence {hj} is locally uniformly convergent t o a holomorphic function h bounded by m and identical with / A on Em n B, which shows t h a t A € "R-mT h e r e are two cases. Case 1: There exists mo such t h a t TZmo has a nonempty interior. Then there exists a n a t u r a l number fco such t h a t

(*)

AeV{N),An[o,k0\

= A0n[o,k0\

=>

AeRmo.

In particular, if A is a finite subset of V(N) such t h a t An{0, ka\ = 0 then A U (AQ n [0,fco])G TZmo a n d t h e function

266 fAu(A0n[o,k0]) i s w e ^ defined and holomorphic o n B U Emo of generality we m a y assume t h a t Emo C C fl). Hence \JA(Z)\ <m0 + M0, for 8U

all finite A

£

V(N)

zeEmo

with A n [0, k0]

P«6B m o I EfcgA0n[o,fc0] /fcWI"

!t follows

J2 \fk(z)\<4(rn0

(Without loss

=

0,

where M 0

:=

that

+ M0),

zeEmo.

k>k0

By Hartogs L e m m a the series Y^y |/fe| i s locally uniformly convergent on Emo. By Proposition this implies t h a t the series J2^° /*. is normally convergent on a neighborhood of a. Case 2: For every m the set 7?™ has empty interior. Then the set M := U i ° ^ m is of t h e first category and (by t h e definition of TZm) for every A £ V(N)\M t h e function JA holomorphic in B h a s no analytic continuation through t h e point a.

• C o r o l l a r y 3 . 3 Let J ^ /fc be a series of holomorphic functions on an open subset fl o / C n convergent at each point z of CI. If its sum f := Y2i fk ls holomorphic on fl, then the following conditions are equivalent: 1°. Y^0 fk is locally normally 2°. Y2i \fk\ is locally uniformly 3°. Y2i \fk\{z)

convergent convergent

converges at each point z

in fl; in fl; offl.

C o r o l l a r y 3 . 4 Let {fk} be a sequence of holomorphic functions on an open set fl C C n . Let D denote the set of all points a in fl such that the series Y2i fk is normally convergent in a neighborhood of a. Assume that D C fl. Then there exists an infinite subset A of N such that the function fA '•= J2keAfk cannot be continued analytically through any boundary point of D. Proof. Let {vjj} be the sequence of all rational points of D (or any countable dense subset of D). Let aj be a point of 3 D such t h a t |aj— wA = dist(aj,dD). By L e m m a for every j there exists a subset Mj of 'P(N) of the first category such t h a t for every A E V(N) \ Mj t h e function / ^ h a s a singular point at aj. T h e set M : = UM,- is again of t h e first category. Therefore for every A 6 P ( N ) \ M t h e function JA has no analytic extension beyond D. •

267 3.1

Polynomial

classes

P r o p o s i t i o n 3 . 1 . 1 Let life be a linear subspace of the space Vk{Cn) of polynomials of n complex variables of degree at most k. Then the following conditions are equivalent: (1) The space life is 9-stable, i.e. peUk^~-eUk

dp

(j =

l,...,n);

(2) The space Hk is shift-invariant, i.e.

Peru,

a e c " =s>p(a + -) e nfe;

(3) There exists a set {a0 £ C n ; \f3\ < k} of ( n + fe ) points of Cn such that det[afl]| a |
£

^V{a\z\

a,zeCn,

PeUk.

\a\
Hence (1) =>• (2). T h e implication (2) =*> (3) is obvious. If (3) is satisfied then the system of linear equations p(ap-\- z) = 5Z|a|<jfc ~a\P a

solution of the form p^ \z)

c

a

iz)

nas a

unique

z

= Yl\e\ where c a / 3 are complex

constants. It follows t h a t (1) is satisfied.

•

By a similar method one can get the following P r o p o s i t i o n 3 . 1 . 2 Let life be a subspace o/T-^C™). Then the following ditions are equivalent. (a) p £ Ilk,

(b) P e nk, (j =

A £ C => p(A •) £ life

(homothety

invariance);

\j € c (i = o, i,..., k), \j ^\i{i¥= j) => P(AJ •) e nfc 0,...,k);

(c) p e rife =*• pj e nfc(j = o,...,fc), ™/iere £,(,?) = Xl|a|=j p "J0''z" j-th

con-

homogeneous

component

of p.

is

the

268 D e f i n i t i o n 3 . 1 . 3 A linear subspace II of T'(Cri) is called a polynomial class if it is invariant with respect to the group of affine automorphisms Ta^ of C " given by Ta,\

• C" 3 z ^ a + Xz e C",

It is clear t h a t Tajx °Ta\y

=

a £ C",

A 6 C \ {0}.

Ta+a>,\\'-

Examples 3.1.4 1. T h e linear space of all polynomials p 6 V(Cn) depending only on k variables z x , . . . , z& with 1 < k < n, is a polynomial class. 2. Let Qj(z) = Y2\a\=k- c<* zOC ( j = 1, • • •, s) be a system of s homogeneous polynomials of n complex variables with constant coefficients. T h e n the linear space of all polynomial solutions of the system of partial differential equations U:={peP(Cn);Qj(^)p(z)

(*)

= Q(j = l,...18)}

is a polynomial class. By specification of Q3•• (j = 1 , . . . , s) we can get here the class of all complex hai'monic polynomials, polyharmonic polynomials etc. Observe t h a t each harmonic polynomial of n real variables has a unique extension to complex harmonic polynomial on C " . In general, if a polynomial p of n real variables Xi,..., xn satisfies a system of constant coefficient partial differential equations Qj{g^)p{x) = 0 (j = 1 , . . . , s) on R", t h e n p has a unique extension t o a polynomial p of n complex variables satisfying t h e system of partial differential equations Qj(J^)p(z) = 0 (j = 1 , . . . , s) on C n . P r o p o s i t i o n 3.1.5 The tensor

product

II 1 ® n 2 := linear span of (p(z)q(w); of two polynomial and moreover

p € II 1 ,

q € IT2}

classes II 1 C V(Cm) and II 2 C V(
class,

n 1 ® n 2 = {P € v(cm x c n ) ; v z£C „ P(z, •) e n 2 , v,„ e c . P{-,w) e n 1 }. Proof. I t is sufficient t o show t h a t every polynomial p £ 'Pfc(Cm X C n ) such t h a t Vzecm p(z, •) € II 2 and V,„ e C " p( - , w) € fl 1 is a member of FI1 (g> fl 2 . P u t

269 m^

:= dimll[, mj:2) = dimIT|. Let {L^j;j

= l,...,m%'}

be a Lagrange

Interpolation Basis of 11]. with nodes {z? ,j = 1, .. ., rn£ }. Then ml(1) mi(2)

p(z,w) = £

^ p ^ z f

(z,w) E Cm x C",

)4V(z)LgV),

which gives the required result. 3.2

•

Functions holomorphic with respect to polynomial classes

Definition 3.2.1 Let II be a class of polynomials of n complex variables. A function / E 0(£2) holomorphic on an open set £2 in C n is called IIholomorphic (or holomorphic with respect to the class II), if for every a E £2 and for every j > 0 the j - t h homogeneous component

\a\=j

of the Taylor expansion of / around a is a member of the class II. Let 0(£2;II) denote the set of all II-holomorphic functions on £2. One can easily prove the following Proposition. Proposition 3.2.2 C(£2; II) is a d-stable Frechet space (with the topology of locally uniform convergence). A function / E 0{D) is Yl-holomorphic in a domain D iff there exists a E D such that fj(a; •) E II for every j > 0 . / / II is given by Example 2 of the former section, then 0(£2; n ) = {/ G O(fi); Qj(fz)f(z)

= 0on£2, j =

l,...,s}.

Corollary 3.2.3 A balanced domain D = {z E
^(z):=limsup^/|QJ(z)|,

z £ C

J-*oo

Proof. Let D be a domain of LT-holomorphy, so that there exists a IIholomorphic function / on D which has no analytic continuation through any boundary point of D. The function / can be written in the form oo

/V*) = £ Q , V 4

ZED,

270 where Qj is a homogeneous polynomial of t h e class II of degree j , and the series is locally normally convergent on D. P u t ip(z) := l i m s u p i/\Qj{z)\. We know t h a t {a 6 C"; ^ ^ ° Qj is locally normally convergent in a neighborhood ofa} = {ze Cn;iP*(z) < 1}. Therefore h = I/J* . Assume now h = tp*, where ip is given by ( + ) . Then D is a domain of local normal convergence of the series of homogeneous Il-polynomials J 2 0 Qj. By Corollary 3.4 there exists A e V(N) such t h a t JA is a ri-holomorphic function on D which cannot be continued analytically through any b o u n d a r y point of

D. 4

• Harmonically holomorphic functions

D e f i n i t i o n 4.1 Let $~2 be an open subset of C". space HO(to)

Elements of the Frechet

:= {/ e O(fi); A c / = ( g ^ ) 2 / + . . . + (^ff

are called functions harmonically

holomorphic

= 0}

on fl.

D e f i n i t i o n 4 . 2 T h e function

L(z) := ^Ikll2 + \ / R F ^ W . 2

2

2

where z := z + . . . + z , is called the Lie

* £ C",

norm.

2 7

It is known ' t h a t L is the maximal complexification of t h e Euclidean norm in R", i.e. L is the norm in C " , L(x) = \\x\\ for x G Rn, and iV(«) < L(z) in C n , if TV is any norm in C " such t h a t N(x) = ||x|| in Rn. T h e set Br := {z £ C n ; L(z) < r } is called Lie ball (classical E. C a r t a n domain of t h e type IV, 3 ) . T h e o r e m 4 . 3 If Q „ is an arbitrary polynomial (harmonic or not harmonic) ofn variables, then ||Qt/||B,- = || Qv Hg • Therefore, if a series of homogeneous polynomials Y^, Qv is convergent in a Euclidean ball Br in W1, it is convergent in the Lie ball Br. In particular, every function f harmonic in the ball Br is continuable to a harmonically holomorphic function in the Lie ball Br. C o r o l l a r y 4 . 4 If D is a domain in M.n (n > 2), and £>:= \J{ze

C"; L(z - a) < dist(a,

3D)},

aeD

then for every function f 6 H(-D) there exists a unique function such that F = f on D. In other words, the mapping HO{D)3F^FlDeH(D)

F 6

HO(D)

271 is an isomorphism

of the Frechet

spaces.

D e f i n i t i o n 4.5 We say t h a t a connected open set D C C " is a domain of harmonic holomorphy, if there exists a function / G HO(D) non-continuable holomorphically beyond D. Corollary 3.2.3 implies the following Theorem. T h e o r e m 4.6 Let D be a balanced domain in C n . Then D is a complex Laplace domain if and only if it is a domain of harmonic holomorphy. Every domain of harmonic holomorphy is a domain of holomorphy, but not v.v. Namely, by M. Kolaf 4 , there is no bounded domain of harmonic holomorphy with smooth b o u n d a r y in C n , n > 2. As a corollary (from the proof) one gets the following theorem. T h e o r e m 4 . 7 ( 4 ) A convex domain fl in Cn (n > 2) is a domain of harmonic holomorphy if and only if from the fact that v = v(b) is a normal vector to dD at a point b G dD it follows that v is a "null" vector, i.e. v2 := v\ + ...+v2 = 0. C o r o l l a r y 4.8 Let B be a convex balanced domain in C n . Then the conditions are equivalent. (1) B is a complex Laplace (2) B is a domain

following

domain;

of harmonic

holomorphy;

(3) If there exists a normal vector v to dB at a point b e dB, then v2 = 0. References 1. N. Aronszajn, M.C. T h o m a s , and J.L. Leonard. Polyharmonic functions. Clarendon Press, Oxford, 1983. 2. V. Avanissian. Cellule d'harmonicite et prolongement analytique complexe. Hermann, 1985. 3. L. K. Hua. Harmonic analysis of functions of several complex variables in the clasical domains. Amer. M a t h . S o c , Providence, 1963. 4. M. Kolaf. Cells of harmonicity. In Proceedings of the Winter School on Geometry and Physics, volume 26, pages 189-196. Rend. Circ. Mat. Palermo (2) (Suppl), 1991. 5. P. Lelong. Prolongement analytique et singularites complexes des fonctions harmoniques. Bull. Soc. Math. Belg., 7:10-23, 1954-1955.

272 6. J. Saint Raymond. Fonctions separement analytiques. Ann. inst. Fourier, 40(1):79-90, 1990. 7. J. Siciak. Holomorphic continuation of harmonic functions. Ann. Polon. Math., 29:67-73, 1974. 8. J. Siciak and S. Kolodziej. Laplace domains. In Tomasz Mazur, editor, Classical Analysis, Proceedings of the 6-th Symposium, 23-29 Sept 1991, pages 247-264, Poland, 1992. World Scientific.

O N T H E SINGULARITIES OF SOLUTIONS OF N O N L I N E A R PARTIAL D I F F E R E N T I A L EQUATIONS IN T H E COMPLEX D O M A I N H. TAHARA Department of Mathematics, Sophia University, Kioicho, Chiyoda-ku, Tokyo 102-8554, JAPAN E-mail: [email protected] This paper considers the following nonlinear first order partial differential equation du/dt = F{t, x, u, du/dx) in the complex domain. The main purpose is to examine whether or not the equation possesses solutions which admit singularities only on the hypersurface S = {t = 0}. This will be done either by examining the possibility of analytic continuation of solutions or by actually constructing solutions that possess singularities only on S.

1

Introduction

Let (t,x) = (t,xu...,xn) E C x C n , y € C, z = (zu...,zn) € C n , d/dx = {d/dxi,..., d/dxn), and let F(t, x, y, z) be a holomorphic function denned in a neighborhood of the origin of Ct x C™ x C„ x C™. In this paper we will consider the following nonlinear first order partial differential equation

a..)

£='(*•** !)•

where u = u(t, x) is the unknown function. (In this paper we will confine ourselves to the study of the first order equation (1.1); but it should be noted that most of the results in this paper can be extended also to the case of higher order equation.) It is well-known by Cauchy-Kowalevsky theorem that for any holomorphic function tp{x) in a neighborhood x — 0 the equation (1.1) has a unique holomorphic solution u(t, x) in a neighborhood of the origin (0,0) G C ( x C" satisfying u(0, x) —
273

274

Problem 1.1. Does (1.1) admit solutions which possess singularities only on the hypersurface S = {t = 0} ? One method of arguing the non-existence of such solutions is by means of analytic continuation. Let fl be an open neighborhood of the origin (0,0) € C t x Q , and let Q+ = {(t,x) £ Cl; Ret > 0}. If the equation (1.1) is linear, then Zerner's Theorem (1971) states that any solution which is holomorphic in Q+ can be analytically extended to some neighborhood of the origin (0,0). In other words, there does not exist a solution with singularities only on S. If the equation (1.1) is nonlinear, we have the following nonlinear analogue of Zerner's theorem due to Tsuno. Theorem 1.2 (Tsuno [10]). / / (1.1) has a solution u(t,x) which is holomorphic in Sl+, and if the solution satisfies u(t,x) = 0(1) (as t —> 0) uniformly in x in some neighborhood of x = 0, then the solution can be analytically continued up to a neighborhood of the origin. The assumption that u(t,x) be bounded in some neighborhood of the origin seemed too strong to other researchers at that time. Some might have believed that Zerner's result can be extended to the nonlinear case without any additional assumption. However, this is not possible if the equation is nonlinear, as can be seen in the following example: Example 1.3. Let (t,x) € C 2 . The equation flit (L2)

-di=U(dx)

/F)it \ m

with m e N * ( = {1,2,...})

has a family of solutions u(t,x) = (-l/m)1^m(x + c)/f 1/,m with an arbitrary c € C. Clearly, this is holomorphic in Q+ but has singularities on {t = 0}. Thus, in the case of equation (1.2) we see the following: (1) singularities on {t = 0} of order u(t, x) = 0(1) (as t —> 0) do not appear in the solutions of (1.2), but (2) there really appear singularities on {t = 0} of order u(t, x) = 0(\t\-l'm) (as t —> 0) in the solutions of (1.2). Hence, for nonlinear equations, it seems better to reformulate our problem into the following form: Problem 1.4. LetCTbe a real number. Does (1.1) admit solutions which possess singularities on S = {t = 0} with growth order 0(|i| CT ) (as t—•())? If a is a non-negative real number, by Tsuno's result we conclude that such singularities do not appear in the solutions of (1.1). Therefore we may

275

assume from now that a is a negative real number. Then, in general the solution may tend to oo (as t —> 0) and so we need to suppose: (A) F(t,x,y,z) 2

is a holomorphic function on il x C y x C™.

Non-existence of singularities

Recently Kobayashi [6] gave a precise result on the non-existence part of the problem 1.4. In this section we will explain Kobayashi's result and then we will give its improvement. Suppose the condition (A). We may expand the function F(t,x,y,z) into the Taylor series with respect to (y, z): F(t,x,y,z)=

^2

ajt0l{t,x)y:>

za

Cj»eNxN"

where a = (a\,... •ya

Z

— r — Zl

a i

, a „ ) £ N", a,jyOI(t,x) are holomorphic functions on fi, and

. . . Z 7an n •

Set A = {(j, a) e N x N"; Oj,a(*, x) ^ 0} and A 2 = {(J, a) € A ; j + \a\ > 2} (where \a\ = a i H h a„). We remark that the equation (1.1) is linear if and only if A 2 = 0; it is nonlinear otherwise. Since we already have Zerner's result for the linear case, we will assume henceforth that (1.1) is nonlinear, that is, A 2 is non-empty. In the following, we will write the coefficients as ajta(t,x)

= tkj'a bjta(t,x)

for (j, a) € A,

where kj>a is a non-negative integer and bjtCl(0,x) r£ 0. Using the above, the equation (1.1) may now be written as

(j,«)eA a

where (du/dx) = {du/dx^ • • • (du/dxn)a». Then the Kobayashi's index c^ is given by (2.2)

/ — k- — 1 ok = (j,a)6A sup '\J + M ^ 2

Note that ok is a non-positive real number and that it is calculated only by looking at the form of the equation. For a neighborhood w of x = 0 £ C™ and a function f(t,x) we define the norm ||/(t)|| w = sapxeu>\u(t,x)\. The following is the result due to Kobayashi. Theorem 2.1 (Kobayashi [6]). Suppose the condition (A) and that A2 is non-empty. If a holomorphic solution u(t,x) of (2.1) defined in fl+

276

satisfies ||u(t)|| w = 0(|i| CT ) (as t —> 0) for some neighborhood UJ of x = 0 and some real number a > ok> then the solution u(t,x) can be extended analytically as a holomorphic solution of (2.1) up to a neighborhood of the origin. More precisely we can prove the following: T h e o r e m 2.2 ([7]). Suppose the condition (A) and that A2 is nonempty. If a holomorphic solution u(t,x) of (2.1) in fi+ satisfies \\u(t)\\w = 0(1*1°") (as t —> 0), then u(t,x) can be extended analytically up to a neighborhood of the origin. Hence we can get the following result on the non-existence of the singularities on S = {t = 0}. Corollary 2.3. Suppose the condition (A) and that A2 is non-empty. Let ok be the real number given in (2.2). Then, there appear no singularities on S with growth order o(\t\^) (as t —> 0) in the solutions of (2.1). In the equation (1.2) the number o\ may be verified to be equal to - 1 / m . Hence, by the above result we see that the singularities of order o ( | i | _ 1 / m ) do not appear in the solutions of (1.2). Note further that the singularities of the solution u(t,x) - (-llm)l/m(x + c)/tl/m has growth order 0 ( | * | _ 1 / m ) (as t —> 0). Thus in the case (1.2) the number o^ = —1/m is just the critical value. 3

Singularities with growth order Odil01*)

In the previous section, we have shown that there appear no singularities on S = {t = 0} with growth order o(\t\^) (as t —• 0) in the solutions of (2.1). But how about the singularities with growth order OGtl0*) (as t —> 0)? In this section, we will study singular solutions with growth order Od^ 0 *) on the hypersurface S. Set (3-1)

M = \(j,a) e A 2 ; ^ " \ =<*}. { j + |a|-l J If M = 0, we have the following result on the singularities with growth order 0(|t|*). T h e o r e m 3.1 ([7]). Suppose the condition (A) and that A2 is nonempty. If Ai = % and if a holomorphic solution u(t,x) of (2.1) in Q.+ satisfies \\u(t)\\u — 0(1^°*) (as t —> 0), then u(t,x) can be extended analytically up to a neighborhood of the origin.

277

This implies that in the case M. = 0 there appear no singularities on S with growth order 0(|*|° k ) (as t —• 0) in the solutions of (2.1). The following equation gives an example with M = 0: let (t, x) G C 2 and consider the first-order nonlinear equation du/dt = eu (du/dx). In this case, it is easily checked that ok = 0 and M. = 0. Therefore by theorem 3.1 we see that this equation has no singular solutions with growth order 0(1) (as t —> 0), which is just the same result as in Tsuno's theorem. Now, suppose the condition M ^ 0 and set

(3.2)

P{x,y,z)=

J2

h<*(0,*)yjza-

(j,a)eM

It is easy to see that P(x,y,z) ^ 0 and that P(x,y,z) is a holomorphic function on {x G C;(0,x) G tt} x Cy x C". Since M ^ 0, we have ok = {—kj,a - l ) / 0 + \a\ _ 1) for any (j, a) G M: this implies that ok is a negative rational number and so ok G Z/L for some L G N*(= {1,2,...}). We remark also that kj
u{t,x)=t'*{(p(x)+w(t,x)),

where 0; c 2 ) there is an a > 0 such that for any 9 > 0 we have s u p ^ ^ ^ \w(t,x)\ = 0 ( | i | a ) (as t —> 0 under | a r g i | < 6). Here H(Ct \ {0}) denotes the universal covering space of Ct \ {0}. Since crK < 0, ip(x) ^ 0 and w(t,x) —> 0 (as this function (3.3) has really singularities of order we can construct such a solution as in (3.3), we can of order Itl* on {t = 0} appear in the solutions of

t —> 0), we easily see that \t\°* on {t = 0}. Hence, if conclude that singularities (2.1).

278 Substituting this to (2.1), we get ta«-1(oK
(i,a)6i

and by canceling the factor i 011-1 we have (3.4)

d ok¥>+(t^+okjiw „/

dw

dw\

(j,a)eA\X

where e,-,a(£, z) = (bj:0l(t,x) — 6j, a (0,x))/i. Since we are now considering a w(t,x) € 0+, we have «;(£,x) = o(l) (as t —> 0) and so by letting t —> 0 in the above equation we have

(!)

P X

ok

dtp N

{ >V,-Q^)-

dx. Then by subtracting subtracti the equation (I) from (3.4) we obtain

(II)

8

(*£+*)•w dP Of (/ =

dw\ Ow\

x

w

^\ ^Tx) (

+

dip

<^dP( ^ar/ +

dtp\ cHp\ dw aw x

^d^\ ^-d-x)-dx-i dw\

Y: ^- +i+ *« + '- | - i) 6i,.(*,*)( V +^(g+g) a . (j,a)eA\M

Here, Gi is the remainder term of the Taylor expansion of P with respect to (w, dw/dx). To summarize our goal, we have the following proposition:

279

Proposition 3.3. / / the equation (I) has a holomorphic solution tp(x) which is not identically zero and the equation (II) has a solution w(t,x) G 0+, then we have succeeded in constructing a solution u(t,x) of (2.1) with singularities of order 1*1°" on S = {t — 0}. Thus, to prove the existence of singularities of order |£|0k on S = {t = 0}, it is sufficient to study about when Proposition 3.3 is valid. 4

Existence of singularities

First let us recall some terminologies on vector fields on C". Let n

r,

2= 1

be a holomorphic vector field with holomorphic coefficients a, (a;) (i = 1 , . . . , n) in a neighborhood of x = 0. i) We write X = 0 if a,i(x) = 0 for all i = 1 , . . . , n. ii) We say that X is non-singular at x = 0 if a;(0) ^ 0 for some i. In this case, it is known that by a suitable holomorphic change of variables x = (xi,...,xn) —• y = (j/i,..., j/ n ) the vector field X is transformed into the form d/dy\. iii) We say that X is singular at x = 0 if a;(0) = 0 for alH = 1 , . . . , n. Then ai(x) is written in the form a,i(x) = J21=iai,jxj + ^ ( M 2 ) - Denote by A the n x n matrix whose (i,j) component is a,j, and by A i , . . . , A„ the eigenvalues of the matrix A. We say that X satisfies the Poincare condition if the following condition holds: (4.1)

liminf m€N" Iml—•oo

Aimi H mi

1- A„m„

+ • • • + m„

> 0

where m = ( m i , . . . , m n ) and \m\ = mi + h mn. It is known that (4.1) is equivalent to the following: the convex hull of the set {X±,..., A n } in C does not contain the origin ofC Now, suppose the conditions (A), A 2 ^ 0, and M. ^ 0. In this section we will give three types of sufficient conditions for the existence of singularities on S with growth order j 11""" . For a function tp(x) we define a vector field X(
^dP( 1=1

dy\

d

280

We have the following result. Theorem 4.1. If the condition (Ci), (C2), or (C3) stated below is satisfied, then the equation (2.1) has really singularities of order ^1°* only on S = {t = 0}. (Ci) There is a holomorphic function ip(x) in a neighborhood of x = 0 € C n which satisfies the following conditions: 1)

(1) The condition 3) of (Ci) means that dP 1 ~dz V'*'

dip\ d~x) ^ °

f

°r

aU % =

1,,

--'n-

(2) The condition 3) of (C2) means that ^(0,^(0),^(0))^0

for some i.

(3) The condition 3) of (C3) means that U (0,
for all i = 1 , . . . , n .

(4) Let A i , . . . , Ara be the eigenvalues of the matrix whose (i, j)-component is given by

'M*^»)

d_ (dP, oxj

x

„~t -

-7

x=0

Then the condition 4) of (C 3 ) is equivalent to the condition that the convex hull of the set {1, —Ai,..., — A m } in C does not contain the origin of C .

281 Sketch of the proof of Theorem 4-1- Suppose the condition (Ci), (C2), or (C3). Then the equation (I) has a holomorphic solution ip(x) satisfying
(««+*).

dP t dtp\ ^^X^dx-)w

+

A0P/ d
„ (

dip

dw\

where R{t1/L,x,y,z) is a holomorphic function in the variables (tl/L,x,y,z). l L Let us find a solution w € 0+ in the form w{t,x) = W{t / ,x) for some W(t, x) € 0+. Then our equation (4.2) is reduced to the equation with respect to W(t,x): dp dy{X^di)W

+

^dP, T;aii{X^dx-)dx1=1

+«

£ *.<«..) <•> + » ? ( £ + £ ) "

(j,a)€M

+ tR(t^ +

w,df^df-).

Note that the right hand side of (4.3) is a holomorphic function with respect to (t,x,W,dW/dx). In the case (Ci) the equation (4.3) is nothing but a Briot-Bouquet type partial differential equation in Gerard-Tahara [3]; therefore by applying the results in [3] and Yamazawa [11] we obtain a solution W(t,x) 6 0+. In the case (C2) the equation (4.3) has a normal form with respect to xf, by applying the Cauchy-Kowalevsky theorem we obtain a holomorphic solution W(t,x) satisfying W(0,x) = 0. In the case (C 3 ) the equation (4.3) is nothing but a totally characteristic type partial differential equation in Chen-Tahara

282

[1]; then by applying the results in [1] and Tahara [9] we obtain a solution W(t,x)eO+. Thus, by Proposition 3.3 we have the conclusion of Theorem 4.1. • Remark 4.3. In the case where the vector field td/dt — X{ip) does not satisfy the Poincare condition, Chen-Luo-Tahara [2] discussed a Maillet type result; but still it is not applicable to the problem discussed in this paper. Example 4.4. Let (t,x) G Ct x Cx. (1) Let us consider du

9

. s/du\2

,, ,

In this case we have aK = - 1 , P = y2 + a(x)z2, and dP/dz = 2a(x)z. By taking if = — 1 we can see that the equation satisfies the condition (Ci). (2) Let us consider du _ fdu\m u ~dt= \dx~) ' where m G N*. In this case we have <% = —1/m, P = yz™, and dP/dz = myzm~l. By taking if = (—\/m)1/m(x + c) with c / O w e can see that the equation satisfies the condition (C2). If we take if = (—l/m) 1/,m x we have X(ip) = —xd/dx and we see that the equation satisfies the condition (C3) (3) Let us consider du

(du\

n/du\3

. .

In this case we have <% = — 1, P = yz, and dP/dz = y. By taking ip = —x + c with c ^ O w e can see that the equation satisfies the condition (C 2 ). If we take ip = -x we have X(ip) = -xd/dx and we see that the equation satisfies the condition (C3) References 1. H. Chen and H. Tahara, On the totally characteristic type non-linear partial differential equations in the complex domain, Publ. RIMS, Kyoto Univ., 35 (1999), 621-636. 2. H. Chen, Z. Luo and H. Tahara, Formal solutions of nonlinear first order totally characteristic type PDE with irregular singularities, Ann. Inst. Fourier, Grenoble, 51 (2001), 1599-1620.

283

3. R. Gerard and H. Tahara, Holomorphic and singular solutions of nonlinear singular first order partial differential equations, Publ. RIMS, Kyoto Univ., 26 (1990), 979-1000. 4. R. Gerard and H. Tahara, Solutions holomorphes et singulieres d'equations aux derivees partielles singulieres non lineaires, Publ. RIMS, Kyoto Univ., 29 (1993), 121-151. 5. R. Gerard and H. Tahara, Singular nonlinear partial differential equations, Aspects of Mathematics, E 28, Vieweg-Verlag, 1996. 6. T. Kobayashi, Singular solutions and prolongation of holomorphic solutions to nonlinear differential equations, Publ. RIMS. Kyoto Univ., 34 (1998), 43-63. 7. J.E.C. Lope and H. Tahara, On the analytic continuation of solutions to nonlinear partial differential equations, to appear in J. Math. Pure et Appl. 8. A. Shirai, Convergence of formal solutions for singular first order nonlinear partial differential equations of totally characteristic type, to appear in Funkcialaj Ekvacioj. 9. H. Tahara, Solvability of nonlinear totally characteristic type partial differential equations when resonances occur, preprint. 10. Y. Tsuno, On the prolongation of local holomorphic solutions of nonlinear partial differential equations, J. Math. Soc. Japan, 27 (1975), 454-466. 11. H. Yamazawa, Singular solutions of the Briot-Bouquet type partial differential equations, to appear in J. Math. Soc. Japan.

EXPONENTIAL POLYNOMIALS A N D THE FOURIER-BOREL T R A N S F O R M S OF A L G E B R A I C LOCAL C O H O M O L O G Y CLASSES

S. TAJIMA Dept. oj Information Engineering, Faculty of Engineering, Niigata University 8050 Ikarashi Niigata, Niigata 950-2181 JAPAN E-mail: [email protected]

1

Introduction

Let M be a holonomic system (i.e., maximally overdetermined system) of linear partial differential equations of the form Pi(dx)u(x)

= P2(dx)u{x)

= ••• = Pt{dx)u(x)

= 0,

where Pi(dx),... , Pi(dx) are linear partial differential operators with constant coefficients. As in the case of ordinary differential equations, the homogeneous solution space is a finite dimensional vector space consisting of exponential polynomial solutions, which are the Fourier-Borel transforms of some analytic functionals supported on a zero-dimensional variety. T h e variety which carries all exponential factors is determined as the common zeros of total symbols to the given holonomic system M. T h u s we have such a lot of information concerning exponential polynomial solutions. Nevertheless there still remains some difficulty in computing explicit forms of exponential polynomial solutions. T h e cause arises from the multiplicity and the problem of determination of Noetherian differential operators involved in Fourier-Borel representation formulas 4 ' 1 4 . In this paper, we study Fourier-Borel integral representation formulas of exponential polynomial solutions to the holonomic system M from an algorithmic point of view. We present a new m e t h o d for treating exponential polynomials and give an algorithm t h a t computes explicit form of exponential polynomial solutions. T h e main ingredient in the present approach is the notion of the Noetherian differential operator representation of an algebraic local cohomology class. In section 2, we consider zero-dimensional algebraic local cohomology classes as analytic functionals 7 , 1 0 and apply the Grothendieck residue theory to define the Fourier-Borel transform of an algebraic local cohomology class. In section 3, we introduce the notion of Noetherian differential operator representation of an algebraic local cohomology class and we present

284

285 their properties. In section 4, we give an algorithm t h a t computes explicitly the exponential polynomial solutions to the constant coefficient holonomic system of linear partial differential equations. In section 5, as an application of the present approach, we consider the complex Cauchy problem for the constant coefficients holonomic system M. We show t h a t the complex Cauchy problem is equivalent to the determination of the dual basis with respect to the global residue pairing of the zero-dimensional Grothendieck duality for the corresponding polynomial system. 2

T h e Fourier-Borel transform of an algebraic local c o h o m o l o g y class

Let X be the complex Euclidean space C* of dimension n with Cartesian coordinates x = (xi,... ,xn), Z the dual space with coordinates £ = (Ci, • • • , Cn)We shall write (-x for Ci^i+C2*2-I hCn^n- Let i f [ g f ^ , . . . , £-] be the ring of linear partial differential operators with constant coefficients, where i f is a field with characteristic zero (e.g., K = C or "practically"if = Q ) . Let i" be the ideal in K[-^-,... , ^ - ] generated by linear partial differential operators Pl(dm), P2(d.), '\ , P / ( 0 . ) " e K[&, . . . , £ ] where fl. = ( £ , . . . , £ ) , I * the ideal in the polynomial ring i f [ £ i , . . . ,£„] generated by pi(Q, • • • ,Pi{(), where Pj(C) stands for the total symbol of Pj(dx) (j = 1, 2 , . . . ,£). Let A denote the affine algebraic variety A = {C G Z | p(C) = 0

for

v

p G

Iz}

of common zeros of polynomials in Iz- To the system M — i f [ g f - , • • • , g f - ] / J of linear partial differential equations with constant coefficients we define the attached left D x - m o d u l e M. to be M. = Vx ®irr,£_»_i M, where X>x is the sheaf of linear partial differential operators with holomorphic coefficients. T h e system M is said to be holonomic if a n d only if the left T>x -module M. is holonomic in the sense of S K K 1 2 . L e m m a 2.1 A system M = K[gf-, • • • , g§~]/-^ °f linear partial differential equations with constant coefficients is holonomic if and only if A is a zero dimensional variety. We assume hereafter t h a t the given system M is holonomic a n d we study the exponential polynomial solutions to M. We p u t

ExpA(X) :={^2 M*)*Xx

I **(*) G C[*i. • • • , *»]}•

286 Since the variety A carries all exponential factors to the exponential polynomial solutions of the system M, the homogeneous solution space S = HomjrbS_i...iBfc](M,0x) becomes a subspace of E x p ^ ( X ) . Let Oz be the sheaf on Z of holomorphic functions. T(Z,7i?AJOz))

Let HV^AOz)

=

be the algebraic local cohomology group supported on the

zero dimensional variety A where 7ifA](0z)

is the sheaf of algebraic local

cohomologies 5 . For an algebraic local cohomology class tp G 11^,(0z), Res^e'-^CK) = 2

we set

ResA(e<*V«K),

where d£ = d£i A • • • A d£n, Res^(*d£) is the Grothendieck local residue at A of the cohomology class *. It is easy to see t h a t KesA(e^'xrf>(Qd(^) is an exponential polynomial in x belonging t o ExpA(X). We define the FourierBorel transform FB(V0 of the algebraic local cohomology class if> 6 HV^JOz) to be the Grothendieck residue FB(V>)(s) = Resa(e<-V(C)dC)

ofe<-"V>(C)T h e Fourier-Borel transformation FB defines an isomorphism between following properties.

:HJlA]{
and E x p j l ( X ) , which enjoys the

P r o p o s i t i o n 2.1 Let P(dx) be a linear partial differential constant coefficients, and p(£) the total symbol of P(dx). $ ^ H\A](®z)> the following formula holds. P(dx)FB(r(>)(x)

=

operator with Then for any

FB(p(CmC))(x).

Set MZ=K{(1,...,CnVh and E = H o m * [ C l i . . . X n ] ( M z , H £ A ] { O z ) ) . have S = {r/> G H?AOz) | W = 0, Vp G Iz}We have the following fundamental result.

We

287 T h e o r e m 2.1 Let M : Pi{dx)u{x)

= ••• = Pt(dx)u(x)

= 0

be a holonomic system of linear partial differential equations coefficients. Let S = {u(x) | P\(dx)u{x) = ••• = Pi(dx)u(x) vector space consisting of the exponential polynomial solutions M. Let

with constant = 0} be the of the system

S = W € Hfc{Oz) I Pi (CM = • • • = Pt(04> = 0}, where Pj(() is the total symbol of Pj{dx), j — 1,2,... Borel transformation F B induces an isomorphism

,i.

Then the

Fourier-

FB : E -»• 5 between S and 5 . 3

N o e t h e r i a n differential o p e r a t o r s

In this section, we introduce the notion of the Noetherian differential operator representation to an algebraic local cohomology class. We show t h a t the computation of an explicit form of polynomial solution a m o u n t s t o the computation of Noetherian differential operators for the algebraic local cohomology class in E. Let Iz = Iz,i f~)Iz,2 H- • -nlz,t be the primary decomposition of the zerodimensional ideal Iz C i f [ & , . . . , f n ] . Let Ai denote the zero dimensional variety V(Iz,i) of common zeros of polynomials in the ideal Iz,%, i.e., At={CGZ\

p(C) = 0, Vp G !*,<}.

We put S i = {rf> 6 S | supp(V>) C At}. We have the following. L e m m a 3.1 (1) The vector space £ has the following

(2) d i m c S i = d i m * i f [ C i , . . • , Cn]/Iz,i-

direct sum

decomposition

288

Let {fl^i,... , <7»,n} be a set of generators of the radical y/lz,ia fundamental class attached to the variety Ai to be

det e Ai

( fc:;:::;f:"))

We define

9i,l(0 • • • 9i,n(C)

£H?Ai](Gz).

Let 8p be the delta function [l/(Ci - /?i) •••{Cn — Ai)] with support at the point j3 = (fli,... ,/9 n )- We have the following lemma : Lemma 3.2 The algebraic local cohomology class Ai has the following properties. (i) Ai e (2) Ai =

E,,

ZpeAth-

To proceed further, we exploit the following property : HF.JOz) is endowed with the structure of Dz module 6 , where Dz is the Weyl algebra K[C1,...,Cn]{dCl,...,d
MC) - *a(C) S VlzTi, for Proof. The annihilator ideal Annz>z(Ai) class A{ is given by AnnDz{Ai)

=

al1

«•

of the algebraic local cohomology Dz^flz^i

where Dz y/Iz,i is the left ideal in Dz generated by \/Iz,iyields the conclusion. •

This immediately

The above lemma implies that to represent an algebraic local cohomology class in HJ^JGz) in terms of differential operators, it it sufficient to consider a linear partial differential operator of the form

n-dd = E(-5<)aMO a

where the coefficients are assumed to be taken from l?[Ci,... ,

Cn]/\/lzti-

289 The algebraic local cohomology class T(—d()Ai in H?A40z) following.

enjoys the

Lemma 3.4 We have the followings : (1) T(-dt)Ai

Ea(-d<)ahaW)Sp,

= ZP€Ai

(2) FBiTi-d^A^x)

= ZpeAi

J2a xaha(J3)ef>-'.

Proposition 3.1 LetT(—d() = ^2a(—d()aha(^) be a linear partial differential operator with ha(C) G K[£i,--- ,Cn]/\/Iz,iThen the cohomology class T{—d^)Ai belongs to 2 ; if and only ifT(—d() satisfies the following condition P(C)T{~d()

e Dgy/I^u

V

P(C)

G iz
Proof. By the definition of Ej, the necessary and sufficient condition for the cohomology class T(-d()At to be in Ei is p(c)T(-dc)Ai = 0, Vp € Iz,i- Since Annoz(Ai) = T>z-\/Iz,i, the above condition is equivalent to p(()T(—d{) E ^Zy/Iz,i, Vp G Iz,i, which yields the proof2. O We set

TAi = {T(-d() = Ea(-d()aha{0 I ha(0

e K[Ci,...

, C n ] / v ^ , p ( C ) T ( - 5 c ) G Dzy/Iz~i

Then we have E,- = C ®K {T(-d()Ai Lemma 3.5 If T ( - d c ) K[Cu...,(n]/Ii.

G TAi,

| T{-dc)

for

v

p G I*,,}.

G TAi}.

then T(-d()h

£

TA. for any h G

Let fa be the multiplicity of the variety vl,-, i.e., dim{K[Q/Iz,i)/dim(JC[C]/\/tz~i)Then we have the foUowing.

p,

=

Theorem 3.1 There exist m linear partial differential operators Titi, Ti>tii in TAi such that, for any T £ TAi, there exist fei,...,/iMi K[
..., G

The set of operators NTi = { T ^ i , . . . ,Tiifii} satisfying the above condition is called the Noetherian differential operators attached to the variety Aj. The pair (Ai,NT{) corresponds to the notion of multiplicity variety 4 ' 14 . The above theorem combined with Proposition 3.1 provides an effective method for computing the Noetherian differential operators.

290. Theorem 3.2 Let Iz = Iz,\ H • • • fl Iz,t be the primary decomposition of the zero dimensional ideal Iz- Let { T ; ^ , . . . ,Tit/ii} be Noetherian differential operators attached to the variety Ai = V{Iz,i), i — 1 , . . . ,s where Tij = i2a(-d()ahij,a(0 with hiJ
, -z—]//

OX\

is given by the following.

OXn

S = S p a n c { ^ tij(X,x)eik(X)eXx where Uj(C> x) — J2a hi,j,a{C)xa mials of K[Ci,... , Cn]/\flz~-

| i = 1 , . . . , s, j = 1 , . . . ,m, k = 1 , . . . ,d{} an

d {e»i(C)i • • • > e«
Example 1 Let us compute exponential polynomial solutions of the following PDE's : d4

fe

+2

d2

d4

4 2 + d + Z 8dy

^W W d2

^ ~ W)u(x'y) = °'

Put Iz = <£4 + 2 £ V + T,4 + 3£T7 - T,4, £2 +

2 V

- 1) C Qfc, rj\. The primary

decomposition of the ideal Iz is Iz = Iz,i H Iz,2 where Iz,i = {i\ V - 1), Iz,3 = (4£2 - 4 7 , - 5 , 4 r , 2 + 4T? + 1) with y/lz^i = (£, T) - 1), y/lz~^ = (4£2 - 3, 2T7 + 1). The fundamental cohomology classes A\ and A% are given by

«fo-l)

,4a

16£ (4^_3)(2r?+l)J

Ti ( i = 1 and Ti )2 = (—d{) are the Noetherian operators for A\. T2 \ — 1 and ^2,2 = {—&{) + (—di?)2£ are the Noetherian operators for A%. Thus we have Au

Tli2A!

£ Ex and Zl2l £A2, T 2 , 2 ^ 2 ) (T 2 , 2 £)4 2 G E 2 .

We have F B ( ^ i ) = e», FB(Ti i 2 ^!) = ase", F B ( ^ 2 ) = e - # * - 5 » , F B ( ^ 2 ) = e ^ — i * . FB(T 2 , 2 Z\ 2 ) = (x - V 3 » ) e - ^ — * * , F B ( T 2 , 2 ^ 2 ) = (* - V 3 y ) e ^ " " * » . The space 5 of exponential polynomial solutions is _^I,_i. S = Span{e 9 , a;es, e ^3 " - f y *, (x-yfty)e-

- ^ S - I

« ""**, (as-V3y)e V — *»}

291 4

Algorithm

We present an outline of the algorithm that computes the exponential polynomial solutions to the holonomic system Pi{dx)u{x)

= ••• = Pt{dx)u{x)

=0

of linear partial differential operators with constant coefficients. We assume that the coefficients of Pj(dx) are taken form K = Q. Input : p i , . . . ,pt e Q[Ci, • • • , Cn] • Compute the primary decomposition of the ideal Iz

iz = h,i n iz,2 n • • • n iz,. • for i = 1 to s — Compute the Grobner basis Gr/#,» of the ideal Izi and the Grobner basis G r y ^ j of the radical ideal ^Iz,i — Compute a monomial basis Mb(Iz,i) of Q [ f t , . . . , (n]/Iz,i monomial basis Mb(y/Iz,i) of Q[&, ••• , Cn]/\/Iz,i

and the

— Compute the Noetherian differential operators NTi = {Titi,... ,Titll.} attached to the variety Ai by solving v p{QT(-d() 6 Dzy/I^i, p G Gilz,i Output •

NTi={Ti>l,...,Tii/ii},i=l,...,s

• eik

: basis monomial of Q[C]/\/^z,»> i = 1, •••,*, 1 < k < dt

~

dim
\ j - 1, 2 , . . . , /i<; k = 1 , . . . , di}

• S = STpsmc{FB{TitjeikAi){x)

\ i = 1,... ,s; j = 1 , . . . ,m;k = 1 , . . .di}.

More precise form of the algorithm which involves computations of the Noetherian differential operators will be described in elsewhere. We give an example of computations for illustration.

292 E x a m p l e 2 Let

1

jP_ d* 3 6 ~~ ~fa* ~ dydx3 6

+

6

d* & 2 2 dy dx

d „ d „ dxe+3dy2dxi+3dyidx2

"

[19

d * dy2dx2

_

ls

d* 6 3 ~~ dy dx

6

6

d

+

d dye

+

d^_ d2 3 dy~* ~ 7~?te2 +

d* * dx*

,„

U

d2 d2 dydx ~ 7~dy2'

d* dydx*

1&

d * _ 4 _#!_ + 23 J?!_2 _ op ^ + 23— 2 - 9. dysdx dy* dx dydx dy

Let Iz = (pi(£,'?),P2(£,»7)) be the corresponding ideal in Q[£,rj\, where Pi (£,*?) a n d P2{Z,v) a r e the total symbols of the operators Pi and Pi respectively. T h e Grobner basis of the ideal Iz is given by {34012224J718 - 337287888?716 + 1482815889»714 - 3793377501»712 + 6223678060??10 - 6791668554»78 + 4929982554»76 - 2295567292r?4 + 622217709»72 - 74805201, (5043831461461474902»72 - 5043831461461474902)^ - 7044046602560066048817024r?17 + 63018276687909803988231264»715 - 245940693464792349823250142»71S + 546927433129698375566380590?7n - 758080425045054486791028899»79 + 670687249976034002805020037??7 - 369893632035381899639182908r75 + 116278640424422398146461645r73 - 15952803070275778203814563f7, 3207431268127889£2 - 6414862536255778^ + 31360879942132329929664»716 - 280515151200743841705264?714 + 1094583842884686142708623»712 - 2433790380590803036761147f710 + 3372954459921918659824641r78 - 2983766273501802323378481»76 + 1645431760447548675949674»74 - 517214672355887172881047»72 + 70955537660381834441226}. Iz = Iz,l^Iz,2^Iz,zC\Izti /z,i

-

is the primary decomposition of the ideal Iz where

(-64T74 + 1 4 4 7 7 2 - 8 1 , 9 | + 8 7 7 3 - 1 8 7 7 ) , / Z ) 2 = (T,3-3772 + 3 7 7 - 1 , 7 7 ^ - ^ -

77 + 1, i2 - 2£ + 772 - 277 + 2), Iz,z

= (-53144177 8 + 2480058?76 - 4294539?74 +

3269322T72-923521,59582£+19683777-91854T75 + 185004773-18162977),J^4 =

{ 7 7 3 + 3 T 7 2 + 3 7 7 + 1 , 77^+^+77+1, £ 2 + 2 £ + 7 7 2 + 2 T 7 + 2 ) , with y ^ = {St?-*, t~v), y/I^ = (77 - 1,£ - 1), y/Tz~z = (729T7 4 - I7OI772 + 961, 31£ + 27r73 - 637?), \JIz,A = (»7 + 1,£ + 1}- T h e Noetherian differential operators JVTj attached to the variety Ai are given by the followings.

NTX = {1, (-de) ~ (-dv)},

NT2 = {1, (-de), (-dv),

(-dtf

NTZ = { 1 , (-fl £ )31 + ( - d „ ) ( - 6 3 + 27T, 2 )},

NT, = {l, (-de), (-dn), (-de f - (-a,) 2 }.

-

(-dv)2},

293 5

Cauchy problem

Let Bi(dx),...

, Br(dx)

£ K[-—,...

, ——] be linear partial differential oper-

OX\

OXn

ators with constant coefficients a n d let 61 ( £ ) , . . . , 6 r (C) €. K[d,... , £„] their total symbols. In this section, we consider the following Cauchy problem for the holonomic system f P!(dx)u(x) = ••• = Pt{dm)u(x) = 0, 1 Bj(dx)u(x)\x=o = CJ, j = 1 , 2 , . . . , r. P r o p o s i t i o n 5.1 Let r G S . Then an exponential polynomial solution to the Cauchy problem ( C P ) if and only if

F B ( T ) ( : E ) is a

Res 4 (6 j (C)r(C)dC) = c i f j = 1, 2 , . . . , r. Proof. Since r G S , we have F B ( T ) G 5 . Furthermore, 5i(a!r)(FB(r))(0) = FB(6,(C)r(CK)(0) = Res i l (6 J (OT(C)e- < «iC)U=o, which is equal t o Res v i(6 J (^)r(C)
•

We say t h a t the Cauchy problem ( C P ) is well-posed, if for any Cj G C, j = 1 , 2 , . . . , r , there exists a unique solution t o the Cauchy problem ( C P ) . Let [6j] G Q[Cij • • • )Cn]/-fz denote the modulo class bj m o d Iz where Iz = {pi,... ,pt). L e m m a 5.1 The following

conditions

(1) The Cauchy problem ( C P ) is

are equivalent

:

well-posed.

(2) { [ 6 1 ] , . . . , [br]} is a basis of the vector space K[C\,...

, Cn]/Iz-

Let 6 1 , . . . , bT be r polynomials. Assume t h a t { [ 6 1 ] , . . . , [K]} constitutes a basis of Mz = K[Ci,... , Cn]/Iz- Let < x i , . . . ,
ReMMCto(CK) = *i.! < *'. i < r holds.

294 T h e o r e m 5.1 Let Bi(dx),...

, Br(dx)

G K[-—,... , - — ] be linear partial axi oxn differential operators. Assume that the modulo class { [ 6 1 ] , . . . , [6T]} form a basis of the vector space Mz = -K"[Ci,--- ,zn]/IzLet {o-\,... ,cr T } be the dual basis of {bi,... ,bT}. Let
= R«4*i(C)I>
=

YtckResA(bj(C)o-k(<;)d()

k=i = Cj.

D T h e above theorem says t h a t solving the Cauchy problem a m o u n t s t o computing the dual basis with respect t o the Grothendieck duality. E x a m p l e 3 Put Q4

Qi

Qi

^+13w+9w

Q3

Q3

~ 2 0 £ - 25w+16w *' y ) =°-

Let Iz = (£ 4 + 2i2V2 + V* + H2V ~ V3,4^4 + (137/2 - 20)£ 2 + 9TI 4 - 25T, 2 + 16). T h e Grobner base of Iz with respect t o the total-lex order (with f >- 7/) is {{-W + 1277 + 20)£ 2 - 5T, 4 - irjs + 25T7 2 - 16, 4 - 5 £ + (-397; - 40)£ 2 + 5T) 4 + 13773 - 5O772 + 32, (-360477 - 3740)£ 2 - 500TJ 5 + I28O774 + 1043TJ 3 - 4575T?2 + 8O77 + 2672}. Basis monomial bj = bj(£, 77), j = l , . . . , 14 of Q[£, T]]/Iz are h = 1, 62 = V, h = £,h = r/ 2 , 65 = 77^, 6 6 = £ 2 , b7 = 77s, b6 = rfi, &10 = £ 3 , 6 n = 774, 612 = 773£, 613 = T/£ 3 , bu = tft.

bQ = 77^,

T h u s one can easily verify t h e well-posedness of t h e Cauchy problem with t h e following initial conditions. du du « ( 0 , 0 ) = c 0 l 0 l ^ - ( 0 , 0 ) = c 0 ,i, ^ ( 0 , 0) = ci, 0 ,

295

0^°' 0) =C0'2' S ( ° ' 0 ) = C1>1' ^ ( ° ' 0 ) = C2'0' 0<°' °>=co-3' w ( 0 , o)=c1-2' aS, ( o ' o ) = c 2 ' u £ ( 0 , o ) = c 3 - ° '

w{0>

0)=C0A

' w ( 0 - 0 ) = C 1 ' 3 ' ^k{0>

0)=C3,1

' w(0'0)=Cl'4-

The primary decomposition of Iz is Iz = Iz,i Fllz,2 r\Iz,3, where J ^ i = (i, - l,e), Ig,2 = ( V + 4T? + 1,4£2 - 4»7 - 5), and' IZfi = <4£2 + 9r)2 16, 25T?4 - 124TI 3 - 160T/ 2 + 192*7+ 256). Let {
/

w

2 7 2

A

^

+

8^5(1°^2

+ 871

((-|)

+

43995308430295040^' V'A*

+

(

- ^

) 2

^

2 )

where h{£, rj) = 8077288162033575£T? 3 - 30339279162874532£T7 2 85867612223694000^ - 513258475722542086 It follows that the Fourier-Borel transform of the local cohomology class
"83W-f " T ^ " * ** + <* T * *

+

43995308430295040 ^ U . ^ e A , M£> f^ "

The exponential polynomial FB(cris)(x,y) ditions. OU

OIL

«(0, 0) = 0, — ( 0 , 0) = 0, — ( 0 , 0) = 0, d2u

,

<92u

<92u

vv

>

•

satisfies the following initial con-

296

0(°' °»= "• W ( 0 ' 0 ) = "• aSi,(0'0) = "• S (0 ' 0) = °' We use the computer algebra system Risa/Asir 1 1 in this computation. References 1. C.A. Berenstein and B.A. Taylor, Interpolation problems in Cn with applications to harmonic analysis, J. d'Analyse m a t h . 3 8 (1980), 188-254. 2. J-E. Bjork, Rings of Differential Operators, North-Holland, 1979. 3. A. M. Dickenstein and C. Sessa, Duality methods for the membership problem, Effective Methods in Algebraic Geometry (eds by T . Mora and C. Traverso), Progress in M a t h . 9 4 , Birkhauser, Boston, (1991) 89-103. 4. L. Ehrenpreis, Fourier Analysis in Several Complex Variables, WileyInterscience, 1970. 5. A. Grothendieck, Local Cohomology, Lecture Notes in M a t h . 4 1 (1967), Springer, Berlin. 6. M. Kashiwara, On the holonomic systems of linear differential equations, II. Inventiones mathematicae 4 9 (1978), 121-135. 7. H. Komatsu, Hyperfunctions and linear partial differential equations, Lecture Notes in M a t h . 2 8 7 , Springer, Berlin, (1973), 180-191. 8. O.Liess, The fundamental principle of Ehrenpreis-Palamodov, Preprint Series in M a t h . 7, Bucharest (1976). 9. A. Martineau, Sur les fonctionnelles analytiques et la transformation de Fourier-Borel, J. Analyse M a t h . 1 1 (1963), 1-164. 10. M. Morimoto, An introduction t o Sato's hyperfunctions, Translations of Mathematical Monographs 1 2 9 , AMS, 1993. 11. M. Noro and T . Takeshima, Risa/Asir - a computer algebra system, Proc. International Symposium on Symbolic and Algebraic C o m p u t a t i o n 1992 (ed. P. S. Wang), ACM (1992), 387-396. 12. M. Sato, T . Kawai and M. Kashiwara, Microfunctions and pseudodifferential equations, Lecture Notes in M a t h . 2 8 7 , Springer, Berlin, (1973), 265-529. 13. S. Tajima, Grothendieck duality and Hermite-Jacobi formulas, Finite or Infinite Dimensional Complex Analysis, Dekker (2000), 503-509. 14. V.P.Palamodov, Linear Differential Operators with Constant Coefficients, Grundlehren M a t h . Wiss. 1 6 8 , Springer-Verlag, 1970.

T H E R E P R O D U C I N G KERNELS OF THE SPACE OF H A R M O N I C POLYNOMIALS IN T H E CASE OF REAL R A N K 1 RYOKO WADA Faculty of Economic Sciences, Hiroshima Shudo-University, Hiroshima 731-3195, Japan E-mail: [email protected] YOSHIO AGAOKA Faculty of Integrated Arts and Sciences, Hiroshima University, Higashi-Hiroshima 739-8521, Japan E-mail: [email protected]. ac.jp

Introduction It is known that classical harmonic polynomials on C p are represented by integral formulas on any 50(p)-orbit except {0} (see, for example, [3], [6], [9], [13]). According to the formulation in [5], these classical harmonic polynomials on Cp can be canonically identified with the harmonic polynomials on p, where p is the complexification of pR in a Cartan decomposition of so(p, 1). In this situation, any SO(p)-orbit in C p corresponds to a KR-orbit in p, where KB, is a Lie subgroup of GL(p) generated by exp ad X (X £ (!R). (In the following, we express this group as exp ad 6R.) Therefore, from the classical integral formulas of classical harmonic polynomials stated in [3], [6], [9], [13], we can obtain the integral representation formulas of harmonic polynomials on each i^R-orbits explicitly (see, for example, Appendix in [10]). As a next case, we consider the Lie algebras su(p, 1), sp(p, 1) and f4(_20)> which are the remaining real rank one cases. Here we also write 6R + pR a Cartan decomposition of su(p, 1), sp(p, 1) or f4(_20) and KR. = exp ad ER. As an extension of the above fact, in our previous papers [10], [11] we obtained integral representation formulas of harmonic polynomials for the cases of su(p, 1) and sp(p, 1). But these formulas are expressed in the form of double integrals on some family of nilpotent ^R-orbits and they are not so simple. In this paper we shall show reproducing formulas of harmonic polynomials on each "single" i^R-orbit KRXO in a unified manner for the cases of real rank 1: i.e. 0 R = so(p, 1) (p > 2), su(p, 1), sp(p, 1) (p > 1) or f4(_2o) and for any X0 € p (Theorem 1.3). Furthermore, we shall give the explicit expressions of

297

298

the reproducing kernels in the cases 0 R = su(p, 1) and sp(p, 1) (Theorems 2.1 and 3.1). Let QR be a real simple Lie algebra with real rank 1. Let "Kn be the space of homogeneous harmonic polynomials on p of degree n, and let 3in = ©feio !Kn,* be the i^R-irreducible decomposition of "Kn. Let P be the generator of the space of KR-invariant polynomials on p and we put E R = {X £ pR.; P{X) = 1}. Remark that E R is a single i^R-orbit and £ R can be identified with the "unit sphere" in pR. The main result in this paper is stated as follows: Theorem. Fix any ZQ € £ R and let KQ be the isotropy group of ZQ in KK. (i) There exists a unique function HnAX,Y) Hn,k(

on p x p such that

,Y) € "Kn,k for any r e p ,

Hn,k{gX,gY) Hn,k(X,Y)

= Hn>k(X,Y) = HnAY,X)

Hn,k{Z0,Zo)

for any g £ KR

and X,Y

£ p,

for any X,Y £ p,

= 1.

(ii) Suppose X0 6 p and Hn:k(X0,X0) ^ 0. i^br any / € IKm,j anrf an?/ X € p we have the following reproducing formula of harmonic polynomials on KRX0: (*)

dim^in,k

[ JKH

f(gX0)HnAX,gX0)dg

=

5n,m6ktlHnAXo,X0)f(X),

where dg is the normalized Haar measure on KR. For explicit expressions of reproducing kernels HnA , Y) in the cases of 0R = su(p, 1) and g R = sp(p, 1), see Theorem 2.1 and Theorem 3.1. Remark. In the case JJR = so(p, 1), (*) is already known as the formulas of the classical harmonic polynomials on C p and HnAX,Y) is expressed by using the Legendre polynomial of degree n and dimension p (see, for example, [3], [6], [9], [13]). The authors would like to thank Professor A. Kowata for his valuable advices.

299 § 0. Notations. In this section we fix some notations which we use in this paper, and recall the definitions and some known results on harmonic polynomials on p. Let g R be a real simple Lie algebra with real rank 1, i.e. g R = so(p, 1) (p > 2), su(p, 1), sp(p, 1) (p > 1), or f4(_20)- Let g R = fiR + p R be a Cartan decomposition of g R and g = t + p be its complexification. Let K be the adjoint group of t, i.e. K = exp ad t C GL(p). Now we define harmonic polynomials on p. We denote by 5 the space of polynomials on p. For / G 5 and g G K, p(g)f G S is defined by p(g)f(X) = f{g~lX) (X G p). Let J denote the ring of AT-invariant polynomials on p and we put J+ = {f £ J; /(0) = 0}. Let B{X,Y) {X,Y G fl) be the Killing form of g. In the case of real rank 1, the generator of J is given by B(X, X) (X G p) and we put P(X) — B(X,X). Let / be the space of if-invariant symmetric algebra over p and we put I+ = {u G / ; d(u)l = 0}. According to the definition in [5], / € S is harmonic if and only if d(u)f = 0 for v u G I+. We denote by Sn the space of homogeneous polynomials on p of degree n, and "Kn the space of homogeneous harmonic polynomials of degree n. Remark that K acts on Sn and CKn. We regard KR C K and p R C p. We put E = {X G p ; P(X) = 1}, E R = S n p R and N = { l G p ; P{X) = 0}. Remark that S R is .Kit-invariant. We put iV p - 1 = {x G C p ; lxx = 0}. When g R = so(p, 1), it is valid that p and p R are naturally identified with C p and R p respectively, and the generator of J is lxx. Then in the case g R = so(p, 1), S R , E and K correspond to the unit sphere in R p , the complex sphere in C p and /V p - 1 , respectively. The following facts are known: Proposition 0.1 ([5]). (i) / / we put (J+S)n Sn=Hn®

= Sn l~l J+S, it is valid that

(J+S)n. n

(ii) The space !Kn is spanned by {B( ,Y)

(0.1) ; Y G X}.

In [5] Kostant-Rallis stated several results on harmonic polynomials on p in the case where g is a general complex reductive Lie algebra. Let Hn(Cp) denote the space of homogeneous harmonic polynomials of degree n on C p . We can identify Hn{Qv) with SKn in a natural way in the case flR = so(p, 1). Therefore, Proposition 0.1 (ii) implies that Hn(Cp) is spanned by the set {ha(x) = ('a;a) n ; a G i V " 1 } .

300

§ 1. The reproducing formula of harmonic polynomials in the case of real rank 1. In this section we give general formulas for all irreducible i^R-submodules Oi„tk of 0in when £)R is a real simple Lie algebra with real rank 1; i.e. £JR, = MP, 1) (P > 2), su(p, 1), sp{p, 1) (p > 1), or f4(-20)Let g be the complexification of gR and let q be the dimension of the vector space pR. Since the Killing form B{,) is positive definite on pR, B(X, Y) gives an inner product on pR. When {X\,X2, • • • ,Xq} is an orthonormal basis of PR with respect to B{ , ), we define the bijective complex linear mapping * : p —> Cq by V(Xj) = eh where we put e^ = *(0 • • • 0 i 0 • • • 0). Then it is clear that the mappings * | s R : £ R —> Sq~l and * | x : Ji —> Nq~1 are bijective. Therefore, we can see that the mapping / i—> f o $ _ 1 is one to one linear mapping from "Kn onto Hn(Cp), and we have dim "K„ = dim Hn{C) =

(2m+

nT( g -2)i g " 3)! ^ S6e fOT G X a m P l e WFor any g 6 KR, there exist aij 6 R (i,j = 1,2, • • • ,q) such that i

2_,ai,jXi-

gXj =

t=i

Since {X\,X2, we have

• • • >-Xg} is an orthonormal basis and B{ , ) is i^R-invariant, i

i=l

From this we can see that M(p) = ( a ^ ) belongs to O(q) and we have 9{gX) = M(g)*{X)

(X € p R ) .

(1.1)

Let OR be a maximal abelian subspace of pR. It is known that pR = .KROR (see for example [2]) and this fact implies that S R = KRZ0 for any Z0 £ X R because dim OR = 1. Let dg be the unique normalized Haar measure on KR. From (1.1), we have the following equality for any integrable function / on /

f(gZ0)dg= [

foy-\s)ds,

(1.2)

where ds is the unique normalized SO(g)-invariant measure on 5 ? _ 1 . Let Pn,q denote the Legendre polynomial of degree n and dimension q (for the definition, see [6]), and we define for x EC and s £ Sq~x

Pn,qM =

(t^r/2Pn,q(^U,)

301

Prom the Funk-Hecke formula on spherical harmonics (cf. [6]) and (1.2) we easily have the following: P r o p o s i t i o n 1.1 (cf. [6], [10], [11]). For any f e "Km, any X € p and any ZQ £ ER. we have Sm,nf(X)

= dim nn

f

f(gZ0)Pn,q(9(X),

Next we consider the KR-irreducible = so(p, 1), su(p, 1), and sp(p, 1), let

^(gZ0))dg.

decomposition of "K„. When JJR

N(n)

M„ = ^£) "Kn,k

(1.3)

fc=0

be the i^R-irreducible decomposition of Jin, where f 0 (flR=«o(pJl)>p>3), N(n) = l n ( a R = B u ( p , l ) , p > 2 ) , Un/2](flR=«P(p,l),P>l), „s _ / 0 (SB = so(2,1) ~ su(l, 1), n = 0), m 1 ' ~ 1 1 ( t e = « ( 2 , 1 ) ~ au(l, 1), n > 1). It is known that 'Kn,k and CKm,; are not equivalent if (n, fc) ^ (m, Z) (see, for example [4], [8] or [12]). The same result holds for the case gp, = f4(_20)L e m m a 1.2. Let 0 R = f4(_2o)

an

d let

N(n) k=0

be the K^-irreducible

decomposition ofKn.

Then we have

N{n) = [n/2]. Furthermore 3in,k o,nd 'Km,i are not equivalent if (n, k) / (m, I). The proof of this lemma will be given in § 4. Now we discuss general formulas of the reproducing kernels of "Kn^ when 0R. is a real simple Lie algebra with real rank 1. In the following, fix any Z 0 G £ R and let K0 be the isotropy group of Z0 in KR. Our main result in this paper is the following

302

Theorem 1.3. (i) There exists a unique function Hn,k{X,Y) such that Hn,k{

, 7 ) G J { n i f e for any Yep,

Hn,k{gX, gY) = Hn
Hn,k(X,Y) Hn,k(Z0,Z0)

on p x p (1.4)

for any g e KR and any X, Y G p, (1.5)

fOTan

y ^,Yep,

(1.6)

= l.

(1-7)

(ii) Suppose X0 S p and iJ ni fc(X 0 ,X 0 ) 7^ 0. For any / € JKm]j and any X € p we Aave £fte following reproducing formula of harmonic polynomials on KRX0: Sn,m6kiif(X)

= Jim3in'k f f(gX0)Hntk(X,gX0)dg. Hn,k{Xo,Xo) JKR

(1.8)

Since it is well known that Hn(Cq) ~ i J n ( C 9 ) | s , - i (cf. [6]), we get easily that !Kn ~ ? < „ | E R and that %n,k — Wn,k\zR (0 < k < N(n)). We shall denote by L 2 ( E R ) the space of square integrable functions on E R , that is, the space of measurable functions / on E R with

11/1,2=

{L

|/(flZo),2d

4

°-

Furthermore we denote by C ( E R ) the space of continuous functions on E R . To prove Theorem 1.3 we prepare the following Lemma 1.4. Suppose that gR is a real simple Lie algebra with real rank 1. (i) There exists a unique Hn
= Hn,k{X)

for any g0 G K0 and X G p,

#n,fc(Z0) = 1(ii) It is valid that

JjM„,M)^=^-~-

(1.9) (1-10)

(Ml)

Proof, (i) Since E R is a compact Hausdorff space, S|s R is dense in C ( E R ) with respect to the sup norm on E R (Stone-Weierstrass' theorem). It is well

303

known that C ( S R ) is dense in L 2 ( £ R ) with respect to the norm of L 2 ( S R ) . Therefore (1.3) and a repeated use of (0.1) give that I/2(ER,)

=

^

^n,fc|s R

(direct sum as Hilbert spaces),

(1-12)

(n,fc)6A

where we put A = {{n,k) ; n = 0,1,2, • • • ,0 < k < N(n)}. Let CK^ be the subspace of !Knifc consisting of elements ft G "Kn,k which is i^o-invariant. Since Jin
= [

hntk(gX)hn,k(gZo)dg.

(1.13)

J KK

It is clear that Hnk G CKn and Hn k is if 0 -invariant. Since !J~Cnj;-l_3"Cm / ((n,fc) 7^ (,m,l)) with respect to the inner product of L 2 ( £ R ) , we have for any / m ,; G J{m,i /

Hntk(gZ0)fm,i(gZ0)dg

JKR

= / / hnik{gigZ0)hn:k{giZ0) J Kn J KB =

[ JKB

\JKR

=0

fm,i(gZ0)dgdg1

hn,k(g1gZ0)fmti{gZ0)dg)hntk(g1Z0)dg1 /

((n,fc)^(m,0).

This shows that #„,& belongs to !>£„,&. The equality (1.10) is a result of the fact ||ft„,fe||2 = 1(ii) Suppose that {/j, f2, • • • , fd(n,k)} is an orthonormal basis of CKn,/t with respect to the inner product of L 2 ( S R ) , where d(n,k) = dim 'Kn^- We put for g £ KR Pij(a) = /

p(g)fj(gxZ0)fi(giZ0)dg1

JKT, IKR

= !

fj(g~1giZ0)fi(g1Z0)dg1

(1 < i,j < d{n,k)).

JKR

Then we have d(n,k)

p(g)fi = E P^9)U

(i-i4)

304 and it is well known that / Pij(g)Pirn(g)dg = ——rr8i,i8j,m JK a{n,K) 'KRR

(1
(1.15)

From (1.14) and (1.15) we get

fi(g-1z0)fi(g-1Z0)dg

= /

JKR d{n,k) d(n,k)

= E

E

/

i=i i=i d(n,k) d(n,k)

= E i=l d(n,k)

Pii(9)fi(Z0)pn(g)fi(Z0)dg

J K R

fi(zo)Wo)

E

/

Pii(g))p^{9~)dg

"K*

1=1

Hence we get d(n,k)

£

\fi(Z0)\2 = d(n,k).

(1.16)

i=l

Now we can choose f\ = hn^- From (1.13) and (1.15) we have /

\Hn^k(g1Z0)\'2dgl

=

[ JKR

(1.17)

hn,k{ggiZo)hn,k{gZo)dg

\JKR

= /

/

/

JKR

JKR

JKR

y

d(n,fc)__

E

Pni9-l)fi{giZ,) Y,

i=1

J=1

U(n,k)

I E r=l

d{n,k) l

Pn{92 )fr{Zo)

Yl s=l

dgi J

JKR

/d(n,k)

x

hnik(g29iZo)hntk{g2Zo)dg2) ^

mti-^MZo) \

P*i^)/«(Si^o)

dgxdg2dg I

305 / =

•.

^ 2 d(n,fcj d(n,fc)

[d(^k))

S

£

d(n,k)

f

/ ^ o ) / ^ ) ^

fi(9lZo)fr(giZo)dgi

2

= E (350,)

I/«>I2-

The equalities (1.16) and (1.17) imply (1.11).

q.e.d.

Proof of Theorem 1.3. (i) We define for X,Y Hn,k(X,Y)

= dim% n , k f

ep

Hn,k(gX)Hn,k(9Y)dg.

Then we have (1.4)-(1.6) clearly. We also have (1.7) from (1.11). Suppose that another function Gn^k(X, Y) on p x p satisfies (1.4)-(1.7). Then from (1.4) and (1,5) H„ik( ,Z0) and Gn,k( ,Zo) belong to !K'n k. Since dim3f^ k = 1, (1.7) implies that Hn,k{ ,ZQ) =G„tk(

,Z0).

(1-18)

From (1.5), (1.6) and (1.18) we get for any g € K R and any y e p Hn,k(gZ0,Y) Since E R = KRZ0

= Gn,k(gZ0,Y).

(1.19)

and "Kn ~ 3£„|s R , by (1.19) we obtain for any X, Y 6 p Hn,k(X,Y)

=

and this shows the uniqueness of Hnik(X, (ii) Since H„tk( from Lemma 1.4

,Z0) belongs to Wnk Hn,k(

Gn,k(X,Y), Y). and Hnik(Z0,Z0)

,Z0)=Hn,k.

= 1, we have (1.20)

For any X0 G p we put hn,k,m,i(X,XQ)=

f

Hm,,(gX0)Hn,k(X,gXo)dg

(Xep).

(1.21)

Since if m) j € IK^, and H„tk( ,Y) € CK„ifc (Y £ p), we see that /i„,/t,m,(( , Z 0 ) belongs to lK'nk for any X 0 G p from (1.5). This and (1.10) give that hn,k,m,l(X,Xo)

= Xn,k,m,l (XQ)Hn,k(X),

(1.22)

306 where (Z0,X0)=

/

Hm^gX0)Hn,k(Z0,gXo)dg.

When {n,k) ^ {m,l), we have from (1.4), (1.5), (1.6) and (1.20) K,k,m,i(xo)

= I

Hm,i(gX0,Z0)Hnik(Z0,gX0)dg

(1.23)

Hm,l(X0,g-1Z0)Hn
= f = [

Hm,i{gZo,X0)Hn,k(gZo,Xo)dg

= 0.

JKR

On the other hand, from (1.6) and (1.20) it is valid A».fc,n.*(*o) = I

Hn>k(gX0)Hn,k{gX0)dg

(1.24)

(dimXn,k)-1Hn,k(X0,X0).

=

The equalities (1.21)-(1.24) give dimCKn,fc /

Hmti(gX0)Hntk(X,gX0)dg

=

6nym5kti5ntk(Xo,X0)Hntk(X).

JKn

(1.25) Prom (1.5) and (1.25) we have for any pi € KB. dim:Kn,fc /

Hm4(gigX0)Hntk(X,gX0)dg

(1.26)

JKn

= dim:H nifc /

Hm,i{gigX0)Hntk(giX,g1gX0)dg

JKn

= dimJ{„iit /

Hm,i(gX0)Hnik(giX,gX0)dg

JKn —

u

n,m

5k,iH„ik(Xo,Xo)Hntk(giX). Since the i^R-irreducible module jim,i is generated by Hm>i, (1.26) implies (1.8). ' ' q.e.d. Remark 1.5. Since we have Hn,k{X0,X0)

= dim Mn,fc I

\Hn,k(gX0)\2dg

the following conditions (a)-(c) are equivalent.

{X0 € p),

307

(a) (b) (c)

Hn,k{Xo,Xo)jL0, Xn,k\KHXo ^ {0}, Mn,k — Wn
Therefore (1.8) gives a reproducing formula on the orbit Hntk(Xo,X0) ^ 0.

KRX0

when

Corollary 1.6. Let Xp be the character of the irreducible representation p:KR-^ GL(Xn,k). (i) When X0 € p and Hnik(X0,Xo) ^ 0, the following formula holds for any g G KR: Hn,k(g-lg1X0,glX0)dg1.

XP(9) = - ^ ^ " t , /

(1.27)

(ii) For any g G i^R, £/ie following formula holds: Hn,k(gZ0,Z0)=

[

xP(9o9~l)dg0,

(1.28)

JKo

where dgo is the normalized Ko-invariant measure. Proof, (i) Let {hj ; 1 < j < d(n, k)} be an orthonormal basis of "Kn^k with respect to the inner product of L 2 ( K R X 0 ) . We put for any X,Y e p d{n,k)

H(X,Y)= Yl hjWh-iYJ. Since H(X, Y) is the reproducing kernel of "Knik on KnX0,

H(X,Y)=

^

we get from (1.8)

HnAX,Y).

Therefore we have for any gi £ i^R.

f; hj(g-1g1X0)h^g~M = -J^^HnAg-l9lX0,giX0). J=1

Hn,k[Xo,Xo)

(1.29)

308 The equality (1.29) gives (1.27) because we have d(n,k)

xP(g) = Y^

Pjj^)

d(n,k)

= Y, (P(9)hi'hi) 3=1 d(n,k)

= Y

hj(g-1g1X0)hj(glX0)dg1

/

= "g—,vn't s / Hn,k(X0,Xo) JKR

^nA9~19ixo,9iX0)dgv

(ii) From (1.5), (1.7) and (1.27) we have for any g0 G KR Xp{9o9~l) = dim34:„,fc /

Hntk(ggolgiZ0,giZ0)dgi

(1.30)

JKR

-dimJ{n^

Hritk(g0~1g1Z0,g~1g1Z0)dg1.

/ JKR

The equality (1.30) implies /

XP(9og~1)dgo =dimDintk

JK0

/

Hn^(gQ~1g1Zo,g~1giZ0)dg0)

(/

JKR

\JK0

J

dg I (l.c (1.31)

= dim'Kntk

(/ JKR

\JK0

Hn^(g0g1Z0,g~1g1Z0)dg0)dg1. J

If we put for any X, Y € p

*n,*pr,y)= f

Hn>k(gQX,Y)dg0,

JKQ

it is clear that *„ |fc ( C(F) G C such that

, 7 ) G I t t ^ (F G p). Hence there exists a constant

9n,k{X,Y)

= C(Y)Hn,k{X,Z0).

(1-32)

309 By (1.7) we have C{Y) = 9n,k(Z0,Y) = / =

(1.33)

Hnik{g0Zo,Y)dgo Hntk(Zo,Y).

The equalities (1.5), (1.7), (1.8) and (1.31)-(1.33) give (1.28) because /

XP{9o9~1)dgo = dimJi^k

H„,k{9izo,Z0)Hn}k{Zo,g~1giZ0)dg1

JKQ

JKR

= dimW„,fc / =

Hntk(giZ0,Z0)Hn
Hn,k(gZo,Zo). q.e.d.

§ 2. Reproducing kernels of the space of harmonic polynomials in su(p, 1) case. In this section we consider the case g = s\(p + 1, C) and 0 R = su(p, 1) (p > 2), and give the reproducing kernel on any i^R-orbit in p. In this case, we have e R =

{(oa) 0 x

PR = U ^x n0 l

={(oa)

; AGU

( P ) ' « e u ( l ) , TrA + a = o J ,

;^ec" M e M ( p , C ) , TrA + a = o | ,

P={(*°S);*.»€C-}, and KR = Ad S(t/(p) x U(l)) = | A d ( ^ J) ; A 6 tf(p) J . For X = ( t ° *) € p, P(X) = {4p + l)-lB(X,X) = ^ T r (X2) = *yx gives the generator of J. We put N = {X e p; P ( X ) = 0 } , S = { I e p ; P ( X ) = 1} and S R = E n p R . Remark that S R ~ S2?-1. 3f„ = { / e S „ ; £ ? = 1 ^ - / ( X ) - 0} is the space of homogeneous harmonic polynomials on p of degree n and dim J{„ = 2(n+p-l)(n+2p-3)! n!(2p-2)!

310

For X = ( t ° *) e p and g = Ad ( £ J) G KR gX = ( t ( ° y ) ^ ) . We put £ r = ( ( 1 _ p 2 °i/2t e 2 7 )

(A G C/(p)) we have € ^ (0 < r < 1), £ , ,

= ( « ( r - i e ? + 9 e 2 ) T ) £ S (r > 0, g > 0), where e

1 =

' ( 1 0 -

0), and

e2 = '(0 1 • • • 0). Then we have i ^ R ^ o = S R and p = N U UA 6 C\{O} K

AE

K

-

Remark that S = Ug>o,r>o B.Er,q and N = U P >o,o
£ pweput Kntk{X,Y)

= (txa)k{tyP)n-k

((n,k) G A). Let 0{n,fc be the

subspace of J{n which is spanned by the elements Kntk(

,Y) ( 7 E N). From

Theorem 14.4 in [8] we can easily see that Jin = © £ = 0 "Kn,k gives the Knirreducible decomposition of 0in and dim Jin,k = We put E0 = (t Hn,k(X,Y)

e

tp+r

V~JkuPZ2)\kUn-kV.~

•

i J. Now, using the above polynomial Kn,k, we define

(X,Yep)by

Hn,k{X,Y)

= dimXn,k("+P~2\(k+P~2')

J

Kn,k(X,gE0)Kn,k(gE0,Y)dg,

where dg is the unique normalized i^R-invariant measure on KR.. For Y G p we have clearly Hn
=dimM„, f c [

f(gX0)Hn,k(X,gX0)dg.

JKR

We can also prove this theorem by obtaining the explicit form of the element of "K'n k, without using Theorem 1.3 in [8]. Remark2.2 (cf. [12]). (i) We have

HnAX,Z)=(n+Pk-2^k+Pk~2^

l

Kn,k{X,Z)

(XGKorZGN).

311

Hence for any Xo 6 N and any / £ Wn,fc we have dim IK,,,* f

f(gXo)Kn,k(X,gX0)dg

=

5ntrn6kilKn,k(X0,X0)f(X).

Kr

(ii) Hn:k(X0,X0) XKRE0 (A £ C).

? 0 for any (n,A) £ A iff X 0 £ A A ^ and X0 $

§ 3. Reproducing kernels of the space of harmonic polynomials in sp(p, 1) case. In this section we give the explicit formula of the reproducing kernel of the harmonic polynomials on each KR-orbit in the sp(p, 1) case. Since the Lie algebra sp(l, 1) is isomorphic to so(4,1), we assume p > 2 in the following argument. Prom now we put g = sp(p + 1, C), 0 R = sp(p, 1), ' / A

0 B 0\

0 o £6

« R = <

A e u(p), a e u(l), foe C Z? is p x p symmetric

kV 0 4

' [Ox PR= <

0 5/ 0

y \

'SO *y

0

Of

-i

0

x, y £ C7

[ VV o -'* o / Then we have (A 0 5

0 \

C 0 -M 0 \0 7 0 - a / /0 x *y 0 0 2 \lz 0

0 w 0 -*x l

. „ „

,,,

„,

a, /3, 7 € C

w\ 0 ; x,y, z,w £ Cp -y 0 /

and

AR

/A 0BO\ 0_ a £ / 3 eAdU(2p Ad -I? 0 ^ 0 \ 0 -/? 0 a /

t

AA + tBB = Ip, t + 2); AB = _BA, t

312

(

Oi ty 0

w\

0

°z '% _°y

6 P, P(X) = j^B(X,X)

= i-TY (X») = ' s y

^ 0 -'x 0 /

+'zw gives the generator of J and "Kn = {/ e S n ; £ ? = 1 ( g ^ j + g ^ j ) / = 0}. We can see that dimM n = 2 ( n + 2 ^ ( " 2 y ~ 3 ) ! • We put N = {X G p • P{X) = 0}, E = {X G p ; P(X) = 1} and E R = E D PR. Remark that E R ~ S4P~ 0 B 0\ 0_ a 0_/3 Let g = Ad -B 0_ A 0 G i^R, and X = \ 0 -/3 0 a / /x\ (A

$(X)

y

z

0 x 0

«i\

'0" I '» _°tf U p . If we put '.z 0 -lx

0 /

G C 4 p , we have

w / A(ax + ~0w) + B(az- ]3y) \ B(-/9x + aw) + A(ay + (3z) $(5X) = -B~(ax + ]5w) + ~X{az - /fy) \A(-0x + aw) - B(ay + fiz)/ rei

We put E r = $-! I

(

\

J

G K (0 < r < 1), £,.,, = $

Ul-r2)1/^ G E (r > 0, g > 0). It is clear that p = N U

\

0

V

UA£C\{O}

rex

_1

^ - Remark that N = |J

>0

0

7

/J7^
>o,r>o Kfi(Er,q) give the i^R-orbit decompositions of 3sf and E respectively. Furthermore we have E R = KREI 0 . /x\ fx>\ x 2/ G p we put For X _= *$ - i z , x' = $-

w

(X,X') = - I r ('XX') = lxx' + lyy' + lzz< + W , K2(X, X') = Cxx1 + hz^Cyy1 Km(X,X')={2^~^f Kn,k{X,X')

=

+ 'tuui7) + ( W - *zy 7 )('yz' - W ) , JK

(gEuXTfrgEirdg,

Kn_2k(X,X'){K2(X,X')}k,

(n = 0,1,2, • • • ,fc= 0,1, • • • , [n/2]). Remark that ( , ) is A^R-invariant and

313

that for any X, X' £ p, g € K R we have Kntk{X,X')

=

Kn,k{X',X),

Knik(X,X')

= Kn,k(gX,gX').

(3.1)

It is valid that Kn>k( ,X') £ %n if X' £ N (see [12]). Let IKnifc denote the subspace of !Kn which is spanned by {#„,*( , Z) ; Z € N}. From (3.1) it is clear that ^Kn,k is i^R-invariant. We also see that %n = ©L=0 -^n,« gives the /^R-irreducible decomposition of Jin (see [12]). From now we put A = {(n, k) ; n = 0,1,2, ••• ,0 < /c < [n/2]} and E0 = $ - J

0

= dim M n , t fn

e

+

X. Now, using tfn>fc, we define for X, Y £ p

^ " 2 ) (2P

+

l ~ 3)

y

^n,fc(X, 9 E 0 )K n ,k(gEo,Y)dg

((n, A;) £ A). Then we can prove that ff„,fc( , Y) belongs to Ji„tk for any Yep and that ^ „ ? i ( Z , y ) satisfies (1.4)-(1.7). (For details, see [12].) Therefore we can see that Hn,k{ ,Y) gives the reproducing kernel of "Kn,k for each .KR-orbit. From Theorem 1.3 we have the following Theorem 3.1. For any / £ ^Km,i and any X§ £ p we have Sn,m6k,iHn,k{Xa,X0)f(X)

= dimJin>k

I

f(gXo)Hn,k(X,gX0)dg.

Remark 3.2 (cf. [12]). (i) We have Hn,k {X, Z)

= (" +2 ;- 2 )( 2p+ *-y'/?„, i( x, Z )

(«».,zeM).

Hence for any X0 € N and any / £ Mm>; we have dim^ n > f c /

f(gX0)kntk(X,gXo)dg

(ii) We have Hn,k(X0,X0) X0 $ XKREo for any A € C.

=

6nim5ktiKn>k(Xo,X0)f(X).

^ 0 for any (n,k) £ A iff X0 $. \KREi

and

314

§ 4. Proof of Lemma 1.2. In this section we give a proof of Lemma 1.2. Since the group .KR is connected, we may consider the problem in the level of Lie algebra. The Lie algebra of KR is ad (so(9)), which is isomorphic to so(9), and it is well known that the action of so(9) on p ~ C 1 6 is equivalent to the spin representation. In the following, we denote by [a, b, c, d] the complex irreducible representation space of so(9) with highest weight aAi + 6A2 + CA3 + (IA4. (We use the standard numbering as in Bourbaki [1].) The spin representation corresponds to [0,0,0,1]. We prove the following proposition. Proposition 4.1. The Kn-irreducible decomposition of 0in is given by Mn = [ 0 , 0 , 0 , n ] e [ l , 0 , 0 , n - 2 ] © [ 2 , 0 , 0 , n - 4 ] ©••• ([m, 0,0,0] \[m, 0,0,1]

(4.1)

(n = 2m), (n = 2m + l).

Then, Lemma 1.2 follows immediately from this proposition. Proof. Since f4(-2o) i s r e a l rank 1, the ring of Kft-invariant polynomials on p is generated by one quadratic polynomial P. Hence, by using Proposition 0.1 (i) repeatedly, we have Sn = -fin © P • -Hn-2 © P

' -Wn-4 © • • • © < pm

nr

l

_ cy

, -i\

(4-2)

We now prove the decomposition (4.1) by induction on n. First, this decomposition clearly holds in the case n = 1 because Jti — S\ = [0,0,0,1]. Assume that (4.1) holds for n < N (N > 2). Then, by using (4.2), we have easily Sn=

0

[A,0,0,n-2fc-2i]

(4.3)

fc,I>0,2fc+21
for n < N. Here, we note that if Sr — ®[a;, 0,0,7* — 2x - 2y] and Ss = ®[,2,0,0, s — 2z — 2w] hold, then the space S r + S contains the irreducible component [x + z, 0,0,r + s — 2x - 2y - 2z - 2w], where x, y, z, w satisfy the conditions x,y,z,w > 0, 2x + 2y < r and 2z + 2w < s. This result follows from the fact that if Sr and Ss contain irreducible components with highest weights A and A', respectively, then the space Sr+S contains the component with highest weight A + A'.

315

By using this fact, we show SN D ®k,i>o, 2k+2i 1, we have S2 D [0,0,0,0] and SN-2 D [k,0,0,N - 2 - 2k - 2(1 - 1)] = [fc,0,0, N — 2k — 21]. And hence, from the above fact, we have SN 3 [A;, 0,0, N - 2k — 21]. In a similar fashion, in case I = 0 and N = odd, the product of [0,0,0,1] C Si and [k,0,0,N - 1 - 2k] C SN-I contains [A;,0,0,iV - 2k]. (Note that 2k < N - 1 because N = odd.) If l = 0, TV = even and 2k < N - 2, then we consider the product of [0,0,0,2] C 5 2 and [fc,0,0,JV - 2 - 2k] C SW-2- Finally in case I = 0 and 2k = N, we consider the product of [1,0,0,0] C S2 and [k - 1,0,0, N - 2 - 2(k - 1)] = [k — 1,0,0, N — 2k] C 5W-2- From these facts, it follows that SND

0

[k,0,0,N

-2k-2l]

k,l>0,2k+2l
=

[k,0,0,N -2k]®P-'HN-2®P2

0

•'KN-i®---

.

k>0,2k
In particular, from (4.2) we have

KND

0

[k,0,0,N-2k].

k>0,2k
We show that the equality holds in this calculate the dimension for both spaces. have dim KN = dim SN - dim 5 W _ 2 = Next, from Weyl's dimension formula, we dim [k,0,0,N

expression. For this purpose, we First, since dim SN = ( ^ 1 5 ) , we ("+ 15 ) - ( ™ ) = *N+X+U» • have

- 2k] = g j j ^ + l)(k + 2){k + 3)(/V - 2k + 1)

x (TV - 2k + 2)(/V -2k + 3)2(/V - 2k + 4)(/V - 2k + 5)(7V - k + 4) x{N-k

+ 5)(N -k + 6)(/V + 7).

This is a polynomial of k of degree 12. By using the well-known formula of Yl kp for p = 1 ~ 12, we can calculate the sum of the above polynomial of k for k — 0 ~ [N/2]. Then, we know that it just equals dim "HN which we calculated above, and hence we obtain the desired equality. (In actual calculations, we used computers.) Thus, by induction on n, we complete the proof of Proposition 4.1. q.e.d. Remark. It is hard to decompose the symmetric power of the spin representation of the Lie algebra so(2k + 1) for large k. For example, it seems that the degrees of generators of invariants are known only in the cases k < 6 (cf. [7; p.262]).

316

References [I] N. Bourbaki, Groupes et Algebres de Lie, Chap. 4, 5 et 6, Hermann, Paris, 1968. [2] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press Inc., Orlando, 1978. [3] K. Ii, On a Bargmann-type transform and a Hilbert space of holomorphic functions, Tohoku Math. J., 38 (1986), 57-69. [4] K. Koike and I. Terada, Young-diagrammatic methods for the representation theory of the classical groups of type Bn, Cn, Dn, J. Algebra, 107 (1987), 466-511. [5] B. Kostant and S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math., 93 (1971), 753-809. [6] C. Muller, Spherical Harmonics, Lecture Notes in Math., 17 (1966), Springer-Verlag. [7] A. N. Parshin and I. R. Shafarevich (eds.), Algebraic Geometry IV {Invariant Theory), Encyclopaedia of Math. Sci. 55, Springer-Verlag, Berlin, 1994. [8] M. Takeuchi, Modern Spherical Functions, Translations of Math. Monographs vol.135, Amer. Math. Soc, 1994. [9] R. Wada, Holomorphic functions on the complex sphere, Tokyo J. Math., 11 (1988), 205-218. [10] R. Wada, The integral representations of harmonic polynomials in the case ofsu(p,l), Tokyo J. Math., 21 (1998), 233-245. [II] R. Wada, The integral representations of harmonic polynomials in the case ofsp(p,l), Tokyo J. Math., 22 (1999), 353-373. [12] R. Wada, Explicit formulas for the reproducing kernels of the space of harmonic polynomials in the case of real rank 1, preprint. [13] R. Wada and M. Morimoto, A uniqueness set for the differential operator A z + A2, Tokyo J. Math., 10 (1987), 93-105.

ON T H E L A M E SERIES REPRESENTATION OF A N A L Y T I C H Y P E R F U N C T I O N S ON A TWO-DIMENSIONAL COMPLEX MANIFOLD AHMED I. ZAYED Department

of Mathematical Sciences DePaul University Chicago, Illinois, 60614 U. S. A [email protected]. depaul. edu

Generalized functions and hyperfunctions have been characterized in terms of their expansions in Fourier series and other series of orthogonal functions, such as Legendre, Jacobi, Laguerre, Hermite, and Faber. In this paper we give a characterization of a class of analytic hyperfunctions on a two-dimensional complex manifold in terms their expansions in series of products of Lame functions.

1

Introduction

The problem of representing generalized functions and hyperfunctions in series of orthogonal functions is an old one. It goes back to the work of L. Schwartz 8 and M. Sato 7 . L. Schwartz gave a characterization of periodic generalized functions, i.e., generalized functions defined on a circle, in terms of their Fourier series expansions. He showed that / is a periodic generalized function with period 2ir if and only if it can be represented in the form oo

f(x)=

f(n)einx,

£ n= — oo

with f(n) — O (np) as \n\ —» oo, where p is an integer. M. Sato and later G. Johnson 4 in a more elaborate setting gave a characterization of hyperfunctions on the circle also in terms of their Fourier series expansions. They showed that / is a hyperfunction on the unit circle if and only if it can be represented in the form oo

/(*)=

]T n— — oo

317

/(n)e^,

318

i/H withlim| n |_ f(n) < 1. Hyperfunctions on the unit circle can be viewed as limits of harmonic functions in the unit disc. That is if / is a hyperfunction on the unit circle, then there is a harmonic function f(x,r) in the unit disc such that

f(x) = lim f(x,r)=

lim

V

/(n)einirM,

| r | < 1.

These results have been generalized in a number of different ways. For example, if the Fourier coefficients < f(n) > grow unrestrictedly, a class of more general types of generalized functions is obtained. G. Walter 14 , T. Boehme 1, and G. Gorbacuk and M. Gorbacuk 3 independently obtained such characterizations in different settings using various techniques. Walter showed that such Fourier series converge to ultradistributions, while Boehme showed that they converge to a class of Mikusinski operators. In a different direction, the orthogonal family of functions, {e m a : } __ , on [—7r,7r] is replaced by other orthogonal family of functions, such as Legendre, Gegenbauer, Jacobi, Laguerre, Hermite , and more generally by a system of orthogonal polynomials, known as Erdos class of orthogonal polynomials. For more on the classical orthogonal polynomials, see 9 . The Gegenbauer polynomials, also known as the ultraspherical polynomials, are special cases of the Erdos class of orthogonal polynomials. The Gegenbauer polynomials are particularly important because they are related to spherical harmonics, which are used in characterizing generalized functions, even Boehmians, on the sphere 5 . In another direction, the unit circle is replaced by more complicated curves and surfaces, e.g., analytic curves and the unit sphere. A curve is said to be analytic if it is a homotopical deformation of the unit circle. A. Zayed 18 obtained a characterization of hyperfunctions on analytic curves in terms of their Faber series expansions. More recently, characterizations of different classes of generalized functions and hyperfunctions in terms of their wavelet expansions have been obtained. One can find several results in that direction in Walter's book "Wavelets and other Orthogonal Systems," 12 . A. Zayed has obtained some results in that direction as well 16 ' 17 . Wavelets are functions formed by dilations and translations of a single function ip(x), called a "mother wavelet," in

319

the form iPjk(x) = 2^(2'x-k),

j,k€Z.

(1)

Under certain conditions, wavelets form an orthonormal basis of L2(R). Hence, generalized functions and hyperfunctions on the real line can be expanded in terms of them. Series representations of generalized functions and hyperfunctions are particularly useful in applications because in many situations one has to deal with divergent series of orthogonal functions. These series, which diverge in the classical sense, may in many cases converge to generalized functions or hyperfunctions. Almost all the work cited above was done on the real line or the complex plane. But many one-dimensional results are not easy to generalize to higher dimensions. For example, although Sato's hyperfunctions were introduced in n dimensions using sheaf theory, which is an algebraic approach, their analytic representations in form of series are not yet known, even in some simple cases like on the complex 2-dimensional unit sphere, say

{(21,22) : zf + z\ = 1, zk = xk + iyk,xk,,yk

G R,k = 1,2}

This article attempts to give a partial answer in that direction. The representation we shall obtain is in terms of series expansions in products of Lame polynomials. Lame polynomials appear in the representation of spherical harmonics in sphero-conal coordinates and they satisfy a second order differential equation, known as Lame's differential equation. One form of this equation involves Jacobi's elliptic functions.

2

Orthogonal Series Representations

In 1862 K. Neumann 6 proved that every function holomorphic inside an ellipse with foci at the points ±1 can be expanded in a series of Legendre polynomials Pn(x); n = 0,1,2, • • • . Actually this is true for Jacobi polynomials p^n (x), a,(3 > —1, n = 0,1,2,....,. The Jacobi polynomials satisfy 2nd order differential equation and form an orthogonal basis for L2[—1,1] with respect to the weight function (1 — x)a(l + x)13. When a = 0 = /?, we obtain the Legendre polynomials. Neumann's result says that if / is analytic in the ellipse {z : \z — l\ + \z +

320 11 < p+ ^, p > 1}, then / has the representation oo f(z)

=

1

,

^2anPn(z),

l i m SUp V l ^ n j < ~ • n

n=0

P

^°°

In particular, if / is analytic on [—1,1], the result holds. On the real line we have a generalization of Neumann's Theorem 19 , but first let us introduce the following class of polynomials, known as Erdos class of orthogonal polynomials. Definition 1 Let {Pn} be a sequence of orthogonal polynomials on the real line with respect to a monotone increasing function a(x), i.e., oo

Pn{x)Pm(x)da(x) /

= 5mtn,

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

-oo

Po(x) = 1 This sequence of polynomials is said to belong to the Erdos class E if da has support in [—1,1], a'{x) exists and is positive a.e. Polynomials in the Erdos class E satisfy the 3-term recurrence relation xPn(x) = -^Pn+1{x) + anPn(x) 7n+l P„(*) = l, P-i(x) = 0.

+ 2nzlpn_l(:r); In

n

= 0,1,2,...

for some constants an and j n . They have associated functions of the 2nd kind defined by 1

Pn(x)

Qn(z) = [ ^^-da(x), J-\ X — Z where z is in the complex plane cut along [—1,1]. Walter and Nevai 13 showed that if an = 0(np) series ^

a n P n (a;)

for some p, then the

where Pn(x) G E,

n=0

converges to a generalized function on [—1,1]. This result was extended to a class of analytic hyperfunctions on [—1,1] in 19 . But before we state this result, we need the following definition. Definition 2 Let I = [a, b] be a bounded interval and H(I) be the space of all functions <j>(z) that are holomorphic in some complex neighborhood U of I.

321

Two functions / and g in H(I) are identified if they agree on some neighborhood of / . For each such U, let A(U) denote the space of functions holomorphic in U and continuous on U. We endow A(U) with the topology defined by the norm

M^supMOI. There is a natural map from A(U) into H(I). The topology of H(I) is defined as the finest locally convex topology on H{I) for which all these maps are continuous. Provided with this topology, the space H(I) is a non-metrizable, complete Montel space. The strong dual H'(I) of H(I) is called the space of hyperfunctions (analytic hyperfunctions) on / . Without loss of generality we take / = [—1,1]. The following result was proved by Zayed and Walter 19 . T h e o r e m 3 Let {Pn(x)} C E and {an} be a sequence of complex numbers such that lim A/|O^| < 1. Then the series Y^=oanPn{x)da converges in H'(I) to say f. Moreover, an = {f,Pn). Furthermore, if f £ H'(I), then (f,Pn) = an with lim \ / | a n | < 1, and n—too oo

/ - YlanPn ?l = 0

in the sense of H'(I). We now extend this result from the real line to the complex plane. But first let B be an open bounded subset of the complex plane with closure B whose complement Bc is simply connected. Let z = x(w) map the domain \w\ > p one-to-one conformally into the domain Bc such that x(oo) = oo. oo

Denote the boundary of B by dB.

Let F(z) = £) bnzn,

b0 ^ 0 be

n=0

analytic in \z\ < 1 and assume that F can be analytically continued to any point outside the unit disc D = D\ by any path not passing through 0,1, oo. oo

Let R{w) = Yl cn/wn,

Co / 0 be analytic in the domain \w\ > p with

n=0

R(w) ^ 0. Assume that z = 0 € B. The generalized Faber polynomials Pn{z) are defined by the generating function

F (-L-)

WX {W)

'

\x(w)J xH

\w\ > p, z e R

R(W)

=

t'o

y^M wn

322

Faber polynomials Pn(z) of a domain B play the role of z", n = 0,1, 2, • • • , in the unit disc. That is if / is analytic in the disc Dr , then OO

-.

f(z) = Y] anzn,

\z\ < lim

^ ^

n—too

— = r I*/ /j

globally in -Dr- Similarly, if / is analytic in B, then OO

n=0

globally in B. Faber polynomials have associated functions, called Faber functions of the 2nd kind, Qn(z), that play the role of z~n, n = 1,2,..., i.e., any function analytic outside B can be written in a series of Qn(z)'s. Theorem 4 Let f be analytic in B where dB is the image of \w\ = p under X- Then OO

..

z

f( ) = ^anPn(z), J

with lim A / K J =r < -.

*•—

n—»oo

71 = 0

o

^

It is worth noting that the Legendre polynomials are Faber polynomials of the ellipse described in Neumann's result. The following theorem extends Sato's result to hyperfunctions on analytic curves. Theorem 5 (A. Zayed m) f G H'(dB), i.e., f is a hyperfunction on the boundary if and only if OO

OO

a

f(z) = J2 nPn + n=0

n=0

J2bnQn^'

limn^oo A/JO^J < j and l i m ^ o o y/\b^~\ < p. In the next section we will try to extend some of these results to higher dimensions. 3

Extension to Higher Dimensions

Let us move to higher dimension. Replace [—1,1] by the 2-dimensional unit sphere S={(x,y,z)eR3:

x2 + y2 + z2 = l}.

It is known that / € L2(S) can be expanded in a series of spherical surface harmonics, which are related to Legendre functions and Gegenbauer polynomials. The domains that will replace the ellipses in Neumann's theorem are

323

subsets of the 2-dimensional manifolds. T={(x,y,z)eC3:x2+y2

+ z2 = l}

and defined by T 7 = {(i,i/,z) G T : \x\2 + \y\2 + \z\2 < cosh(2 7 )} . We have T 0 = S c Tp C T 7 c T if 0 < (3 < 7 . T 7 is open and pathwise connected. 5. i

Spherical Harmonics

A spherical harmonic of degree n is a polynomial in x, y, z, with complex coefficients, which is homogeneous in x,y, z, of degree n and which is harmonic, i.e., it satisfies Laplace's equation. A spherical surface harmonic of degree n is a function defined on the two-dimensional unit sphere S by restricting a spherical harmonic of degree n onto S={(x,y,z)£R3:x2

+ y2 + z2 = l}.

Spherical surface harmonics of degree n form a complex linear space of dimension In + 1, and they can be normalized to be orthonormal. Spherical surface harmonics of different degrees are orthogonal with respect to the inner product

h,h 6 L2(S),

5 = -^ J fi(P)72(P)dP,

and they form a complete orthogonal basis for L2(S). L2(S), we have

Hence, for every / e

oo

/0P) = £/n(P),

P&S,

n-0

where /„ is the orthogonal projection of / onto the linear space of spherical surface harmonics of degree n. The series is called the Laplace series of / . The harmonic radius of / is defined as ( 11 ).

PU) = flimsuPll/n|li \

If p(f) > 1, then the function

n—>oo

324

/V 3 ) = E ^ / n ^ ) ,

°< r

P S

^

n

is harmonic on the ball with center zero and radius one; hence it solves the Dirichlet problem on the unit ball because it is equal to / on S. Can the series expansion in spherical surface harmonics of an analytic function / on 5" be extended to some domain T 7 ? Theorem 6 (Volkmer n ) Let f € L2(S), 0 e 7 for some 7 > 0. 2. f admits a holomorphic extension onto T 7 . 3. The expansion

f(P) =

^fn(P), n

of f converges uniformly on compact subsets ofT-y. Corollary 7 / £ L2(S) is analytic on S if and only if p(f) > 1. Now we shift our attention to rectangles of the form [a, b] x \b, c] . 3.2

Sphero-Conal Coordinates and Lame Polynomials

The sphero-conal coordinates form an orthogonal systems of coordinates on the 2-dimensional sphere S. They are introduced as follows. Let (x, y,z) G S with x,y,z > 0, the sphero-conal coordinates (s, t) are defined as the solutions of the quadratic equation in u : x2

,

y2

z2

u—a u—b u where 0 < a < b < c are given constants It can be shown (s — a)(t "«) x2 (b — o)(c --a ) -b) {s-b){tV2 (0 — b)(c--b) (3 - c)(t -

c)(b-c)

=

o

325

where a<s 0 one-to-one onto the open rectangle (a, b) x (b,c). Let k be a non-negative integer and 6\,...,Qf. be real numbers different from a,b,c, and consider the polynomial f(x,y,z) defined by V2

fl«.'i=ni-^+

,

z2

—c

which is homogeneous in x,y,z of degree 2k. If we restrict it to the unit sphere S, we obtain for ( i , y,z) £ S f(x,y,z)

=

Ek(s)Ek(t)

in sphero-conal coordinates, where Ek is a polynomial of degree k with zeros at 6\, ...,9k- The polynomial f{x, y, z) is harmonic if and only if Ek satisfies the differential equation

& +1 (-L- + > + _L) E> - r ± ^ ± ^ i - , = o 2 \u — a u — b u — cj [u — a)(u — o)(u — c) for some A. So let us consider the differential equation

x +-i-W-7— -^—-E=o, 2 \u — a u — b u — cj (u — a)(u — b){u — c) which has 3 singular points. This differential equation is called Lame differential equation. The following theorem summarizes the main properties of Lame polynomials that we will need. Theorem 8 / / Lame differential equation has a non-trivial solution which is holomorphic at each of the 3 singular points, then A2 = — " „ , where n is an even non-negative integer and the solution is a polynomial (Lame polynomial) of degree n/2. And for each such n, there are precisely § + 1 different real values of \\ such that Lame's equation has a solution which is a polynomial of degree n/2. The system of ^ + 1 Lame polynomials of a given degree ^ , E™(w), m = 0,1,..., ^, can be arranged in such a way that E™ has m simple zeros between a and b and \ — m zeros between b and c. Properties of Lames differential equation, Lames polynomials, and sphero-conal coordinates can be found in Whittaker and Watson l s . Let f™(x,y,z) denote those spherical surface harmonic which are in the form f™(x,y,z) = E™(s)E™(t). These are spherical surface harmonics

^a(-j_+

*

326

of degree n that are even with respect to x, y, z. We normalize the Lame polynomials in such a way that for every non negative integer n the system f™{x, y, z),m = 0,..., T| form an orthogonal basis in the (^ + l) dimensional space of spherical surface harmonics. By transforming spherical coordinates into sphero-conal coordinates, we have

{9l 92)

' * = ^JJb

W(s)W(t)

^'

where W{u) = y/\u ~a\\u-

b\ \u - c\,

gi(s,t) = fi(x,y,z),

i = 1,2.

The Lame products E™(s,t) = E™(s)E™{t),n = 0,2,4,...., m = 0,1,..., f, form a complete orthogonal system on L2(Q), with respect to the inner product ( , )Q , where Q = (a, b) x (b, c). Recall that T={(x,y,z)£C3:

x2+y2

+ z2 = l}

and T 7 = { ( i , j , z ) e T : \x\2 + \y\2 + \z\2 < cosh(2 7 )} . In sphero-conal coordinates T 7 —> G 7 ^ f, ,N ^,2 Is — al 16 — a.1 Is —6116 —61 Is — cl 16 — cl , .„ , G1 = \ (s,t) e C2 : \U J J J. + U^ < cosh 2 + 7 [_ |6 — a||c —a| | a - 6 | | c — b\ \a — c\\b — c\ Therefore, if g is holomorphic and symmetric on G 7 , then 10 oo

n/2

71=0

m=0

n even

with g? = (g,E?)Q and

lim VW\ = P<^ for some 7 > 0 . And the same conclusion holds as in Theorem 6, except that S -> Q and T 7 -> G 7 . Note G 0 = Q = [a, 6] x [6, c] U [6, c] x [a, 6]. First let us observe that since E™(s,t) are polynomials in s and t, they belong to tf(G7). Therefore, if F(s,t) e # ' ( G 7 ) , its action on E™(s,t), denoted

327

by (F,E™), is well defined. Thus, it now readily follows that F(s,t) analytic hyperfunction on G 7 , i.e., F £ H'(Gy) if oo

is an

n/2

F s

J

( >t)= £

E(F,E?)K(s,t)dW(s,t)

71=0

7Tl = 0

n even

where dW(s,t)

= (t - s)/W(S)W(t)dtds, lim

with

y/\F™\<e\

n—>oo

and F™ — (F,E™). The convergence is in the sense of H'(Gy). similar to that given in 19 .

The proof is

328

References 1. T. Boehme, The support of Mikusinski operators, Trans. Amer. Math. Soc, Vol. 176 (1973), pp. 319—334. 2. A. Erdely, W. Magnus, F. Oberhettinger, and F. Tricomi, " Higher Transcendental Functions," Vol III, McGraw Hill (1955). 3. V. Gorbacuk and M. Gorbacuk, Trigonometric series and generalized periodic functions, Soviet Math. Dokl., Vol. 2 (1981), pp. 342—346. 4. G. Johnson, "Harmonic functions on the unit die," Illinois J Math., 12 (1968), 366-385. 5. M. Morimoto, " Analytic Functionals on the Sphere, Transl. Amer. math. Soc, Vol 178, Providence, Rhode Island (1998). 6. K. Neumann, "Uber die Entwicklung einer Funktion nach den Kugelfunktionen," Halle (1862). 7. M. Sato, "The theory of hyperfunctions," I, II, J. Fac. Sci. Univ. Tokyo, 8 (1959), 139-193, 387-437. 8. L. Schwartz, Theorie des distributions, Hermann, Paris (1966). 9. G. Szego, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., Vol. 23, Providence, Rhode Island (1939). 10. H. Volkmer, "The expansion of a holomorphic function in a series of Lame products," J. Approx. theory, vol. 75 (1993), 167-174. 11. H. Volkmer, "The expansion of a holomorphic function in a Laplace series," SIAM J. Math. Analy., Vol. 21 (1990), 771-781. 12. G. Walter, "Wavelets and other Orthogonal Systems," CRC Press, 2nd Edition (2001). 13. G. Walter and P. Nevai, "Series of orthogonal polynomials as boundary values, " SIAM J. Math. Anal., Vol.l2(1981), 502-513. 14. G. Walter , "Series de Fourier, Tipo de convergencia, " Bol. Ciencias Univ. Cat. Peru., Vol. 2(1969), 40-53. 15. E. Whittaker and G. Watson, A Course on Modern Analysis, Cambridge University Press, Cambridge, England (1965). 16. A. Zayed, "Wavelet Expansions of Analytic Hyperfunctions," the Journal of Integral Transforms and Special Functions, Vol. 3 (1995), pp. 305-320. 17. A. Zayed " Wavelet transforms of periodic generalized functions, " the Journal of Mathematical Analysis and Applications., Vol. 183, 2(1994), pp. 391-412. 18. A. Zayed, "Generalized Faber expansions of hyperfunctions on analytic curves," J. Math. Soc. Japan, Vol 42 (1990), 155-170. 19. A. Zayed and G. Walter, Series of orthogonal polynomials as hyperfunctions, SIAM J. Math Anal., Vol 13 (1982), 664-675.

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