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
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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.
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ISBN 9812381619
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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 InfraRed 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 PseudoDifferential 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 FourierBorel 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 TwoDimensional 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, HigashiOsaka, 5778502 Japan Email:
[email protected] TATSUYA KOIKE Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto, 6068502 Japan Email:
[email protected] YOSHITSUGU TAKEI Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 6068502 Japan Email:
[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) ("nonadjacent 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 nonapparent 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 nonadjacent 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 nonadiabatic 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 singlevalued 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 nonadjacent 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 1i, 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 nonadjacent 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 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 nonadjacent 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 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
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 nonadjacent 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 nonadjacent 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 GrantinAid (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. 6984. [2] : On the exact WKB analysis for the third order ordinary differential equations with a large parameter. Asian J. Math., 2(1998), 625640. [3] : On the exact steepest descent method — a new method for the description of Stokes curves. J. Math. Phys., 42(2001), 36913713. [4] : Exact WKB analysis of nonadiabatic 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), 9881002. [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. 8092. [8] : An explicit description of the connection formula for the first Painleve equation. Toward the Exact WKB Analysis of Differential Equations, Linear or NonLinear, Kyoto Univ. Press, 2000, pp. 271296. [9] K. Uchiyama: On examples of Voros analysis of complex WKB theory. Analyse algebrique des perturbations singulieres, I, Methodes resurgentes, Hermann, 1994, pp. 104109. [10] : Graphical illustration of Stokes phenomenon of integrals with saddles. Toward the Exact WKB Analysis of Differential Equations, Linear or NonLinear, Kyoto Univ. Press, 2000, pp. 8796. [II] A. Voros: The return of the quartic oscillator — The complex WKB method. Ann. Inst. Henri Poincare, 39(1983), 211338.
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 341, Ohkubo, Shinjyuuku, Tokyo 1698555, 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 + ' . .. (11*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 blowingup 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\q2u (E)
Ut
=
ApU
+
(lg)° '
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* = /
^0. Set a
E(u(t)) = \\Vu(t)\\lP  JB
Theorem 2 (Blowup) Let p = q > 2,
{
^
{ ) a
0 < a
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
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) = u5f)
b(u) = f
u n
'
d
dx.
Here we define the socalled potential depth d by d=
inf supJ(Au). «ex\{o}A>o
Furthermore, if 0 < a < — 2p
(which is weaker than 0 < a < P ) and q < — , then Lemma 1 with s = a and 3 = a N  p
assures that there exists a constant C such that b(u)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) 0}.
In the case that 2 < p < q, we state our results in the following two theorems for the local and global existence of radially symmetric solutions of (E).
28
T h e o r e m 5 (Local Existence)
Np Let 1 < q < —— r + 1 for N > p, and 2{N  p)
'^~ ' + P . Then for any 2p UQ G X, there exists a positive number To = To(uojf) such that (E) has a radially symmetric strong solution u(x, t) satisfying 1 < q < oo for N < p. Let 0 < a
0, then Um[7.(p2)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• 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 WL 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 !—^ pr;— 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 < p < o c , l < < 7 < 2 and T > 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}, /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 '
(7)
l/l=*
where dj is a (0,n) current. What one does next is to relate this current dj to the currentvalued holomorphic map in C A ^ /i • • • fnYXdh
2A
A • • • A df„
=  / r a / , Re A »
0
Using the BernsteinSato functional equation one sees that the holomorphic function A K» A f  /  2 ( A _ 1 a / A 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), (len)D 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\ 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(zw)) = ( z x + w i , . . .
,zn+wn,t+s+2lm(zw)),
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 lefttranslation 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\ — 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) = e2"xlz\2z»"Li»^{4Tr\\z\2) 2
v l
if
2
WttV(z) = e "Wz ' L%ri(4K\\z\ )
z
if n 0, A > 0,

defined by
n
where C*
is a positive constant chosen so that W^^i 2 (C") = 1> ie> x
"•"
TT f 7T (ma3c{/ij,i/j})l)~g / i I (47rA)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 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{2K\r2Y
t=o
m+n
1
£
~ (27rA)n 1
)dp
n+\v\+kl 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 V A + m >) n n i/e(z+) fie(z+)
61 we have £
W
A
> ) = Z2^ITy
1 £ ( £
/,(%,(A))
2p2n+2e1 \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+Mi(27rAr2)^
All in all for w(p2) = amp2m + amip2{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)