COMPLEX RNniVSIS AND APPUCRTIONS
ng • Shengjian Wu Jasi Wulan • LoYang editors
coMPiec nNnivsis RND nppucnnoNs
COMPL6X flNALVSIS FIND APPLICATIONS Proceedings of the 13th International Conference on Finite or Infinite Dimensional Complex Analysis and Applications Shantou University, China
8 - 1 2 August 2005
editors
Yuefei Wang
Shengjian Wu
Academy of Mathematics and
Peking University, Beijing, China
System Sciences, Beijing, China
Hasi Wulan
LoYang
Shantou University,
Academy of Mathematics and
Guangdong, China
System Sciences, Beijing, China
\[p World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • S H A N G H A I • HONG KONG • TAIPEI • CHENNAI
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British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
COMPLEX ANALYSIS AND APPLICATIONS Proceedings of the 13th International Conference on Finite or Infinite Dimensional Complex Analysis and Applications Copyright © 2006 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 981-256-868-9
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PREFACE
This proceedings consists of articles by participants in the 13th International Conference on Finite or Infinite Dimensional Complex Analysis and Applications, which was held at Shantou University, China, August 8-12, 2005. More than 90 participants from 17 countries or regions attended this conference. The conference was supported by the LI KA SHING Foundation, National Natural Science Foundation of China, Chinese Mathematical Society, Academy of Mathematics and Systems Science of Chinese Academy of Sciences and Shantou University. The proceedings contains all plenary lectures and some selected talks at the conference. These contributions mainly concentrated in the following topics: • • • • • •
Applied Complex Analysis Clifford Analysis Complex Dynamical Systems Complex Function Spaces and Operator Theory Complex Numerical Analysis Qusiconformal Mapping, Teichmuller Theory and Klainian Groups • Several Complex Variables • Value Distribution Theory and Complex Differential Equations • Related Topics The book offers not only reviews of these fields for researches but also perspectives on ways they may proceed, together with a comprehensive bibliography of books and papers. The book also serves to inform readers of the main research directions pursued outside their own institutions. We would like to thank all of the contributors and referees. We thank all members of The International Advisory Board and Organizing committee of the conference for their efforts. We wish to thank the listed institutions for their support. On this basis, many participants did get financial help including receiving a copy of the proceedings.
v
vi
Thanks are due to Ms. Yali Liu who was heavily involved with the conference organization and to all the other helpers from the Department of Mathematics, Shantou University. The Editors
CONTENTS
Preface
v
Complex Boundary Value Problems in a Quarter Plane H. Begehr and G. Harutyunyan
1
Some Equations over 4-Dimensional Clifford Algebra W. Cao
11
A Change of Scale Formula for Wiener Integrals of Unbounded Functions over Wiener Paths in Abstract Wiener Space K. S. Chang, D. H. Cho, B. S. Kim, T. S. Song and I. Yoo
22
The Complex Oscillation of Solutions for Differential Equations with Periodic Coefficients Z.-X. Chen and S.-A. Gao
44
Qp-Spaces: Generalizations to Bounded Symmetric Domains M. Englis
53
Order of Growth of Painleve Transcendents A. Hinkkanen and I. Laine
72
Uniqueness of Meromorphic Functions that Share Four Values B. Huang
78
On Two Perturbation Results of Second Order Linear Differential Equations with Periodic Coefficients Z.-B. Huang and Z. Chen
89
A Remark on Holomorphic Sections of Certain Holomorphic Families of Riemann Surfaces Y. Imayoshi and T. Nogi
101
a-Asymptotically Conformal Fixed Points and Holomorphic Motions Y. Jiang
109
The Hyper-Order of Solutions of Certain High Order Differential Equation C. Li and Y. Gu
130
Sharing Values and Normal Families J.-T. Li
137
The Bloch Type Spaces and the Cesaro Means J. Lin
149
Uniqueness of Meromorphic Functions Concerning Weakly Weighted-Sharing S. Lin and W. Lin
159
Uniqueness Theory of Meromorphic Functions in an Angular Domain W. Lin and S. Mori
169
The Convergence of Laplace-Stieltjes Transforms X. Luo and D. Sun
178
A New Decomposition for the Hardy Space on Domains Z. Lou
185
Transversality on Coordinated Manifolds T.-W. Ma
189
On Logarithmic (a, /3)-Bloch Space X. Meng
197
Infinite Series in Japanese Mathematics of the 18th Century M. Morimoto
203
Meromorphic Functions that Share Four Small Functions G. Qiu
218
Integration Operators on the BMOA Type Spaces D. Qu
225
The Growth of Analytic Functions of Infinite Order Represented by Laplace-Stieltjes Transformations L. Shang and Z. Gao
231
On T-direction of Meromorphic Function M. Shu and C. Yi
242
On Nevanlinna Type Classes ./V. Sukantamala and Z. Wu
249
Fredholm Module and Cauchy Integral Operator J. Tao
261
On ^-Modulus and ^-Capacities Equalities in Metric Measure Spaces J.-Q. Wu
267
A Criterion of Bloch Functions and Little Bloch Functions P. Wu
276
Some Results of Uniqueness for Algebroid Functions Z.-X. Xuan and Z. Gao
281
On Non-Existence of Teichmiiller Extremal G. Yao
293
Two Meromorphic Functions Sharing Four Small Functions in the Sense of EK)(0, F) = EK)(0, G) W. Yao
300
X
Multiplication Operators in the a-Bloch Spaces S. Ye
309
An Improved Inequality for Meromorphic Functions with Few Simple Zeros Q. Zhang
320
The Mobius Invariance of Besov Spaces on the Unit Ball of C" K. Zhu
328
COMPLEX B O U N D A R Y VALUE P R O B L E M S IN A QUARTER P L A N E
H. B E G E H R / . Math. Inst. FU Berlin Arnimallee 3 14195 Berlin, Germany Email:
[email protected] G. HARUTYUNYAN Fac. Computer Sci. Appl. Math. Yerevan State Univ. Alex Manoogian 375049 Yerevan, Armenia Email:
[email protected] 1
Dirichlet and Neumann boundary value problems are considered for the inhomogeneous Cauchy-Riemann equation in a quarter plane. Solvability conditions and solutions are given in explicit form.
Keywords: Inhomogeneous Cauchy-Riemann equation, quarter plane, Dirichlet boundary value problem, Neumann boundary value problem. 2000 Mathematics
Subject Classification:
30E25, 35C15, 35F15
1. Introduction The Cauchy-Pompeiu representation formula for functions in the upper right quarter plane Qi of the complex plane C follows from the Gauss theorem for regular domains and a limiting process, see 1. Cauchy-Pompeiu representation where
Any w € C ^ Q i j C ) D C(Qi;C),
Qi = {z £ C : 0 < R e z , 0 < l m z } , l
2
for which for some 0 < 6 the function (1 + r)sM(r,w)
with
M{r,w) = max{|to(z)| : \z\ = r,0 < Re z,0 < Im z} is bounded in M + and iuf e Li (Qi; C) is representable as +oo
+oo
w(z) = — / w(t)w 27rz y £-z
—- / iu(ii)-— : y 2m J 't + iz
0
-K J
0
wdO^—1CVs; C,- z
(1) '
K
Q!
Introducing the harmonic Green function for Qi G i ( z , 0 = log
C2-z2 2
log
2
C -z
(C-z)(C (C-z)(S
+ z) + z)
this representation can be altered into 1 (C + z)
1 2
(C-^)2
^dtj - I J Wf(QdzGi (z, Qd$dr),
Qi
(2) see 1 . For respective representations in the upper half plane see 8 , in the unit disc see e.g. 2 ' 3 - 4 ' 5 and in arbitrary regular domains see 9 . In x formula (1) is adjusted to Schwarz boundary data and the solution to the Schwarz boundary value problem is given for the inhomogeneous Cauchy-Riemann and Bitsadze equations. Here the Dirichlet and Neumann boundary value problems are treated for the Cauchy-Riemann equation. These problems are known to be overdetermined so that solvability conditins have to be determined, see e.g 4 ' 5 ' 6 ' 7 .
2. The Dirichlet problem The complex Gauss theorem in the form -—: / w(z)dz = — 2TTI J dD
•K J D
Wz{z)dxdy
for functions w £ Cl(D; C) D C(D; C) for bounded domains D in the complex plane C with piecewise smooth boundary dD, see e.g. 2 , applied for »1R
= {z = x + iy : \z\ < R, 0 < x, 0 < y}
3
besides leading to (1) also supplies for w e C ^ Q i j C ) f~l C(Qi;C) with in M + bounded (1 + r)sM{r, w) and Wg G L P , 2 (Q; C), 2 < p, and z G Qi + 00
1 f , s dt — / w(t) K m J 't-z
+ 00
1 f ,., dt / w(it) K 2m J 't + iz
o
If ,^d£dn S L h"c(QT — =0, Q ir J ^'£ - z
„
,n. (3) w
o
+oo
+00
1
M dt /f w(t) KJ 2m J t +z
l
f/ w(it) C*\ dt v 2m J 't-iz
+ 0O
l
f/ wAC)fs\dtdr> = n0, (A\ (4) cvs; nj C + -*
+ OO
1
f , , dt / wt K 2m J 't + z 0
1
f , , dt / «;(it) K 2m J ' t - iz
1 f ,,. d^dn / tUffO-r2—^ = 0. CK TT J ^\ + z
0
(5) w
Qi
Properly combined with (1) they lead to solutions to Schwarz problems, see 1. But they together with (1) also lead to solvability conditions and solutions to the Dirichlet problem for the Cauchy-Riemann equation Theorem 1 The Dirichlet problem Wz = f in Qi, w = 7i for 0 < x, y = 0, w = 72 for 0 < y, x — 0 for f G Lp i 2 (Qi;C),2 < p,7i,72 G C(R;C) aucfc i/iat (1 + t)*7l(*),(! + i)l572(^) are bounded for some 0 < 6 and satisfying the compatibility condition 71(0) = 72(0) is uniquely weakly solvable in the class C 1 (Qi;C) D C(Qi; C) if and only if + + 00 00
1 2m
f
+ + 00 00
, . dt
1
/•
0
0
+00
+00
0 +00
0 +00
1
/"
, . dt
1
/"
..
dt
If
_,,.. d£d??
„
. ,
* * z = 0,
(7)
Qi
,
N
dt
If
r,^d4dn
n
,N
0
The solution is +00
+00
«'>-hhMih-*if«&;-kJm&
