Finite or Infinite Dimensional Complex Analysis 9781482270594, 1482270595, 978-0-8247-0442-1

Table of contents :
Content: Cover
Half Title
Series Page
Lecture Notes In Pure and Applied Mathematics
Title Page
Copyright Page
Contents
Preface
Contributors
1 Modular Space for Complete Intersection Curve-Singularities
2 Quasinormability of Vector Valued Sequence Spaces
3 Holomorphic Mappings and Cardinality
4 Approximation Numbers for Polynomials
5 Applications of Laguerre Calculus to Dirichlet Problem of the Heisenberg Laplacian
6 The Pisier-Schütt Theorem for Spaces of Polynomials
7 Canonical v. Functional Extensions of Holomorphic Functions 8 On Convergence of Trigonometric Interpolation for Periodic Analytic Functions9 On the Double Series Expansion with Harmonic Components
10 Extension of Pluriharmonic Functions in Locally Convex Spaces
11 Regeneration in Complex, Quaternionic, and Clifford Analysis
12 Schauder Decompositions of Weighted Spaces of Holomorphic Functions
13 Spaces of Banach-Valued Holomorphic Functions in the Polydisk in Connection with Their Boundary Values
14 Univalent C[sup(1)] Mappings on the Unit Ball in C[sup(n)]
15 The Growth Theorem of Biholomorphic Mappings on a Banach Space 16 Stability of Solutions for Singular Integral Equations for Two Classes in Locally Convex Spaces17 The Nevanlinna's First Main Theorem for Holomorphic Hermitian Line Bundles
18 On Distortion Theorem for N-Set Quasiconformal Mappings
19 Monodromy of a Holomorphic Family of Riemann Surfaces: A Remark on Monodromy of a Holomorphic Family of Riemann Surfaces Induced by a Kodaira Surface and the Nielsen-Thurston-Bers Classification of Surface Automorphisms
20 Characterizations of Holomorphy of Domains through Validity of Theorem A, B or Oka's Principle 21 Envelope of Biregularity and Ƒ-Convexity in Clifford Analysis22 On a Representative Domain in Matrix Space
23 A New Approximation of the Navier-Stokes Equations
24 Généralisation du Produit de Blaschke dans le Bidisque-Unité
25 P-Spaces
26 On the Arithmetic-Geometric Mean Inequality
27 Weakly Sufficient Sequences in the Space of Functions of Polynomial Growth
28 Some New Results of Clifford Analysis in Higher Dimensions
29 Matrix Criteria for the Extension of Solutions of the General Linear System of Partial Differential Equations with Function Coefficients 30 Existence of the Solutions of the Cauchy-Riemann Equations31 Solution of a Class of Periodic Hypersingular Integral Equations
32 On Two Questions in Clifford Analysis and Octonion Analysis
33 An Inequality of Teichmüller's Distance
34 Cyclically Symmetric Circular Crack Problems of Different Media
35 The Grassmannian Manifold Associated to a Bounded Symmetric Domain
36 Holomorphic Mappings and Weak Continuity
37 A Characterization of Analytic Functionals on the Sphere I
38 Ideals of Holomorphic Functions on Infinite Dimensional Spaces

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finite on infinite dimensional complex analysis

PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes

EXECUTIVE EDITORS Zuhair Nashed University of Delaware Newark, Delaware

Earl J. Taft Rutgers University New Brunswick, New Jersey

EDITORIAL BOARD M. S. Baouendi University o f California, San Diego Jane Cronin Rutgers University JackK. Hale Georgia Institute of Technology

Anil Nerode Cornell University Donald Passman University o f Wisconsin, Madison Fred S. Roberts Rutgers University

David L. Russell S. Kobayashi Virginia Polytechnic Institute University of California, and State University Berkeley Marvin Marcus University of California, Santa Barbara W. S. Massey Yale University

Walter Schempp Universitat Siegen Mark Teply University o f Wisconsin, Milwaukee

LECTURE NOTES IN PURE AND APPLIED MATHEMATICS

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N. Jacobson, Exceptional Lie Algebras L -A Lindahl and F. Poulsen, Thin Sets in Harmonic Analysis I. Satake, Classification Theory of Sem i-Sim ple Algebraic Groups F Hirzebruch et at., Differentiable Manifolds and Quadratic Forms I. Chavel, Riemannian Symmetric Spaces of Rank One R. B. Bunckel, Characterization of C (X) Among Its Subalgebras B. R. McDonald et a l, Ring Theory y.-T. Siu, Techniques of Extension on Analytic Objects S. R. Caradus et al., Calkin Algebras and Algebras of Operators on Banach Spaces E. O. Roxin et al., Differential Gam es and Control Theory M. Orzech and C. Small, The Brauer Group of Commutative Rings S. Thornier; Topology and Its Applications J. M. Lopez and K. A. Ross, Sidon Sets W. W. Comfort and S. Negrepontis, Continuous Pseudometrics K. McKennon and J. M. Robertson, Locally Convex Spaces M. Carmeli and S. Matin, Representations of the Rotation and Lorentz Groups G. B. Setigman, Rational Methods in Lie Algebras D. G. de Figueiredo, Functional Analysis L Cesari et al., Nonlinear Functional Analysis and Differential Equations J. J. Schaffer, Geom etry of Spheres in Normed Spaces K. Yano and M. Kon, Anti-Invariant Submanifolds W. V. Vasconcelos, The Rings of Dimension Two R. E. Chandler, Hausdorff Compactifications S. P. Franklin and B. V. S. Thomas, Topology S. K. Jain, Ring Theory B. R. McDonald and R. A. Morris, Ring Theory II R. B. Mura and A. Rhemtulla, Orderable Groups J. R. Graef, Stability of Dynam ical Systems H.-C. Wang, Homogeneous Branch Algebras E. O. Roxin et al., Differential Gam es and Control Theory II R. D. Porter, Introduction to Fibre Bundles M. Altman, Contractors and Contractor Directions Theory and Applications J. S. Golan, Decomposition and Dimension in Module Categories G. Fairweather, Finite Elem ent Galerkin Methods for Differential Equations J. D. Sally, Numbers of Generators of Ideals in Local Rings S. S. Miller, Complex Analysis R. Gordon, Representation Theory of Algebras M. Goto and F. D. Grosshans, Semisimple Lie Algebras A. I. Arruda et al., M athem atical Logic F. Van Oystaeyen, Ring Theory F. Van Oystaeyen and A. Verschoren, Reflectors and Localization M. Satyanarayana, Positively Ordered Semigroups D. L Russell, Mathematics of Finite-Dimensional Control Systems P.-T. Liu and E. Roxin, Differential Games and Control Theory III A. Geramita and J. Seberry, Orthogonal Designs J. Cigler, V. Losert, and P. Michor, Banach Modules and Functors on Categories of Banach Spaces P.-T. Liu and J. G. Sutinen, Control Theory in Mathematical Economics C. Byrnes, Partial Differential Equations and Geometry G. Klambauer, Problems and Propositions in Analysis J. Knopfmacher, Analytic Arithmetic of Algebraic Function Fields F. Van Oystaeyen, Ring Theory B. Kadem, Binary Tim e Series J. Barros-Neto and R. A. Artino, Hypoelliptic Boundary-Value Problems R. L Sternberg et al., Nonlinear Partial Differential Equations in Engineering and Applied Science B. R. McDonald, Ring Theory and Algebra III J. S. Golan, Structure Sheaves Over a Noncommutative Ring T. V. Narayana et al., Combinatorics, Representation Theory and Statistical Methods in Groups T. A. Burton, Modeling and Differential Equations in Biology K. H. Kim and F. W. Roush, Introduction to Mathematical Consensus Theory

60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100. 101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111. 112. 113. 114. 115. 116. 117. 118. 119. 120.

J. Banas and K. Goebel', M easures of Noncompactness in Banach Spaces O. A Nielson, Direct Integral Theory J. E. Smith eta!., Ordered Groups J. Cronin, Mathematics of Cell Electrophysiology J. W. Brewer, Power Series Over Commutative Rings P. K. Kamthan and M. Gupta, Sequence Spaces and Series T. G. McLaughlin, Regressive Sets and the Theory of Isols T. L Herdman et al., Integral and Functional Differential Equations R. Draper, Commutative Algebra W. G. McKay and J. Patera, Tables of Dimensions, Indices, and Branching Rules sentations of Simple Lie Algebras R. L Devaney and Z. H. Nitecki, Classical Mechanics and Dynamical Systems J. Van Geel, Places and Valuations in Noncommutative Ring Theory C. Faith, Injective Modules and Injective Quotient Rings A Fiacco, Mathematical Programming with Data Perturbations I P. Schultz et al., Algebraic Structures and Applications L Bican et al., Rings, Modules, and Preradicals D. C. Kay and M. Breen, Convexity and Related Combinatorial Geom etry P. Fletcher and W. F. Undgren, Quasi-Uniform Spaces C.-C. Yang, Factorization Theory of Meromorphic Functions O. Taussky, Ternary Quadratic Forms and Norms S. P. Singh and J. H. Burry, Nonlinear Analysis and Applications K. B. Hannsgen et al., Volterra and Functional Differential Equations N. L. Johnson et al., Finite Geom etries G. I. Zapata, Functional Analysis, Holomorphy, and Approximation Theory S. Greco and G. Valla, Commutative Algebra A. V. Fiacco, Mathematical Programming with Data Perturbations II J.-B. Hiriart-Urruty et al., Optimization A. Figa Talamanca and M. A. Picardello, Harmonic Analysis on Free Groups M. Harada, Factor Categories with Applications to Direct Decomposition of Modules V. I. IstrStescu, Strict Convexity and Complex Strict Convexity V. Lakshmikantham, Trends in Theory and Practice of Nonlinear Differential Equations H. L. Manocha and J. B. Srivastava, Algebra and Its Applications D. V. Chudnovsky and G. V. Chudnovsky, Classical and Quantum Models and Problems J. W. Longley, Least Squares Computations Using Orthogonalization Methods L. P. de Alcantara, Mathem atical Logic and Formal Systems C. E. Aull, Rings of Continuous Functions R. Chuaqui, Analysis, Geom etry, and Probability L. Fuchs and L. Salce, Modules O vervaluation Domains P. Fischer and W. R. Smith, Chaos, Fractals, and Dynamics W. B. Powell and C. Tsinakis, Ordered Algebraic Structures G. M. Rassias and T. M. Rassias, Differential Geometry, Calculus of Variations, Applications R -E . Hoffmann and K. H. Hofmann, Continuous Lattices and Their Applications J. H. Lightboume III and S. M. Rankin III, Physical Mathematics and Nonlinear Partial Equations C. A. Baker and L. M. Batten, Finite Geometries J. W. Brewer et al., Linear Systems O ver Commutative Rings C. McCrory and T. Shifrin, Geom etry and Topology D. W. Kueke et al., Mathem atical Logic and Theoretical Computer Science B.-L. Lin and S. Simons, Nonlinear and Convex Analysis S. J. Lee, Operator Methods for Optimal Control Problems V. Lakshmikantham, Nonlinear Analysis and Applications S. F. McCormick, Multigrid Methods M .C . Tangora, Computers in Algebra D. V. Chudnovsky and G. V. Chudnovsky, Search Theory D. V. Chudnovsky and R. D. Jenks, Computer Algebra M. C. Tangora, Computers in Geom etry and Topology P. Nelson et al., Transport Theory, Invariant Imbedding, and Integral Equations P. Clement et al., Semigroup Theory and Applications J. Vinuesa, Orthogonal Polynomials and Their Applications C. M. Dafermos et al., Differential Equations E. O. Roxin, Modem Optimal Control J. C. Diaz, Mathematics for Large Scale Computing

for Repre­

Arithmetic

and Their

Differential

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P. S. Milojevi£ Nonlinear Functional Analysis C. Sadosky, Analysis and Partial Differential Equations R. M. Shortt, General Topology and Applications R. Wong, Asymptotic and Computational Analysis D. V. Chudnovsky and R. D. Jenks, Computers in Mathem atics W. D. Wallis et at., Combinatorial Designs and Applications S. Elaydi, Differential Equations G. Chen et al., Distributed Param eter Control Systems W. N. Everitt, Inequalities H . G. Kaper and M. Garbey, Asymptotic Analysis and the Numerical Solution of Partial Differ­ ential Equations O. Anno et al., Mathem atical Population Dynamics S. Coen, Geom etry and Complex Variables J. A. Goldstein et al., Differential Equations with Applications in Biology, Physics, and Engineering S. J. Andima et al., General Topology and Applications P Ctement et al., Semigroup Theory and Evolution Equations K. Jarosz, Function Spaces J. M. Bayodet al., p-adic Functional Analysis G. A. Anastassiou, Approximation Theory R. S. Rees, Graphs, Matrices, and Designs G. Abrams et al., Methods in Module Theory G. L Mullen and P. J .S . Shiue, Finite Fields, Coding Theory, and Advances in Communications and Computing M. C. Joshi and A. V. Balakrishnan, M athem atical Theory of Control G. Komatsu and Y. Sakane, Complex Geometry I. J. Bakelman, Geometric Analysis and Nonlinear Partial Differential Equations T. Mabuchi and S. Mukai, Einstein Metrics and Yang-M ills Connections L. Fuchs and R. Gdbel, Abelian Groups A. D. Pollington and W. Moran, Number Theory with an Emphasis on the Markoff Spectrum G. Dore et al., Differential Equations in Banach Spaces T. West, Continuum Theory and Dynamical Systems K. D. Bierstedt et al., Functional Analysis K. G. Fischer et al., Computational Algebra K. D. Elworthy et al., Differential Equations, Dynamical Systems, and Control Science P.-J. Cahen, et al., Commutative Ring Theory S. C. Cooperand W. J. Thron, Continued Fractions and Orthogonal Functions P. Ctement and G. Lumer, Evolution Equations, Control Theory, and Biomathematics M. Gyltenberg and L. Persson, Analysis, Algebra, and Computers in Mathematical Research W. O. Bray et at., Fourier Analysis J. Bergen and S. Montgomery, Advances in Hopf Algebras A. R. Magid, Rings, Extensions, and Cohomology N. H. Pavel, Optimal Control of Differential Equations M. Ikawa, Spectral and Scattering Theory X. Liu and D. Siegel, Comparison Methods and Stability Theory J.-P. Zotdsio, Boundary Control and Variation M. K fiz e k e ta !., Finite Elem ent Methods G. Da Prato and L. Tubaro, Control of Partial Differential Equations E. Ballico, Projective Geometry with Applications M. Costabel et al., Boundary Value Problems and Integral Equations in Nonsmooth Domains G. Ferreyra, G. R. Goldstein, and F. Neubrander, Evolution Equations S. Huggett, Twistor Theory H. Cook et al., Continua D .F . Anderson and D. E. Dobbs, Zero-Dimensional Commutative Rings K. Jarosz, Function Spaces V. Ancona et al., Complex Analysis and Geometry E. Casas, Control of Partial Differential Equations and Applications N. Kalton etal., Interaction Between Functional Analysis, Harmonic Analysis, and Probability Z. Deng et al.. Differential Equations and Control Theory P. Marcellini et al. Partial Differential Equations and Applications A. Kartsatos, Theory and Applications of Nonlinear Operators of Accretive and Monotone Type M. Maruyama, Moduli of Vector Bundles A. Ursini and P. Agliand, Logic and Algebra X. H. Cao et al., Rings, Groups, and Algebras D. Arnold and R. M. Rangaswamy, Abelian Groups and Modules S. R. Chakravarthy and A. S. Alfa, Matrix-Analytic Methods in Stochastic Models

184. 185. 186. 187. 188. 189. 190. 191. 192. 193. 194. 195. 196. 197. 198. 199. 200. 201. 202. 203. 204. 205. 206. 207. 208. 209. 210. 211. 212. 213. 214.

J. E. Andersen et al., Geom etry and Physics P.-J. Cahen et a!.. Commutative Ring Theory J. A. Goldstein et a/., Stochastic Processes and Functional Analysis A. Sorbi, Complexity, Logic, and Recursion Theory G. D a Prato and J.-P. Zolesio, Partial Differential Equation Methods in Control and Shape Analysis D. D. Anderson, Factorization in Integral Domains N. L Johnson, Mostly Finite Geom etries D. Hinton and P. W. Schaefer, Spectral Theory and Computational Methods of Sturm -Liouviile Problems W. H. Schikhofet al., p-adic Functional Analysis S. Sertdz, Algebraic Geometry G. Caristi and E. Mitidieri, Reaction Diffusion Systems A. V. Fiacco, Mathem atical Programming with Data Perturbations M. KfiZek et al., Finite Elem ent Methods: Superconvergence, Post-Processing, and A Posteriori Estim ates S. Caenepeel and A. Verschoren, Rings, Hopf Algebras, and Brauer Groups V. Drensky et al., Methods in Ring Theory W. B. Jones and A. Sri Ranga, Orthogonal Functions, Moment Theory, and Continued Fractions P. E. Newstead, Algebraic Geometry D. Dikranjan and L. Salce, Abelian Groups, Module Theory, and Topology Z. Chen et al., Advances in Computational Mathematics X. Caicedo and C. H. Montenegro, Models, Algebras, and Proofs C. Y. Yildirim and S. A. Stepanov, Number Theory and Its Applications D. E. Dobbs et al., Advances in Commutative Ring Theory F. Van Oystaeyen, Commutative Algebra and Algebraic Geom etry J. Kakol et al., p-adic Functional Analysis M. Boulagouaz and J.-P. Tignol, Algebra and Number Theory S. Caenepeel and F. Van Oystaeyen, Hopf Algebras and Quantum Groups F. Van Oystaeyen and M. Saorin, Interactions Between Ring Theory and Representations of Algebras R. Costa et al., Nonassodative Algebra and Its Applications T.-X. He, W avelet Analysis and Multiresolution Methods H. Hudzik and L. Skrzypczak, Function Spaces: The Fifth Conference J. Kajiwara et al., Finite or Infinite Dimensional Complex Analysis

Additional Volumes in Preparation

finite or infinite dimensional complex analysis proceedings of the Seventh International Colloquium

edited by .Jaji Kajiwara Kyushu University Fukuoka,Japan Zhong Li Peking University Beijing, People's Republic of China Kwang Ho Shon Pusan National University Pusan, Korea

This project has been executed with a grant from the Specific Research, Computer Science Laboratory, Fukuoka Institute of Technology.

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business

First published 2000 by Marcel Dekker Published 2019 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2000 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works ISBN-13: 978-0-8247-0442-1 (pbk) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com(http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

Preface

This Proceedings of the Seventh International Colloquium on Finite or Infinite Dimensional Complex Analysis collects excellent up-to-date research results on function theory of 1. One complex variable 2. Several complex variables 3. Infinitely many complex variables as the name of the colloquium indicates. Also featured are 4. Quatemionic and octonionic variables and variables in Clifford algebra and related topics in 5. Hyperfunctions 6. Differential equations 7. Numerical analysis and associated 8. Applied sciences, e.g., acoustics Hence, this book offers not only reviews of these fields for researchers, 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. The colloquia originated when many Korean and Japanese mathematicians recognized that they drew only upon the research done in Europe and the United States, without looking at the nearest neighbor countries. With Joji Kajiwara of Kyushu University in Japan as Chairman, they established the Organizing Committee of the Korean-Japanese Colloquium on Finite or Infinite Dimensional Complex Analysis. Kwang Ho Shon of Pusan University in Korea held the First Korean-Japanese Colloquium on Finite or Infinite Dimensional Complex Analysis at Pusan University in July 1993. Hideaki Kazama of Kyushu University held the Second Colloquium at Kyushu University in July 1994. The committee invited their Chinese colleague Zhong Li of Peking University and extended the name from Korean-Japanese to International. Suk Young Lee held the Third International Colloquium on Finite or Infinite Dimensional Complex Analysis at Iwha University in Seoul, Korea, July/August iii

iv

Preface

1995. Tadayoshi Kanemaru held the Fourth Colloquium at Kumamoto Univer­ sity in Japan, August 1996. Zhong Li and Lo Yang of Academia Sinica held the Fifth Colloquium at Peking University in Beijing, August 1997. Em Gun Kwon of Andong University held the Sixth Colloquium at Andong University in Korea, August 1998. The Seventh Colloquium was held at the Socioeducational Center of Fukuoka Prefecture by Masaru Nishihara of Fukuoka Institute of Technology in Japan. The Eighth will be held August, 2000 at Shandong, China, by Lo Yang, Zhong Li, Shengjian Wu, and Hong-Xun Yi {e-mail: [email protected]). The Ninth will be held August 2001 at Hanoi, Vietnam, by Hung Son Le {e-mail: [email protected]) and Hai Khoi Le. We would like to express our heartfelt gratitude to all contributors. The papers were selected carefully, not only for their mathematical quality, but also for their much-needed practical applicability in the contributors’ countries: Japan, Korea, Russia, Ukraine, and Vietnam. We would like to express sincere appreciation to all referees, especially to Professor Sean Dineen of University College Dublin, Ireland, who not only selected the best research on infinite dimensional complex analysis throughout the world but also refereed papers submitted in his fields and cooperated to make this book the best collection of papers on infinite dimensional complex analysis to date. We would like to thank the Fukuoka Institute of Technology, which endowed Professor Masaru Nishihara with the G rant from the Specific Research, Computer Science Laboratory, Fukuoka Institute of Technology and enabled Marcel Dekker, Inc. to provide a complimentary copy of this book to each registered contributor except in Japan. We thank Professor Nishihara, who not only chaired the Seventh Colloquium at Fukuoka, but who also obtained this grant. Joji Kajiwara Zhong Li Kwang Ho Shon

Contents

Preface Contributors

iii xi

i.

Modular Space for Complete Intersection Curve-Singularities A. G. Aleksandrov

2.

Quasinormability of Vector Valued Sequence Spaces F. Blasco

21

3.

Holomorphic Mappings and Cardinality C. Boyd

29

4.

Approximation Numbers for Polynomials H.-A. Braunss, H. Junek, andE. Plewnia

35

5.

Applications of Laguerre Calculus to Dirichlet Problem of the Heisenberg Laplacian D.-C. Chang and J. Tie

1

47

6.

The Pisier-Schtitt Theorem for Spaces of Polynomials A. Defant, J. C. Diaz, D. Garcia, and M. Maestre

55

7.

Canonical v. Functional Extensions of Holomorphic Functions S. Dineen

63

8.

On Convergence of Trigonometric Interpolation for Periodic Analytic Functions J. Du and H. Liu

69

9.

On the Double Series Expansion with Harmonic Components K. Fujita

77

10.

Extension of Pluriharmonic Functions in Locally Convex Spaces Y. Fukushima

87

V

vi

Contents

11.

Regeneration in Complex, Quatemionic, and Clifford Analysis Y. Fukushima

12.

Schauder Decompositions of Weighted Spaces of Holomorphic Functions D. Garcia, M. Maestre, and P. Rueda

13.

Spaces of Banach-Valued Holomorphic Functions in the Polydisk in Connection with Their Boundary Values V. S. Guliev

95

103

109

14.

Univalent C1 Mappings on the Unit Ball in Cn H. Hamada and G. Kohr

125

15.

The Growth Theorem of Biholomorphic Mappings on a Banach Space T. Honda

133

16.

Stability of Solutions for Singular Integral Equations for Two Classes in Locally Convex Spaces C.-G. Hu, J.-H. Jia, andC. Liu

139

The Nevanlinna’s First Main Theorem for Holomorphic Hermitian Line Bundles C.-G. HuandX.-F. Ye

149

17.

18.

On Distortion Theorem for TV-Set Quasiconformal Mappings X. Huang

19.

Monodromy of a Holomorphic Family of Riemann Surfaces: A Remark on Monodromy of a Holomorphic Family of Riemann Surfaces Induced by a Kodaira Surface and the Nielsen-Thurston-Bers Classification of Surface Automorphisms Y. Imayoshi, M. Ito, and H. Yamamoto

20.

Characterizations of Holomorphy of Domains through Validity of Theorem A, B or Oka’s Principle J. Kajiwara, Y. Fukushima, L. Li, andX. D. Li

157

169

179

21.

Envelope of Biregularity and ^ “-Convexity in Clifford Analysis J. Kajiwara and D.-G. Zhou

201

22.

On a Representative Domain in Matrix Space T. Kanemaru

209

23.

A New Approximation of the Navier-Stokes Equations H. Kato

213

Contents

vii

24.

Generalisation du Produit de Blaschke dans le Bidisque-Unite K. Kato

219

25.

P- Spaces B. S. Khots

227

26.

On the Arithmetic-Geometric Mean Inequality E. G. Kwon and K. H. Short

233

27.

Weakly Sufficient Sequences in the Space of Functions of Polynomial Growth H. K. Le

28.

Some New Results of Clifford Analysis in Higher Dimensions H S. Le

29.

Matrix Criteria for the Extension of Solutions of the General Linear System of Partial Differential Equations with Function Coefficients H. S. Le and T. V. Nguyen

237

245

267

30.

Existence of the Solutions of the Cauchy-Riemann Equations J. Lee and K. H. Shon

283

31.

Solution of a Class of Periodic Hypersingular Integral Equations X Li

289

32.

On Two Questions in Clifford Analysis and Octonion Analysis X.-M. Li

293

33.

An Inequality of Teichmiiller’s Distance Z. Li

301

34.

Cyclically Symmetric Circular Crack Problems of Different Media J.Lu

309

35.

The Grassmannian Manifold Associated to a Bounded Symmetric Domain M. Mackey

317

36.

Holomorphic Mappings and Weak Continuity L. A. de Moraes

325

37.

A Characterization of Analytic Functionals on the Sphere I M. Morimoto and M. Suwa

331

38.

Ideals of Holomorphic Functions on Infinite Dimensional Spaces J. Mujica

337

v iii

Contents

39.

Local Cohomology and Ideal of Partial Differential Operators Y. Nakamura

40.

On the Large Time Behavior of Solutions of Some Nonlinear Degenerate Parabolic System T. Nanbu

41.

The Extension Problem for the Maxwell System with Additional Conditions T. V. Nguyen

42.

Application of Extension Theorem of the Solution of a First Order General Linear System of Partial Differential Equations T. V. Nguyen

345

353

363

373

43.

The Extension of Holomorphic Functions on a Nuclear Space M. Nishihara

389

44.

Characterization of Clifford Differentiable Functions K. Nono

395

45.

Hamiltonian Algorithm for Sound Synthesis K. Ohya and K. Shinjo

401

46.

Proper Inclusions of Function Spaces and Ryll-Wojtaszczyk Polynomials on die Unit Ball of C" C. Ouyang

409

Boundary Behavior of Nonlocally Convex Vector-Valued Hardy Classes in the Complex Ball C. Ouyang and Z. Chen

427

Boundary Properties of Axial-Symmetrical Potential and Stokes Flow Function S. Plaksa

443

Bessel Function by Parallel Integration of ODEs under Parallel Virtual Machine Environment V. M. Raffee

457

Numerical Simulation of Air Core Magnet in 3 Dimensions under PVM Environment V. M. Raffee, R. Rajput, N Dhoble, A. Rawat, and S. P. Mhaskar

465

47.

48.

49.

50.

51.

Some Results for Modified Kontorovitch-Lebedev Integral Transforms J. Rappoport

473

Contents

ix

52.

Dynamics of Transcendental Meromorphic Functions A. P, Singh

479

53.

Exponential Forms with Exponential Growth R. L. Soraggi

495

54.

Grothendieck Duality and Hermite-Jacobi Formula S. Tajima

503

55.

Finely Holomorphic Functions in Contour-Solid and Cluster Problems P. M Tamrazov

511

56.

Infinitesimal Parameter Spaces of Locally Trivial Deformations of Compact Complex Surfaces with Ordinary Singularities S. Tsuboi

523

Round-off Errors of Residual in Boundary Value Problems by Chebyshev Series K. Tsuji

533

58.

Special Reflexive Modules and the First Chem Class M. Tsuji

549

59.

Normality and Julia Sets Y. Wang

583

60.

Some Results on Extremal Quasiconformal Mappings S. Wu

591

61.

a-Normal Functions and Complex Differential Equations H Wulan

605

62.

Entire Functions that Share One Value with One of Their Derivatives L.-Z Yang

617

63.

On Estimate for Some Bilinear Forms M. Yoshida

625

57.

Contributors

A. G. Aleksandrov Fernando Blasco

Polytechnical University of Madrid, Madrid, Spain

Christopher Boyd H.-A. Braunss

Max Planck Institute for Mathematics, Bonn, Germany

University College Dublin, Belfield, Dublin, Ireland

University of Potsdam, Potsdam, Germany

Der-Chen Chang

Georgetown University, Washington, DC

Zeqian Chen Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan, P.R. China Andreas Defant Nikhil Dhoble

University of Valencia, Buijasot (Valencia), Spain Computer Centre, Centre for Advanced Technology, Indore, India

Juan Carlos Diaz Sean Dineen Jinyuan Du Kelko Fujita

University of Valencia, Buijasot (Valencia), Spain

University College Dublin, Belfield, Dublin, Ireland Wuhan University, Wuhan, Hubei, P. R. China Saga University, Saga, Japan

Yuldo Fukushima Domingo Garcia Vagif S. Guliev

Fukuoka University, Fukuoka, Japan University of Valencia, Buijasot (Valencia), Spain

Baku State University, Baku, Azerbaijan

Hidetaka Hamada

Kyushu Kyoritsu University, Kitakyushu, Japan

xi

Contributors

x ii

Tatsuhiro Honda

Ariake National College of Technology, Fukuoka, Japan

Chuan-Gan Hu Nankai University, Tianjin, P. R. China Xinzhong Huang

Huaqiao University, Fujian, P. R. China

Yoichi Imayoshi

Osaka City University, Osaka, Japan

Osaka City University, Osaka, Japan

Manabu Ito

Nankai University, Tianjin, P. R. China

Jian-Hua Jia H. Junek

University of Potsdam, Potsdam, Germany Kyushu University, Fukuoka, Japan

Joji Kajiwara

Tadayoshi Kanemaru

Kumamoto University, Kumamoto, Japan

Kyushu University, Fukuoka, Japan

Hfsako Kato

University of Ryukoku, Ootsu, Japan

Kazuko Kato Boris S. Khots

Compressor Controls Corporation, Des Moines, Iowa

Gabriela Kohr Babe§-Bolyai University, Cluj-Napoca, Romania Ern Gun Kwon

Andong National University, Andong, Korea

Hanoi Institute of Information Technology, Hanoi, Vietnam

Hai Khoi Le Hung Son Le

Hanoi University of Technology, Hanoi, Vietnam

Pusan National University, Pusan, Korea

Jinkee Lee

Lin Li Jinan Television University, Jinan, P. R. China Xiao Dong Li R. China

University of Electronic Science and Technology of China, Sichuan, P.

Xing Li Ningxia University, Ningxia, P. R. China Xing-min Li Zhong Li

Guangzhou Normal University, Guangzhou, P. R. China

Peking University, Beijing, P. R. China

Contributors

Chao Liu

xiii

Nankai University, Tianjin, P. R. China

Hua Liu Wuhan University, Wuhan, Hubei, P. R. China Jianke Lu Wiihan University, Wuhan, P. R. China Michael Mackey

University College Dublin, Belfield, Dublin, Ireland

Manuel Maestre

University of Valencia, Buijasot (Valencia), Spain

Shriram Padmakar Mhaskar Magnet Development Laboratory, Centre for Advanced Technology, Indore, India Luiza Amalia de Moraes Federal University of Rio de Janeiro, Rio de Janeiro, R.J., Brasil International Christian University, Tokyo, Japan

Mitsuo Morimoto Jorge Mujica

IMECC-UNICAMP, Campinas, SP, Brazil

Yayoi Nakamura

Ochanomizu University, Tokyo, Japan

Tokumori Nanbu

Toyama Medical and Pharmaceutical University, Toyama, Japan

Thanh Van Nguyen Masaru Nishihara Kiyoharu Nono Ken’ichi Ohya

Hanoi University of Technology, Hanoi, Vietnam Fukuoka Institute of Technology, Fukuoka, Japan

Fukuoka University of Education, Munakata, Japan Nagano National College of Technology, Nagano, Japan

Calheng Ouyang Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan, P. R. China Sergly Plaksa Institute of Mathematics of the National Academy of Sciences of Ukraine, Kiev, Ukraine E. Plewnia

Diisseldorf, Germany

Vadla Mohammad Raffee Indore, India

Computer Centre, Centre for Advanced Technology,

Renuka Rajput Magnet Development Laboratory, Centre for Advanced Technology, Indore, India Juri Rappoport Russian Academy of Sciences, Moscow, Russian Federation

x iv

Contributors

Anil Rawat Computer Centre, Centre for Advanced Technology, Indore, India University of Valencia, Buijasot (Valencia), Spain

Pilar Rueda

ATR Adaptive Communications Research Labs, Kyoto, Japan

Kazumasa Shinjo Kwang Ho Shon

Pusan National University, Pusan, Korea

University of Jammu, Jammu, India

A. P. Singh

Rio de Janeiro - RJ, Brazil

Roberto Luiz Soraggi Masanori Suwa

Sophia University, Tokyo, Japan

Shinichi Tajima

Niigata University, Niigata, Japan

Promarz M. Tamrazov Sciences, Kiev, Ukraine

University of Georgia, Athens, Georgia

Jingzhi Tie

Kagoshima University, Kagoshima, Japan

Shoji Tsuboi Kumiko Tsuji Miki Tsuji

Institute of Mathematics, National Ukrainian Academy of

Teikyo University, Fukuoka Junior College, Ohmuta, Japan

Kyushu University, Fukuoka, Japan

Yuefei Wang

Institute of Mathematics, Academia Sinica, Beijing, P. R. China

Shengjian Wu Peking University, Beijing, P. R. China Hasi Wulan

Inner Mongolia Normal University, Hohhot, P. R. China

Hiroshi Yamamoto Lian-Zhong Yang Xiu-Fang Ye

Osaka, Japan Shandong University, Shandong, P. R. China

Nankai University, Tianjin, P. R. China

Mamoru Yoshida Dong-Guo Zhou

Fukuoka University, Fukuoka, Japan Dalian Railway Medical School, Dalian, P. R. China

Modular Space for Complete Intersection Curve-Singularities A. G. Aleksandrov*

Max-Planck-Institut fur Mathematik Vivatsgasse 7 P.O.Box: 7280 D-53072, Bonn Germany E-mail: [email protected]

Institute of Control Sciences Russian Academy of Sciences Profsojunaja ul. 65 GSP-7, Moscow, 117806 Russian Federation E-mail: [email protected]

A b str a c t: Modular families and modular spaces for quasi-homogeneous iso­ lated complete intersection curve-singularities with modularity one are com­ puted. In particular, it is proved that the modular space of any unimodular singularity has a covering which is biholomorphically equivalent to a punc­ tured or twice punctured Riemann sphere. Some related questions concerning singularities of higher modularity are also discussed. AMS Classification: Primary 32S15,14H15, 32J05, 14M10, 32G13; Secondary 14H30, 32S05, 32H20, 14D22, 30C45, 30C55 Key words and phrases: modular space; moduli space; complete intersection curves; unimodular singularities; versal deformation; adjacency

In tro d u c tio n The problem of classification of singularities is a significant part of general de­ formation theory. In principal, one may consider different equivalence relations between singularities such as the right equivalence (change of coordinates in the source of defining mapping), contact equivalence (change of coordinates and multi­ plication with a unit, that is, preserving the isomorphism class of the corresponding germ), and others (see e.g. [19]). In the framework of the corresponding deforma­ tion theory, the first step in solving the classification problem is the description of simple singularities. As a rule, one can write down a finite or at least a perceptible list of normal forms or similar data. However a scheme of classification of more *Partially supported by the Russian Foundation of Fundamental Investigations RFFI (Grant No. 99-01-01067)

1

2

A.G. Aleksandrov

complicated singularities seems to be a rather nontrivial and hard problem since such phenomenon as moduli or parametric families may occur. Among different approaches to further classification, the most fruitful and useful are probably those which are based on the analytical and geometrical methods. We shall discuss here some aspects of the approach originated in the work of V.P.Palamodov [13]. Given a singularity X q, the base space of its minimal versal deformation contains a remarkable subspace M consisting of points that satisfy a certain condition of (semi)universality. It is called the maximal modular germ, and its dimension is called the modularity of X 0. Some fibers of the family induced over M can be isomorphic, though nearby fibers are non-isomorphic in virtue of the universality. In many interesting cases this allows to consider a natural action of a discrete group G on M. More precisely, the group G is the quotient of the automorphism group of the distinguished fiber X q of the modular family by the connected component of a unit. Thus for any two points M close enough to the distinguished one the corresponding fibers are isomorphic, if and only if these points belong to the same orbit of the action of the discrete group G on M. It does mean that the quotient M jG may be regarded as the germ of a modular space, i.e. as a local prototype of the corresponding moduli space. In fact, for certain singularities all fibers X t over M, as well as the distinguished one, have the same (the greatest possible) dimensions of their first tangent cohomology groups. It is called the Tjurina number of X q and denoted by r = t ( X q). For example, one can prove (see [3]) that for quasi-homogeneous isolated complete intersection singularities the germ M is reduced and coincides with the stratum {rmax = const}. It consists of points s e S for which r ( X s) — t (X q). The notions of the modular deformation, maximal modular germ, and modular space in the category of complex spaces have been introduced and investigated by V.P.Palamodov [13] and somewhat later, in a more general context, by O.Laudal [9]. In the special case of an isolated complete intersection singularity a closely related notion of the stratum {rmax — const} has been introduced and studied in the earlier works of B.Teissier (see [19], (4.8)) and M.Lindner [12]. The further development of the theory enables one to consider a special strat­ ification of the base space of the versal deformation. Its strata M r correspond to singular fibers with fixed Tjurina numbers Tm\n < r < r max. This leads directly to the analysis of the structure of the coarse moduli space of a given algebraic or analytical variety. It turns out that, in general, the coarse moduli space does not lie in the categories of algebraic or analytical varieties. In particular, it may be non-Hausdorff and can possesses many other unexpected properties. The reason is th at the kernel of Kodaira-Spencer map has a highly complicated and complex structure. A series of works of O.Laudal, G.-M.Greuel and their followers (see [11], [7], [8], and others) are devoted to the study of different aspects of this problematic. Thus they construct a (coarse) moduli space for isolated hypersurface singularities with certain fixed invariants.

Modular space for complete intersection curve-singularities

3

The present paper deals with the problem of explicit calculation of modular families, modular and moduli spaces. In the first section we briefly discuss the notion of modular germ, due to V.P.Palamodov [13]. The second section is devoted to the study of the properties of modular germs for graded singularities. In the next two sections we compute the modular spaces for unimodular quasi-homogeneous germs of curves, which are complete intersection with Milnor numbers less than seventeen and modularity one. In fact, our computational results give a partial answer on a question posed in [16], Problem 4.4. In the final section singularities of higher modularity are considered. We also discuss some questions, relating to the notion of adjacency and to computing modular spaces for non-complete intersection singularities. Acknowledgment. The author would like to thank the Max-Planck-Institut fiir Mathematik for its support and hospitality during the preparation of the paper.

1. M o d u la r g erm a n d m o d u la rity First, we recall the notion of modular germ and modular deformation, following [13], [14]. Let Xo = (X0, 0) be the germ of an isolated singularity, and let (p : (X, X 0) —►(Y, *) be a flat deformation of X Q. D efin itio n 1.1. A subgerm M c Y is called modular if for any two morphisms of germs g : Z —►M and h : Z —> Y the following condition holds: If there is an isomorphism g*(f) — h*(f) of the induced deformations, then the germs g(Z) — h(Z) coincide. The restriction of the deformation (j) to the modular germ M is called the modular deformation of X0. For simplicity, in what follows, we only consider the case when is the minimal versal deformation of X q; it is denoted by ip : (X,Xo) —> (S',*). Let T°(Xo) be the module of holomorphic vector fields on Xo, and let T ° ( X / S ) be the module of ’’relative” vector fields on X /5 . The latter consists of the so-called vertical vector fields on X, i.e. of those vector fields that are tangent to any fiber of the minimal versal deformation at each of its points (see [15]). In the case when Xo is a complete intersection, they are usually called hamiltonian fields (see [4]). Then the following natural mapping is defined. The quotient module

is called the truncated zero-dimensional tangent cohomology of Xo- One then gets the following conditions of modularity.

4

A.G. Aleksandrov

T h e o re m 1.2 ([14], (6.2)). Let ip : X —►S be the versal deformation of X 0. A subgerm M C S is modular if for any artinian subgerm Z C M the restriction

is surjective. In other words, any element ofT°(Xo) or, equivalently, ofT®(Xo) can be extended to a vertical vector field on X / S . In fact, it follows from this theorem that the structure sheaf of a modular germ M may contain nilpotent elements. Thus, in general, a modular germ may be nonreduced: M ^ Mred- A priory, there is also another reason for a modular germ to have singularities: it is the existence of a nontrivial obstruction in T 2( X o) for the extension of infinitesimal deformations. However, in the hypersurface case or, more generally, for complete intersections T 2( X o) = 0. T h e o re m 1.3 ([13], [14]). Let cp : (X , X 0) —> (S, *) be the minimal versal defor­ mation of an isolated singularity X q. Then there exists a subgerm M C S with the following properties: 1) The germ M is modular and contains all modular subgerms of (S , *). 2) The support Mred of the germ M is the set of points s € S such that

(S, *) be the minimal versal deformation of the germ X0. For any s e S the fiber X s is a multigerm consisting of finite number of germs X*, i ~ 1 ,2 ,... , and the space T l (Xs) is the direct sum of T l {Xi) :

C o ro lla ry 1.7. Let X 0 be the germ of an isolated complete intersection singularity. Then the support of the maximal modular germ Mred coincides with the support of the stratum {rmax = const} c S, i.e. with the set of points s e S for which

P ro o f. In the case of complete intersection T 2(X0) = 0. The statement follows directly from the condition 4) of Theorem (1.3) and the properties of the minimal versal deformation. D efin itio n 1.8. The dimension of the maximal modular germ M C S is called the modularity of the singularity Xo and denoted by p = p(X0). R e m a rk 1.9. Let /i(Xo) be the Milnor number of the singularity X0. Then the dimension of the stratum {/zmax = const} of the minimal versal deformation of the singularity Xo is usually called the modality of the singularity X q (see e.g. [5]). It consists of points s € S for which /i(Xs) — p (X q).

2. M ax im al m o d u la r g erm for g ra d e d sin g u la ritie s We now assume that the germ X q is Z +-graded or, equivalently, there is a good C*-action on X q. Any C*-action on the germ X q induces the C*-action on the tangent cohomology T* (X o). In particular, in the case of an isolated singularity we have a decomposition T 1(Xo) = ®I/€z ^ 1(Xo)J/ into a direct sum of the homogeneous components of the finite-dimensional space T 1(X q). Further, the tangent cohomology T*(X q) is endowed with a natural Lie algebra structure [14], and T°(Xo) acts on T a(Xo) via the Lie bracket. Let £ € T°(X 0) be the Euler vector field and let £% : T l ( X o) —►T x(Xo) denote the linear operator defined by £^(t) = [£,£]. It is easy to prove the following three assertions (see [1]). L em m a 2.1. Let Xo be the -graded germ of isolated singularity. Then the Zgrading on T x(Xo) defined by the C *-action coincides with the Z-grading defined by the Euler vector field. P ro o f. It is easy to check that T 1(X0)l/ are the eigenspaces of 1$ corresponding to the eigenvalues v G Z, so th at the dimension of T 1(Xo)t/ is equal to the multiplicity of v in the spectrum of this linear operator.

6

A.G. Aleksandrov

P ro p o s itio n 2.2. Let ip : ( X , X 0) —►(5,*) be the minimal versal deformation of the graded isolated singularity X q, and let M c S be the maximal modular germ with respect to (p. Then there exist canonical embeddings

where the tangent spaces of the germs M and S are denoted by T*(M) and %(S), respectively. In particular, p (X o) < d im c T 1(Xo)oP ro o f. Our statement follows immediately from the bijectivity of the differential D f and the functoriality of the notion of maximal modular germ M . P ro p o s itio n 2.3. Let X q be a graded singularity. Then the obstruction to ex­ tending an infinitesimal deformation of the (n — 1)-order, n > 2, corresponding to an element from T 1(Xo)I/, v G Z, to an infinitesimal deformation of the n-order belongs to T 2(X0)niy. P ro o f. Recall that the cohomology H*(C) of any graded Lie algebra C whose differential is a differentiation of degree one inherits the Lie algebra structure. There exist a partial cohomological operation H l (C) —> H 2(C) defined by the formula h i— > [h,h\ and called Massey square. In the deformation theory the tangent complex of X q plays the role of £, so that the Massey square T 1(X q) —>T 2(X 0) is defined on the tangent cohomology. Further, it is well known that the obstruction to extending an infinitesimal deformation of the first order corresponding to an element t € T 1(X q), to an infinitesimal deformation of the second order is equal to i[£,£] 6 T 2( X0) (see e.g. [9],(5.1), or [18]). In the graded case it is easy to verify the equality If the Massey square is equal to zero, then a desirable extension exists. In such a case the obstruction to extending an infinitesimal deformation of the second order corresponding to the element t e T 2(X0) to an infinitesimal deformation of the third order is equal to

[t,t]] € T 2(X q), deg[£, [t,t]} = 3v, and so on.

R e m a rk 2.4. Thus the obstruction to extending infinitesimal deformations of the first order, corresponding to an element from T 1(Xo)o? to complete ones lies in T 2( X q) q. If there are no such obstructions (for instance, if T 2(X q)q — 0), then it is easy to see that the dimension of the tangent space % (M ) is equal to the multiplicity of zero in the spectrum of the linear operator on T x(X0) defined by the Euler vector field f e T°(X0) (cf. Proposition (2.2)). For complete intersections, this remark can be made more precise (see [2], (1.3)).

Modular space for complete intersection curve-singularities

7

T h e o re m 2.5. For a graded isolated complete intersection singularity the maximal modular germ M is reduced and smooth. The modularity p of such a germ is equal to the multiplicity of zero in the spectrum of the linear operator on T x( X o) defined by the Euler vector field £ e T ° ( X o). P ro o f. Prom the theorem in [3], (6.1) it follows that for such a singularity the image of T°( X/ S) —* T°( Xo) is generated over Ox 0 by Hamiltonian derivations, while the finite-dimensional vector space T®(X0) is generated by fields which are multipliers of £. The Euler field £ acts on T 1(X q) with no kernel, so that under our assumptions there is an isomorphism 7 l(M ) ~ T 1(X q)0. N o w let Z c Y be any Artinian subgerm such that T*(Z) C T*(M). Since 7l (M ) ~ T 1(Xo)o, it is evident th at the field £ (and, consequently, any element of T®(Ao)) extends to a vertical vector field on the restriction / |z • For complete intersections T 2(X0) = T 2(Xo)o = 0, so it is clear th at the above considerations imply

th at is, M is smooth. C o ro lla ry 2.6. For a graded isolated complete intersection singularity the stratum {rmax = const} is reduced and smooth. R e m a rk 2.7. The assertion of the proposition was conjectured by V.P. Palamodov, and it was proved for hypersurfaces in [13], Theorem 9. In the case when X q is semiquasihomogeneous isolated hypersurfaces singularity the maximal modular germ may be non-reduced. The corresponding examples have been considered by V.P. Palamodov in 1976 (see [16], Ex. 3.2). Among them there are TPjgjr-singularities given by the equation x p + yq + 2 r + x yz = 0, where 1/p + 1/q -f 1 /r < 1.

3. M o d u la r space Let us recall some notions from the deformation theory of complex spaces (see [14], [16]). For the construction of modular spaces we have to consider families of germs over arbitrary complex base spaces. D efin itio n 3.1. A modular family is a proper flat mapping of complex analytic spaces / : X —* M such that for every point s 6 M the restriction of / to the germ (M, s) is a modular deformation of the fiber X 3. Let X 0 be an admissible representative of the germ of a complex analytic space and / : X —» M be its modular deformation. By definition, for every point s € M there is an open neighbourhood s € V C M satisfying the following condition: For any v E V the fibers X v and X 3 are not isomorphic.

8

A.G. Aleksandrov

However, some fibers over V may be isomorphic to each other. Assume that such fibers belong to one-dimensional families. Then there are distinct morphisms of germs M such th at the induced deformation g ^ i f ) and the deformation / with the fixed isomorphism a of the distinguished fiber are isomorphic. It is clear that gM is a biholomorphic mapping from the germ M into itself. Thus, the action of the group Aut(Ao) lifts to an action in the bundle / : X —►M. It turns out that the subgroup Aute(X0) C A ut(X 0) as well as the discrete group r(X0) = Aut(X0)/A u te(X0) act on the base M. Let us also assume that the modular deformation of X q satisfies the following condition (see [16], Problem 4.3). For any points s,£ G M which are sufficiently near the distinguished point of M the following is true: The fibers X s and X t are isomorphic if and only if both points s ,t G M lie in the same orbit of the action T on M . D efin itio n 3.2. Let X q be the germ of an isolated singularity satisfied the above assumptions. Let M be the maximal modular germ, and let T = T( X q) be the discrete group as defined before. Then the quotient germ A4 = M / T is called the germ of a modular space for X q. It seems that in many cases quite a clear description of modular spaces is given by the collections of charts or finite coverings, that is, by proper multivalent mappings Vu : U —> A4, where U C C m are the open subsets. The next sections aim at computing modular spaces for quasi-homogeneous space curve-singularities.

4. U n im o d u la r cu rv e-sin g u laritie s According to the classification theorems, [2], Theorem 5.4 and Theorem 5.10, there are, in all the 14 types of unimodular families, three 2-modular ones and only one 3-modular family of graded curve singularities, which are complete intersection with Milnor numbers less than seventeen (cf. also [6], [20]). We denote the normal form of such a germ and the corresponding one-parameter family by the same symbol. Any member of such a family X \ is embedded in (C 3,0). In the notations [2], we obtain the following assertion.

Modular space for complete intersection curve-singularities

9

T h e o re m 4.1. Let X \ be an unimodular family of a graded complete intersection curve singularity. Then there exists a modular space M. containing the base of such a family. Moreover, there is a covering M ! —* A4 such that A i r is biholomorphically equivalent to the complex plane C 1, except for three cases of families T}5, Q \6, Q\6. In the latter cases M ! is biholomorphically equivalent to the punctured complex plane. P ro o f. The method of the proof is the case-by-case checking. 1) The family Hg. Let e € C*, so that e4 = 1/4. Set U = C 1 \ {e, €2, e3, e4}. Then the family IZg is given by the mapping / : X —►U, so that the fiber X a C C 3, a e U , is defined by the following system of equations:

Let us define a chart ipu • U —>C P 1 by ipu(a) = a4. It is clear that the chart ipu covers the whole Riemann sphere C P 1, except for two points a4 = 1/4 and oo. The latter may be covered by another chart. To see this, let us consider the family Xg of hypersurfaces given by the equation y4 -f z4 + by2z 2 = 0, b e V = C 1 \ {±2}. Then (pv : V —> C P 1, defined by tpv(b) — will be the desirable chart. Besides ipv maps the member of the family Xg with the parameter b = 0 to oo. It is not difficult to verify that both charts are glued along ) and (fyl (N), where N = ) = ^6- Then the chart ipu covers the whole projective line C P 1, except for two points a6 = —4/27 and oo. The latter may be covered by another chart corresponding to the modular family J\g of hypersurfaces {y3 + z6 + 6y2z2 = 0}, where 6 e V = C 1 \ 1/3 = —27/4. While ipy : V —> C P 1 is defined by C P 1, defined by (pu(o) = a3, covers the whole projective line C P 1, except for two points a3 = 27/4 and oo. The latter may be covered by another chart. Consider an open set V = C 1 \ {±2/3\/3}, the modular family over this open set looks as follows:

Then ipy : V —» C P 1, defined by ipv(b) = 62, covers the point oo by the element corresponding to 6 = 0. As before, we see that M ! = C 1 and the gluing isomorphism is given by the formula a3 = 1/62. R e m a rk 4.2. In fact, there exists the third chart corresponding to the family U\\ from [2], Table 2) of Theorem 5.4. In order to show this, let us consider an open set W = C 1 \ {±2}, so that the modular family U\\ over W is given by the system of equations:

Then ipw • W —> C P 1, defined by the formula C P 1, acting by formula ipuip) — a3, covers the whole projective line C P 1 except for two points a3 = —27/4. and oo. The latter may be covered by

Modular space for complete intersection curve-singularities

11

another chart ipy • V —> C P 1, where V = C 1 \ { ± 2 \/—1/27}. The corresponding modular family over V is defined by the following system of equations:

and (C 2,0) have been recounted. 6) The family M \ \ . Let U — C 1 \ { ± y / ^ l } and / : X -* U be the modular family M .\\ defined over U by the system of equations:

Then ipu : U C P 1, acting by the formula c/(a) = a2, covers the whole Riemann sphere C P 1, except for two points a2 = —1 and oo. The latter point may be covered by another chart. Set V = C 1 \ {±\/~T } and consider the modular family over V, given by the equations:

where b € V. Then (fy : V —* C P 1 is defined by the formula (pv(b) = 62. Besides, (py maps the member of the family M>\\ with the parameter b — 0 to oo. The gluing isomorphism is given by the formula a = 1/6.

12

A.G. Aleksandrov

It should be noted that one can construct the third chart, corresponding to the family from the list [5], p.132. In fact, this family is given by the system:

where c e W = C 1 \ {=L2}. Then (pw • W —> C P 1 is defined by the formula v and the gluing isomorphism are given by exactly the same formulae as in 4) and 5). Thus, M ! = C 1. 8) The family Again, in the same notations, the first chart is constructed as the modular family Vi6 given over U by the equations:

Modular space for complete intersection curve-singularities

13

while the second one is given as the modular family over V defined by the system of equations:

The charts ifu and (fy and the gluing isomorphism are given by exactly the same formulae as in the preceding case. 9) The family Qiq. Once more, in the same notations, the first chart is con­ structed as the modular family Qiq given over U by the system of equations:

while the second one is given as the modular family over V defined by the system of equations:

The charts (pu and C P 1, acting by formula (pu(a) = a2, covers the whole projective line C P 1, except for two points a 2 = —1 and oo. The latter may be covered by another chart, corresponding to the modular family W 15 of hypersurfaces defined by the equation y*+z 6 +by2 z 2 = 0, where b G V = C 1\{±2}. The chart ipy : V —►C P 1 is described by the formula ipv(b) = b2, while the gluing isomorphism is described by the formula b2 = 4/(1 4- a 2). As a result M ! = C 1. 11) The family A/15. Set U = C 1 \ { ± v ^ l} and consider the modular family A/15 defined over U by the system of equations:

Then ipu : U —> C P 1, acting by formula 2, whereas r 0 < = f3 < on V [}E ) — E '. Proof’ By Proposition 6 we have that (V(nE),Tbj is quasinormable for every n € N. The ideas of the rest of the proof are similar to the ideas used in the proof of Theorem 3 in [lj: To < T5 because E is not Montel (contains a complemented copy of a nondistinguished space). 71 < (3 (for n > 2). Since F does not verify the (B B ) 2,a property and F is complemented in E we have that E does not verify i t , so E does not verify (B B )niS either ([4]). Since having the ( £ £ ) n>a property is equivalent to the coincidence of these topologies ([8 ]) we get the result. n

(J < To, Since E • • • E contains a complemented copy of a non-distinguished «,ir

S,7T

Ai(A), itself can not be distinguished, so its strong dual (V (nE),f3^ is not barrelled and (V (nE),T„) is, so they are different. R eferences [1] J. M. Ansemil, F. Blasco, S. Ponte, About quasinorm ability and topologies on spaces o f polynom ials , J. Math. Ann. Appl. 2 1 3 (1997), 534-539. [2] K. D. Bierstedt, J. Bonet, Stefan H ein rich ’s D en sity C ondition fo r FrSchet Spaces and the C haracterization o f the D istinguished K oth e Echelon Spaces , Math. Nach. 135 (1988), 149-180. [3] K. Bierstedt, R. Meise and B. Summers. K oth e S ets and K oth e Sequence Spaces. Functional Analysis, Holomorphy and Approximation Theory. North Holland Math. Stud. 71 (1982), 27-91. [4] F. Blasco, C om plem en tation in spaces o f sym m etric ten so r products and polynom ials , Studia Math. 123 (1997), 165-173. [5] J. Bonet and S. Dierolf, On (L B )-spaces o f M oscatelli type, Doga Tr. J. Math. 13 (1989), 9-33. [6] S. Dierolf, On Spaces o f Continuous L inear M appings B etw een Locally C onvex Spaces , Note Mat. 5 (1985), 147-255. [7] S. Dineen, Q uasinorm able Spaces o f Holom orphic Functions, Note Mat. 12 (1993), 155-195. [8] S. Dineen, H olom orphic Functions and the (B B )-property, Math. Scand. 74 (1994), 215-236. [9] S. Dineen, C om plex A n alysis on Infinite D im ensioned Spaces, Springer-Verlag Monograph Series, 1999. [10] E. Dubinski, P erfect F rich et Spaces, Math. Annalen 1 74 (1967), 186-194. [11] L. Frerick, On Vector-valued Sequence Spaces, Note Mat. 14 (1994), 1-43. [12] A. Grothendieck, P roduits tensoriels topologiques et espaces nucliaires, Mem. Amer. Math. Soc. 16, 1955. [13] H. Jarchow, Locally Convex Spaces, B. G. Teubner, Stuttgart, 1981. [14] G. Kothe Topological Vector Spaces I, Springer-Verlag, 1969.

28

F. Blasco

[15] A. Peris, P roductos T ensoriales de Espacios Localm ente Convexos C asinorm ables y otras C loses R elacionadas. Ph. D. Thesis, Universitat de Valencia, Spain, 1992. [16] A. Peris, Topological ten so r products o f a P rechet-Schw artz space and a B anach space , Studia Math. 106 (1993), 189-196. [17] J. Taskinen, Counterexam ples to “Problem e des Topologies” o f G rothendieck , Ann. Acad. Sci. Fenn. Math. Diss. 63, 1986. [18] J. Taskinen, The P rojective T ensor P roduct o f P rech et-M on tel Spaces , Stdia Math. 91 (1988), 17-30. [19] J. Taskinen, E xam ples o f n on-distinguised P rechet spaces , Ann. Acad. Sci. Fenn. 14 (1989), 75-88.

Holomorphic Mappings and Cardinality Christopher BOYD Department of M athematics, University College Dublinf Belfield, Dublin 4, Ireland.

1

In tro d u c tio n

Let E be a complex locally convex space. A function / : E —> C is said to be holomorphic if / is continuous and / \ f is holomorphic for each finite dimensional subspace F of E. We denote by H (E ) the vector space of all holomorphic func­ tions on E . An important subspace of H (E ) is V (nE) the space of all (contin­ uous) n-homogeneous polynomials on E. A (continuous) function P : E —> C is said to be an n-homogeneous polynomial if there is a continuous linear mapping L p \ E x E x . .. x E C such that P (x) = L p o A(x) where A(x) — (x ,... , x). Sl ...... ’V 1 y n-times Indeed, a continuous functions / : I? —►C is holomorphic if and only if for each x in E we can find a neighbourhood Ux of x and a sequence {Pn(x)}£L0 with Pn (x) € P (nE) so th at f{ x 4-y) = p n(x)(y) for all y e Ux . The space Tt(E) becomes a locally convex space when endowed with r OJ the topology of uniform convergence on compact subsets of E . This topology is gen­ erated by the semi-norms {p x - K compact in E } where p x ( f ) — s u P x e K \f( x )\A semi-norm p on locally convex space E is said to be lower semi-continuous if {x : p(x) > a} is open in E for every a > 0. A locally convex space in which all lower semi-continuous semi-norms are continuous is called a barrelled space. Seminorms on a barrelled locally convex space which axe the sum or supremum of an arbitrary family of continuous semi-norms axe themselves continuous. Given a set I we shall use C 7 to denote the cartesian product of / copies of C endowed with the product topology and use to denote the locally convex direct sum of / copies of C. In this article we present a survey of the conditions on I that 1991 M ath em atics Subject Classification: Primary 46G20; Secondary 03E55. K eyw ords: Holomorphic Function, Product spaces, Direct sums, Measurable Cardinals.

29

30

C. Boyd

will ensure that the spaces (H (C 7), r0) and (W (C ^ ), r 0) are barrelled. We shall see that ('H(C i ) , t0) or (W ( C ^ ) , t0) being barrelled locally convex spaces depends on the cardinality of /. We refer the reader to [8] for further details on locally convex spaces and to [5, 6] for further information on infinite dimensional holomorphy.

2

H o lo m o rp h ic fu n ctio n s on p ro d u c t spaces

Let us begin our survey by considering the locally convex space (H (C 7) ,r 0) when / is a countable set. W ithout loss of generality we shall assume that / is the set of natural numbers, N. The result we shall prove was first announced by L. Nachbin in [11]. In what follows however we shall use a specific case of the proof given to a more general result given in Proposition 3.5 of [4] (see also Example 3.24 (a) of [6]). We shall use ej for the point of C N which has the value 1 in the +h co-ordinate and which is 0 in all other co-ordinates. The set {e j } < ^ =1 is an absolute basis for C N We begin with the following observation (see [5] for further information). If / G 7 i( C ^ ) then there is an integer nOJ depending on / , so that if y = Yl™=n0 Vn then f ( x + y) = f( x ) for all x in C ^ . i.e. Each holomorphic function on through a finite dimensional subspace of C ^ . Fix x in and consider the semi-norm p on H (C ^ ) defined by

factor

Our previous remark will ensure that p ( f ) is finite for each / G H{C ^ ) . For each integer n the semi-norm

is r 0-continuous. Therefore if (TC(C ), r0) were barrelled p would be a r 0-continuous semi-norm on H (C ^ ) . Let us see that this is not the case. Suppose that p was T0-continuous. Then we can find a compact subset K of and C > 0 so that

for all / G H{C ^ ) . Choose a sequence { 0 and (j>n{^m) ^ 0 if and only if n = m. Letting / = (j>Q R + by

We claim that there is no function t : B —+R + such that

for all i , j in J . Suppose just a function t exists. Since B is uncountable we can find a positive integer n 0 so that B no — {i e J : t(bi) < n 0} is uncountable. For all t, j e B Uo, t(bi)t(bj) < n^. However, since B Uo is uncountable, {^(6i)}ieBTlo has an accumulation point an so can find i , j G B no with \bi — bj\ < 4^, which is a contradiction and proves our claim. T v Given a 2-homogeneous polynomial P on C we shall use P to denote the unique symmetric bilinear form on C t x C t such that P{x) = Pv o A ( x ) for all r 6 C T. Define a semi-norm p on P ( 2 C 1) by

C. Boyd

32

Since each polynomial on C 7 depends on only finitely many variables, v

is finite

p(P)

for every P £ 'P(2C i ). For each z, j £ J the mapping P —> |P(ei, e j ) \ r ( b i , b j ) is r 0 continuous. Therefore (P (2C 7) , t0) being a barrelled space will imply th at p is r0-continuous. Let us prove this is not so. A fundamental system of compact subset in C 7 is given by sets of the form {(xi)i£i : \xi\ < ^}(t .).€j€(R +)/- Thus if p is r 0-continuous we can find £ (R + )7 so that Letting P ij = XiXj this will imply that

for all z and j in R, which, as we have seen, is impossible. Thus we have shown that (P (nC 7) ,r 0) is barrelled if and only if I is countable.

3

H o lo m o rp h ic fu n c tio n s o n d ire c t sum s

Let us now discuss under what conditions (7i(Cv*), r0) is barralled. To study this problem we need the concept of Ulam measures introduced by Ulam [14]. An U la m m e a su r e (m e a su ra b le c a rd in a l) on a set I is a {0, l}-valued measure p defined on 27 with p ( i ) = 0 for each i £ / and p ( I ) = 1. It is a longstanding open problem of set theory as to whether there is a set I which supports an Ulam measure. The existence of Ulam measures is related to the concept of strongly inaccessible cardinals. A cardinal number d is str o n g ly in a c c e ssib le if (i)

d

> «0,

(ii) If

d

(iii) If

{ d x } X£A is a family of cardinals numbers such that > cardA then d > Y l x t A

d

>

c

then

d

> 2C, d

>

dx

all

x

£

A

and

Any set which admits an Ulam measure has strongly inaccessible cardinality. As with Ulam measures it is an open problem as to whether sets with strongly in­ accessible cardinality exist. The existence (or non-existence) of Ulam measures is equivalent to the solution of some classical open problems in functional analysis. A continuous linear operator between locally convex spaces maps bounded sets to bounded sets. A locally convex space is said to be bornological if the converse is true. More formaly, a locally convex space E is b o rn o lo g ica l if given any locally convex space F any linear operator T : E —> F which maps bounded sets to bounded sets is continuous. Mackey, [10], shows that given a set I R 7 is bornological if and

Holomorphic mappings and cardinality

33

only if / does not admit an Ulam measure. This result is know as the Mackey-Ulam Theorem. Let U be an ultrafilter on a set /. We shall say that I is countably incomplete if there is a sequence {/nJJJLj in U such that P£Li Ai = 0- Given a cardinal N we say th at U is N-good if for each set A with card(A) < N and any function / from Pu,(A), the sets of all finite subsets of A , into U with the property that f(u) D f(v) whenever u C v, there is a function g : Pu,(A) —►U such th at g(u) C f(u) and g(u Uv) = g(u) D g(v) for all u and v in Pw(A). Given a locally convex space E #(E) will denote maximum of the least cardinality of a neighbourhood basis of 0 in E and the least cardinality of fundamental system of bounded subsets of E. Let E be a locally convex space and U be a countably incomplete N(E)+-good ultrafilter on a set I. We denote fin(E 1 /U ) the subset of E 1 with the property that lim^ p(xi) < oo for all continuous semi-norms p on E and by p (E T/U ) the subset of E 1 with the property that lim^ p(xi) = 0 for all continuous semi-norms p on E. The locally convex space fin(E 1 jlX) j ^ ( E 1 /U ) endowed with the topology generated by the semi-norms pu((xi)iei) = limup(^i) is called the uttrapower of E and is denote by (E)u- A locally convex space E is invariant if (E)u is isomorphic to E for every countably incomplete K(i£)+-good ultrafilter U. Every Schwartz space is invariant while a Frechet space is invariant if and only if it is Montel. Necessary and sufficient conditions for the invariance of a locally convex space are given in [7] and [2]. In particular, Henson and Moore, [7], show that is invariant if and only if I is not Ulam measurable. Using the Mackey-Ulam Theorem and the result of Henson and Moore the fol­ lowing result is obtained in [3].

Theorem 1 su ra b le .

T h e sp a c e

(W (C ^ ),r0)

is b a rre lle d i f a n d o n ly i f I is n o t U la m m e a ­

We refer the reader to [3] for the technical details.

R eferen ces [1] B arroso J.A. & N achbin L., Some topological properties of spaces of holomor­ phic mappings in infinitely many variables, Advances in Holomorphy, (ed J.A. Barroso), North-Holland Math. Studies, 34, (1979), 67-91.2 [2] B oyd C. Sc D ineen S., Compact sets of holomorphic mappings, Math Nadir., 193, (1998), 27-36.

34

C. Boyd

[3] B oyd C., Holomorphic mappings on C (/), I uncountable, 36, (1999), 21-25.

R e su lts in M a th e m a tic s ,

[4] D ineen S., Holomorphic functions on locally convex topological vector spaces: 1. Locally convex topologies on H i U ) , A n n . Inst. Fourier G renoble, 23, (1), (1973), 19-54. [5] D ineen S., Complex analysis on locally convex spaces, ies, 57, (1981).

N o rth -H o lla n d M ath . S tu d ­

[6] D ineen S., Complex analysis on infinite dimensional spaces, em a tics, Springer, (1999).

M on ographs in M a th ­

[7] Henson C.W. &; M oore J r . L.C., Invariance of the nonstandard hulls of a locally convex space, D u ke M ath . J ., 40, (1973), 193-205. [8] J archow H., Locally convex spaces,

B .G . Teubner,

[9] J ech T.J., On a problem of L. Nachbin, 341-342.

Stuttgart, (1981).

P roc. A m er. M ath . S o c .,

79, (2), (1980),

[10] M ackey G.W., Equivalence of a problem in measure theory to a problem in the theory of vector lattices, Bull. A m er. M ath . S oc., 50, (1944), 719-722. [11] N achbin L., Sur les espaces vectoriels topologiques d u p lic a tio n s continues, C. R . A cad. Sci. Paris, S er A , 271, (1970), 596-598. [12] N achbin L., When does finite holomorphy imply holomorphy? 51, (4), (1994), 525-528. [13] R udin M.E., Letter to L. Nachbin,

Ja n u a ry

&h,

P ortu gal. M a th .,

1979.

[14] U lam S., Zur Masstheorie in der allgemeinen Mengenlehre, (1930), 140-150.

Fund. M a th .,

16,

Approximation Numbers for Polynomials H.-A. B raunss H. J unek University of Potsdam Institute of Mathematics 14469 Potsdam, Germany

E. P lewnia Kaiserswerther Str. 35 40477 Diisseldorf Germany

Mathematics Subject Classification (1991): Primary 46E50; Secondary 46G20, 47B10. E-mail: [email protected] A bstract. Approximation numbers of linear operators are a very useful tool in order to understand the structure and the numerical behaviour of the oper­ ators. In this paper, this concept is extended to polynomials on Banach spaces and the approximation numbers of diagonal polynomials are estimated. As a main tool the rank of polynomials as a graduation of finite type polynomials is introduced and studied. 1. R a n k s of polynom ials. Let E and F be Banach spaces and let S e C(E, F) be a linear operator of finite rank. Then it is well known that the rank rk(S) of S can be computed by each of the following quantities:

1. dim(S'(£,))> 2. min 3. dim 4. min If we consider multilinear forms or m-homogeneous polynomials, there are more than one natural setting to measure the degree of finiteness, and an axiomatic approach seems to be convinient. Ranks for multilinear mappings were defined and studied by A. PlETSCH [4] and S. B rademann [1]. The third named author treated this question in [6] for m-homogeneous polynomials. We recall the basic notions. Let E be any Banach space over K (K = R or C). By C(mE) we denote the Banach space of all continuous m-linear forms A on E x . . . x E with the norm ||A|| = s u p flA f o ,. . . ,arm)| | ||an ||, • • •, !l*m|| < 1} and by V(mE) we denote the space of all m-homogeneous polynomials P on E with the norm ||P|| = sup{|P(x)| | ||x|| < 1}. Remember th at for each P e P(mE) there is a

35

H.-A. Braunss, H. Junek and E. Plewnia

36

unique symmetric m-linear form P G C ^ E ) satisfying P(x) = P ( x ,. . . , x) for each x € E. This form P is given by the polarization formula. For details see [2,1.6]. By Cf (mE) = {ai 0 ... am | Oj € £ '} lin and Vf{mE) = {aj • ... • am \ aj € £ '} lin we denote the linear subspaces of m-linear forms and polynomials of finite type, respectively. Finally, for an element x € t p we denote its norm by \\x\\p. Let us recall now the axiomatic definition of a rank used by E. P lewnia in [6]. D efin itio n 1.1 Let V m be the class of all m-homogeneou$ polynomials on Banach spaces. A functional p : V m —>N 0 U {oo} is called to be a rank on V m if it satisfies the following conditions:

As an easy consequence one obtains: L e m m a 1.2 For each rank p on V m the following hold true:

Important examples are the following functions: 1. The maximal rank

2. The factorization rank.

P ro p o s itio n 1.3 Each rank p on V m satisfies p < v. P ro o f: It suffices to suppose

By the triangle inequality we get p(P) < oo for each P G Vf. W hat about the converse? Preparing the answer we need the following lemma:

37

Approximation numbers for polynomials

L em m a 1.4 For each P = a\ • ... •am with aj € E f there is a power representation

P ro o f: We consider the symmetric ra-linear operator L G £ (m-E/,P ( m£ !)) given by L ( a i,. . . , am) = a\ • ... • am. Then Q(a) = am is the associated polynomial, and by the polarization formula we have

Now we can answer the question stated above. P ro p o s itio n 1.5 For m > 1 and P

6

P ( mE) the following conditions hold true:

P ro o f: 1 : This estimation follows from Lemma 1.4 and the triangle inequality for the rank. 2 : Any m-homogeneous polynomial P 0 on K n admits a representation

So, if P

6

P ( mF ) is of factorization rank 17(F) = n, then P admits a representation

for some aj € E '. This implies v(P) < rcm • max v(aix • . .. • aim) < nm • 2m = (2 r7(P ))m by the triangle inequality and by 1 . 3 : For p > 77 the condition p(P) < 00 is equivalent to ^(P ) < Proposition 1.3. But v(P) < 00 is equivalent to P € V f by 1 . Let us close this section with an example showing that

77 and

00

by

2

v may differ.

and

38

H.-A. Braunss, H. Junek and E. Plewnia

E x a m p le 1.6 For E = K 2 and P = 3e2 e 2 we have rj(P) =

2

< v(P) = 3.

Indeed, 77(F) = 2 follows directly from the definition of 77. Assume now v(P) = 2 . Then there would exist vectors a,b e K 2 with P = a3-f b3. In coordinates this means th at the equality 3f ^£2 = ( a i£ i+ a 2£2 ) 3 + (/?ifi+ /? 2C2 )3 is satisfied for all £ i,f 2 € AT. This leads to the non-linear system a? -f = 0, c*ia2 4 * = 1, oc\a2 4 - Pi 0 2 = 0, a ! 4 - /?2 = 0 which does not have a solution neither in R nor in C. This shows v{P) > 3, and the equality proves jy(P) = 3.

2. A p p ro x im a tio n n u m b ers. Let T : E — ►F be a bounded linear op­ erator. The value ak(T) — inf{||T - 5|| | rk(5) < k} is called to be the kth approximation number of T. The approximation numbers first appeared in a pa­ per of E. S chmidt in the context of integral operators on L 2. They have been extended to Banach spaces and intensively studied by A. P ietsch in [3] and [5]. The approximation numbers of a given operator T are of great importance in two directions: they reflect mapping properties of T (e.g. the asymptotic behaviour of the eigenvalue sequence) and they include information about the complexity and the approximation error in numerical procedures. In this section we transfer the concept of approximation numbers to m -homogeneous polynomials. In contrast to the linear case, these approximation numbers depend on the chosen rank p. D efin itio n 2.1 Let E be a Banach space, p a rank. For P G P ( mF ) and k € No the number

is called to be k th approximation number of P with respect to p. P ro p o s itio n 2.2 Let E and F be Banach spaces, P ,Q € F ( mF ) 7 T € C(F, E), k,l € No, and p ,a ranks. Then we have

Approximation numbers for polynomials

39

P ro o f: 1, 4 j and 6 by definition. 2 : For a given e > 0 w e choose R ,S € V ^ E ) with p(R) < k, p(S) < l and a£(P) + e > \\P - P ||, af(Q) + e > \\Q - S||. Since p(R + 5) < p(R) + p(S) we get

7 : For any polynomial Q e P ( mP ) with r](Q) < k we have v{Q) < (2k)m by Proposition 1.5.2, and this implies 7. W hat about the converse of statement 4 in Proposition 2.2 ? It comes out that this question is connected with the existence of a polynomial of “best approxima­ tion” . In preparation of an affirmative answer we define: D efinition 2.3 A rank p is called to be closed, if P e V ^ E ) and (Pn) C P ( mE) with ||P —Pn || —►0 and p(Pn) < k implies p(P) < k. L e m m a 2.4 Let p be a rank. Then the following conditions are equivalent 1.

p is closed,

2. ak(P) = 0 implies p(P) < k. Moreover, for finite dimensional Banach spaces E the conditions (1) and (2) are equivalent to: 3. For any polynomial P 6 P ( mP ) there is a polynomial Q € P ( mE) with p(Q) < k and \\P —Q|| —a£(P). Proof: 1 => 2 : Let a£(P) = 0. Then there is a sequence (Pn) with p(Pn) < k and ||P —Pn || —►0. Now, 1 implies p(P) < k. 2 => 1 : Let P ,P n 6 P ( mE) be polynomials with ||P —Pn || —> 0 and p(Pn) < k. Then Proposition 2.2 implies af (P) = lim a?(Pn) = 0. By 2 we get p(P) < k. n —KX>

1 => 3 : Now, let E be a finite dimensional Banach space and P € P ( mP ), k € No. Since ak(P) < ||P|| we have a£(P) = inf{||P —R\\ | p{R) < k , \\R\\ < 2||P||}. The space P ( mE) is finite dimensional. Therefore, K = {P € V ( mE) | p(R) < k , ||P||
2 : Let a£(P) = 0. By assumption there is a polynomial R € V ( mE) with ||P —P || = a£(P) = 0 and P(R) < fc. This shows P = R and p(P) = p(R) < k . Now we can prove that the factorization rank rj is closed. The following con­ struction yields also a new characterization of rj. D efin ition 2.5 Let E be a Banach space and P

e P ( mP ). The set

is called the kernel of P . P rop osition 2.6 Let E be a Banach space and P e P ( mE). closed subspace of P ~ l (0).

Then ker(P) is a

Proof: Let t/, t/i,t /2 € ker(P),x e P , and 0 ^ A € K. Then we have P( x 4(Vi 4- 2/2 )) = P ((s + 2/1 ) + 2/2 ) = P{x + 2/i) = P(x) and P(x 4- At/) = AmP ( f 4y) — AmP (^ ) = P(x). Therefore ker(P) is a linear subspace. The closedness follows directly from the continuity of P . Finally, P (0 4- y) = P ( 0) = 0 shows ker(P) C P-HO). P roposition 2.7 Let E be a Banach space and P e P ( mP ). Then r)(P) = d im P /k e r(P ) = codim ker(P). Moreover, if r}(P) < 0 0 , then there is a projec­ tion 7r € £ (P , E) with rk(7r) = t/(P) and P — P o 7r. Proof: Suppose l — tj(P) < 0 0 . Then there are T € £ (P , i f 2) and Q e P ( mi f 2) with P — Q o T. Define E q = T -1 (0) C E. For x e E and y € E q we get

Therefore y 6 ker(P), i.e. Po C ker(P). This proves codim ker(P) < codim P 0 < Z= 7/(P). Conversely, we now assume l = codim ker(P) < 0 0 . Choose any subspace P i of P with P = P i 0 ker(P). Let 7r be the projection onto P i and let $ : P i —» i f 12 be an isomorphism. With Q = P o 3>-1 , T = $ o n and xi € P i, t/ € ker(P) we have

hence Q o T = P o n = P and therefore rj(P) < dim T (P ) — dim7r(P) — l — codim ker(P). This completes the proof. The proof of the following lemma is evident.

41

Approximation numbers for polynomials

L em m a 2.8 Let (Pn) be a convergent sequence in V( mE) with limit P. xn x implies Pn (xn) —►P(x).

Then

Now we axe ready to verify: T h e o re m 2.9 The rank rj is closed. P ro o f: Let Pn , P € P ( mF ) with Pn —* P and r)(Pn) < k be given and let us assume 77(F) > k. We choose a subspace F of E with dim F = k 4- 1 and F fl ker(F) = {0}. By Proposition 2.7 there are projections 7rn € L(E) with Pn = Pn o nn and rk( 7rn) < k for all n € N. Since ker(7rn) fl F ^ {0}, there is a sequence (yn) C F with ||yn || = 1 and 7rn (yn) — 0 for each n G N. W ithout loss of generality we may assume that yn —>y € F. By Lemma 2.8 we have

for each x e E. Therefore, y e F n ker(F) = {0}, and this contradicts ||i/|| = 1. The maximal rank v is not closed. The following counterexample was found by E. B ehrends. Prom Example 1.6 we know i/(P) = 3 for P = 3eie 2 6 P ( 3K 2). On the other hand, the sequence Pn e P (3K 2) given by Pn = (ne 1 -f e2 ) 3 4- (—ne 1 + —l ) e2 )3 —3efe 2 -I- (—^ 4- - )e ie | 4+ ^ ) e 2 tends to P in norm and satisfies v{Pn) — 2 for each n e N . We will close this section by establishing a relation between the approximation numbers a]? and the approximation numbers of linear operators. P ro p o s itio n 2.10 Let F and F be Banach spaces, P Then a ^ ( P o T ) < {2e)rn\\P\\\\T\\m- 1ak(T).

e V ( mE), and T e £ ( P ,F ) .

P ro o f: For any given e > 0, we choose Tk G £(F , F ) with rk(Tfc) < k and ||T — Tfcll < ttfc(T) 4* e. For any z € F with ||z|| < 1 we set x = T 2: and y = T^z —Tz. Let P denote the symmetric m-linear form associated to P . Then we have

H.-A. Braunss, H. Junek and E. Plewnia

42

and this implies

3. E stim a tio n s for d iagonal polynom ials. It is very difficult to compute the approximation numbers even for simple polynomials on sequence spaces. Here we obtain estimations from above and from below for diagonal polynomials on £p. D efin itio n 3.1 A decreasing sequence A = (A*) of non-negative numbers is called to be admissible (with respect to m > 1 and 1 < p < oo) if it satisfies X G £ for p < m and X 6 t p/(p-m) f 0T p > m. The polynomial

is called a diagonal polynomial. From the generalized Holder’s inequality it follows easily: L em m a 3.2 For each admissible sequence X we have ||P\|€p|| < ||A||p/(p_m) in the case p > m , and ||P aKp|| = ||A||oo for p < m. Let us start with upper estimations of the approximation numbers with respect to the maximal rank v. P ro p o s itio n 3.3 Let X — (A*) be admissible with respect to m > 1 and 1 < p < oo and let A^fc+1^ = (0 ,. .., 0, A^+i,...) be the tail of X. Then we have

oo P ro o f: The standard polynomial S m =

eT € is well defined and holds i=i ||5m|| = 1. Let Oi — A^m and a — (ai). We define a linear operator D 6

Approximation numbers for polynomials

43

thus we axe done. We will use the estimations obtained in the proposition stated above to get nice asymptotic results for the sequences of approximation numbers. To avoid a refer to Lorentz spaces, we will state the following lemma in preparation: L e m m a 3.4 For any decreasing zero sequence

P ro o f: By partitioning the index set into pieces of the form [2fc,2 fc+1) and using the monotonicity of (/x^) we get

44

H.-A. Braunss, H. Junek and E. Plewnia

Now we can state the asymptotic result: T h e o re m 3.5 For any rank p and any admissible sequence A the following hold true:

P ro o f: By Proposition 2.2.6 it is sufficient to prove the statements only for the rank za The parts 1 and 3 follow easily from Proposition 3.3. Let us turn now to part 2. We first suppose From Proposition 3.3 we get Hence

by Lemma 3.4. This yields ||(a£(P;0)||2r — 8 ^ ||A ||r . In the general case \ < r < 2\p-m) we use fact that the real function f(q) = 2(q-m) *s decreasing on (m, oo) with limit / ( oo) = | . Therefore, given r, there is some q > p with r = 2(q-m) • From the particular case we already know (a%(P\\iq)) G £2r, and from Px \£p = P\\iq O (Id : ip -> i q) we finally get a^(Px \£p) < a£(PA\£q) • ||Id||m.

4. E s tim a tio n fro m below . In this section we are interested in lower estimates of the approximation numbers a£(P) for standard and diagonal polynomials Px of degree m > 2 on the Hilbert spaces £% and £2 . From 3.2 we know ||P a|^2|| = ||A||oo for diagonal polynomials of degree m > 2. This shows ao(P\\£ 2 ) = Halloo* P ro p o s itio n 4.1 The linear operators R : i 1^ 5^(x,ei)eJri and T : i —1

satisfying T o R = I d ^ .

—►P ( m^?) defined by R x —

—* £^o defined by T(P) = (P(ei))™=1 are contractions

Approximation numbers for polynomials

45

Proof: We have R x = P \ for A = ((x,ei)). This implies ||Pr|| = ||Pa|| = IIA||oo = Halloo• Therefore, R is even an isometry. Concerning T we have ||T(P)|| = ||(P(ei))||00 - max|P(e*)| < ||P|| hence ||T|| < 1. Finally, we have (T o R )ej = Te™ = (eyi(e»))i = ej for each ej, and this shows T o R = Id. P roposition 4,2 For any sequence A = (A*) let us consider the linear operators D™ : ^5 ~ * P (m^2 ) 9iven by D ^ x = YjX ^x.e^e™ and D \ : i £ 9™™ by

i D \x = 53 A$J* = D \, and this implies a,k(D™) < ||i?||afc(£)J) = afc(Z)JJ and a.k(D\) < ||T||afc(Z?r) = ak(D ? ). n P roposition 4,3 Let PA = 53

€ P (m"l'1^2 ) be a diagonal polynomial of n degree m + 1 and let D™ : ££ —►PC71#?) be given by D™x = 53 X^x^e^e ™. TTien i=l

we Aave < ( P A|^ ) >

: l 2 -> P (m^ ) ).

Proof: The case /c = 0 is shown by ||Pa|£*|| = Halloo = \\F>\\\ = II-CTII- Suppose now k > 1. Let {a* G : 1 < 2 < A;} be any family of linear functionals. Using the fact that the polarization constant of the Hilbert space ££ is 1 (cf.[2, Proposition 1.44]), we obtain

since

is an operator of rank


2

and A = (Ai)™=1 we have ^ ( P aI ^ ) > min \\i\ • l\ : Pf ^oo)* Let / : ^5 -> be the inclusion map and define 5 a : t%o -» *£) bY ^Ae* = Aiei and 5 A-i : ^ ^ by 5 A-ie< = A^e*. Then we have I = S \ - i S \ I = S \ - i D \ , and therefore ak(I : £% —> ^£o) < ||5 a- i ||afc(D^) = max |A^-11•ak(D \). So we get min |A < ak(D \). Finally, we have ak(I *£%

^oo) = \ J rl7r Ly [3, 11.11.8 and 11.7.4]. This proves the statement.

P ro p o s itio n 4.5 Let A = (A*) be admissible for m >

2

and p — 2. Then we have

In particular, we have a£(5m |^ ) = 1 /o r p < p < v , m > 2 and k € N . P ro o f: We fix n > k. Let Jn : ££

^2

be the canonical embedding and define

PA n = P a o Jn. By 4.4 we get An • = a£(PA o J n) < a j'( t t) • || J n ||m — ak(P*)i and iLe claim follows by putting n = k + 1 oi n = 2 k, respectively. The supplement follows now from Proposition 2.2.7. R eferences [1] S. B rademann , Approximationsideale homogener Polynome und holomorpher Funktionen, Thesis, Potsdam 1993.23456 [2] S. D ineen , Complex Analysis on Infinite Dimensional Spaces, New YorkHeidelberg-Berlin, 1999. [3] A. PiETSCH, Operator Ideals, Berlin, 1978. [4] A. P ietsch, Ideals of multilinear functionals, Proceedings of the Second In­ ternational Conference on Operator Algebras, Ideals and Their Applications in Theoretical Physics, Leipzig, 1983. [5] A. P ietsch, Eigenvalues and s-Numbers, Leipzig, 1987. [6] E. P lewnia , Approximationstheorie nachraumen, Thesis, Potsdam 1997.

homogener Polynome

auf Ba-

Applications of Laguerre Calculus to Dirichlet Problem of the Heisenberg Laplacian Der-Chen CHANG 1* and Jingzhi TIE 2 1. Department of Mathematics, Georgetown University, Washington D.C. 20057, USA E-mail: [email protected] 2.

Department of Mathematics, University of Georgia, A thens, GA 30602, USA E-mail: [email protected] The purpose of this paper is to describe research we have been doing recently in an area of analysis lies at the confluence of two directions of research: the study of Laguerre calculus on the Heisenberg group, and the theory of subelliptic boundary value problems. Here we intend only to outline part of the results we have obtained. Detailed proofs will appear elsewhere. The non-isotropic Heisenberg group H n is the Lie group with underlying manifold

and multiplication law

where a i ,a 2, •*,an are positive numbers. The sub-Laplacian on the Heisenberg group is

where

1 This first author is partially supported by a grant from National Science Foundation and a Competitive Research Grant at the Georgetown University.

47

D.-C. Chang and and J. Tie

48

It is convenient in our setting to use the real coordinates for the Heisenberg group. So we let Zj = Xj 4 - i£j+ n , and

for j = 1,2, • • •, n. Then the sub-Laplacian becomes

Let us consider the following Dirichlet problem associated with Ca : (i)

This problem was studied by several mathematicians (see e.g., Jerison [1], [2] and Capogna, Garofalo and Nhieu [3]). It is known that the operator Ca is hypoelliptic for all a £ C with exceptions ± a = 5^?_i(2fcj-fl)oj where k = (fci,. . . , kn) £ (Z+)n (see e.g., Folland-Stein [4] and Chang-Tie [5]). It is striking that the Dirichlet problem ( 1 ) can not be solved in general for Ca while a < —n or a > n even when fi = {(z,t) £ H n : t 2 4- ]C j= iai l zjl 4 < 1}? “u11^ ball” on H n (see e.g., Dunkl [6 ]). In this paper, we concentrate on the construction of the explicit solving operator for the problem ( 1 ) by using Laguerre calculus. We assume that fi is a bounded connected open set in H n = R 2n+1 whose boundary d fi is smooth, say C 2, hypersurface. We make the hypothesis that Q lies on one side of its boundary. Because of the Riemannian structure on R 2n+1 we may talk about the normal to this hypersurface. On the other hand, dfi divides R 2n+1 into two regions: an inner region, ft, and an outer one, the interior of R 2n+1 \ Q. The former is relatively compact; but the latter is not. The normal line to dQ at some point consists oi two well-defined half line: the exterior normal, and the interior one. We may also define the unit vector field n at each point of dfi, normal to dfi and pointing into R 2n+1 \ fi. We first introduce Green’s theorem and one of its corollary. Here we assume that fi is a bounded domain in R n T H E O R E M 0 .1 Let lf • • •, (j>n b e n complex-valued functions which are defined and C 1 in fi, i.e., in a neighborhood of fi. Denote $(x) = (0i(x), • • •, n (x)). Then we have

where

Laguerre Calculus to Dirichlet Problem of the Heisenberg Laplacian

49

We denote d u /d n the partial differentiation in the direction of the outer normal to dCt:

As usual, < z, w > = Z\W\ + • • • -f znwn denotes the real scalar product between z € C n and w G C n. C o ro lla ry 0.2 Let (x) G C 1^ ) , v G

We have

We now return to the boundary value problem ( 1 ). Let x n denote the charac­ teristic function of the bounded open set Q. By the Leibniz formula we have ( 2)

By Laguerre calculus (see e.g., Chapter 2 in Berenstein-Chang-Tie [7]), the funda­ mental solution E a of Ca has the following form:

Here

is the complex distance and

is the volume element defined by Beals, Gaveau and Greiner [8 ]. Since all members in (2) have compact support, we may apply E a* to them. It follows that

(3) Now the problem reduces to understand the following distribution

50

D.-C. Chang and and J. Tie

Let 0 be a test function in H n supported in a neighborhood U of Q. By a detailed computation, we have

Notice that, when x G f i, ^ ( y - 1 *x) is C°° in y in a neighborhood of dQ. Plugging the above indentity into (3). We may treat $ as a distributions in the variable y. On the other hand, E a is locally integrable and therefore, E a * ( x n f) can be computed by integration over f2. Thus, for x £ ft, we obtain the following representation for u:

Here the vector fields X act on the variable y which means we have to replace x by y in X. We now define the Green's function G(x, y) of the Dirichlet problem ( 1 ) with the homogeneous boundary condition, i . e G (x,y) = 0 when the argument point y is located on dQ.. Thus the Green’s function has the form

where ^ ( x , y) is a compensating operator of ( 1 ) which reduces the fundamental solution E q to the boundary We will decide the condition on S a . Since G(x, y) is the fundamental solution, i.e.,

therefore, 5(x, y) has to satisfy

Laguerre Calculus to Dirichlet Problem of the Heisenberg Laplacian

51

The above identity follows integration by parts. This in turn yeilds Sa (x,y) is the solution of the boundary value problem: (4) The superscript y means the differential equation acts on the variable y. Note th at when we insert the Green’s function G into the above integral rep­ resentation, it plays the role of E a . The contribution from the normal derivative < X (u ),n > disappears since G (x,y) = 0 when y G dfl. Therefore, we obtain the specific representation (5) for the solution u of the Dirichlet problem (1) as a definite integral of the prescribed boundary values, multiplied by the kernel < X a [G (x,y)],n >. Having developed the general theory, we turn our attention to the explicit cal­ culation of the Green’s function associated to the sub-Laplacian on special domains. Consider first the “upper” half space Q, — {[z,t] : t > 0}. We might expect that G (x,y) is odd in its dependence on s. Therefore, it must vanish when s = 0. Thus, corresponding to its singularity at y = [w, s] in the upper half space s > 0 , the Green’s function ought to posses an opposite singularity in the lower half space s < 0 at the point y = [w, - s] which is the reflected image of y in the hyperplane s = 0. These remarks suggest that G(x, y) can be expressed as the difference ( 6)

between the two fundamental solutions of CQ with singularities located at y and y. Indeed, (6 ) represents the Green’s function of the half-space t > 0 for a = 0 follows from direct observation. Furthermore, simple calculation yields:

where we have used the property of the fundamental solution E a (x) = E - a (x *) = £■_ 0 } can written as ( 8)

where G(x, y) and P (x ,w ) are given by (6 ) and (7) respectively. For the isotropic Heisenberg group, i.e., aj = a > 0 for j = 1, • • • ,n , the kernel P (x , w) can be computed explicitly. More precisely,

Here the variables p and (3 satisfy the following identity:

Moreover, we have the following theorem for the Poisson kernel P (x , w). T H E O R E M 0.3 I f f e L l {C n) and P is given by (7), set

Then C qu = 0 on O. I f f is continuous, u extends continuously to Q and u = / on dflj If f e L p(C n), (1 < p < oo) then u —* f in the IP norm as t —>0+ . Let ^ be a bump function supported in a small neighborhodd of x = (z,£) ^ (0,0). By using the formula (5), we may obtain the following estimates for the solution ( 1 ).

Laguerre Calculus to Dirichlet Problem of the Heisenberg Laplacian

53

T H E O R E M 0.4 Let Q = {(z,£) G H n : t > 0} is the upper half space of H n . For f € and g G r / 3+2 (dfi), £/iene is a unique solution u given by ( 8 ) to the Dirichlet problem ( 1 ). Moreover, ipu € Yp+2 ((i). Here Yp, 0 < (3 < oo, is the non-isotropic Lipschitz space introduced by Folland and Stein (see [4]). Acknowledgement. We would like to thank the organizing committee, es­ pecially Professor Shanzhen Lu, for his invitation to participate the second ISAAC Congress at Fukuoka Institute of Technology, Fukuoka, Japan.

R eferen ces [1 ]

D. Jerison; The Dirichlet problem for the Kohn Laplacian on the Heisen­ berg Group, Parts I and II, J. Funct. Analysis, 43 (1981), 97-142.

[2 ]

D. Jerison; Boundary regularity in the Dirichlet problem for manifolds, Comm. Pure. Appl. Math., 36 (1983), 143-181.

[3]

L. Capogna, N. Garofalo and D.-M. Nhieu; The Dirichelt Problem for sub-Laplacians, preprint, (1999).

[4]

G.B. Folland and E.M. Stein; Estimates for the db complex and analysis on the Heisenberg group, Comm. Pure. Appl. Math., 27 (1974), 429-522.

[5 ]

D.C. Chang and J. Tie; Estimates for spectral projection operators of the sub-Laplacian on the Heisenberg group, J. d 5Analyse Math., 71 (1997), 315-347.

[6 ]

C.F. Dunkl; Boundary value problems for harmonic functions on the Heisenberg group, Canadian J. Math., 38 (1986), 478-512.

[7]

C. Berenstein, D.C. Chang and J. Tie; Laguerre Calculus on the Heisen­ berg Group, AM S/IP series in Advanced Mathematics, will published by International Press, (1999).

[8 ]

R. Beals, B. Gaveau and P. C. Greiner: Complex Hamiltonian mechanics and parametrices for subelliptic Laplacian, Bull. Sci. Math., 1 2 1 (1997), 1-36, 97-149, 195-259.

on C R

The Pisier-Schiitt Theorem for Spaces of Polynomials Andreas DEFANT, Juan Carlos DIAZ 1 , Domingo GARCIA 2 and Manuel MAESTRE 2 Manuel Maestre, Dpto. Anaiisis Matematico, Universidad de Valencia, Spain E-mail: [email protected]

Introduction. Pisier [21] and Schiitt [24] proved independently that for any tensor norm a on the tensor product X 0 Y of two Banach spaces X and Y with unconditional basis (x*) and (Vj)i respectively, the following are equivalent: (xi Xj)ij is an unconditional basis of if and only if X (Q aY has unconditional basis if and only if X 0 Q7 has a property isolated in the local Banach theory called the Gordon-Lewis property (see the definition below and a exposition in [22]). W ith this kind of approach we attach the open problem of the existence of an unconditional basis in spaces of m-homogeneous polynomials on infinite dimensional Banach spaces. To dos so, we prove an analogue of the Pisier-Schiitt theorem for symmetric tensor products. We calculate for each m an asymptotically optimal estimate of the unconditional basic constants of all m-homogeneous polynomials on €£, 1 < p < oo and we give various applications for nuclear and compact polynomials. We show that for a wide class of Banach spaces both the space of all m-homogeneous and of all compact m-homogenous polynomials do not have unconditional basis for m > 2 . In our purpose of studying the existence of unconditional basis in spaces of polyno­ mials we axe going to use some tools that come from Banach spaces theory and that we define to provide an easier understanding of the proofs of this paper. (For the notations and notions needed from Banach spaces theory and the theory of operator ideals see [19], [8],[25] and [11 ], and for symmetric tensor products and polynomials 1 The second author was supported by DGES pr. no. P.B.97-0447 2 The third and fourth authors were supported by DGES pr. no. P.B.96-0758.

55

A. Defant, J.C. Diaz, D. G arda and M. Maestre

56

we refer to [14].) Given a Banach space X with a basis (Xi) we are going the define the unconditional constant of that basis, denoted x ((xi);X )

35

x((x 0 and dm > 0 such that for every symmetric m-tensor norm (3 there exist a m-tensor norm a such that for every

57

The Pisier-Schutt theorem for spaces of polynomials

Banach space X :

Moreover the dual norms a* and (3* satisfy that a* is a tensor norm, (3* is a symmetric tensor norm and are also equivalent in the above sense.

Given m ,n G JN,m > 1, we will denote by G {1, ...,n}}, J ( m , n )

{I =

:= { l= (ii,...,z m)>

G M { m ,n ) ,i\ < ... < im}. When there

is no risk of confusion we will write M and J . If (X j)jLx are Banach spaces of dimension n G IN with basis (xJJJLj and dual orthogonal basis (x^ ’JJLj , j = 1,..., m we will denote by xT := x\ 0 . . . 0

I G M ( m , n ) the basis of X \ 0 . . . (8) X m

and x$ := xj* 0 ... 0 x?™, I G M ( m , n) its dual basis in X { 0 ... 0 X

If (X j) 1JL1

are infinite dimensional Banach spaces with basis (x jjg ^ and biorthogonal system 3 = 1’ •••>m and a is a tensor norm, then by [?] and [23] {xj := xj10 . . . 0 x ^ , I

g

lNm} endowed with the square order is a basis of X i 0 a ...0 a -X’m with biorthog­

onal system {x^ := x ** 0 ... 0 xj™, IG IN171} T h e o re m 1 (P is e r-S c h iitt th e o re m for s y m m e tric te n s o r n o rm s).- Given a symmetric tensor norm (3, a natural number m and a Banach space X with unconditional basis (x i)? ^ , the following are equivalent (a) where C" is a constant. Thus for / G E x p (E ;(r,L )) taking r ' with r < r ' < r / ( l —e), we have |(T2 ,/( z ) ) | < C^II/llxCr'.L) < °°i which implies T G E xp'(E ; (r, L)). By (10), the (k,k — 21)harmonic component of T equals to the given T k ^ - 21q.e.d.

2.1

F o u rier-B o rel tr a n sfo r m a tio n

The Fourier-Borel transform T T of T e Exp'(E5; (0)) is defined by

where C is sufficiently small. We call the mapping T : T »—►T T the Fourier-Borel transformation. Since

for T G E xp'(E ; (0)) and sufficiently small w we have

(13) A.Martineau ([3]) proved the following theorem:

Double series expansion with harmonic components

85

T heorem 2.3 Let N(z) be a norm on E. The Fourier-Borel transformation T establishes the follomng topological linear isomorphisms:

3

H o lo m o rp h ic fu n c tio n s o n th e d u a l Lie b all

Note th at limp_ 00( ^ ^ ) 1/ p = 1 for any constant q € R . By means of the FourierBorel transformation, for the dual Lie norm L*(z), we have the following theorems: T heorem 3.1 (14)

T heorem 3,2 ([2, T h e o re m 5.21)

P roof.

We prove only (14).

and we have

Therefore Theorem 2.2 implies limsup^QQ (\\Tktk - 2i\\c{Si)/kl)1^k < 1 /r and we have (14).

86

K. Fujita

Conversely, suppose that a sequence {fkyk - 2i} of homogeneous harmonic poly­ nomials satisfies

Since TT{ z) = f ( z ), Theorem 2.3 implies f (z ) e 0(B *[r]).

q.e.d.

R eferen ces [1] K.Fujita and M.Morimoto, Holomorphic functions on the dual Lie ball, to appear in the proceedings of Second ISAAC congress. [2] Y.Iwahara, Entire functions of exponential type related to the Lie ball, M aster’s thesis, Sophia University, 1998. [3] A.Martineau, Sur les fonctionnelles analytiques et la transformation de FourierBorel, J. d’analyse Math, de Jerusalem 11, 1963, 1-164. [4] M.Morimoto, Analytic functionals on the Lie sphere, Tokyo J.M ath., 3(1980), 1-35 [5] M.Morimoto, A generalization of the Cauchy-Hua integral formula on the Lie ball, Tokyo J. Math., 22(1999), 177-192.

Extension of Pluriharmonic Functions in Locally Convex Spaces Yukio FUKUSHIMA Department of Applied Mathematics, Fukuoka University FUKUOKA, 814-0180, Japan E-mail: [email protected] 1. Introduction The present paper concerns the pluriharmonic extension on a domain over a complex manifold, the dimension of which may be infinite and for which the Levi problem may not have an affirmative solution. In the present paper, we give the following result : Let E be a sequentially complete locally convex C-linear space, (ft, p) be a domain over the space E and (ft, p) be the pseudoconvex hull of (ft, ip) in the sense of Matsuda[24]-[25], We prove that any pluriharmonic function / on ft can be extended to a pluriharmonic function on f 2.

2. Envelopes o f pluriharm ony Let E be a sequentially complete locally convex C-linear space. A pair (ft, p) is said to be a open set over E , if p : ft —>E is a locally biholomorphic mapping of a Hausdorff space ft in E. Moreover, if ft is connected, (ft, p) is said to be a domain over E. Let (ft, 0. P ro p o s itio n 2 I f V is a countable family of radial weights on X then each element of V is rapidly decreasing if and only i f V ( mX ) is contained in 7iV(X) for all m G IN U {0}. In that case the topology ry coincides with the norm topology on V V { mX ) for all m e IN U {0}. D efin itio n 3 Let V be a countable family of continuous radial weights on U . We say that V satisfies Condition II if for each v in V there exists R > 1 and w in V such that

Schauder decompositions of weighted spaces of holomorphic functions

105

Theorem 4 I f V is a countable family of radial weights on U satisfying Conditions I and II. Then the sequence (7>Vr(mX ))m (resp. ( W o ( mX ) ) m) is an S-absolute and decomposition ofHV( U) (resp. HVo(U)).

7 -complete

Corollary 5 I f V is a countable family of rapidly decreasing radial weights on X satisfying Conditions I and II, then TiV(X) = 7iV0 (X).

Corollary 6 I f V is a countable family of radial weights on U satisfying Conditions I and II, then (a) HV{U) (resp. TlVo(U)) is a reflexive space if and only if each space of the sequence (VV(rnX ) ,T y ) rn, (resp. (VVo(mX ) , T y ) rn) is a. reflexive space. (b) I f U = X and the weights are rapidly decreasing, then H V ( X ) is reflexive if and only if each element of the sequence (P(mX ), ||.||)m is a reflexive Banach space. This corollary is a generalization to weighted spaces of the one given by AnsemilPonte in [1 ], for the space (7ib(X),Tb). We refer to [4] where several interesting (even non-radial) examples of weights on an open subset of - } , n — 1,2... is a fundamental system of U-bounded subsets of U. For each n in IN consider vn : U — ►[0,11 defined as V := (vn)n is an increasing sequence of continuous radial weights on U such that HV(U) == 7ib(U) algebraically and topologically and satisfies Conditions I and II.

Example 8 a) I f v : X — >]0, +oo[ a radial continuous weight such that inf{u(rr) : ||x|| < n} > 0 , for all n = 1 , 2 .., then the families of continuous radial weights V := (v„)£Li and W := (wn)™=l, defined as vn(x) := u(2±ix),ion (x) := u(£x), x in X , n = 1,2... satisfy Condition I and I I and V V ( mX ) = Vv{mX ) = V W { mX ) algebraically and topologically. b) I f additionally H V ( X ) / ® „ =0 V V ( mX ), for all k in IN and v (\x ) < v(x), for all A > 1, x in IN, then H W { X ) C H v( X) C n v ( x ) c) A concrete example fulfilling all the hypothesis of a) and b) would be v(x) := e“ ll«llp? x in X , p > 1. V = (e“' 2 J i H*MP) ^ =1, W — m that case by Proposition 2, (P (mX ), ||.||) = (Pv(mX ), Tv), for all m = 0,1,.... d) Another one satisfying only the hypothesis a) would be v(x) := 1+| a.||p, p > 1. In that case (Pv(mX ) , r v) = (P (mX ), ||.||), if m < \p] and ('Pv(mX ) ,r t,) = {0} if m > [p].

106

D. Garcia, M. Maestre and P. Rueda

The notation of v, V , W of Example examples.

8

will be used again in the next three

E x a m p le 9 Let Y be a complemented subspace of X , let Z be its complement and 7r : X — ► Y the projection of X onto Y . I f t : Y — >]0, -fo o [ and u : Z — ►JO, -foo[ are two continuous radial weights as in Example 8 .a), then v : X — >]0 , - f oo[ defined as v{x) := t(n(x))u(x—ir(x)), for all x i n X , is a continuous radial weight satisfying the hypothesis of Example 8 .a). Concrete examples would be:

E x a m p le 10 Let X be an infinite dimensional complex Banach space, and let (x* )n be a sequence of continuous linear forms weakly-star convergent to 0 in X* with ||rr*|| = 1, n = 1,2.... The sequence of radial continuous weights V = (un)^=i defined by

E x a m p le 11 Consider in t \ the continuous weight v : t\ — >]0, -hoo[ defined as

Schauder decompositions of weighted spaces of holomorphic functions

107

Let

endowed with the topology of the uniform convergence on the sets D a. By Corollary 5 of [2], it is a complete barrelled (DF)-space such th at its strong dual is topologi­ cally isomorphic to (‘H V (U ), t v ). In particular, when V = {v}, Gv(U) := GV(U) is a Banach space such th at its strong dual is topologically isomorphic to the Banach space Hv(U). T h e o re m 1 2 Let V = (vn)n be an increasing sequence of radial rapidly decreas­ ing continuous weights satysfying Conditions I, II and such that for each n 6 IN vn (tx) / 0 for allt e (D with |£| < 1 whenever vn(x) ^ 0. Then GV{X) = ind nGvn (Un) holds algebraically and topologically, where Un = {x € X : vn(x) > 0}. This inductive limit is also boundedly retractive. C o ro lla ry 13 Under the hypotheses of Theorem 12 TiV(X) is a quasinormable Frechet space. The proofs of the results stated here are included in [7]. A cknow ledgem ents: We warmly thank to Klaus Bierstedt and Jose Bonet for introducing us in the subject and for the many helpful discussions we have held with them.

R eferen ces [1 ] J. M. Ansemil and S. Ponte, An example of quasinormable Frechet function space which is not a Schwartz space, in: Functional Analysis, Holomorphy and Approximation theory, S. Machado (ed.), Lecture Notes in Math. 843 (1981) 1- 8 . [2] K.D. Bierstedt and J. Bonet, Biduality in Frechet and (LB)-spaces, in: Progress in Functional Analysis, K.D. Bierstedt et alt.(ed.), North-Holland Math. Stud­ ies 170, North-Holland, Amsterdam, 1992, pp. 113-133. [3] K.D. Bierstedt, J. Bonet and A. Galbis, Weighted spaces of holomorphic func­ tions on balanced domains, Michigan Math. J. 40 (1993) 271-297. [4] K.D. Bierstedt, J. Bonet and J. Taskinen, Associated weights and spaces of holomorphic functions, Studia Math. 127 (1998) 137-168. [5] K.D. Bierstedt, R.G. Meise and W.H. Summers, A projective description of weighted inductive limits, TVans. Amer. Math. Soc. 272 (1982) 107-160. [6 ] K.D. Bierstedt and W.H. Summers, Biduals of weighted Banach spaces of an­ alytic functions, J. Austral. Math. Soc. Ser. A 54 (1993) 70-79. [7] D. Garcia, M. Maestre and P. Rueda, Weighted spaces of holomorphic functions on Banach spaces, 1999, to appear in Studia Mathematica.

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[8 ] L.A. Rubel and A.L. Shields, The second duals of certain spaces of analytic functions, J. Austral. Math. Soc. 11 (1970) 276-280. [9] P. Rueda, On the Banach-Dieudonne theorem for spaces of holomorphic func­ tions, Quaestiones Mathematicae, 19 (1996) 341-352. [10] D.L. Williams, Some Banach spaces of entire functions, Ph. D. Thesis, Uni­ versity of Michigan, 1967.

Spaces of Banach-Valued Holomorphic Functions in the Polydisk in Connection with Their Boundary Values GULIEV Vagif S. Faculty of Mechanics and Mathematics, Baku State University 23 Lumumba St., Baku 370148, Azerbaijan E-mail: [email protected]

Recently, much attention in mathematical literature has been concentrated on the Hardy spaces Hp(Un, E) of the functions w(z) = w (rQ , analytic in the polydisk

with values in a Banach space E . This attention is explained by the fact that, in the study of the Hardy spaces, the geometry of space E plays an important role. Hence, many properties of the scalar-valued (i.e., for E = C) Hardy spaces Hp change and, to some degree, characterisize the structure of E. Our results are the necessary and sufficient conditions th at connect E-valued holomorphic functions in the polydisk Un with their boundary values on the sceleton t/ n (i.e. on the torus T "). Under certain restrictions on the parameters p and q, we give the description of the boundary values in the simplest terms of harmonic analysis on T n. The applied technique is new and of independent interest. It is based on the estimates connected with the Poisson kernel and on the use of the theory of vector multipliers. Let Cn be the n-dimensional complex space of points z = ( zi ,. . . , zn), Zj — Xj + ipj- A E -valued holomorphic function cj ( z ) = a>(zi,... ,z n) on Un can be represented as a power series: (i)

In the last expression we have used the compact notation k = (fci,. . . , fcn), N q = No x ... x N q, Z n — Z x ... x Z (n times), N q = iV (J{0}, N denotes the 109

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collection of all natural numbers, Z is the set of integers and z k — z \ x ... z kn. Recall th at if A;! = k \ \ ... kn\ and |fc| = A* + ... + kni then

Let T n be the ’’distinguished boundary” of Un . This is the n-dimensional torus:

i.e., the collection of points C = (etil, •. •, eltn) , t = ( t i , . . . , tn) . In this article we make essential use of the fact that both T™ and Un can be represented as direct products

where U and T are one-dimensional disk and torus, respectively. The properties of holomorphic function on u on Un are completely determined by its properties on the ’’subdistinguished boundary” T n given by

Since the subdistinguished boundary can be represented as direct product

we can characterize various classes of holomorphic functions by their radial (i.e., with respect to the variable r) and/or tangential (i.e.,with respect to the variables t i , ... , t n) properties. This is especially useful in considering n-harmonic functions on t / n . In the holomorphic case it is expedient to resort to an operator 1Z that mixes the radial and tangential properties and acts according to the formula

Its role and its significance are determined by the fact that it generates a simple multiplier transformation of the series ( 1 ):

Let O (£/n , E) denotes the set of valued functioins holomorphic in t/n , llu;(r *)llp,£ denotes the nom of the function uj (r£ ), £ € T*1, in the space Lp(Tn , E ) of functions that are Bochner summable on T n .

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Using the operator TZ (more precisely, its restriction to T "), we study the Evalued holomorphic Nikol’skii-Besov space B+* (J7n, E ) , for which the behavior of averages is defined for s > a by the law

Here, a is a real number, s is a positive integer, s > a , and 1 < p < oo. The expression after the colon (in braces) is denoted by the symbol b^q{ in J3+* (t/n, E). The seminorms of (2), (3) are equivalent for distinct s. Note that, for p = q and in the case E — C and a = 0, the space ([/n , E) coincides with the known weighted Bergman space. For distinct p and g, we obtain a richer classification of function spaces, since the tangential and radial characteristics of the functions are, in a sense, taken into account separately. If E ^ C , then, as mentioned above, additional effects appear that are connected with the geometry of E. Unfortunately, the Hardy space Hp(Un,E ) is not contained in the scale (I/n, E) (due to the requirement s > a , which is important for a number of reasons). Its nearest ’’relative” in this scale is the space B*®Q(Un,E ) [which coincides for p — oo and E = C , n = 1 with the known Bloch space B consisting of the functions co(z) e O (U) with the property | ^1 —|z|2^ u/(z)| < oo, z e U ]. We denote by Bp q(Tn , E) a Besov type space of the valued functions on T n (see [2],[3]). Recall that a Banach space E is called Burkholder convex if there exists a function £(£, 77),defined and symmetric on E x E and convex with respect to each of its arguments, such that

Such a space E is often called UMD-space and is written as E € U M D (see, e.g., references in [4]). Denote by V = V (T n ,E ) and V* = V t(Tn ,E ) the spaces of test E -valued functions and F?-valued distributions on 7™ with the corresponding topologies. The spectrum of an / G 7? is defined to be the collection of k £ Z* such th at the Foureir coefficients f(k ) ± 0. One of our main results is T h e o re m A Let E e U M D . Then, in order that a E-valued function u>(z), z € C/n , holomorphic in the polydisk t/n, belong to the space B+* (Un;E ) a > 0 , 1 < p < 0 0 , and 1 < q < 0 0 , it is necessary and sufficient that the boundary value f ( t ),

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t e T n , of the function u;(z) on the skeleton T 71 that is regarded in the sense

belong to the space B ^ ^ T 71, E) and have a nonnegative spectrum. In this case, the corresponding norms are equivalent. The method of proving this theorem is of independent interest. It is based on various theorems about Fourier multipliers of the vector-valued functions of the class Lp (T 71, E ).

1

S paces Bp q( T n , E ) a n d C h a ra c te riz a tio n o f T h e ir D eco m p o sitio n s

Staying on a torus, we assume, as is customary, for the kih difference of the function

(i.i) For F-valued function F, the derivatives and integrals are meant in the strong and Bochner senses, respectively. Thus, let E be a Banach space. We denote by Lp(Tn , F ), 1 < p < oo, a set of the functions / (£), strongly measurable on T n, with the finite norm

where do(Q is standard Lebesgue measure on T 71 . In this case, the functions th at differ only on a set of measure zero are identified. D efin itio n 1 The linear space of the functions f e L P(T71, E) for which the quan­ tity ( 1. 2)

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is finite for any 0 < a < oo (where a > 0 , is a positive integer, and k > a ) is denoted by E). The linear space Bp QO(T n 1E) is defined similarly; however, in this case ( 1. 2')

The number of elements of B p q(Tn , E) does not depend on a (cf [5] ). For defi­ niteness, we consider a = oo. As known (see [2] ), that seminorms (1.2) and (1.29) unessentially depend on k (remain equivalent for distinct k > a). For definiteness, one might take k = \a\ + 1 . D e fin itio n 2 The class of the functions B p q(fTn, E), a > 0 and 1 < p, q < oo that are normed by using (1.3) is called a E-valued (abstract) Besov space. The Besov space is a Banach space. In this section, our goal is to describe the spaces Bp ^T * 1, E) in terms of the Fourier transform of the functions contained in it. This description is especially simple in the case when E is Burkholder convex, i.e., E € U M D . We largely deal with this case. Let Q n be a ” discrete cube” in Z n, i.e., a collection of multi-indices

m m—1 Denote by \ A the characteristic function of a set A C Z n . The set K m —Q \ Q m (where Q— Q 2^ ) is called a corridor. If a function f{ t), t e T 71, is expanded in a Fourier series

then the polynomial

is called the packet of / along the corridor Km, m = 1 ,2 ,__ Further, let fo(t) = Co. The decomposition of / with respect to corridors is defined to be the represen­ tation of / by the weakly convergent series

V. S. Guliev

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Theorem 1 Let E G U M D , 1 < p < oo, 1 < 0 < oo and a > 0. The function f G L p(T n,E ) belongs to the space B p 0(Tn, E) if and only if

(1 4 ) In this case, (1.5) in this sense of weak convergence, and the expression on the left-hand side of (1-4) is equivalent to the norm of f in B ^ e {Tn, E) (for 0 = oo, it is meant as

suPm 2 - “ ||/m||PiB;. The theorem 1.1 is proved in the case of n = 1 in [2] and in the n- valued case in [3]. D efinition 3 The set of the generalized functions f G D f(Tn, E) = D ' for which the sym m etric batches fm of the Fourier series have the property that

( 1 . 6)

is called the abstract (E-valued ) Besov space

It is clear that, for a > 0 and E G U M D , the latter definition coincides with that used above. Indeed, the fact that the function / belongs to D '(T n, E) provides the weak convergence of its Fourier series and the corresponding series of the batches / m. Thus, / G Lp(Tn,£') and the hypotheses of Theorem 1 hold, i.e., we return to the initial definition (under the condition E G U M D ). It is also clear that Bp ^T * 1, E) remains a Banach space not only for a > 0 but for a < 0 with the norm defined by relation ( 1 .6 ) (for a > 0 , this norm is equivalent to the initial and, hence, we use for it the previous designation).

2

B o u n d a ry V alues o f F u n ctio n s F rom

In this section we consider estimates connected in one way or another with the Poisson kernel. Subsection 2 .1 is devoted to the one-dimensional case, and §2.2 to the multidimensional case.

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2.1. T h e re m a in d e r o f th e T aylor series, a n d re la te d e s tim a te s (one­ d im e n sio n a l case). The method of dissecting decomposition, which we used to define the spaces Bp^T™ , E) , gives rise to the necessity of dissecting Fourier series by the E —valued function w(z), holomorphic in U, into the batches wrn(z):

where

wq(z )

= Co, and

We need estimates of wm (z) = wm (re1*) in terms of the boundary function

Generalizing the situation somewhat, we consider a holomorphic function g(z) hav­ ing a zero of order l at z = 0 . P ro p o s itio n 1 Let, for E e U M D , the function g e 0{U yE) have on dU Bochner p-summable radial boundary value g (e1*) = h(t) and be expanded into the series (2. 1)

Then, for 0 < r < 1 , the following estimate holds: ( 2. 2 )

P ro o f. Recall that if

and E G U M D , then the E -valued function g(z) is restored by the boundary values of h(t) [1 0 , 11 ] by the formula

For this, as shown in [11 ], it is sufficient that E possess the Radon-Nicodym property, which holds in our case as a consequence of the reflexivity of the U M D space.

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Rewrite (2.1) in the form (2.3) The fraction in (2.3) is a E'-valued function analytic in U that has the boundary values e~tHh(t) on T by the hypotheses of the mentioned theorem and, hence, can be restored from these boundary values. Thus,

Taking the boundary Lp-norm of both sides of the equality and using the Jung inequality for convolutions give

C orollary. The following inequality holds: (2.4) 2.2. T h e m u ltid im e n sio n al case. We consider a /^-valued holomorphic function g(z) = g ( zi ,. . . , zn) on Un having expansion

with coefficients vanishing for k £ Q\. We have the relation (2.5)

where for

Under these conditions we can write the products

Spaces of Banach-valued holomorphic functions in the polydisk

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where the fractional factors remain E -valued holomorphic functions on Un . If g(z) has integrable boundary value h(t) = h ( t\y. . . ,t n) on t/ n, then the indicated frac­ tional factors can be recovered from their boundary values, i.e.,

( 2 . 6)

Here

is the Poisson kernel on T*1:

We are interested in estimates of g(z) on the subdistinguished boundary T™ — n x [0 , 1) (when z = Cp, C € T**, 0 < p < 1 ) . Prom the foregoing we get P ro p o s itio n 2. Suppose for E € U M D , that a E -valued holomorphic function g(z) on Un has a boundary value h e 1^(7™, 2£), l < p < o o , a s p ^ l , with spectrum outside Qi. Then

T

(2.7) where the constants C\ and Ci w(z) can be represented by the series

(3.2) and we must estimate the quantity

in terms of the norm

(3.3)

Spaces of Banach-valued holomorphic functions in the polydisk

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We regard the relation (3.2) as a multiple Fourier series whose coefficients depend on the parameter p, and we break up this series into segments with respect to the corridors K m : (3.4) The segment (3.4) is obtained from (3.3) under the action of the multiplier trans­ formation by the factors

For p 6 [1/2,1], the norm of this transformation does not exceed esssup const x (this is verified by the Martsinkiewicz criterion). Using the analog of Martsinkiewicz criterion for the U M D -valued functions, which was obtained by Zimmerman [13], produces 2 ~ma

(3.5) where p e [1 / 2 , 1 ) and the constant c depend on neither m nor p. If we multiply (3.5) by 2ma, raise to the power g, and integrate from (1 —2 "m) to ( l —2 ~m_1) , we get

Summing over rrt from 1 to oo (in this case, we can write p in the integrand on the right because p > 1 / 2 ) yields

(3.6) Recall that rp = (p,. . . , p) here. By our definitions, the quantity / m (£ )|m =0 = fo(t) coincides with the zero coefficient Co of the Fourier series of the E —valued function / ,

and the latter coincides in turn with w( 0 ) (because w(z) — $2 Ckrke%kt, f (t ) = linv - ,1 w(z)). Therefore, adding HColl# on the left-hand side in (3.6) and ||to(0)||#

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on the right-hand side, we conclude finally that (for

1

< q < oo)

An analogous conclusion for q = oo follows immediately from (3.5) after replacement of by (1 —p)(s“ Q) on the right-hand side. The necessity is proved (since the nonnegativity of the spectrum of / is obvious). Sufficiency. Now assume that a i£-valued function / has a nonnegative spec­ trum and belongs to B * q (T, E ) , i.e., it is represented by the series

(for a > 0 , we can speak about the convergence of the series in the metric Lp(Tn , E) and, for a = 0, we can speak about weak convergence), for which the quantity (3.7) is finite. We need to give an estimate for the expression

in terms of the quintity (3.7). Recall that

We have (the condition k > 0 is the same as k e N q ) The inner sum is obtained from the expresion (3.8) by means of the multiplier transforation with factor \k \8 ~ |fc|£o, where the quantity 1*1^ = max {fci,. . . , kn} varies from 2m _1 to 2 m — 1 . The norm of this multiplier transformation is majorized by the number 2 ms(in essence, a vector inequality of the Bemstein-type inequality [9] is used here). Therefore, from (3.8) we get

(3.9)

Spaces of Banach-valued holomorphic functions in the polydisk

12 1

The inequality (2.8) is used in the last step. The rest of the computations are of a tecnnical nature and axe accompanied by minimal explanations. By (3.9), for 1 < q < oo

(3.10)

(apply the Holder inequality to the sum). We use the following, inequalities, which are not very hard to derive ( 0 ): (3.11)

(3.12) Continuing (3.10), we get

by using (3.11) and then (3.12), and this is what was required. Since, fo(t) — / (0 ) = Co, the proof is complete for 1 < q < oo. For q = oo we proceed similarly. We have

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In the last step we use the inequality (3.12). The theorem is proved. Now note the extention of the above results to the case of n-harmonic E- valued functions. It is known [15] that, for E — C , the real part u (as well as the imaginary part v) of the holomorphic function w(z) = u{z) + iv(z), z G C n, is an n-harmonic function, i.e., the function u ( zi , . . . , zn) is harmonic in each pair of the variables Xj and yj(xj + iyj = zj). Of course, not every real n-harmonic function is the real part of a holomorphic function for n > 1 . The collection of the n-harmonic E-valued function w(z) = u(z) + iv{z) in Un is designated by H( Un, E). E- valued functions, n-harmonic in t / n, are closely connected with their boundary values on the skeleton T n of the polydisk Un. Under certain conditions (if E e R N P , i.e., E posseses the Radon-Nicodym property [10,11]), this conection is realized by the Poisson integral. The function w e H( U n 1E) has the boundary values on T n ,

and is uniquely restored from these boundary values by the Poisson integral

Starting from this, one succeeded in developing the theory of the theory of the B-space of n-harmonic E-valued functions in Un in the connection with their boundary values in T71, which is quite similar to the holomorphic case. A key point is the possibly of representing a valued function w on H{Un, E) by the Tailor series of the form

The corresponding spaces have to be introduced on the basis of the differention operator with respect to r (because differention with respect to Zj does not make sense)

[here, rp = (p, . . . , p) and rpeH = (peil, p e u )}. The following theorem holds for the space B“ q(Un, E): T h e o re m 3 (A ’) Let E e U M D , 1 < p, q < oo. In order that an n-harmonic E-valued function u) belong to the space B">g(l7n , E1), it is necessary and sufficient that its boundary value F (t) belong to the space B ^ q(Tn yE). In this case, the cor­ responding norms are equivalent

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T h e p ro o f of this theorem in considerable measure repeats the proof of Theorem A; however it requires the use of new considerations. Note that, in the case E — C , Flett [16] studied B-spaces of holomorphic and harmonic functions on the one-dimensional disc for 1 < p, q < oo. Oswald [17] investigated these spaces in the holomorphic case for 0 < p, q < oo when E — C .

R eferen ces [1] S.V .S vedenko , H ardy classes and the corresponding spaces o f analytic fu n ction s on the unit disk and ball, Itogi Nauki i Tekhniki, Mat. Anal., vol. 21, VINITY, Moscow, 1983, pp. 3-124, English transl. in J. Soviet Math., vol. 30, 1985. [2] V.S. G uliev and P.I. Lisorkin , Spaces o f B anach-valued analytic and periodic fu n ctio n s , Trudy Mat. Inst. Steklov, 1995, v.210, pp. 101-119; English transl. in Proc. Steclov Inst. Math., 1995, v.210, pp. 74-88. [3] V.S. G uliev , B anach-valued B esov fu n ction spaces , Proc. Azerbaijan Math. Soc., 1996, v.2, p.33-49. [4] D.L. B urkholder, M artingales and F ourier analis in B anach space , Lect. Notes Math., 1985, vol. 1206,pp. 61-108. [5] O.V. B esov , V .P. Il’in and S.M. N ikol’skii, Integral R epresen tation s o f Functions and Em bedding Theorems, Moscow, Nauka, 1975. [6] S.M. N ikol’skii, A pproxim ation o f Functions o f Several Variables and Embedding Theorem s, Moscow, Nauka, 1969. [7] O.V. B esov, S tu dy o f a Family o f Function Spaces in C onnection w ith Em bedding and E xten sion Theorem s, Trudy Mat. Inst. Steklov, Akad. Nauk USSR, 1961, vol.60, pp.42-81. [8] A .P. T erekhin, D ifferential Equations and C om putational M athem atics, Saratov, Saratov Univ., 1973, vol.2, pp. 3-28. [9] P.I. Lisorkin , G eneralized H older Spaces and T h eir Connection w ith Sobolev Spaces LT p . Sib. Mat. Sb., 1968, vol. 9, no. 5, pp. 1127-1152. [10] W. Hensgen , On the D ual Space o f H p ( X ) , 1 < p < oo, J. Funct. Anal., 1990, vol. 92, pp. 348-371. [11] A.V. B ukhvalov, H ardy Spaces o f Vector-Valued Functions , Zap. Nauchn. Semin. LOMI Akad. Nauk USSR, 1976, vol. 65, pp. 5-16. [12] T.R. M cC onnel, F ourier M ultiplier T ransform ations o f Banach-Valued F unctions, Trans. Am. Math. Soc., 1984, vol. 285, no. 2, pp. 739-757. [13] F. Zimmerman, On Vector-Valued F ourier M ultiplier Theorem s, Stud. Math., 1989, vol. 53, pp. 201-222. [14] V.S. G uliev , To the theory o f m vltiplicators o f F ourier integrals fo r Banach-valued fu n c­ tions i, Trudy Mat. Inst. Steklov, 1997, v.214, pp. 164-181; English transl. in Proc. Steclov Inst. Math., 1997, v.214, pp. 157-174.

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[15] W. R udin , F unctional theory in the u n it ball o f C n , New York; Springer, 1980. [16] T.M. F lett , L ipsch its spaces o f fu n ction son the circle and disc , J. Math. Anal. Appl., 1972, vol. 39, pp. 125-158. [17] P. O swald, On Besov-Sobolev spaces o f analytic fu n ction s in the u n it disc , Czech. Math. J., 1983, vol. 33, pp. 408-426.

Univalent C1 Mappings on the Unit Ball in C” Hidetaka HAMADAa and Gabriela KOHRb a. Faculty of Engineering, Kyushu Kyoritsu University, 1-8, Jiyugaoka, Yahatanishi-ku, Kitakyushu 807-8585, Japan E-mail: [email protected] b. Faculty of Mathematics Babe§-Bolyai University 3400 Cluj-Napoca, Romania E-mail: [email protected]

1

M. Kogalniceanu Str.

A b s tra c t: Let B be the unit ball with respect to an arbitrary norm on C n . We give analytic sufficient conditions for a local diffeomorphism of C l class on B to be univalent and to have a 4>-like image. When $ is holomorphic and / is a locally biholomorphic mapping with /(0 ) = 0, D f ( 0) = / , the conditions are also necessary conditions. We also give analytic sufficient conditions to be univalent and to have an Archimedean spirallike image or a hyperbolic spirallike image.

1

In tro d u c tio n

Let / be a holomorphic function on the unit disc A in C with /(0 ) = 0, /'(0 ) ^ 0. Then / is univalent and starlike if and only if $l[zff(z)/f(z)] > 0 for z e A. As a generalization of this result to several complex variables, Matsuno [11 ] and Suffridge [15], [16] gave an analytic condition for locally biholomorphic mappings to be biholomorphic and to have starlike images. More generally, a holomorphic function / on A with /(0 ) = 0, /'(O) = 1 is spirallike of type a if and only if ^i[exotz f f(z)/f(z)] > 0 for z e A. As a generalization of this result to several com­ plex variables, Gurganus [5] and Suffridge [17] gave an analytic condition for locally biholomorphic mappings to be biholomorphic and to have spirallike images. As a complete generalization of a starlike or spirallike domain, Brickman [3] introduced the concept of a $-like domain in C and he gave an analytic condition for a holo­ morphic function / on A with /(0) = 0, /'(0 ) ^ 0 to be biholomorphic and to have a 3>-like image. Gurganus [5] introduced the concept of a $-like domain in Cn 125

H. Hamada and G. Kohr

126

and he gave an analytic condition for normalized locally biholomorphic mappings on the unit ball in C n with respect to an arbitrary norm to be biholomorphic and to have $-like images. Hamada-Kohr-Liczberski [8 ] generalized the above result for normalized locally biholomorphic mappings on a bounded balanced domain in C n with C l plurisubharmonic defining functions. On the other hand, Amiri-Mocanu [1], [2 ] gave analytic sufficient conditions for a local diffeomorphism / of C 1 class on A to be univalent and to have a spirallike image. The second author [9] extended this result to a local diffeomorphism f of C 1 class on the domain in C n for which the Kernel function becomes infinite everywhere on the boundary. The authors [7] gave an analytic sufficient condition for a local diffeomorphism / of C l class on a bounded balanced domain in C n with C l plurisubharmonic defining functions to be univalent and to have a spirallike image and also gave a necessary and sufficient condition for locally biholomorphic mappings on such a domain to be biholomorphic and to have spirallike images. In this paper, we give analytic sufficient conditions for a local diffeomorphism of C 1 class on the unit ball B with respect to an arbitrary norm on C n to be univalent and to have a 3>-like image. When $ is a holomorphic mapping with 3>(0) = 0 and / is a locally biholomorphic mapping with / ( 0 ) = 0 , the conditions are necessary conditions for / to be univalent and to have a 3>-like image. We also give analytic sufficient conditions for a local diffeomorphism of C 1 class on B to be univalent and to have an Archimedean spirallike image or a hyperbolic spirallike image. This result generalizes results on A due to Amiri-Mocanu [1 ], [2 ] and Rakhmanov [12], [13]-

2

P re lim in a rie s

Let C n denote the space of n complex variables z = (z i , . . . , zn) with the Euclidean n inner product (z,w) = ]T) ZjWj and the norm \z\ = (z^z)1' 2. j

=i

For a holomorphic mapping / from a domain G in C n into C n , we define

For a C 1 class mapping / from a domain G in C n into C n, we define

where z j X j + y/—lyj and f j — U j -1- y/—l Vj . Brickman [3] introduced the notion of a $-iike domain in C. Gurganus [5] introduced the notion of a 3>-like domain in a complex Banach space X . We will —

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give a new definition of a $-like subset of C n. D efin itio n 2.1 Let E be a subset of C n containing the origin 0 and let $ : E —►C n be a uniformly Lipschitz continuous mapping on every compact subset of E such that $(0) = 0. Then we say that E is $-like if for any z G E , the initial value problem (1)

has a solution for all t > 0 such that v(t) G E for all t > 0. Remark 2 A If there exists a solution for ( 1 ), it is unique. Remark 2.2 In Brickman [3] and Gurganus [5], their definition need the solution to the initial value problem ( 1 ) tend to 0 as t —* oo. As usual, by £ ( C n, C m) we denote the space of all continuous linear operators from C n into C m with the standard operator norm. D efin itio n 2.2 Let A G £ (C n ,C n). Let E be a subset of C n . We say th at E is spirallike relative to A, if e~aAE C E for all s > 0. When E contains the origin and A = e~xyI ( 7 € R , |7 | < 7t/ 2 ), we say that E is spirallike of type 7 . When E is spirallike of type 0, we say that E is starlike. Let (a2) be the parametric family of spiral axes defined by oz : w — wz (t) = exp (—t -f ie~i —i)z,

t G (—00,00), z G C n , \z\ = 1.

Then, through each point w € C n \ {0} passes only one spiral of the family (crz ). D efin itio n 2.3 Let E be a subset of C n containing the origin 0. We say th at E is an Archimedean spirallike set, if for each w e E \ {0}, the part of the spiral axe oz joining the point w to the origin lies entirely in E . Let (tz) be the parametric family of spiral arcs defined by

Then,through each point w G C n \ {0} passes only one spiral of the family ( r z ). D efin itio n 2.4 Let E be a subset of C n containing the origin 0. We say th at E is a hyperbolic spirallike set, if for each w £ E \ {0}, the part of the spiral arc rz joining the point w to the origin lies entirely in E. Let X be a complex Banach space with norm || • ||. Let X* be the dual of X . For each z G X \ {0}, we define

By the Hahn-Banach theorem, T(z) is nonempty.

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3

M a in re s u lts

Let B be the unit ball in C n with respect to an arbitrary norm || • || on C n and let B r = rB , for 0 < r < 1 . In this section, we will give a sufficient condition for a C 1 mapping / from B into C n with /(0 ) = 0, J r /(z ) ^ 0 for all z G B to be univalent and to have a $-like image. If Jrf ( z ) 0, for all z G B, there exists a neighborhood Wz of z such th at / is a diffeomorphism of C l class between W z and f ( W z) y so there exist matrices

and

where T h e o re m 3*1 Let f G C 1 (B) such that /(0 ) = 0 and Jrf ( z ) 7^ 0 for all z 6 B. I f there exists a mapping $ from /(B ) into C n which is locally uniformly Lipschitz continuous and $ ( 0 ) = 0 such that

for all z G B \ {0}, z* G T(z), then f ( Br) is $>-like for any r with 0 < r < 1. Remark 3.1 If $ is C 1 on /(B ), then $ is locally uniformly Lipschitz continuous on /(B ). Using Theorem 3.1, we obtain the following theorem as in Gong-Wang-Yu [4]. T h e o re m 3.2 Let f and $ be as in Theorem 3.1. Assume that f satisfies one of the following two conditions:

for all z e B \ {0}, z* G T(z) and the solution v(t) to

tends to 0 as t —>00 for any w G /(B ) \ {0};

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Then f is univalent on B. Remark 3.2 Let $ be a holomorphic mapping with $(0) = 0, 5Rz*(O(0)z) > 0 for all z G B \ {0}, z* G T(z). Let / be a locally biholomorphic mapping with /(0 ) = 0, 0 / ( 0 ) = I. If / is univalent and /(B ) is #-like, then we have

for all z G B \ {0}, z*

4

6

T(z) (Gurganus [5]).

E x am p les

Throughout this section, we assume th at / G C'1 (B), /(0 ) = 0 and Jrf ( z ) ^ 0 for all z € B. E x a m p le 4.1 When $(z) = Az, where A e £ (C n,C n), we obtain the following result from Theorems 3.1 and 3.2 (cf. [7], [9]). Assume th at / satisfies one of the following two conditions: (i)

(ii) for all z G B \ {0}, z* G T(z). Then / is univalent on B and /(B ) is spirallike relative to A. When / is a holomorphic mapping with /(0 ) = 0, -0/(0) = / , we obtain the finite dimensional case of Theorem 11 of Sufirridge [17] from Remark 3.2. E x a m p le 4.2 If we put A = e~i7/ the following result. Assume that / satisfies

(7 G R ,

H < n/2) in Example 4.1, we obtain

for all z G B \ {0}, z* G T(z). Then / is univalent on B and /(B ) is spirallike of type 7 . When n = 1 , the above condition reduces to

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for all z G A \ {0} (cf. Amiri-Mocanu [1, Theorem 1], [2]). When / is holomorphic on A, we obtain the result of Spacek [14]. E x a m p le 4.3 If we put 7 = 0 in Example 4.2, we obtain the following result (cf. [ 10] ) . Assume that / satisfies for all 2 € B \ { 0 } , z * g T ( z ) . Then / is univalent on B and /(B ) is a starlike domain. When / is holomorphic, we obtain the finite dimensional case of Theorems 1 and 2 of Suffridge [16] using Remark 3.2. E x a m p le 4.4 Let 4>(z) = (1 + i|z|)z. Then for each z G C n with \z\ = 1, v(z,t) = exp{ —t + i(e~t — 1 )}z is the unique solution to the initial value problem dv/d t = —$(u), v ( z , 0 ) = z. Therefore, we obtain the following result from Theorems 3.1 and 3.2. Assume that / satisfies for all z G B \ {0}, z* G T(z), where w = /(z ). Then / is univalent on B and /(B ) is an Archimedean spirallike domain. When n — 1 , the above condition reduces to

for all z G A \ {0} (cf. Amiri-Mocanu [1, Theorem 2], [2]). When / is holomorphic on A, we obtain the result of Rakhmanov [12]. E x a m p le 4.5 Let 4>(z) = (|z| —i)z/\z\. Then for each z G C n with |z| = 1 , v(z,t) = exp{ —t 4 *i{el — l)}z is a solution to the initial value problem dv/dt = —$(v), u(z, 0) = z. $ is not locally uniformly Lipschitz continuous. However, since we already know that u(z, t) is a global solution, we obtain the following result by a similar argument to section 3. Assume that / satisfies for all z G B \ {0}, z* G T(z), where w = /(z ). Then / is univalent on B and /(B ) is a hyperbolic spirallike domain. When n — 1 , the above condition reduces to

for all z G A \ {0} (cf. Amiri-Mocanu [1, Theorem 3], [2]). When / is holomorphic on A, we obtain the result of Rakhmanov [13].

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R eferen ces [1] H. AI. Amiri and P. T. Mocanu, Spirallike, nonanalytic functions, Proc. Amer. Math. Soc. 82 (1981), 61-65. [2] H. AI. Amiri and P. T. Mocanu, Certain sufficient conditions for univalency of the class C 1, J. Math. Anal. A ppl 80 (1981), 387-392. [3] L. Brickman, $-like analytic functions.! B u ll Amer. Math. Soc. 79 (1973), 555-558. [4] S. Gong, S. Wang and Q. Yu, A necessary and sufficient condition th at biholomorphic mappings are starlike on a class of Reinhardt domains, Chin. Ann. Math. 13B:1 (1992), 95-104. [5] K. R. Gurganus, $-like holomorphic functions in C n and Banach spaces, Trans. Am er. Math. Soc. 205 (1975), 389-406. [6 ] H. Hamada, Univalent holomorphic mappings on a complex manifold with a C 1 exhaustion function. Manuscripta Math. 99 (1999), 359-369. [7] H. Hamada and G. Kohr, Spirallike non-holomorphic mappings on balanced pseudoconvex domains. Complex Variables Theory Apply to appear. [8 ] H. Hamada, G. Kohr and P. Liczberski, $-like holomorphic mappings on balanced pseudoconvex domains. Complex Variables Theory Appl. 39 (1999), 279-290. [9] G. Kohr, On some conditions of spirallikeness in C n. Complex Variables The­ ory Appl. 32 (1997), 79-88. [10 ] G. Kohr, Some sufficient conditions of starlikeness for mappings of C l class, Complex Variables Theory Appl. 36 (1998), 1-9. [11 ] T. Matsuno, Star-like theorems and convex-like theorems in the complex vec­ tor space, Sci. Rep. Tokyo Kyoiku Daigahu, Sect. A 5 (1955), 88-95. [12] B. N. Rakhmanov, On the theory of schlicht functions(Russian), Dold. Akad. Nauk SSSR 91 (1953), 729-732. [13] B. N. Rakhmanov, On the theory of schlicht functions (Russian), Dokl. Akad. Nauk SSSR 97 (1954), 973-976. [14] L. Spacek, Contribution a la theorie des fonctions univalentes, Casopis Pest. Mat. 62 (1933), 12-19. [15] T. J. Suffridge, The principle of subordination applied to functions of several complex variables, Pacific. J. Math. 33 (1970), 241-248.

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[16] T. J. Suffridge, Starlike and convex maps in Banach spaces, Pacific. J. Math. 46 (1973), 575-589. [17] T. J. Suffridge, Starlikeness, convexity and other geometric properties ofholomorphic maps in higher dimensions, Lecture Notes in Math., 599 New York: Springer-Verlag 1976, pp. 146-159

The Growth Theorem of Biholomorphic Mappings on a Banach Space Tatsuhiro HONDA Ariake National College of Technology 150 Higashihagio-machi, Omuta, Fukuoka, 836-8585, Japan E-mail: [email protected]

1991 Mathematics Subject Classification. Primary 32H02. Secondary 30C45. A b s tra c t Let || • || be an arbitrary norm on a Banach space E. Let B be the open unit ball of E for the norm || • ||, and let / : B —» E be a biholomorphic convex mapping such th at /(0 ) = 0 and df(0) is identity. We will give an upper bound of the growth of /•

1

IN T R O D U C T IO N

Let A = {z e C; |z| < 1 } denote the open unit disc in the complex plane C. Let / : A —> C be a biholomorphic convex mapping with /(0) = 0 and /'(0 ) = 1. Then the following inequality holds :

It is natural to consider a generalization of the above growth theorem to C n . Let fi be a domain in C n which contains the origin in C n . A holomorphic mapping / : Q, —►C n is said to be normalized, if /(0 ) = 0 and the Jacobian m atrix D f ( 0) at the origin is identity. Let B n denote the Euclidean unit ball in C n. Let / : B n —> C n be a normalized biholomorphic convex mapping. Then C.H.FitzGerald and C.R.Thomas [7], T.Liu [11 ] and T.J.SufEridge [12 ] extended the above upper bound of the growth of / to B n in C n by using different methods and showed that

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134

where || • H2 denotes the Euclidean norm. Let

with P i , . . . , p n > 1. S.Gong and T.Liu [4] gave the above upper bound of the growth of normalized biholomorphic convex mappings on B p and D( pi , . . . , pn). H.Hamada [8 ] proved on the unit ball in C n with respect to an arbitrary norm. In this paper, we give the upper bound of the growth of a biholomorphic convex mapping on the unit ball in a complex Banach space as follows : M a in T h e o rem . Let E be a complex Banach space with the norm || • ||. Let B be the open unit ball of E for the norm || • ||, and let f : B —> E be a biholomorphic convex mapping such that /(0 ) = 0 and df{0) is identity. Then

for all z e B .

2

N O T A T IO N S A N D P R E L IM IN A R IE S

Let U be an open set in a complex normed space E , let F be a complex Banach space. Let / be a holomorphic mapping from U to F. Then the following equation holds in a neighborhood V of x in U for x G U : ( 2 . 1)

where

for any y € E \ { 0} such that x + (y G U for all £ G C with |C| < 1. The series (2.1) is called the Taylor expansion of / by n-homogeneous polynomials Pn at x. Let E be a complex Banach space. Let D be a convex domain of E. A mapping f : D —►E is said to be convex if f ( D) is convex. The following theorem ( the Maximum Modulus Principle ) is well-known (cf. Dunford and Schwartz [1 ], etc). T h e o re m 1 Let E be a complex Banach space with the norm || • ||. Let A be the unit disc in C, and let f : A —>E be a holomorphic mapping. I f there exists a point Co 6 A such that ||/(Co)|| = sup{||/(C)||;C £ A}> then ||/(C)II w constant on A.

Growth theorem of biholomorphic mappings on a Banach space

3

135

P R O O F O F M A IN T H E O R E M

Let A be the unit disc in C. We take a boundary point w € d B of B and fix. oo

Let f ( z ) =

Pn (z) be the expansion of f by n-homogeneous polynomials Pn in a n=l

oo

neighborhood V of 0 in E. ^Prom the condition of / , we have f ( z ) — z +

Pn (z). n —2

Then for C € A, (3-1)

Let m > 2, m € Z be fixed. Let a = exp(27r\/=T /m ). Then

This is holomorphic with respect to £

6

A. Since f ( B ) is convex, we have

Then /i(C) is holomorphic on A. ^From the conditions of We have

Therefore small,

is a holomorphic mapping from A into E . If e > 0 is enough to is continuous and holomorphic on {£; |£| < 1 —e}. Since /i(A) C B, by

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Theorem 1, we obtain

on

Letting e tends to 0, we have

for all m > 2. Then we have

and the proof of Main Theorem is complete. Let D be a bounded convex balanced domain in a complex Banach space E . The Minkowski function N d ( z ) of D is defined by

for all z e E. Then N&(z) is a norm on E. By Main Theorem, we obtain the following corollary. C o ro lla ry 1 Let D be a bounded convex balanced domain in a complex Banach space E . let f : D —►E be a biholomorphic convex mapping such that /(0 ) = 0 and df(0) is identity. Then

for all z € D.

R eferen ces [1] S. Dineen, Complex Analysis in Locally Convex Spaces, North-Holland Math. Studies 57, 1981.

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[2] S. Dineen, The Schwarz Lemma, Oxford mathematical monographs, 1989. [3] N. Dunford and J. Schwartz, Linear operators ( 1 ), Interscience, New York, 1958. [4 ] C. H. FitzGerald and C. R. Thomas, Some bounds on convex mappings in several complex variables, Pacific J. Math. 165, (1994) 295-320. [5] T.Franzoni and E.Vesentini, Holomorphic maps and invariant distances, NorthHolland Math. Studies 40, 1980. [6 ] S. Gong, Biholomorphic mappings in several complex variables, Contemporary Math. 142, (1993) 15-48. [7] S. Gong and T. Liu, The growth theorem of biholomorphic convex mappings on B p, Chin. Quar. Jour. Math. 6 , (1991) 78-82. [8 ] H. Hamada, The growth theorem of convex mappings on the unit ball in C n, Proc. Amer. Math. Soc. 127(4), (1999) 1075-1077. [9] M.Herve, Analyticity in Infinite Dimensional Space, Walter de Gruyter, 1989. [10] M. Jarnicki and P. Pflug, Invariant distances and metrics in complex analysis, de Gruyter, Berlin-New York, 1993. [11] T. Liu, The growth theorems, covering theorems and distortion theorems for biholomorphic mappings on classical domains, University of Science and Tech­ nology of China Thesis, 1989. [12 ] T. J. SufEridge, Biholomorphic mappings of the ball onto convex domains, Ab­ stracts of papers presented to American Mathematical Society 11(66), (1990) 46.

Stability of Solutions for Singular Integral Equations for Two Classes in Locally Convex Spaces

Chan-Gan HU, Jian-Hua JIA and Chao LIU Department of Mathematics, Nankai University, Tianjin, 300071, P. R. C. E-mail: [email protected]

1

In t r o d u c t io n

Theory of the Cauchy type integral and singular integral equations with a com­ plex variable or several complex vaxiables(CTISIE for short) have been established with range in a complex plane C(see [3,9-16]). Later, the CTISEE on C to Bar* nach(or locally convex) spaces have been studied with a complex variable(see [5-8]). In this paper, the solutions of singular integral equations for two classes on finite dimensional complex manifolds to C are extended to vector-valued cases with range in a complete Hausdorff locally multiplicatively convex algebra E and the stabil­ ity of solutions is presented in order to lay a foundation of theory of CTISIE and boundary value problems with domains and ranges in infinite dimensional spaces. 2

V e c t o r - v a l u e d p r in c ip a l v a l u e

Assume that P is the sufficient directed set of seminorms which generates the topology of E, and that the Bochner-Martinelli kernel K(z, Q is denoted by1

1The project was supported by the National Natural Science Foundation of China

139

C.-G. Hu, J.-H. Jia and C. Liu

140

where a = (n —2)!/(27ri)n , £, 2 € C n , and the sign dC[fc] expresses to omit the element d£k with subscript k y i.e. d^ kj = A • • • A dCfc-i A A • • • A dCn. Let D C C n be a bounded domain, and 12(or 12$) a boundary of D via the origin. Let £