Complex Potential Theory

Complex Potential Theory

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Series C: Mathematical and Physical Sciences - Vol. 439

Complex Potential Theory

edited Ьу

Paul М. Gauthier Departement de Mathematiques et de Statistique, and Centre de Recherches Mathematiques, Universite de Montreal, Montreal, Quebec, Canada

and Technical Editor

Gert Sabidussi Departement de Mathematiques et de Statistique, Faculte des Arts et des Sciences, Universite de Montreal, Montreal, Quebec, Canada

Sргiпgег-Sсiепсе+Вusiпеss Media, B.V.

Proceedings of the NATD Advanced Study Institute and Seminaire de mathematiques superieures on Complex Potential Theory Montreal, Canada July 26-August 6, 1993

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-94-010-4403-5 ISBN 978-94-011-0934-5 (eBook) DOI 10.1007/978-94-011-0934-5

Printed on acid-free paper

AII Rights Reserved © 1994 Springer Science+Business Media Dordrecht Driginally published by Kluwer Academic Publishers in 1994 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photo­copying, recording or by any information storage and retrieval system, without wriUen permission from the copyright owner.

Table of Contents

Preface

Participants

Contributors

Bernard AU PETIT Analytic multifunctions and their applications

Thomas BAGBY, Paul M. GAUTHIER Harmonic approximation on closed subsets of Riemannian manifolds

Brian J. COLE, John WERMER Pick interpolation, Von Neumann inequalities, and hyperconvex sets

John Erik FORNlESS, Nessim SIBONY Complex dynamics in higher dimensions

Theodore W. GAMELIN Analytic functions on Banach spaces

Paul M. GAUTHIER Uniform approximation

Christer O. KISELMAN Plurisubharmonic functions and their singularities

Jacob KOREVAAR Chebyshev-type quadratures: use of complex analysis and potential theory

Nikolai N. TARKHANOV General aspects of potential theory with respect to problems of differential equations

Joan VERDERA Removability, capacity and approximation

Edoardo VESENTINI Semigroups of holomorphic isometries

Index

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1

75

89

131

187

235

273

325

365

419

475

549

Preface

The objective of this ASI was to bring together specialists in several complex variables (many of whom have contributed to complex potential theory) and specialists in potential theory (all of whom have contributed to several complex variables) together with young researchers and graduate students for an interchange of ideas and techniques. Not only was the status of current research presented, but also the relevant background, much of which is not yet available in books. The following topics and interconnections among them were discussed:

1. Real and Complex Potential Theory. Capacity and approximation, basic prop­erties of plurisubharmonic functions and methods to manipulate their singularities and study their growth, Green functions, Chebyshev-type quadratures, electrostatic fields and potentials, propagation of smallness.

2. Complex Dynamics. Review of complex dynamics in one variable, Julia sets, Fatou sets, background in several variables, Henon maps, ergodicity, use of potential theory and multifunctions.

3. Banach Algebras and Infinite Dimensional Holomorphy. Analytic multi­functions, spectral theory, analytic functions on a Banach space, semigroups of holomor­phic isometries, Pick interpolation on uniform algebras and Von Neumann inequalities for operators on a Hilbert space.

The basic notion of complex potential theory is that of a plurisubharmonic function. In his lectures, C.O. Kiselman begins by comparing convex, subharmonic, and plurisub­harmonic functions. He goes on to show that certain sets associated to plurisubharmonic functions are analytic varieties. One of the important attributes of an entire function is its rate of growth. Kiselman studies, more generally, the growth of plurisubharmonic functions and generalizes the notions of order and type of an entire function of finite order to functions of arbitrarily fast growth.

A major theme of several of the lecturers was approximation. N.N. Tarkhanov considers the general problem of approximation of a function defined on a compact set by solutions of a partial differential equation Pu = 0, where P is a linear elliptic partial differential operator with analytic coefficients. J. Verdera considers finer problems by restricting his attention to the case where the operator P is homogeneous with constant coefficients. He devotes particular attention to the case of the Cauchy-Riemann operator - that is, holo­morphic approximation in one complex variable. In this setting, P.M. Gauthier considers the approximation problem when the set on which the approximation occurs is no longer necessarily compact but is rather allowed to be a (possibly unbounded) closed set. This same problem is investigated by T. Bagby and Gauthier, but in the context of harmonic

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approximation. The problem of approximation on unbounded sets by solutions of more general elliptic equations has been considered elsewhere and is mentioned in the lectures of Tarkhanov. Both Verdera and Tarkhanov treat the relation between approximation and re­movability of singularities for solutions of PDEs. Tarkhanov's lectures are greatly motivated by boundary value problems in PDEs.

The subject of complex dynamics, that is, iteration of holomorphic mappings, has at­tracted a lot of attention in recent years from a wide public, in part (but not only) because of its beautiful pictures and connections with chaos. The dynamics for a function of a single complex variable have been the subject of a large number of studies. Recently, however, new methods from pluripotential theory have produced many new interesting results in the higher dimensional case. J .E. Fornress and N. Sibony present an overview of this timely topic.

B. Aupetit surveys the subject of analytic multifunctions. This new theory which has its origins in both several complex variables and spectral theory grew out of such problems as the following. How do the eigenvalues of a family of matrices behave if the coefficients of these matrices depend analytically on a parameter? Aupetit presents a remarkable array of applications of this theory: to spectral theory, to the joint spectrum, to uniform algebras in connection with approximation, to spectral interpolation, to local spectrum, to non­associative Jordan algebras, and to complex dynamics.

The lectures of E. Vesentini on semigroups of holomorphic isometries and hyperbolic domains begin with a review of finite-dimensional hyperbolic complex analysis, but mainly, treat infinite-dimensional complex analysis. In fact, infinite-dimensional complex analy­sis arises naturally in finite-dimensional complex analysis, since, for example, spaces of holomorphic functions (of even a single variable) are infinite-dimensional. The Kobayashi pseudodistance is a very natural pseudodistance on a domain of Cn which is invariant for automorphisms. If it is a distance, the domain is said to be hyperbolic. In the theory of a single complex variable, there are two domains which are of outstanding importance: the plane itself, C, and the unit disc. Hyperbolic domains are a higher dimensional analog of the unit disc. Vesentini discusses holomorphic mappings on infinite-dimensional hyperbolic domains in complex Banach spaces, devoting particular attention to automorphisms of a domain. For these the basic algebraic operation is composition.

T.W. Gamelin lectures on analytic functions on a Banach space. Here, the target space is usually one-dimensional, the complex plane C. However, he also occasionally discusses analytic functions with values in a normed space. The basic algebraic operations on func­tions to C are addition and multiplication. These functions form an algebra. The spectrum of a uniform algebra, which consists of the non-zero complex-valued homomorphisms of the algebra, has played an important role in various problems in analysis. Gamelin studies the spectra of various algebras of holomorphic functions. An interesting aspect of the subject is that natural problems of approximation that are trivial in the plane become difficult in

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the infinite-dimensional setting.

The paper by B. Cole and J. Wermer on Pick interpolation, Von Neumann inequalities and hyperconvex sets, was presented by Wermer. The authors investigate a class of convex bodies in en which they call hyperconvex. These arise naturally in many interpolation problems - for example, in the problem of interpolating by bounded holomorphic functions in the unit disc. They also arise in problems in operator theory on Hilbert space. Von Neumann proved the following inequality: if T is a contraction on a Hilbert space and if P is a polynomial, then

IIP(T)II :5 sup IIP(z)ll· 1 .. 19

D. Sarason has shown that these two beautiful topics (complex interpolation and Von Neu­mann inequalities) are in fact related. Cole and Wermer embellish this relationship for us.

J. Korevaar's lectures were on Chebyshev-type quadratures: use of complex analysis and potential theory. A Chebyshev-type quadrature formula with nodes (1, ... ,(N for a set E and a measure u on E is an approximation formula

N 1 f(x)du(x) ~ (liN) L f«(i) E ;=1

for integrals over E. In his lectures, Korevaar surveys and extends fundamental quadrature formulas. Of course, the choice of nodes is crucial and Korevaar shows how this is related to electrostatics (distribution of point charges), potential theory, and complex analysis (one and several variables). As an offshoot of his investigation on the "social habits of electrons" Korevaar rediscovered the phenomenon of "propagation of smallness" of harmonic functions. This phenomenon had been observed by Armitage, Bagby, and Gauthier but in a purely qualitative way. Now, Korevaar presents a very elegant quantitative formulation. Indeed, he shows that if n is a domain in Rn, no a non-empty open subset of n, and E a compact subset of n, then there is a constant a in (0,1] such that for any harmonic function u on n,

sup lui :5 (sup luDa(sup IuD I-a • E flo fl

Notice the striking resemblance to the Nevanlinna two-constants theorem. Since it is not assumed that the domain is bounded, the phenomenon of propagation of smallness has an impact on the possibility of approximation on unbounded sets, the theme of the lectures of Bagby and Gauthier.

Open problems were also a major component of the conference. All speakers formulated such problems and the very last event of the conference was a problem session at which all participants were invited to submit and discuss their favorite problems.

x Preface

I wish to express my sincere thanks to all the lecturers and participants for having helped to make this ASI a success. Special thanks are due to Aubert Daigneault, director of the ASI, and to Ghislaine David, secretary of the SMS, both of whom contributed immeasurably to the preparation, mise-en-scene, and "aftermath". Also, my thanks go to Gert Sabidussi and Guogang Gao for their excellent work in editing the present volume.

Last not least, I wish to express on behalf of the Organizing Committee our gratitude to NATO whose financial support has made this ASI possible, and especially to Dr. L. Veiga da Cunha, the Director of the ASI programme, for his help, advice, and understanding.

Paul M. Gauthier

Scientific Director of the ASI

Montreal, April 21st, 1994.

Participants

Kuzman ADZIEVSKI Department of Mathematics University of South Carolina Columbia, SC 29208 USA

John T. ANDERSON Department of Mathematics College of the Holy Cross Worcester, MA 01610-2395 USA

Federica ANDREANO Department of Mathematics Brown University Box 1917 Providence, RI 02912 USA

Ayse Z. AROGUZ Department of Chemistry Faculty of Engineering Istanbul University 34459 A vcilar - Istanbul Turkey

Jonas AVELIN Matematiska Institutionen Uppsala Universitet Box 480 S-751 06 Uppsala Sweden

Ruben AVETISYAN 402 Ocean Parkway, Apt. 309 Brooklyn, NY 11218 USA

Sahbi AYARI Departement de matMmatiques

et de statistique Universite de Montreal C.P. 6128-A, Montreal, Que., H3C 3J7 Canada

Aydin AYTUNA Department of Mathematics Middle East Technical University 06531 Ankara Turkey

Ulf BACKLUND Department of Mathematics University of Michigan Ann Arbor, MI 48109-1003 USA

Esther BARRABES VERA Dept. de Matematica Aplicada i Ana.J.isi Universitat de Barcelona Gran Via 585 E-08071 Barcelona Spain

Riadh BEN GHANEM Departement de matMmatiques

et de statistique Universite de Montreal C.P. 6128-A, Montreal, Que., H3C 3J7 Canada

Charaf BENSOUDA Departement de matMmatiques

et de statistique Universite de Montreal C.P. 6128-A, Montreal, Que., H3C 3J7 Canada

Anders BJORN Department of Mathematics Linkoping University S-581 83 Linkoping Sweden

Pierre BLANCHET 19 rue Ste-Catherine Lauzon (Comte Levis) Quebec, Que., G6V 2W4 Canada

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Zbigniew BLOCKI Institute of Mathematics Jagiellonian University ul. Reymonta 4 PL-30059 Krakow Poland

Andre BOIVIN Department of Mathematics University of Western Ontario London, Ont., N6A 5B7 Canada

James BRENNAN Department of Mathematics 715 Patterson Office Tower University of Kentucky Lexington, KY 40506-0027 USA

Gregory T. BUZZARD Department of Mathematics University of Michigan Ann Arbor, MI 48109-1003 USA

Jean-Paul CALVI Departement de Mathematiques U.F.R. - M.I.G. Universite Paul Sabatier 118, route de Narbonne F-31062 Toulouse Cedex France

Seddik CHACRONE Departement de mathematiques

et de statistique Universite de Montreal C.P. 6128-A, Montreal, Que., H3C 3J7 Canada

RamonCOVA Dept. of Mathematical Sciences Science Laboratory University of Durham South Road Durham, DHI 3LE UK

Chiara DE FABRITIIS SISSA-ISAS Via Beirut 2/4 1-34014 Trieste Italy

Driss DRISSI Departement de mathematiques

et de statistique Universite Laval Cite Universitaire Quebec, Que., GIK 7P4 Canada

EI Kettani M. ECH-CHERIF Departement de mathematiques

et de statistique Universite Laval Cite Universitaire Quebec, Que., GlK 7P4 Canada

Abdelkrim EZZlRANI Lab. de mathematiques appliquees Universite de Pau Ave de l'Universite F-64000 Pau France

Juan Carlos FARINA GIL Dpto. de Analisis Matematico Universidad de La Laguna E-38271 La Laguna-Tenerife Spain

Manuel FLORES MEDEROS Dpto. de Analisis Matematico Universidad de La Laguna E-38271 La Laguna-Tenerife Spain

Jacques FORTIN Departement de mathematiques

et de statistique Universite Laval Cite Universitaire Quebec, Que., GIK 7P4 Canada

Participants

Participants

EI Mostapha FRIH Departement de Mathematiques Faculte des Sciences Universite Mohammed V B.P.1014 Rabat Morocco

Estela GAVOSTO Department of Mathematics University of Michigan Ann Arbor, MI 48109-1003 USA

Louis-Philippe GIROUX Departement de mathematiques

et de statistique Universite de Montreal C.P. 6128-A, Montreal, Que., H3C 3J7 Canada

Ian GRAHAM Department of Mathematics University of Toronto Toronto, Ont., M5S 1A1 Canada

Sandrine GRELLIER Mathematiques-Batiment 425 Universite de Paris-Sud F-91405 Orsay Cedex France

Allal GUESSAB Lab. de mathematiques appliquees Universite de Pau Ave de i'Universite F-64000 Pau France

Stefan HALVARSSON Matematiska Institutionen Uppsala Universitet Box 480 S-751 06 Uppsala Sweden

Osvaldo HOSSIAN Departement de mathematiques

et de statistique Universite Laval Cite Universitaire Quebec, Que., G1K 7P4 Canada

Alexander IZZO Department of Mathematics Brown University Box 1917 Providence, RI 02912 USA

Hakki T. KAPTANOGLU Department of Mathematics Middle East Technical University 06531 Ankara Turkey

Oleg KAREPOV Institute of Physics Siberian Section Russian Academy of Sciences Akademgorodok 660036 Krasnoyarsk Russia

Ognyan KOUNCHEV FB 11 Mathematik Universitat Duisburg D-4100 Duisburg 1 Germany

Arno KUIJLAARS Faculteit der Wiskunde en Informatica Universiteit van Amsterdam PI. Muidergracht 24 NL-1018TV Amsterdam The Netherlands

Per E. MANNE Department of Mathematics University of Oslo P.O. Box 1053 Blindern N-0316 Oslo 3 Norway

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Abdelaziz MAOUCHE Departement de matMmatiques

et de statistique Universite Laval Cite Universitaire Quebec, Que., G1K 7P4 Canada

Joan MATEU Dept. de Matema.tica Aplicada ETSEJB Univ. Politecnica de Catalunya Diagonal 647 E-08028 Barcelona Spain

Thanh Van NGUYEN Departement de Mathematiques U.F.R. - M.I.G. Universite Paul Sabatier 118, route de Narbonne F-31062 Toulouse Cedex France

Marco PELOSO Dpto. di Matematica Politecnico di Torino Corso Duca degli Abruzzi 24 1-10129 Torino Italy

Karen PINNEY Department of Mathematics 715 Patterson Office Tower University of Kentucky Lexington, KY 40506-0027 USA

Wieslaw PLESNIAK Institute of Mathematics Jagiellonian University ul. Reymonta 4 PL-30059 Krakow Poland

Eugeny POLETSKY Department of Mathematics Syracuse University Syracuse, NY 13244-1150 USA

Analogyj PRYKARPATSKYJ Department of Nonlinear

Mathematical Analysis Ukrainian Academy of Sciences 290052 Lviv Ukraine

Alexander RASHKOVSKII Mathematics Division Institute for Low Temperature

Physics apd Engineering 47 Lenin Ave. 310164 Kharkov Ukraine

Alexander RUSSAKOVSKII Mathematics Division Institute for Low Temperature

Physics and Engineering 47 Lenin Ave. 310164 Kharkov Ukraine

Leszek RZEPECKI Department of Mathematics University of South Carolina Columbia, SC 29208 USA

Selim SEKER Department of Electrical and

Electronic Engineering Bogazi<;i University 80815 Bebek - Istanbul Turkey

Participants

Nikolay SHCHERBINA Departament de Matema.tiques Universitat Autonoma de Barcelona E-08193 Bellaterra (Barcelona) Spain

Rafat N. SIDDIQI Department of Mathematics Kuwait University P.O. Box 5969 13060 Safat Kuwait

Participants

Ragnar SIGURDSSON Science Institute University of Iceland Dunhaga 3 107 Reykjavik Iceland

Sankhata SINGH Department of Mathematics

and Statistics Memorial University St. John's, Newfoundland, A1C 5S7 Canada

Dan SIRBU Str. Lirei, Mr15-Bll-Sc A-Et 4 Bucure§ti Sect II Roumania

Mikhail M. SMIRNOV Department of Mathematics Princeton University Princeton, NJ 08544 USA

Manfred STOLL Department of Mathematics University of South Carolina Columbia, SC 29208 USA

Synne STORLIEN Department of Mathematics University of Oslo P.O. Box 1053 Blindern N-0316 Oslo 3 Norway

Jerzy SZCZEPANSKI Institute of Mathematics Jagiellonian University ul. Reymonta 4 PL-30059 Krakow Poland

Jan SZYNAL Institute of Mathematics M. Curie-Sklodowska University PL-20031 Lublin Poland

Roberto TAURASO Classe di Scienze Scuola Normale Superiore Piazza dei Cavalieri 1-56100 Pisa Italy

B. Alan TAYLOR Mathematics Department University of Michigan Ann Arbor, MI 48109-1003 USA

Adnan TAYMAZ Department of Nuclear Physics Faculty of Science University ofIstanbul, Vezniceler Campus 34459 Istanbul Turkey

Giilsen TOKAT Department of Mathematics Faculty of Sciences Istanbul Technical University 80626 Maslak - Istanbul Turkey

TomaTONEV Dept. of Mathematical Sciences University of Montana Missoula, MT 59812-1032 USA

Daniel TURCOTTE Departement de matMmatiques

et de statistique Universite Laval Cite Universitaire Quebec, Que., GIK 7P4 Canada

M. van FRANKENHUYSEN Mathematisch Inst. Katholieke Universiteit Toernooiveld NL-6525 ED Nijmwegen The Netherlands

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Dror VAROLIN Department of Mathematics University of Toronto Toronto, Ont., M5S lAI Canada

Bert G. WACHSMUTH Department of Mathematics

and Computer Science Seton Hall University South Orange, NJ 07079-2696 USA

James Li-Ming WANG Department of Mathematics University of Alabama Box 870350 Tuscaloosa, AL 35487 USA

Abdoul O. WATT :Ecole Poly technique B.P.lO Thies Senegal

Georges WEILL Department of Mathematics Polytechnic University 6 Metrotech Center Brooklyn, NY 11201 USA

Tim WILKINS Department of Pure Mathematics

and Mathematical Statistics University of Cambridge 16 Mill Lane Cambridge CB2 ISB UK

Vyaceslav ZAHARIUTA Department of Mathematics Middle East Technical University 06531 Ankara Turkey

Participants

Contributors

Bernard AUPETIT Departement de mathematiques

et de statistique Universite Laval Cite Universitaire Quebec, Que., GIK 7P4 Canada

S. Thomas BAGBY Department of Mathematics Rawles Hall Indiana University Bloomington, IN 47405 USA

Brian J. COLE Department of Mathematics Brown University Box 1917 Providence, Rl 02912 USA

John Erik FORNlESS Department of Mathematics University of Michigan Ann Arbor, M148109-1003 USA

Theodore W. GAMELIN Department of Mathematics 405 Hilgard Ave. University of California, Los Angeles Los Angeles, CA 90024-1555 USA

Paul M. GAUTHIER Departement de mathematiques

et de statistique Universite de Montreal C.P. 6128-A, Montreal, Que., H3C 3J7 Canada

Christer O. KISELMAN Matematiska Institutionen Uppsala Universitet Box 480 8-75106 Uppsala Sweden

Jacob KOREVAAR Faculteit der Wiskunde en Informatica Universiteit van Amsterdam PI. Muidergracht 24 NL-1018TV Amsterdam The Netherlands

Nessim SIBONY Mathematiques-Bitiment 425 Universite de Paris-Sud F-91405 Orsay Cooex France

Nikolay N. TARKHANOV Max-Planck-Gesellschaft Arbeitsgruppe Analysis Universitiit Potsdam Pf. 60 1553 D-14415 Potsdam Germany

Joan VERDERA Departament de Matematiques Universitat Autonoma de Barcelona E-08193 Bellaterra (Barcelona) Spain

Edoardo VESENTINI Istituto Matematico Scuola Normale Superiore Piazza dei Cavalieri 1-56100 Pisa Italy

John WERMER Department of Mathematics Brown University Box 1917 Providence, Rl 02912 USA

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