Analytic and Plurisubharmonic Functions in Finite and Infinite Dimen
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198 Michel Herve Universite de Paris, Paris/France
Analytic and Plurisubharmonic Functions in Finite and Infinite Dimensional Spaces Course Given at the University of Maryland, Spring 1970
Springer-Verlag Berlin . Heidelberg· New York 1971
AMS Subject Classifications (1970): 31B05, 31 C 10, 58C 20
ISBN 3-540-05472-3 Springer-Verlag Berlin' Heidelberg· New York ISBN 0-387-05472-3 Springer-Verlag New York· Heidelberg· Berlin This work is subject to copyright. All rigbts are reserved, whetber the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, tbe amount of the fee to be determined by agreement WIth the publisher. © by Springer-Verlag Berlin' Heidelberg 1971. Library of Congress Catalog Card Number 77-161475. Offsetdruck: Beltz, 6944 HemsbachlWeinheim.
The following lectures were given at the University of Maryland, U. S. A., in the year on complex analysis (1969-70) organized by the Department of Mathematics of that University. The visiting professorship conferred upon me during the Spring semester of 1970 was a great honor and an enriching experience, for which I am deeply grateful to the Department of Mathematics, especially to Professor J. K. Goldhaber, chairman, and Professor J. Horvath. To the latter I am also indebted for his friendly help and advice throughout my stay at the University of Maryland.
To him I dedi-
cate these lecture notes as a token of my wholehearted thankfulness.
September 30, 1970 Michel A. Herve
CONTENTS Chapter I. §l.
§2.
§3.
Classical potential theory
Harmonic functions
1
1.1.
The class H(U) of harmonic functions
1
1.2.
Harmonic distributions
3
1.3.
The Poisson kernel
Hyper-, super-, nearly super-harmonic functions Hyperharmonic functions
2.2.
The class
2.3.
Superharmonic distributions and Newton potentials
11
2.4.
The class N(U) of nearly superharmonic functions
13
S(U)
6
of superharmonic functions
Sweeping-out theory, polar sets and quasisuperharmonic functions
17
3.1.
Sweeping-out
17
3.2.
Choquet capacities related to the sweeping-out process
19
3.3.
Polar sets
23
3.4.
The class Q(U) of quasisuperharmonic functions
26
30
Plurisuperharmonicity and separate analyticity with respect to a finite number of variables
31
Plurihyperharmonic and separately hyperharmonic functions
31
Chapter II.
§2.
6
2.1.
References for Chapter I
§l.
1
1.1.
Separately hyperharmonic functions
31
1.2.
Plurihyperharmonic and plurisuperharmonic functions
35
1.3.
The class functions
37
NP(~)
of nearly plurisuperharmonic
Separate analyticity 2.1.
Hartogs' theorems for the complex case
41 41
VI 2.2.
Siciak's theorem for the real case
44
2.3.
The real case: proofs of Propositions 1 and 2 stated in the preceding section
48
2.4.
Lelong's theorem for the
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