On Topologies and Boundaries in Potential Theory Enlarged edition of
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175
Marcel Brelot Universite de Paris, Paris / France
On Topologies and Boundaries in Potential Theory (Enlarged edition of a course of lectures delivered in 1966)
Springer-Verlag Berlin' Heidelberg' New York 1971
Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Oold, Heidelberg and B. Eckmann, ZOrich Series: Tata Institute of Fundamental Research, Bombay Adviser: M. S. Narasimhan
175
Marcel Brelot Universite de Paris, Paris / France
On Topologies and Boundaries in Potential Theory (Enlarged edition of a course of lectures delivered in 1966)
Springer-Verlag Berlin' Heidelberg' New York 1971
ISBN 3-540-05327-1 Springer-Verlag Berlin . Heidelberg . New York ISBN 0-387-05327-1 Springer-Verlag New York· Heidelberg· Berlin This work is subject to copyright. All rights are reserved, whether 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, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin' Heidelberg 1971. Library of Congress Catalog Card Number 70-147403. Printed in Germany.
Offsetdruck: Julius Beltz, Weinheim/Bergstr.
Introduction Potential theory, whose rich structure has provoked much research in such fields as capacity, distributions, extreme elements, Dirichlet spaces, Hunt's kernels and semi-groups, probability theory etc., has also led to the introduction of new topologies and boundaries. Among these notions' chose which enable us to express general results on the behaviour of potential functions will be especially studied here. Their success is due to the fact that they are adapted to the nature of the functions studied. Classical potential theory introduced a notion of thinness in Euclidean n-space (Brelot
[4,5})
Rn
and a corresponding "fine" topology (H. Cartan [2J) for which
the potential functions are continuous. It also introduced several boundaries, chiefly the Martin boundary (see the book of Constantinescu-Cornea [1J) and a notion of thinness at this boundary (Naim
[1 J),
which led Doob
I} ,4J
to de-
finitive results generalizing the famous Fatou theorems on radial or nontangential limits. It was possible to extend these notions to axiomatic potential theories, which include the theory of second-order partial differential equations of elliptic type and to some extent also those of parabolic type. Probabilistic interpretations and connections with Markov processes were also developed. The importance of such topological researches in potential theory justifies their further study. This course of lectures will be an introduction to this vast subject. The rich book of Constantinescu-Cornea constituted an important synthesis of classical potential theory on Riemann surfaces, particularly regarding boundary question
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