Hardy Classes on Infinitely Connected Riemann Surfaces
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1027 Morisuke Hasumi
Hardy Classes on Infinitely Connected Riemann Surfaces
Springer-Verlag Berlin Heidelberg New York Tokyo 1983
Author
Morisuke Hasumi Department of Mathematics, Ibaraki University Mito, Ibaraki 310, Japan
AMS Subject Classifications (1980): 30F99, 30F25, 46J15, 46J20, 31A20, 30055 ISBN 3-540-12729-1 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-12729-1 Springer-Verlag New York Heidelberg Berlin Tokyo 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 "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1983 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
PREFACE
The purpose of these notes is to give an account of Hardy classes on infinitely connected open Riemann surfaces and some related topics. Already in this Lecture Notes series we have a beautiful monograph "Hardy Classes on Riemann Surfaces" by Maurice Heins, which appeared in print in 1969.
It is therefore natural that our stress should now be
placed on some new advances we have seen during subsequent years. As generally recognized, Hardy classes made their debut in the literature in 1915, when G. H. Hardy discussed the mean growth of functions analytic on the unit disk in his paper [15J.
The theory of these
very useful classes of functions was laid its foundation in the work of Hardy himself, J. E. Littlewood, F. and M. Riesz, G. Szego among others. And we now have a large and still growing amount of literature in this area.
Speaking roughly, Hardy classes have been studied most inten-
sively in the case of the unit disk from the cradle for its importance as well as simplicity.
The case of finitely connected surfaces, planar
or not, has drawn much attention and enjoyed considerable progress in recent years. Opposed to this, our knowledge seems to be relatively small in the case of infinitely connected surfaces.
The classical theory of Hardy
classes deals mostly with the unit disk and does not have much direct bearing on our present problem.
From 1950's downwards, functional-
analytic methods have found their successful applications in the field of complex function theory including Hardy classes and the abstract Hardy class theory thus created has grown to form the core of the newlyborn theory of function algebras, as evidenced from Gamelin's book [lOJ for example.
Nevertheless, the case of infinitely connected surfaces,
as I understand it, still lies beyond the reach of the new theory and needs an independent study as in the case of polydisks and balls.
The
structure of general Riemann surfaces is not yet very well known and so we should begin with this basic question:
"For which class of in
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