Marek Jarnicki - Extension of Holomorphic Functions

De Gruyter Expositions in Mathematics

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ISBN 9783110627664

e-ISBN (PDF) 9783110630275

e-ISBN (EPUB) 9783110627695

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The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de.

2020 Walter de Gruyter GmbH, Berlin/Boston

Mathematics Subject Classification 2010: 32-02, 32Axx, 32Dxx, 32Exx, 32Txx, 32Uxx, 32Wxx,

To Mariola and Rosel

At the beginning of 2018 de Gruyter asked us to write a second edition of the book. After a longer discussion, we decided to write under the conditions that we maintain the structure of the old book but extend the text by important results in this area that have appeared after 2000.

The main statements in the preface for the first edition are still true and the reader is encouraged to consult them.

We would like to thank:

• the de Gruyter company for giving us the opportunity to update the 2000 version;

• the National Science Centre (Poland) for the support given during the time we were working on this book (OPUS grant 2015/17/B/ST1/00996).

P.S. According to our experience with other books we are sure that a lot of errors can still be found in this text. We appreciate any remark or comment on that which may be sent to one of the e-mail addresses:

marek.jarnicki@im.uj.edu.pl

peter.pflug@uni-oldenburg.de

KrakwOldenburg, February 21, 2020.

Marek Jarnicki

Peter Pflug

Starting from the discussion of the holomorphic logarithm in the plane, it is necessary to introduce Riemann surfaces in order to be able to deal with the maximal domain of existence of that function, but without being bothered about its possible multivaluedness. On the other hand, any domain in the complex plane is the existence domain of at least one holomorphic function, i.e. there exists a holomorphic function that is not the restriction of another holomorphic function having a strictly larger domain of definition.

In the n-dimensional situation ( n2 ) it was already observed by Hartogs that there are pairs of domains GG , GG , such that any holomorphic function on G extends holomorphically to G . Moreover, simultaneously one has to handle the problem that the extended functions may be multivalued. This phenomenon has led to the notion of Riemann domains over Cn . Exactly this category of objects and the theory of holomorphic functions on them is the subject of our book. We try to give a systematic representation of domains of holomorphy and envelopes of holomorphy in that category. The authors feel that a lot of results they are presenting have never been published in book form.

We are not touching the theory of holomorphic functions on complex spaces or even on complex manifolds. Instead, we continue investigating domains of holomorphy for special classes of holomorphic functions on special types of domains.

Our interest in this area of complex analysis started directly after our studies when both of us were working on the continuation of holomorphic functions. Although we had changed our fields of interest, we were attracted by such questions all the time, and we were following the development in that direction. During the years we got the impression that there is a need of a source where the main results are collected. We hope this book can serve as such a source. The choice of topics obviously reflects our personal preferences. For example, we will solve the Levi problem via the -problem and functions of restricted growth.

Our idea is to address this book to everybody who likes to extend her/his knowledge beyond the standard course in several complex variables in Cn . We tried to make the book as complete as possible and to keep the results used without proving them as limited as possible. Nevertheless, for some parts we will have to use certain facts (for example consequences of Theorems A and B) without giving proofs. The same will happen also with some facts on plurisubharmonic functions (related to the MongeAmpre operator), although we present an extended section on plurisubharmonic functions. Textbooks that support the reader can be found in the first part of the bibliography.

We should point out that the bibliography is far from being complete. We included only the papers that we had studied during the preparing of the book or before. So we have to apologize if the reader is interested in historical developments of presented results.

The theory of extension of holomorphic functions contains a lot of questions which are still waiting for being solved. We have put many of them into the text (marking them by ? ). The reader is encouraged to work on some of them.

During the process of proof reading we got an impression that this became a never-ending story and there had to be a time for us to stop. We would be pleased if the reader would inform us of any errors he/she may have detected while studying the text.

It is our deep pleasure to thank our teachers, Professors H. Grauert and J. Siciak, who taught us the beauty of complex analysis. Next, we would like to thank Dr. habil. W. Zwonek who helped us in corrections of the text.

We thank the following institutions: Committee of Scientific Research (KBN), Warsaw (PB 2 P03A 060 08, PB 2 P03A 017 14), Volkswagen Stiftung (Az. I/71 062, R i P program at Oberwolfach), and Niederschsisches Ministerium fr Wissenschaft und Kunst (Az. 15.350 113(55) PL); without their financial support this work would never be possible. We would like to thank also our universities for support before and during the preparation of the book.

Finally, we are deeply indebted to the Walter de Gruyter Publishing Company, especially to Dr. M. Karbe, for having encouraged us to write this book.

KrakwOldenburg, May 2000.

Marek Jarnicki

Peter Pflug

Roughly speaking, a Riemann domain over Cn is a domain spread over Cn , i.e. a connected complex manifold X having a global projection p:XCn such that p is locally homeomorphic. The class of Riemann domains over Cn extends the class of subdomains of Cn . Riemann domains appear in a very natural way while discussing problems related to holomorphic continuation. Consider the following classical example. Let G:=CR and let Log denote the principal branch of the logarithm. Then the function Log extends holomorphically to a domain X which is no longer a plane domain but a so-called Riemann surface of the function Log . The same phenomenon appears if we consider on G a branch of the kth root. This means that the maximal domain of existence of a holomorphic function defined in a plane domain may be non-univalentit may be a Riemann domain over C .

More generally, given a domain GCn , we can consider the maximal domain of existence of a family SO(G) , where O(G) denotes the space of all functions holomorphic on G. The most interesting case is the one when S=O(G) . From the point of view of the theory of holomorphic functions the following three fundamental questions are the most important ones:

Does the maximal domain of existence always exist in the category of Riemann domains over Cn ?

What is a characterization of those domains which cannot be holomorphically extended to any larger Riemann domain?