# Several Complex Variables II

## Function Theory in Classical Domains Complex Potential Theory

• G. M. Khenkin
• A. G. Vitushkin
Book

Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 8)

1. Front Matter
Pages i-vii
2. L. A. Aizenberg, A. K. Tsikh, A. P. Yuzhakov
Pages 1-58
Pages 59-106
4. A. B. Aleksandrov
Pages 107-178
5. A. G. Sergeev, V. S. Vladimirov
Pages 179-253
6. Back Matter
Pages 255-262

### Introduction

Plurisubharmonic functions playa major role in the theory of functions of several complex variables. The extensiveness of plurisubharmonic functions, the simplicity of their definition together with the richness of their properties and. most importantly, their close connection with holomorphic functions have assured plurisubharmonic functions a lasting place in multidimensional complex analysis. (Pluri)subharmonic functions first made their appearance in the works of Hartogs at the beginning of the century. They figure in an essential way, for example, in the proof of the famous theorem of Hartogs (1906) on joint holomorphicity. Defined at first on the complex plane IC, the class of subharmonic functions became thereafter one of the most fundamental tools in the investigation of analytic functions of one or several variables. The theory of subharmonic functions was developed and generalized in various directions: subharmonic functions in Euclidean space IRn, plurisubharmonic functions in complex space en and others. Subharmonic functions and the foundations ofthe associated classical poten­ tial theory are sufficiently well exposed in the literature, and so we introduce here only a few fundamental results which we require. More detailed expositions can be found in the monographs of Privalov (1937), Brelot (1961), and Landkof (1966). See also Brelot (1972), where a history of the development of the theory of subharmonic functions is given.

### Keywords

Complex analysis Dimension Grad Hardy-Räume Leraysche Theorie Monge-Ampere-Operator Nevanlinna theory Plurisubharmonische Funktionen Potential Potential theory Residuen Satz von Noether-Lasker Twistortransformation Zukunftstubus mathematical physics

### Editors and affiliations

• G. M. Khenkin
• 1
• A. G. Vitushkin
• 2
1. 1.Central Economic and Mathematical Institute of the Russian Academy of SciencesMoscowRussia
2. 2.Steklov Mathematical InstituteMoscowRussia

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-642-57882-3
• Copyright Information Springer-Verlag Berlin Heidelberg 1994
• Publisher Name Springer, Berlin, Heidelberg
• eBook Packages
• Print ISBN 978-3-642-63391-1
• Online ISBN 978-3-642-57882-3
• Series Print ISSN 0938-0396
• Buy this book on publisher's site