Capacities in Complex Analysis

✍ Scribed by Urban Cegrell

• Publisher: Vieweg+Teubner Verlag
• Year: 1988
• Tongue: English
• Leaves: 165
• Series: Aspects of Mathematics; 14
• Category: Library

✦ Synopsis

The purpose of this book is to study plurisubharmonic and analytic functions in n using capacity theory. The case n=l has been studied for a long time and is very well understood. The theory has been generalized to mn and the results are in many cases similar to the situation in . However, these results are not so well adapted to complex analysis in several variables - they are more related to harmonic than plurihar- monic functions. Capacities can be thought of as a non-linear generali- zation of measures; capacities are set functions and many of the capacities considered here can be obtained as envelopes of measures. In the mn theory, the link between functions and capa- cities is often the Laplace operator - the corresponding link in the n theory is the complex Monge-Ampere operator. This operator is non-linear (it is n-linear) while the Laplace operator is linear. This explains why the theories in mn and n differ considerably. For example, the sum of two harmonic functions is harmonic, but it can happen that the sum of two plurisubharmonic functions has positive Monge-Ampere mass while each of the two functions has vanishing Monge-Ampere mass. To give an example of similarities and differences, consider the following statements. Assume first that is an open subset VIII of n and that K is a closed subset of Q. Consider the following properties that K mayor may not have.

✦ Table of Contents

Title
Contents
Introduction
General references
List of Notations
I. Capacities
II. Capacitability
III.a Outer regularity-
III.b Outer regularity (cont.)
IV. Subharmonic functions in Â
V. Plurisubharmonic functions in Cn - the Monge-Ampere capacity
VI. Further properties of the Monge-Ampere operator
VII. Green's function
VIII. The global extremal function
IX. Gamma capacity
X. Capacities on the boundary
XI. Szego kernels
XII. Complex homomorphisms
English-language subseries (E)