Capacities in Complex Analysis

by Urban Cegrell

Overview

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.

Product Details

ISBN-13:	9783528063351
Publisher:	Vieweg+Teubner Verlag
Publication date:	01/01/1988
Series:	Aspects of Mathematics , #14
Edition description:	1988
Pages:	156
Product dimensions:	6.10(w) x 9.25(h) x 0.01(d)
Language:	German

Table of Contents

I. Capacities.- II. Capacitability.- III.a Outer regularity.- III.b Outer regularity (cont.).- IV. Subharmonic functions in—n.- V. Plurisubharmonic functions in— n — the Monge-Ampère capacity.- VI. Further properties of the Monge-Ampère operator.- VII. Green’s function.- VIII. The global extremal function.- IX. Gamma capacity.- X. Capacities on the boundary.- XI. Szegö kernels.- XII. Complex homomorphisms.