Capacities in Complex Analysis

ISBN-13: 9783528063351 / Niemiecki / Miękka / 1988 / 168 str.

Urban Cegrell;Urban Cegrell

Opis

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.

Spis treści

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.