Capacities in Complex Analysis

Urban Cegrell

ISBN 10: 3528063351 / ISBN 13: 9783528063351

Published by Vieweg+Teubner Verlag, Vieweg+Teubner Verlag, 1988

About this Item

Druck auf Anfrage Neuware - Printed after ordering - 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.

Seller Inventory # 9783528063351

Bibliographic Details

• Title: Capacities in Complex Analysis
• Publisher: Vieweg+Teubner Verlag, Vieweg+Teubner Verlag
• Publication Date: 1988
• Language: German
• ISBN 10: 3528063351
• ISBN 13: 9783528063351
• Binding: Taschenbuch
• Condition: Neu
• Dimensions: 235W x 155H x 10D millimeters 235 millimeters width by 155 millimeters height by 10 millimeters depth
• Weight: 271 grams

About this title

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.