Capacities in Complex Anaylsis (Aspects of Mathematics): E 14 - Tapa blanda

Cegrell, Urban

 
9783528063351: Capacities in Complex Anaylsis (Aspects of Mathematics): E 14
Tapa blanda
ISBN 10: 3528063351 ISBN 13: 9783528063351
Editorial: Friedrich Vieweg & Sohn Verlagsgesellschaft mbH, 1988

Sinopsis

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.

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Reseña del editor

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.

Reseña del editor

"Notes ... prepared in November 1983 for a couple of seminars at the University of Uppsala. These notes were made into a more complete form during a series of lectures at the University of Umea in the fall of 1985 and at Universite Paul Sabatier, Toulouse in December 1986"--P. ix.

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Editorial
Friedrich Vieweg & Sohn Verlagsgesellschaft mbH
Año de publicación
1988
Idioma
Alemán
ISBN 10 
3528063351
ISBN 13 
9783528063351
Encuadernación
Tapa blanda
Número de páginas
172
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