Capacities on the Boundary
Let U be an open, bounded and connected subset of ₵n containing zero. Then H(U) (H∞(U)) is the class of (bounded) analytic functions on U and A(U) consists of the functions in H(U) that extends continuously to Ū. If μ is a positive measure on ∂U we define
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Capacities in Complex Analysis
Aspects of Mathana1ics Aspekte der Mathema1ik Editor: Klas Diederich
All volumes of the series are listed on pages 154-155.
Urban Cegrell
Capacities in Complex Analysis
Springer Fachmedien Wiesbaden GmbH
CIP-Titelaufnahme der Deutschen Bibliothek Cegrell, Urban: Capacities in complex analysis/Urban Cegrell. — Braunschweig; Wiesbaden: Vieweg, 1988 (Aspects of mathematics: E; Vol. 14) ISBN 978-3-528-06335-1
NE: Aspects of mathematics/E
Prof. Dr. Urban Cegrell Department of Mathematics, University of Umea, Sweden
AMS Subject Classification: 32 F 05, 31 B 1 5 , 3 0 C 85, 32 H 10, 35 J 60
Vieweg is a subsidiary company of the Bertelsmann Publishing Group. All rights reserved © Springer Fachmedien Wiesbaden,
1988
Originally published by Friedr. Vieweg & Sohn Verlagsgesellschaft m b H , Braunschweig in 1988
No part of this publication may be reproduced, stored in a retrieval system or transmitted in any f o r m or by any means, electronic, mechanical, photocopying, recording or otherwise, w i t h o u t prior permission of the copyright holder.
Produced by Lengericher Handelsdruckerei, Lengerich ISBN 978-3-528-06335-1 DOI 10.1007/978-3-663-14203-4
ISBN 978-3-663-14203-4 (eBook)
Contents VII
Introduction
XI
List of notations I.
Capacities
II.
Capacitability
4
III.a Outer regularity
11
III.b Outer regularity (cont.)
22
IV.
Subharmonic functions in ]Rn
30
V.
Plurisubharmonic functions in
~n_
the Monge-Ampere capacity VI.
VII.
32
Further properties of the Monge-Ampere operator
56
Green's function
66
VIII. The global extremal function
73
IX.
Gamma capacity
81
X.
Capacities on the boundary
99
XI.
Szeg6 kernels
116
XII.
Complex homomorphisms
148
Introduction 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
many cases similar to the situation in
and the results are in ~.
However, these
results are not so well adapted to complex analysis in several variables - they are more related to harmonic than pluriharmonic functions. Capacities can be thought of as a non-linear generalization 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 ~n
differ
mn
and
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 sub
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