Order and Convexity in Potential Theory: H-Cones
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853 Nieu Boboe Gheorghe Bueur Aurel Cornea In Collaboration with Herbert Hollein
Order and Convexity in Potential Theory: H-Cones
Springer-Verlag Berlin Heidelberg New York 1981
Authors Nicu Boboc Department of Mathematics, University of Bucharest Str. Academiei 14, Bucharest, Romania Gheorghe Bucur Department of Mathematics, INCREST Bdul Pacii 220, Bucharest, Romania Aurel Cornea Department of Mathematics, INCREST Bdul Pacii 220, Bucharest, Romania and Fachbereich Mathematik, Universitat Frankfurt/M. Robert-Mayer-Str. 6-8, 6000 Frankfurt/M. Federal Republic of Germany Herbert Hollein Fachbereich Mathematik, Universitat Frankfurt/Main Robert-Mayer-Str. 6-8, 6000 Frankfurt/Main Federal Republic of Germany
AMS Subject Classifications (1980): 31 D05, 46A20, 46A40, 46A55, 60J45 ISBN 3-540-10692-8 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-10692-8 Springer-Verlag New York Heidelberg Berlin
Library of Congress Cataloging in Publication Data. Boboc, Nicu, 1933- H-cones: order and convexity in potential theory. (Lecture notes in mathematics; v. 853) Bibliography; p. Includes indexes. 1. Potential, Theory of. 2. Cone. 3. Convex domains. I. Bucur, Gheorghe. II. Cornea, Aurel. Ill. Title. IV. Series: Lecture notes in mathematics (Springer-Verlag); v. 835. QA3.L28 vol. 853 [QA404] 510s [515.7J81-5241 AACR2
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© by Springer-Verlag Berlin Heidelberg 1981 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
CON TEN T S
Page Introduction
1
Terminology and notations
6
1. Resolvents 1.1. Excessive functions with respect to a resolvent
7
1.2. Resolvents in duality and energy form
23
Exercises
33
2. H-Cones 2.1. Definition and first results
35
2.2. H-morphisms
43
2.3. Dual and bidual of an H-cone
56
Exercises
62
3. H-Cones of functions 3.1. Definition and first results
68
3.2. Balayages
75
3.3. Thinness and base
83
3.4. Harmonic carrier on an H-cone
88
of functions 4. Standard H-Cones 4.1. Weak units, continuous and universally
96
continuous elements 4.2. Standard H-cones, natural topology
104
on the dual 4.3. Standard H-cones of functions
113
4.4. Standard H-cones of excessive functions
123
4.5. The natural topology
141
Exercises
148
IV
5. Potential theory on standard H-Cones of functions
Page
5.1. Localization
152
5.2. Balayages on standard H-cones of functions
163
5.3. Thinness, essential base and fine topology
171
5.4. Negligible and polar sets
177
5.5. Carrier theory on standard H-cones of functions
184
5.6. Convergence properties and sheaf properties for standard H-cones
202
Exercises
223
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