Finely Harmonic Functions
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289 Bent Fuglede Kebenhavns Universitet, K0benhavn/Danmark
Finely Harmonic Functions
Springer-Verlag Berlin· Heidelberg· NewYork 1972
AMS Subject Classifications (1970): 3102, 31B05, 31B15, 31B99, 31C05, 31D05
ISBN 3-540-06005-7 Springer-Verlag Berlin . Heidelberg . New York ISBN 0-387-06005-7 Springer-Verlag New York . Heidelberg· Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by phorocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined hy agreement with the publisher.
© by Springer-Verlag Berlin - Heidelberg 1972. Library of Congress Catalog Card Number 72-90194. Printed in Germany. Otlsetdruck: Julius Beltz, HemsbachlBergstr.
1
Introduction
14
Chapter I. Preliminaries 1. The cone of positive hyperharmonic functions
14
2. Semibounded potentials ..••. 3. Balayage of measures. Base, thinness, and fine topology . . . .
19
Chapter II. Capacity in axiomatic potential theory 4. 5. 6. 7.
The domination axiom . The set functions dm, and (:.co) The functionals R. Jk d-, and R. (x o) Balayage on intersections of finely closed sets
, 5
Chapter III. Finely harmonic and finely hyperharmonic functions 8. Definitions and examples 9. Fundamental properties 10. Finely superharmonic functions and fine potentials 11. Balayage and specific multiplication relative to a finely open set Chapter IV. Applications 12. Properties involving fine connectivity and balayage of measures . . . • . • . • 13. On the balayage of semibounded potentials 14. The fine Dirichlet problem . 15. An application to the study of the Choquet property References
25 32 32 40 48 61 67 67 76 100 118 146 146 162 170 178 186
Introduction
0.1.
Definitions.
a harmonic space
12
harmonic measure
permits us, in particular, to define the
V
relative to a finely 1) open set
V
and a point x
The theory of balayage of measures on
as follows
[V
e:c
that is, the swept-out of the Dirac measure
ex
on the com-
V .
plement of
It is therefore natural to introduce corresponding notions of "fine harmonicity" and "fine hyperharmonicity" for numerical functions
f
L1 ciL
defined in a finely open set
as follows
(Definitions 8.1 and 8.3 below):
f
A function
j
is finely lower semicontinuous and
the induced fine topology on finely open sets
V
f
[1
'> -
00
in
U ,
if
and if
has a basis consisting of
V
of fine closure
[t x) A function
LT
is called finely hyperharmonic in
[V
f dex
C [1 such that for every
is called finely harmonic in U
if
f
is
finite and finely continuous in [1, and if the finely open sets
....
V with V
C
U such that f
CV .
is Ex -lntegrable and
1) The qualification fine(1Y) refers to the fine topology on 12 , that is, the coarsest topology on 12 making all hyp
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