Contemporary Mathematics
Volume 32, 1984
CAPACITIES IN tn
*
H. Alexander
1. INTRODUCTION. Various notions of capacity in higher dimensional complex
spaces have been studied during the last few years. In the classical case of
logarithmic capacity in the complex plane, it turns out that several different
possible definitions of a capacity do in fact yield the logarithmic capacity.
In the higher dimensional case one obtains different capacities and the problem
of relating them to each other.
There is naturally a close connection between capacities, pluripolar sets
and plurisubharmonic functions. In particular, Bedford and Taylor have applied
their "local" capacity to obtain results which add considerably to our know-
ledge about plurisubharmonic functions.
The object of this paper is to give a survey of some of these recent
developments. We begin with a list of several equivalent definitions of loga-
rithmic capacity; in higher dimensions these lead to different capacities. Then
we consider a connection between equilibrium measure and Jensen measure; the
local capacity of Bedford and Taylor; Siciak
1
s capacity; the capacities defined
from Tchebychef polynomials; and some relationships among the various capacities.
2. CAPACITY IN t
1
We shall recall some of the several equivalent definitions
of logarithmic capacity. The book of Tsuji [12] is a good reference for this.
2.1. Potential theory. Let K be a compact subset of
t
1
If
~
is a pro-
bability measure on K, the energy integral is defined by
1
I(~)
=" log /a-bj
d~(a)d~(b).
Put V = inf
I(~).
If V
~
~
measure
~
on K such that
defined to be e-V
*
one shows that there is a unique probability
V
=I(~).
Now the capacity of K, cap(K), is
Supported in part by the National Science Foundation
1
© 1984 American Mathematical Society
0271-4132/84 $1.00
+
$.25 per page
http://dx.doi.org/10.1090/conm/032/769493
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