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."
General
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