A new capacity for plurisubharmonic functions. (English) Zbl 0547.32012

This is an important paper in which the complex Monge-Ampère operator \((dd^ c)^ n\) is used to replace the Laplacian and to prove, in the category of complex analysis, the analogues of some well known results of classical potential theory. Let us mention here the following ones: (1\(\circ)\) continuity of the operator \((dd^ c)^ k\) (1\(\leq k\leq n)\) under decreasing limits; (2\(\circ)\) quasicontinuity of plurisubharmonic function with respect to the relative capacity \(c(K,\Omega):=\sup\{\int_{K}(dd^ cv)^ n;\quad v\in PSH(\Omega),\quad 0<v<1\},\) where \(\Omega\subset {\mathbb{C}}^ n\) is a fixed open set and \(K\subset\Omega \) is compact; (3\(\circ)\) domination principle for plurisubharmonic functions; (4\(\circ)\) negligible sets are pluripolar (solution of an old problem due to P. Lelong). As a consequence one finds that various capacities considered in complex analysis satisfy the axioms of Choquet capacity and hence Borel sets are capacitable.
Reviewer: J.Siciak


32U05 Plurisubharmonic functions and generalizations
31C15 Potentials and capacities on other spaces
31C10 Pluriharmonic and plurisubharmonic functions
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