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Zbl 0547.32012
Bedford, Eric; Taylor, B.A.
A new capacity for plurisubharmonic functions. (English)
[J] Acta Math. 149, 1-40 (1982). ISSN 0001-5962; ISSN 1871-2509/e

This is an important paper in which the complex Monge-Ampère operator $(dd\sp c)\sp n$ 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 $(dd\sp c)\sp k$ (1$\le k\le n)$ under decreasing limits; (2$\circ)$ quasicontinuity of plurisubharmonic function with respect to the relative capacity $c(K,\Omega):=\sup\{\int\sb{K}(dd\sp cv)\sp n;\quad v\in PSH(\Omega),\quad 0<v<1\},$ where $\Omega\subset {\bbfC}\sp n$ 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.
[J.Siciak]

MSC 2000:: *32U05 Plurisubharmonic functions and generalizations
31C15 Generalizations of potentials, etc.
31C10 Pluriharmonic and plurisubharmonic functions

Keywords: pluripolar set; complex Monge-Ampère operator; plurisubharmonic function; relative capacity; negligible sets

Cited in: Zbl 1084.32027 Zbl 0963.14009 Zbl 1045.34056 Zbl 0883.32010 Zbl 0849.31010 Zbl 0748.31006 Zbl 0655.32001 Zbl 0677.31005 Zbl 0637.32013 Zbl 0629.32011 Zbl 0624.31004 Zbl 0602.31006 Zbl 0562.35021 Zbl 0554.32016 Zbl 0574.32026

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