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Continuity of the complex Monge-Ampère operator. (English) Zbl 0849.31010
Let \(\Omega\) be an open set in \(\mathbb{C}^n\) and let \(\text{PSH} (\Omega)\) be the set of all plurisubharmonic functions on \(\Omega\). The complex Monge-Ampère operator \((dd^c)^n\) is well defined on \(\text{PSH} (\Omega) \cap L^\infty_{\text{loc}} (\Omega)\), where \(d= \partial+ \overline {\partial}\) and \(d^c= i(\partial- \overline {\partial})\) is continuous under monotone limits. For every Borel set \(E \subset \Omega\) the capacity \(C_n\) is defined by \[ C_n= \sup\{ (dd^c)^n,\;u\in \text{PSH} (\Omega),\;0< u< 1\} \] [see E. Bedford and B. A. Taylor, Acta Math. 149, 1-40 (1982; Zbl 0547.32012)]. The author gives sufficient conditions for the weak convergence \((dd^c u_j)^n\to (dd^c u)^n\) (in the sense of currents) when \(u_j\) tends to \(u\) in \(C_{n-1}\)-capacity on each \(E\subset \subset \Omega\).

MSC:
31C10 Pluriharmonic and plurisubharmonic functions
32W20 Complex Monge-Ampère operators
32U05 Plurisubharmonic functions and generalizations
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