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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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##### References:
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