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On the definition of the Monge-Ampère operator in $$\mathbb{C}^2$$. (English) Zbl 1060.32018
Given an open set $$\Omega$$ in $$\mathbb C^n$$, let $$\mathcal D(\Omega)$$ be the family of all $$u\in PSH(\Omega)$$ such that there exists a non-negative Radon measure $$\mu$$ on $$\Omega$$ such that if $$\Omega'$$ is an open subset of $$\Omega$$, and a sequence $$u_j\in PSH\cap\mathcal C^{\infty}(\Omega')$$ decreases to $$u$$ in $$\Omega'$$, then $$(dd^cu)^n$$ tends weakly to $$\mu$$ on $$\Omega'$$.
The aim of the paper is to prove the following complete description of the class $$\mathcal D$$ for $$n=2$$: If $$\Omega$$ is an open set in $$\mathbb C^2$$ then $$\mathcal D(\Omega ) = PSH\cap W^{1,2}_{loc}(\Omega )$$.

##### MSC:
 32U05 Plurisubharmonic functions and generalizations 31C10 Pluriharmonic and plurisubharmonic functions 32Uxx Pluripotential theory
##### Keywords:
Monge-Ampère operator; Sobolev space
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