Błocki, Zbigniew On the definition of the Monge-Ampère operator in \(\mathbb{C}^2\). (English) Zbl 1060.32018 Math. Ann. 328, No. 3, 415-423 (2004). 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 \(\mathcal D(\Omega ) = PSH\cap W^{1,2}_{loc}(\Omega )\). Reviewer: Józef Siciak (Kraków) Cited in 2 ReviewsCited in 34 Documents MSC: 32U05 Plurisubharmonic functions and generalizations 31C10 Pluriharmonic and plurisubharmonic functions 32Uxx Pluripotential theory Keywords:Monge-Ampère operator; Sobolev space PDF BibTeX XML Cite \textit{Z. Błocki}, Math. Ann. 328, No. 3, 415--423 (2004; Zbl 1060.32018) Full Text: DOI