On the definition of the Monge-Ampère operator in \(\mathbb{C}^2\). (English) Zbl 1060.32018

Błocki, Zbigniew

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)

MSC:

32U05	Plurisubharmonic functions and generalizations
31C10	Pluriharmonic and plurisubharmonic functions
32Uxx	Pluripotential theory

Keywords:

Monge-Ampère operator; Sobolev space