On the definition of the Monge-Ampère operator in \(\mathbb{C}^2\)
type: Item
instance of: scholarly article
title: On the definition of the Monge-Ampère operator in \(\mathbb{C}^2\)
zbMATH Open document ID: 1060.32018
author: Zbigniew Błocki
published in: Mathematische Annalen
publication date: 2004-07-01T00:00:00Z
review text: 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 \(\mathcal D(\Omega ) = PSH\cap W^{1,2}_{loc}(\Omega )\).
reviewed by: Józef Siciak
Mathematics Subject Classification ID: 32U05
Mathematics Subject Classification ID: 31C10
Mathematics Subject Classification ID: 32Uxx
zbMATH DE Number: 2078023
zbMATH Keywords: Monge-Ampère operator
zbMATH Keywords: Sobolev space
MaRDI profile type: Publication
full work available at URL: s00208-003-0491-0
OpenAlex ID: W2070601356
DOI: 10.1007/S00208-003-0491-0
description: scientific article; zbMATH DE number 2078023