The general definition of the complex Monge-Ampère operator.
type: Item
instance of: scholarly article
title: The general definition of the complex Monge-Ampère operator.
zbMATH Open document ID: 1065.32020
author: Urban Cegrell
publication date: 2004-12-20T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/116103
review text: Denote by PSH\((\Omega)\) the plurisubharmonic functions on a hyperconvex domain \(\Omega\) and by PSH\(^-(\Omega)\) the subclass of negative functions. The author considers global approximation of negative plurisubharmonic functions by decreasing sequences of negative plurisubharmonic functions that are continuous on \(\overline\Omega\), equal to zero on \(\partial\Omega\) and with Monge-Ampère mass. The elements of this class of functions serves as ``test functions''. The author proves that global approximation is possible in PSH\(^-(\Omega)\). Furthermore, the author discusses a general definition of the complex Monge-Ampère operator. This is done by introducing a class \(\mathcal E\) = \({\mathcal E}(\Omega)\) of plurisubharmonic functions which consists of all functions that are locally equal to decreasing limits of test functions. The Monge-Ampère operator can be extended to \(\mathcal E\), and this is the most general definition if we require the operator to be continuous under decreasing limits. Consider a class \(\mathcal K\) (=\({\mathcal K}(\Omega)\)) \(\subset \text{PSH}^-(\Omega)\) such that: (1) If \(u\in\mathcal K\), \(v\in PSH^-(\Omega)\) then \(\max(u,v)\in\mathcal K\). (2) \(u\in\mathcal K\), \(\varphi_j \in PSH^-(\Omega)\cap L^\infty_{\text{loc}}\), \(\varphi_j\searrow u\), \(j\to +\infty\), then \(((dd^c\varphi_j)^n)^\infty_{j=1}\) is weak*-convergent. The author proves that \(\mathcal E\) is the largest class for which (1) and (2) holds true. Thus, \(\mathcal E\) is optimal in this sense. In the remaining part of the paper, the author studies the Monge-Ampère operator using this general definition and solves a Dirichlet problem. Let \({\mathcal E}_0(\Omega)\) be the convex cone of bounded plurisubharmonic functions \(\varphi\) with \(\lim_{z\to\xi}\varphi(z)=0\), for all \(\xi\in\partial\Omega\) and \(\int_\Omega(dd^c\varphi)^n<+\infty\). We denote by \({\mathcal F}(\Omega)\) the subclass of functions \(u\) in \(\mathcal E\) such that there exis…
Mathematics Subject Classification ID: 32U15
Mathematics Subject Classification ID: 32W20
zbMATH DE Number: 2123563
zbMATH Keywords: Monge-Ampère operator
zbMATH Keywords: plurisubharmonic function
zbMATH Keywords: test functions
zbMATH Keywords: weak*-convergence
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: pluripolar set
MaRDI profile type: Publication
OpenAlex ID: W1496640230
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DOI: 10.5802/AIF.2014
published in: Annales de l’institut Fourier
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description: scientific article; zbMATH DE number 2123563