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The general definition of the complex Monge-Ampère operator. (English) Zbl 1065.32020

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 exists a decreasing sequence \(u_j\in{\mathcal E}_0(\Omega)\) such that \(u_j\searrow u\) on \(\Omega\) and \(\sup_j\int_\Omega(dd^cu_j)^n<+\infty\). The author considers the Dirichlet problem in \(\mathcal F\). Suppose \(\psi\in{\mathcal E}_0\), \(v\in{\mathcal F}\) where \((dd^cv)^n\) is carried by a pluripolar set. Then there is a \(g\in{\mathcal F}\) with \[ (dd^cg)^n=(dd^c\psi)^n+(dd^cv)^n\,. \]

MSC:

32U15 General pluripotential theory
32W20 Complex Monge-Ampère operators
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References:

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