A class of delta-plurisubharmonic functions and the complex Monge-Ampère operator (Q1002453)

For a bounded domain \(\varOmega\subset\mathbb C^n\), let \(\delta^\ast\mathcal{PSH}(\varOmega)\) stands for set of all functions \(u\) that are locally the difference of two bounded plurisubharmonic functions. The authors present various properties of the class \(\delta^\ast\mathcal{PSH}(\varOmega)\). The main result of the paper is the following \textit{comparison principle}. Let \(u, v \in \delta^\ast\mathcal{PSH}(\varOmega)\) be such that: (a) \(\liminf_{z\to\partial\varOmega}(u(z)-v(z))\geq0\), (b) \(u\) and \(v\) are locally differences of two continuous plurisubharmonic functions, (c) for every open set \(\varOmega'\Subset\varOmega\) there exists an \(h\in(0,\operatorname{dist}(\partial\varOmega', \partial\varOmega))\) such that for arbitrary \(h_j\in\mathbb C^n\) with \(\|h_j\|<h\), \(j=1,\dots,n\), we have \(dd^cu_{h_1}\wedge\cdots\wedge dd^cu_{h_n}\geq0\), \(dd^cv_{h_1}\wedge\cdots\wedge dd^cv_{h_n}\geq0\), \(dd^c(u+v)_{h_1}\wedge\cdots\wedge dd^c(u+v)_{h_n}\geq0\), \(dd^c(u+v)_{h_1}\wedge\cdots\wedge dd^c(u+v)_{h_{n-1}}\geq0\) on \(\varOmega'\), where \(w_\xi:=w(z-\xi)\). Then \(\int_{\{u<v\}}(dd^cu)^n\geq\int_{\{u<v\}}(dd^cv)^n\).