Pluricomplex energy
In the paper the author defines certain classes \(\mathcal E_p\) and \(\mathcal F_p\) of plurisubharmonic functions and studies the complex Monge-Ampère operator \((dd^c)^n\) on them. Let \(\Omega\) be a bounded hyperconvex domain in \(\mathbb C^n\), \(n\geq 2\). Denote by \(\mathcal E_0\) the class of negative, bounded, plurisubharmonic functions \(\varphi\) on \(\Omega\) such that \(\lim_{z\to\zeta}\varphi(z)=0\) for all \(\zeta\in\partial\Omega\), and \(\int (dd^c\varphi)^n <\infty\). For \(p\geq 1\), \(\mathcal E_p\) denotes the class of plurisubharmonic functions \(\varphi\) on \(\Omega\) such that there exists a sequence \(\varphi_j\in\mathcal E_0\) with \(\varphi_j\downarrow\varphi\) and \(\sup_j\int (-\varphi_j)^p (dd^c\varphi_j)^n <\infty\). If in addition the \(\varphi_j\) can be chosen so that \(\sup_j\int(dd^c\varphi_j)^n<\infty\), then \(\varphi\) is said to be in \(\mathcal F_p\). The author proves that \(\mathcal E_p\) and \(\mathcal F_p\) are convex cones and that \((dd^c)^n\) is well defined on \(\mathcal E_p\), i.e., if \(\varphi\in \mathcal E_p\) and \(\varphi_j\downarrow \varphi, \varphi_j\in \mathcal E_0\), with \(\sup_j\int(-\varphi_j)^p (dd^c\varphi_j)^n\) finite, then \((dd^c\varphi_j)^n\) is weakly convergent and the limit is independent of the choice of \(\varphi_j\). The main result of the paper is the following. Theorem 5.1. Let \(\Omega\) be a bounded and hyperconvex set in \(\mathbb C^n\), \(n\geq 2\), \(p\geq 1\), and \(\mu\) a positive measure with finite total mass on \(\Omega\). Then there is a (uniquely determined) function \(u\in\mathcal F_p\) with \((dd^cu)^n=\mu\) if and only if there is a constant \(A\) such that \[ \int(-\varphi)^pd\mu \leq A \left(\int (-\varphi)^p (dd^c\varphi)^n\right)^{\frac{p}{n+p}},\quad \forall \varphi\in \mathcal E_0. \] In Section 6 the author considers the Dirichlet problem for \(\mathcal E_p\), and the remaining two sections are devoted to the Dirichlet problem with continuous boundary data.
scientific article; zbMATH DE number 1207288