A strong comparison principle of plurisubharmonic functions with finite pluricomplex energy
Let \(\varOmega\Subset\mathbb C^n\) be a hyperconvex domain. Let \(\mathcal E_0(\varOmega):=\{u\in\mathcal{PSH}^-(\varOmega)\cap L^\infty(\varOmega): \forall_{\zeta\in\partial\varOmega}: \lim_{z\to\zeta}u(z)=0,\; \int_{\varOmega}(dd^cu)^n<+\infty\}\) and let \(\mathcal E_p(\varOmega)\) stands for the Cegrell class \(\{u\in\mathcal{PSH}(\varOmega): \exists_{(u_j)_{j=1}^\infty\subset\mathcal E_0(\varOmega)}: u_j\searrow u,\;\sup_{j\in\mathbb N}\int_{\varOmega}(-u_j)^p(dd^cu_j)^n<+\infty\}\). The main result of the paper is the following strong comparison principle for functions of class \(\mathcal E_p(\varOmega)\). Let \(u, v\in\mathcal E_p(\varOmega)\) (\(p>0\)). Then for all \(r\geq0\), \(w_j\in\mathcal{PSH}^-(\varOmega)\cap L^\infty(\varOmega)\) with \(w_j\geq-1\), \(j=2,3,\dots,n\), we have \(A_{n,p}\int_{u<v}(v-u)^pdd^cw_1\wedge\dots\wedge dd^cw_n+ \int_{u<v}(r-w_1)(v-u)^p(dd^cv)^n\leq\int_{u<v}(r-w_1)(v-u)^p(dd^cu)^n\), where \(A_{n,p}:=(n!(n+p)\cdots(p+1))^{-1}\). As a corollary the author gets the following result. Let \(u, u_j\in\mathcal E_p(\varOmega)\) (\(p>0\)) be such that \(u_j\) converges monotonically to \(u\) in \(\varOmega\). Then for any \(0\leq p_1<+\infty\) and \(\psi\in\mathcal C_0^\infty(\varOmega)\) we have \(\int_{\varOmega}\psi(-\varphi)^{p_1}(dd^cu_j)^n\longrightarrow \int_{\varOmega}\psi(-\varphi)^{p_1}(dd^cu)^n\) uniformly with respect to all \(\varphi\in\mathcal{PSH}(\varOmega)\) with \(-1\leq\varphi\leq0\).
scientific article; zbMATH DE number 5509673