A note on maximal subextensions of plurisubharmonic functions
The authors study the convergence in capacity of maximal subextensions of plurisubharmonic functions from the Cegrell's class \(\mathcal E\). Let \(\Omega\subset\widehat\Omega\subset\subset\mathbb C^n\) be hyperconvex domains and let \(u\in\mathcal{PSH}(\Omega)\). A function \(\widehat u\in\mathcal{PSH}(\widehat\Omega)\) is called a subextension of \(u\) if \(\widehat u|_{\Omega}\leq u\). For \(f\in\mathcal E(\Omega)\) let \(\mathcal F^a(\Omega,f)\) denote the family of all \(u\in\mathcal{PSH}(\Omega)\) such that \(\psi+f\leq u\leq f\) for a function \(\psi\in\mathcal{PSH}(\Omega)\) such that \((dd^c\psi)^n=0\) on all pluripolar subsets of \(\Omega\). Let \(g\in\mathcal E(\widehat\Omega)\), \(g|_\Omega\leq f\), and \(u\in\mathcal F(\Omega,f)\). The function \(S_{u,g}:=\sup\{\varphi\in\mathcal{PSH}(\widehat\Omega): \varphi\leq g,\;\varphi|_\Omega\leq u\}\) is called the maximal subextension of \(u\) with boundary value \(g\). The main result of the paper is the following theorem. Let \(w\in\mathcal F^a(\Omega,f)\) be such that \(\int_{\widehat\Omega}(dd^cg)^n+\int_\Omega(dd^cw)^n<+\infty\). Assume that \((u_j)_{j=1}^\infty\subset\mathcal F^a(\Omega,f)\) is such that \(u_j\geq w\), \(j\in\mathbb N\), and \(u_j\longrightarrow u\) in capacity. Then \(S_{u_j,g}\longrightarrow S_{u,g}\) in capacity.
subextension of plurisubharmonic functions
scientific article; zbMATH DE number 6838516
scientific article