On an extension problem for plurisubharmonic functions, the Riesz mass of which is locally finite at a closed obstacle (Q1303870)

Let \(G\) be an open subset of \(\mathbb C^{n}\), let \(E\subset G\) be relatively closed with \(\Gamma_{n}(E)=0\), where \(\Gamma_{n}(E):=\sup\{\gamma_{n}(\beta(E)): \beta\) is a unitary transformation of \(\mathbb C^{n}\}\), \(\gamma_1:=c\) is the logarithmic capacity in \(\mathbb C\), \(\gamma_{n}(E):=c(\{z_{n}\in\mathbb C: \gamma_{n-1}(\{z'\in\mathbb C^{n-1} : (z',z_{n})\in E\})>0\})\). Let \(u\) be plurisubharmonic (psh) on \(G\setminus E\) and let \(p>0\). The author studies the problem of the psh continuation of \(u\) to \(G\). He proves that \(u\) is continuable in each of the following three cases: (i) \(\exp(pu)\) has a psh extension; (ii) \(\Delta(\exp(pu))\) has locally finite mass near \(E\); (iii) for any \(z_0\in G\) there exist a neighborhood \(U\subset G\) of \(z_0\) and a function \(v\) psh on \(U\) such that \(\Delta(\exp(pu))\leq\Delta(v)\) on \(U\setminus E\).