Cone of maximal subextensions of the plurisubharmonic functions (Q6563079)

The aim of this paper is to study the subextension problem for plurisubharmonic functions in bounded hyperconvex domains. Let \(\Omega\subset \widetilde {\Omega}\subset \subset \mathbb C^n\) be bounded hyperconvex domains. Assume that \(u\in \mathcal {PSH}^-(\Omega)\). A plurisubharmonic function \(v\in\mathcal {PSH}(\widetilde{\Omega})\) is said to be a subextension of \(u\) if \(v\leq u\) on \(\Omega\). Moreover, the maximal subextension of \(u\) is defined by\N\[\N\tilde u=\sup\left\{v\in \mathcal {PSH}(\widetilde{\Omega}): v|_{\Omega}\leq u \right\}.\N\]\NThe authors prove the following characterization of maximal subextensions. Assume that \(u\) belongs to the Cegrell class \(\mathcal F^a(\Omega)\). Then \(\tilde u\) is the unique function with the following properties: \(\tilde u\in \mathcal F(\widetilde {\Omega})\), \(\tilde u\leq u\) on \(\Omega\) and\N\[\N\int_{\{\tilde u<u\}\cap \Omega\cup(\widetilde{\Omega}\setminus \Omega)}(dd^c\tilde u)^n=0.\N\]\NIf \(u\) belongs to the Cegrell class \(u\in \mathcal E(\Omega)\) and \(\tilde u\in \mathcal E(\widetilde\Omega)\) then for all \(w\in \Omega\) the log canonical thresholds are equal \(c_{\tilde u}(w)=c_u(w)\), where\N\[\Nc_u(z)=\sup \{c>0: \exists \, r>0 \ \text{such that} \ \exp(-2cu)\in L^1(B(w,r))\}.\N\]\NMoreover, the intersection numbers are equal too, \(e_j(u)=e_j(\tilde u)\), where\N\[\Ne_j(u)=\int_{\{w\}}(dd^cu)^j\wedge (\log ||z-w||)^{n-j}.\N\]\NIn particular similar equality holds also for Lelong number \(\nu_u(w)=\nu_{\tilde u}(w)\), where \(\nu_u(w)=e_1(u)\).\N\NBy \(\operatorname {Sub}(\Omega,\tilde{\Omega})\) denote the class of plurisubharmonic functions on \(\Omega\) which have maximal subextensions to \(\tilde{\Omega}\). \textit{U. Cegrell} and \textit{A. Zeriahi} [C. R., Math., Acad. Sci. Paris 336, No. 4, 305--308 (2003)]\N proved that \(\mathcal F(\Omega)\subset \operatorname {Sub}(\Omega,\tilde{\Omega})\). The last part of the paper is devoted to study the structure of the set \(\operatorname {Sub}(\Omega,\tilde{\Omega})\). It was proved that \(\operatorname {Sub}(\Omega,\tilde{\Omega})\) is positive cone which has local property and which is invariant under proper holomorphic surjections.