A generalization of the annulus formula for the relative extremal function (Q2925741)

Let \(X\) be a complex space (reduced, countable at infinity, pure dimensional), let \(D\subset X\) be a domain, and let \(A\subset D\). Then the \textit{relative extremal function of \(A\) in \(D\)} is defined as \(h^\ast_{A,D}\), where NEWLINENEWLINE\[NEWLINE h_{A,D}(z):=\sup\Big\{u(z): u\in\mathcal{PSH}(D),\; u\leq1,\;u|_A\leq 0\Big\},\quad z\in D, NEWLINE\]NEWLINE and \({}^\ast\) stands for the upper semicontinuous regularization. Define \(\Delta(r):=\{z\in D: h^\ast_{A,D}(z)<r\}\), \(0<r\leq1\).NEWLINENEWLINEThe main result of the paper is the following theorem. Assume that \(D\subset\subset X\) and: NEWLINENEWLINENEWLINE - either \(X\) is a Stein space and \(D\) is the union of an increasing sequence of irreducible, locally irreducible, weakly parabolic Stein spaces;NEWLINENEWLINENEWLINENEWLINE - or \(X\) is a Josefson manifold and \(D\) is the union of an increasing sequence of Stein manifolds. NEWLINENEWLINENEWLINENEWLINE Then for any non-pluripolar set \(A\subset D\) and \(0<r<s\leq1\) we haveNEWLINE NEWLINE\[NEWLINE h^\ast_{\Delta(r), \Delta(s)}(z)=\max\bigg\{0, \frac{h^\ast_{A,D}(z)-r}{s-r}\bigg\},\quad z\in\Delta(s). NEWLINE\]NEWLINE In the case where \(X\) is a Riemann domain of holomorphy over \(\mathbb C^n\) the above formula was proved in the paper [the reviewer and \textit{P. Pflug}, Proc. Am. Math. Soc. 138, No. 11, 3923--3932 (2010; Zbl 1220.32002)]. It is still an open problem whether the above formula occurs in the general case.NEWLINENEWLINEAs an application, the author proves a general extension theorem for separately holomorphic functions defined on \((N,k)\) crosses in Stein manifolds.