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Value distribution of harmonic and finely harmonic morphisms and applications in complex analysis. (English) Zbl 0607.31002
If X is a harmonic space, we recall that the fine topology on X is the weakeset topology which makes all subharmonic functions on X continuous. A polar subset of an open subset U of X is given locally as the set where a subharmonic function takes on the value \(-\infty\). Let \(V\subset X\) be a finely open set and u a finely continuous morphism of X into a second harmonic space X’ equipped with its usual topology. For x an element of the fine boundary of V, one defines the fine cluster set of u at x, \(C_ f(u,x)\), as the set of all x’\(\in X'\cup \{\infty \}\) such that there exists a filter \(\Gamma\) converging finely to x with \(u(\Gamma)\to x'\). For \(F\subset X-V\), the fine cluster set of u at F, \(C_ f(u,F)\), is the union of \(C_ f(u,x)\) for all \(x\in F\). For F a finely closed proper subset of V, the author studies conditions which guarantee an extension of u as a finely continuous harmonic morphism across F in terms of the polar properties of F and \(C_ f(u,F)\). These results generalize well known results for the usual topology on X.
Reviewer: L.Gruman

31B25 Boundary behavior of harmonic functions in higher dimensions
31C10 Pluriharmonic and plurisubharmonic functions
32U05 Plurisubharmonic functions and generalizations
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