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Harmonic morphisms applied to classical potential theory. (English) Zbl 1235.31002
Let \(X\) and \(Y\) be Brelot harmonic spaces. A continuous map \(\phi:X\rightarrow Y\) is called a harmonic morphism if, for any open subset \(V\) of \(Y\) and any harmonic function \(f\) on \(V\), the function \(f\circ \phi \) is harmonic on \(\phi ^{-1}(V)\). Further, a harmonic morphism \(\phi \) is said to be of type Bl if, for any open \(V\subset Y\) and any locally bounded potential \(p\) on \(V\), the function \(p\circ \phi \) is a potential on \(\phi^{-1}(V)\). The author investigates the interplay between harmonic morphisms and the fine topology, thinness, polarity, superharmonicity and fine superharmonicity (in the last two cases, Bl harmonic morphisms are considered). He then applies his general theory to two specific cases in classical potential theory, namely where \(\phi \) is a projection from \(\mathbb{R}^{N}\) to a lower dimensional space \(\mathbb{R}^{n}\), and where \(\phi \) is the radial projection from \(\mathbb{R}^{N}\backslash \{0\}\) to the unit sphere. This leads to the recovery and generalization of results of several authors. The paper concludes with a consideration of the more general notion of a finely harmonic morphism.

MSC:
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
31C05 Harmonic, subharmonic, superharmonic functions on other spaces
31C12 Potential theory on Riemannian manifolds and other spaces
31C40 Fine potential theory; fine properties of sets and functions
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