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Harmonic morphisms applied to classical potential theory

Published online by Cambridge University Press:  11 January 2016

Bent Fuglede
Bent Fuglede*
Affiliation:
Department of Mathematical Sciences, University of Copenhagen, 2200 Copenhagen Ø, Denmark fuglede@math.ku.dk
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Abstract

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It is shown that if ϕ denotes a harmonic morphism of type Bl between suitable Brelot harmonic spaces X and Y, then a function f, defined on an open set V ⊂ Y, is superharmonic if and only if f ∘ ϕ is superharmonic on ϕ–1(V) ⊂ X. The “only if” part is due to Constantinescu and Cornea, with ϕ denoting any harmonic morphism between two Brelot spaces. A similar result is obtained for finely superharmonic functions defined on finely open sets. These results apply, for example, to the case where ϕ is the projection from ℝN to ℝn (N > n ≥ 1) or where ϕ is the radial projection from ℝN \ {0} to the unit sphere in ℝN (N ≥ 2).

Keywords

31B0531C0531C1231C40

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 202 , June 2011 , pp. 107 - 126
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2011

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