Harmonic morphisms applied to classical potential theory

Abstract

It is shown that if $\phi$ denotes a harmonic morphism of type Bl between suitable Brelot harmonic spaces $X$ and $Y$, then a function $f$, defined on an open set $V\subset Y$, is superharmonic if and only if $f\circ \phi$ is superharmonic on ${\phi }^{-1}\left(V\right)\subset X$. The “only if” part is due to Constantinescu and Cornea, with $\phi$ 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 $\phi$ is the projection from ${\mathbb{R}}^N$ to ${\mathbb{R}}^n$ ($N>n\ge 1$) or where $\phi$ is the radial projection from ${\mathbb{R}}^N\setminus \left\{0\right\}$ to the unit sphere in ${\mathbb{R}}^N$ ($N\ge 2$).