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Harmonic morphisms applied to classical potential theory

Abstract 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 ℝ N to ℝ n (N > n ≥ 1) or where ϕ is the radial projection from ℝ N \ {0} to the unit sphere in ℝ N (N ≥ 2).