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Let ${\mathcal {D}}$ and $T$ be, respectively, a $C^1$ distribution of $k$-planes and a normal $k$-current on ${\mathbb {R}}^n$. Then ${\mathcal {D}}$ has to be involutive at almost every superdensity point of the tangency set of $T$ with respect to ${\mathcal {D}}$.

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).

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Involutivity of distributions at points of superdense tangency with respect to normal currents

Harmonic morphisms applied to classical potential theory

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Involutivity of distributions at points of superdense tangency with respect to normal currents

Harmonic morphisms applied to classical potential theory