The Dirichlet problem and the inverse mean‐value theorem for a class of divergence form operators

Abstract

The aim of this paper is to study some classes of second-order divergence-form partial differential operators ℒ of sub-Riemannian type. Our main assumption is the C∞-hypoellipticity of ℒ, together with the existence of a well-behaved fundamental solution Γ(x, y) for ℒ. We consider the mean-integral operator Mr naturally associated to the mean-value theorem for the ℒ-harmonic functions and we investigate the following topics: the positivity set of the kernel associated to Mr; the role of Mr in solving the homogeneous Dirichlet problem related to ℒ in the Perron–Wiener–Brelot sense; the existence of an inverse mean-value theorem characterizing the sub-Riemannian ‘balls’ Ωr(x), superlevel sets of Γ(x,·). This last result extends a previous theorem by Kuran [Bull. London Math. Soc. 1972]. As side-results, we provide a short proof of the Strong Maximum Principle for ℒ using Mr, a Poisson–Jensen formula for the ℒ-subharmonic functions and several results concerning the geometry of the sets Ωr(x).