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

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^infinity-hypoellipticity of L, together with the existence of a well-behaved fundamental solution Gamma(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 M_r; the role of M_r 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’ Omega_r(x), superlevel sets of Gamma(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 M_r, a Poisson–Jensen formula for the L-subharmonic functions and several results concerning the geometry of the sets Omega_r(x).