Wiener-type tests from a two-sided Gaussian bound

Abstract

In this paper, we are concerned with hypoelliptic diffusion operators \(\mathcal {H}\). Our main aim is to show, with an axiomatic approach, that a Wiener-type test of \(\mathcal {H}\)-regularity of boundary points can be derived starting from the following basic assumptions: Gaussian bounds of the fundamental solution of \(\mathcal {H}\) with respect to a distance satisfying doubling condition and segment property. As a main step toward this result, we establish some estimates at the boundary of the continuity modulus for the generalized Perron–Wiener solution to the relevant Dirichlet problem. The estimates involve Wiener-type series, with the capacities modeled on the Gaussian bounds. We finally prove boundary Holder estimates of the solution under a suitable exterior cone condition.