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 towards 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 Hölder estimates of the solution under a suitable exterior cone-condition.