Wiener-type tests from a two-sided Gaussian bound

In this paper, we are concerned with hypoelliptic diffusion operators H. Our main aim is to show, with an axiomatic approach, that a Wiener-type test of H-regularity of boundary points can be derived starting from the following basic assumptions: Gaussian bounds of the fundamental solution of 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 Hölder estimates of the solution under a suitable exterior cone condition.