Wiener-type tests from a two-sided Gaussian bound

TL;DR

This paper develops Wiener-type tests for boundary regularity of hypoelliptic diffusion operators using Gaussian bounds, boundary estimates, and capacity conditions, advancing the understanding of boundary behavior in such PDEs.

Contribution

It introduces an axiomatic framework for Wiener-type boundary regularity tests based on Gaussian bounds and doubling conditions, with new boundary estimates and capacity models.

Findings

Established Wiener-type boundary regularity tests for hypoelliptic operators.

Derived boundary H"older estimates under cone conditions.

Connected Gaussian bounds with boundary behavior via capacity estimates.

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 H\"older estimates of the solution under a suitable exterior cone-condition.