Wiener criterion for X-elliptic operators

TL;DR

This paper establishes a Wiener criterion for boundary point regularity in the Dirichlet problem related to X-elliptic operators, using capacity and cone-type conditions, under doubling and Poincaré assumptions.

Contribution

It introduces a Wiener criterion for X-elliptic operators and characterizes boundary regularity via capacitary potentials and cone conditions.

Findings

Wiener criterion for boundary regularity proved

Characterizations of regularity via capacitary potentials provided

Cone-type criterion established for X-elliptic operators

Abstract

In this note we prove a Wiener criterion of regularity of boundary points for the Dirichlet problem related to $X$-elliptic operators in divergence form enjoying the doubling condition and the Poincar\'e inequality. As a step towards this result, we exhibit some other characterizations of regularity in terms of the capacitary potentials. Finally, we also show that a cone-type criterion holds true in our setting.