The doubling property of the elliptic measure, for elliptic operators with drifts satisfying an average diverging condition

TL;DR

This paper proves the doubling property of elliptic measures for operators with drifts that diverge on average near the boundary, extending previous results by relaxing pointwise smallness conditions to average conditions.

Contribution

It introduces a new approach to establish doubling of elliptic measures under average divergence conditions on the drift, generalizing earlier pointwise smallness assumptions.

Findings

Doubling of elliptic measure under average diverging drift conditions.

Extension of results to domains with Alhfors-David regular boundaries.

Proof of Hardy inequalities in 1-sided chord arc domains.

Abstract

We show doubling of the elliptic measure corresponding to the operator with an elliptic principal term and a drift that diverges, on average on Whitney cubes, like the inverse distance to the boundary, with a small constant. Essentially a small Carleson constant assumption on the drift, this generalizes earlier results with the hypothesis of pointwise smallness of such a drift. This relates to recent perturbative results of rough Dirichlet solvability in domains with drifts or potentials that satisfy a Carleson measure condition, which have also been considered earlier by Hofmann-Lewis and Kenig-Pipher. While we work in 1-sided chord arc domains, these results are new even for the half-space. In the process, we also prove Hardy inequalities in such domains with Alhfors-David regular boundary, using a stopping time argument.