BMO solvability and absolute continuity of harmonic measure

TL;DR

This paper demonstrates that BMO-solvability of elliptic operators in domains with Ahlfors-David regular boundaries ensures the absolute continuity of elliptic-harmonic measure, even without connectivity assumptions, with a converse established under additional conditions.

Contribution

It establishes a new link between BMO-solvability and absolute continuity of harmonic measure without connectivity assumptions, extending results even for the Laplacian.

Findings

BMO-solvability implies weak-$A_ Infty$ absolute continuity of harmonic measure.

Results hold without connectivity assumptions, even for the Laplacian.

A converse result is proved under interior Corkscrew condition for the Laplacian.

Abstract

We show that for a uniformly elliptic divergence form operator $L$, defined in an open set $\Omega$ with Ahlfors-David regular boundary, BMO-solvability implies scale invariant quantitative absolute continuity (the weak-$A_\infty$ property) of elliptic-harmonic measure with respect to surface measure on $\partial \Omega$. We do not impose any connectivity hypothesis, qualitative or quantitative; in particular, we do not assume the Harnack Chain condition, even within individual connected components of $\Omega$. In this generality, our results are new even for the Laplacian. Moreover, we obtain a converse, under the additional assumption that $\Omega$ satisfies an interior Corkscrew condition, in the special case that $L$ is the Laplacian.