BMO solvability and absolute continuity of harmonic measure

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.

References in corpus (2)

• Uniform Rectifiability and harmonic measure IV: Ahlfors regularity plus Poisson kernels in $L^p$ implies uniform rectifiability
• Square Functions and the $A_{\infty }$ Property of Elliptic Measures