Sets of absolute continuity for harmonic measure in NTA domains

Abstract

We show that if $\Omega$ is an NTA domain with harmonic measure $w$ and $E\subseteq \partial \Omega$ is contained in an Ahlfors regular set, then $w{|}_E\ll {\mathcal{H}}^d{|}_E$. Moreover, this holds quantitatively in the sense that for all $\tau >0$ $w$ obeys an $A_{\infty }$-type condition with respect to ${\mathcal{H}}^d{|}_{E^{\prime }}$, where $E^{\prime }\subseteq E$ is so that $w(E\backslash E^{\prime })<\tau w(E)$, even though $\partial \Omega$ may not even be locally ${\mathcal{H}}^d$-finite. We also show that, for uniform domains with uniform complements, if $E\subseteq \partial \Omega$ is the Lipschitz image of a subset of ${\mathbb{R}}^d$, then there is $E^{\prime }\subseteq E$ with ${\mathcal{H}}^d(E\backslash E^{\prime })<\tau {\mathcal{H}}^d(E)$ upon which a similar $A_{\infty }$-type condition holds.

Made referee's recommended corrections, edited introduction, added a figure, removed a section in the appendix that is essentially known. To appear in Potential Analysis