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 $w|_{E}\ll \mathscr{H}^{d}|_{E}$. Moreover, this holds quantitatively in the sense that for all $\tau>0$ $w$ obeys an $A_{\infty}$-type condition with respect to $\mathscr{H}^{d}|_{E'}$, where $E'\subseteq E$ is so that $w(E\backslash E')<\tau w(E)$, even though $\partial\Omega$ may not even be locally $\mathscr{H}^{d}$-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 $E'\subseteq E$ with $\mathscr{H}^{d}(E\backslash E')<\tau \mathscr{H}^{d}(E)$ upon which a similar $A_{\infty}$-type condition holds.