Sets of absolute continuity for harmonic measure in NTA domains

We show that if $Ω$ is an NTA domain with harmonic measure $w$ and $E\subseteq \partialΩ$ 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 $τ>0$ $w$ obeys an $A_{\infty}$-type condition with respect to $\mathscr{H}^{d}|_{E'}$, where $E'\subseteq E$ is so that $w(E\backslash E')<τw(E)$, even though $\partialΩ$ may not even be locally $\mathscr{H}^{d}$-finite. We also show that, for uniform domains with uniform complements, if $E\subseteq\partialΩ$ is the Lipschitz image of a subset of $\mathbb{R}^{d}$, then there is $E'\subseteq E$ with $\mathscr{H}^{d}(E\backslash E')<τ\mathscr{H}^{d}(E)$ upon which a similar $A_{\infty}$-type condition holds.

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