Sets of absolute continuity for harmonic measure in NTA domains

Jonas Azzam

Published 2014-10-10Version 1

If $\Omega\subseteq\mathbb{R}^{d+1}$ is an NTA domain with harmonic measure $w$ and $E\subseteq \partial\Omega$ is contained in an Ahlfors regular set, then for all $\tau>0$ there is $E'\subseteq E$ $d$-rectifiable with $w(E\backslash E')<\tau w(E)$ and $w|_{E'}\ll \mathscr{H}^{d}|_{E'}\ll w|_{E'}$, so in particular, $w|_{E}\ll \mathscr{H}^{d}|_{E}$. Moreover, this holds quantitatively in the sense that $w$ obeys an $A_{\infty}$-type condition with respect to $\mathscr{H}^{d}$ on $E'$, even though $\partial\Omega$ may not even be locally $\mathscr{H}^{d}$-finite. We also show that if $(\Omega^{c})^{\circ}$ is also NTA, and if $E\subseteq\partial \Omega$ is in a Lipschitz image of $\mathbb{R}^{d}$, then for all $\tau>0$ there is $E'\subseteq E$ with $\mathscr{H}^{d}(E\backslash E')<\tau\mathscr{H}^{d}(E)$, $w|_{E'}\ll\mathscr{H}^{d}|_{E'}\ll w|_{E'}$, and such that a similar $A_{\infty}$-condition holds.