Approximate tangents, harmonic measure, and domains with rectifiable boundaries

Abstract

Let $\Omega \subset {\mathbb{R}}^{n+1}$, $n\ge 1$, be an open and connected set. Set ${\mathcal{T}}_n$ to be the set of points $\xi \in \partial \Omega$ so that there exists an approximate tangent $n$-plane for $\partial \Omega$ at $\xi$ and $\partial \Omega$ satisfies the weak lower Ahlfors-David $n$-regularity condition at $\xi$. We first show that ${\mathcal{T}}_n$ can be covered by a countable union of boundaries of bounded Lipschitz domains. Then, letting ${\partial }^{\star }\Omega$ be a subset of ${\mathcal{T}}_n$ where $\Omega$ satisfies an appropriate thickness condition, we prove that ${\partial }^{\star }\Omega$ can be covered by a countable union of boundaries of bounded Lipschitz domains contained in $\Omega$. As a corollary we obtain that if $\Omega$ has locally finite perimeter, $\partial \Omega$ is weakly lower Ahlfors-David $n$-regular, and the measure-theoretic boundary coincides with the topological boundary of $\Omega$ up to a set of ${\mathcal{H}}^n$-measure zero, then $\partial \Omega$ can be covered, up to a set of ${\mathcal{H}}^n$-measure zero, by a countable union of boundaries of bounded Lipschitz domains that are contained in $\Omega$. This implies that in such domains, ${\mathcal{H}}^n{|}_{\partial \Omega }$ is absolutely continuous with respect to harmonic measure.

arXiv commentAccepted for publication in Proceedings of the AMS; 12 pages