Rectifiability, interior approximation and Harmonic Measure

TL;DR

This paper proves a structure theorem for rectifiable sets with finite Hausdorff measure, showing they can be covered by Lipschitz domain boundaries, and establishes absolute continuity between Hausdorff measure and harmonic measure under certain conditions.

Contribution

It introduces a new structure theorem for rectifiable sets with weak ADR conditions and connects geometric properties with harmonic measure behavior.

Findings

Almost all of the rectifiable set can be covered by Lipschitz domain boundaries.

Harmonic measure is absolutely continuous with respect to Hausdorff measure on rectifiable parts.

Counterexamples show the harmonic measure result is optimal.

Abstract

We prove a structure theorem for any $n$-rectifiable set $E\subset \mathbb{R}^{n+1}$, $n\ge 1$, satisfying a weak version of the lower ADR condition, and having locally finite $H^n$ ($n$-dimensional Hausdorff) measure. Namely, that $H^n$-almost all of $E$ can be covered by a countable union of boundaries of bounded Lipschitz domains contained in $\mathbb{R}^{n+1}\setminus E$. As a consequence, for harmonic measure in the complement of such a set $E$, we establish a non-degeneracy condition which amounts to saying that $H^n|_E$ is "absolutely continuous" with respect to harmonic measure in the sense that any Borel subset of $E$ with strictly positive $H^n$ measure has strictly positive harmonic measure in some connected component of $\mathbb{R}^{n+1}\setminus E$. We also provide some counterexamples showing that our result for harmonic measure is optimal. Moreover, we show that if, in addition, a set $E$ as above is the boundary of a connected domain $\Omega \subset \mathbb{R}^{n+1}$ which satisfies an infinitesimal interior thickness condition, then $H^n|_{\partial\Omega}$ is absolutely continuous (in the usual sense) with respect to harmonic measure for $\Omega$. Local versions of these results are also proved: if just some piece of the boundary is $n$-rectifiable then we get the corresponding absolute continuity on that piece. As a consequence of this and recent results by Azzam-Hofmann-Martell-Mayboroda-Mourgoglou-Tolsa-Volberg, we can decompose the boundary of any open connected set satisfying the previous conditions in two disjoint pieces: one that is $n$-rectifiable where Hausdorff measure is absolutely continuous with respect to harmonic measure and another purely $n$-unrectifiable piece having vanishing harmonic measure.