Absolute continuity of the harmonic measure on low dimensional rectifiable sets

TL;DR

This paper proves that harmonic measure on certain low-dimensional rectifiable sets in Euclidean space is mutually absolutely continuous with the Hausdorff measure, extending previous results to sets of codimension greater than one with a simpler proof.

Contribution

It provides an alternative, simpler proof that harmonic measure on low-dimensional uniformly rectifiable sets satisfies the $A^ abla$-property, extending prior work to higher codimension cases.

Findings

Harmonic measure and Hausdorff measure are mutually absolutely continuous on these sets.

The proof is simpler and specific to the case when the set's dimension is less than n-1.

The result extends previous theorems to higher codimension uniformly rectifiable sets.

Abstract

We consider a uniformly rectifiable set $\Gamma \subset \mathbb R^n$ of dimension $d<n-1$. By using degenerate elliptic operators on the complement $\Omega = \mathbb R^n \setminus \Gamma$, Guy David, Svitlana Mayboroda, and the author introduced a notion of harmonic measure on $\Gamma$. We prove in the present article that this harmonic measure on $\Gamma$ satisfies the $A^\infty$-property, that is the harmonic measure and the $d$-dimension Hausdorff measure on $\Gamma$ are mutually absolutely continuous in a quantitative and scale invariant way. Thus, we give an alternate proof of a recent theorem of David and Mayboroda, which itself extends a result of Hofmann and Martell to the case where the uniformly rectifiable set $\Gamma$ is not of codimension 1. The proof is surprisingly simple - in particular does not follow the route used by David and Mayboroda, or by Hofmann and Martell - but is specific to the case when $d<n-1$.