Absolute continuity of the harmonic measure on low dimensional 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 $d<n-1$.