Rectifiability of harmonic measure

Abstract

In the present paper we prove that for any open connected set $\Omega \subset {\mathbb{R}}^{n+1}$, $n\ge 1$, and any $E\subset \partial \Omega$ with ${\mathcal{H}}^n(E)<\infty$, absolute continuity of the harmonic measure $\omega$ with respect to the Hausdorff measure on $E$ implies that $\omega {|}_E$ is rectifiable. This solves an open problem on harmonic measure which turns out to be an old conjecture even in the planar case $n=1$.

arXiv admin note: text overlap with arXiv:1505.06088, arXiv:1507.04409

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