Rectifiability of harmonic measure

By Jonas Azzam, S. Hofmann, J. M. Martell, Svitlana Mayboroda, Mihalis Mourgoglou, Xavier Tolsa, Alexander Volberg. 2015

84 citations recorded by the source corpus.

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

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/75c93148468e8309b9d4451b288f49f994b97a19