Abstract
In the present paper we prove that for any open connected set \({\Omega\subset\mathbb{R}^{n+1}}\), \({n\geq 1}\), 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 \({n=1}\).
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Azzam, J., Hofmann, S., Martell, J.M. et al. Rectifiability of harmonic measure. Geom. Funct. Anal. 26, 703–728 (2016). https://doi.org/10.1007/s00039-016-0371-x
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DOI: https://doi.org/10.1007/s00039-016-0371-x