Rectifiability of harmonic measure |
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| Authors: | Jonas Azzam Steve Hofmann José María Martell Svitlana Mayboroda Mihalis Mourgoglou Xavier Tolsa Alexander Volberg |
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| Affiliation: | 1.Departament de Matemàtiques,Universitat Autònoma de Barcelona,Bellaterra (Barcelona),Spain;2.Department of Mathematics,University of Missouri,Columbia,USA;3.Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, Consejo Superior de Investigaciones Científicas,Madrid,Spain;4.Department of Mathematics,University of Minnesota,Minneapolis,USA;5.Departament de Matemàtiques,Universitat Autònoma de Barcelona and Centre de Recerca Matemàtica,Bellaterra (Barcelona),Spain;6.ICREA, Universitat Autònoma de Barcelona, and BGSMath. Departament de Matemàtiques. Edifici C Facultat de Ciències,Bellaterra (Barcelona),Spain;7.Department of Mathematics,Michigan State University,East Lansing,USA |
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| Abstract: | In the present paper we prove that for any open connected set ({Omegasubsetmathbb{R}^{n+1}}), ({ngeq 1}), and any ({Esubset partial Omega}) with ({mathcal{H}^n(E), 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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