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Rectifiability of harmonic measure

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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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Authors and Affiliations

  1. Departament de Matemàtiques, Universitat Autònoma de Barcelona, Edifici C Facultat de Ciències, 08193, Bellaterra (Barcelona), Catalonia, Spain

    Jonas Azzam

  2. Department of Mathematics, University of Missouri, Columbia, MO, 65211, USA

    Steve Hofmann

  3. Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, Consejo Superior de Investigaciones Científicas, C/Nicolás Cabrera,13-15, 28049, Madrid, Spain

    José María Martell

  4. Department of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Svitlana Mayboroda

  5. Departament de Matemàtiques, Universitat Autònoma de Barcelona and Centre de Recerca Matemàtica, Edifici C Facultat de Ciències, 08193, Bellaterra (Barcelona), Catalonia, Spain

    Mihalis Mourgoglou

  6. ICREA, Universitat Autònoma de Barcelona, and BGSMath. Departament de Matemàtiques. Edifici C Facultat de Ciències, 08193, Bellaterra (Barcelona), Catalonia, Spain

    Xavier Tolsa

  7. Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    Alexander Volberg

Authors
  1. Jonas Azzam
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  2. Steve Hofmann
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  3. José María Martell
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  4. Svitlana Mayboroda
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  5. Mihalis Mourgoglou
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  6. Xavier Tolsa
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Correspondence to Xavier Tolsa.

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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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