The Journal of Geometric Analysis

, Volume 29, Issue 2, pp 1193–1205 | Cite as

On Harmonic Measure and Rectifiability in Uniform Domains

  • Mihalis MourgoglouEmail author


Let \(\Omega \subset \mathbb {R}^{d+1}\), \(d \ge 1\), be a uniform domain with lower d-Ahlfors–David regular and d-rectifiable boundary. We show that if the d-Hausdorff measure \(\mathcal {H}^d|_{\partial \Omega }\) is locally finite, then \(\mathcal {H}^d|_{\partial \Omega }\) is absolutely continuous with respect to harmonic measure for \(\Omega \).

Mathematics Subject Classification

31A15 28A75 28A78 



We warmly thank J. Azzam and X. Tolsa for their encouragement and several discussions pertaining to this work and rectifiability. We are particularly grateful to J. Azzam for explaining the techniques developed in his earlier work on the same topic. We would also like to thank the anonymous referees for their valuable comments that helped us improve the paper. The current manuscript was finished and uploaded on ArXiv in mid-2015 when the author was a post-doc of X. Tolsa at Universitat Autònoma de Barcelona supported by the ERC Grant 320501 of the European Research Council (FP7/2007–2013).


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Authors and Affiliations

  1. 1.Departament de MatemàtiquesUniversitat Autònoma de Barcelona and Centre de Reserca Matemàtica, Edifici C Facultat de CiènciesBellaterraCatalonia
  2. 2.Departamento de MatemáticasUniversidad del País VascoLeioaSpain
  3. 3.Ikerbasque, Basque Foundation for ScienceBilbaoSpain

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