Harmonic measure is rectifiable if it is absolutely continuous with respect to the co-dimension-one Hausdorff measure

Jonas Azzam; Steve Hofmann; José María Martell; Svitlana Mayboroda; Mihalis Mourgoglou; Xavier Tolsa; Alexander Volberg

doi:10.1016/j.crma.2016.01.012

Mathematical analysis/Potential theory

Harmonic measure is rectifiable if it is absolutely continuous with respect to the co-dimension-one Hausdorff measure

Presented by: Gilles Pisier

Jonas Azzam ¹; Steve Hofmann ²; José María Martell ³; Svitlana Mayboroda ⁴; Mihalis Mourgoglou ¹; Xavier Tolsa ¹; Alexander Volberg ⁵

¹ Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Catalunya
² Department of Mathematics, University of Missouri, Columbia, MO 65211, USA
³ Instituto de Ciencias Matemáticas CSIC–UAM–UC3M–UCM, Consejo Superior de Investigaciones Científicas, 28049 Madrid, Spain
⁴ Department of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA
⁵ Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

Comptes Rendus. Mathématique, Volume 354 (2016) no. 4, pp. 351-355.

Abstract
Résumé

In the present paper, we sketch the proof of the fact that for any open connected set $Ω \subset R^{n + 1}$ , $n \geq 1$ , and any $E \subset \partial Ω$ with $0 < H^{n} (E) < \infty$ , absolute continuity of the harmonic measure ω with respect to the Hausdorff measure on E implies that $ω |_{E}$ is rectifiable.

Dans cet article, nous présentons les grandes lignes de la démonstration prouvant que, pour tout ensemble connexe ouvert $Ω \subset R^{n + 1}$ , $n \geq 1$ , et pour tout $E \subset \partial Ω$ avec $0 < H^{n} (E) < \infty$ , la continuité absolue de la mesure harmonique ω par rapport à la mesure de Hausdorff sur E implique la rectifiabilité de $ω |_{E}$ .

Received: 2015-09-22
Accepted: 2016-01-21
Published online: 2016-03-03

DOI: 10.1016/j.crma.2016.01.012

Author's affiliations:

Jonas Azzam ¹; Steve Hofmann ²; José María Martell ³; Svitlana Mayboroda ⁴; Mihalis Mourgoglou ¹; Xavier Tolsa ¹; Alexander Volberg ⁵

¹ Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Catalunya
² Department of Mathematics, University of Missouri, Columbia, MO 65211, USA
³ Instituto de Ciencias Matemáticas CSIC–UAM–UC3M–UCM, Consejo Superior de Investigaciones Científicas, 28049 Madrid, Spain
⁴ Department of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA
⁵ Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

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     author = {Jonas Azzam and Steve Hofmann and Jos\'e Mar{\'\i}a Martell and Svitlana Mayboroda and Mihalis Mourgoglou and Xavier Tolsa and Alexander Volberg},
     title = {Harmonic measure is rectifiable if it is absolutely continuous with respect to the co-dimension-one {Hausdorff} measure},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {351--355},
     publisher = {Elsevier},
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     number = {4},
     year = {2016},
     doi = {10.1016/j.crma.2016.01.012},
     language = {en},
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%A Mihalis Mourgoglou
%A Xavier Tolsa
%A Alexander Volberg
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Jonas Azzam; Steve Hofmann; José María Martell; Svitlana Mayboroda; Mihalis Mourgoglou; Xavier Tolsa; Alexander Volberg. Harmonic measure is rectifiable if it is absolutely continuous with respect to the co-dimension-one Hausdorff measure. Comptes Rendus. Mathématique, Volume 354 (2016) no. 4, pp. 351-355. doi : 10.1016/j.crma.2016.01.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.01.012/

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Cited by Sources:

^☆ We acknowledge the support of the following grants: the second author – NSF DMS 1361701, the third author – SEV-2011-0087 and (FP7/2007-2013)/ ERC agreement no. 615112 HAPDEGMT, the fourth author – Sloan Fellowship, NSF DMS 1344235, NSF DMS 1220089, NSF DMR 0212302, the sixth author – ERC 320501 FP7/2007-2013 (which also funded the first and fifth authors), 2014-SGR-75, MTM2013-44304-P, ITN MAnET (FP7-607647), the last author – NSF DMS 1265549. The results of this paper were obtained at the Institut Henri Poincaré, and at the 2015 ICMAT program Analysis and Geometry in Metric Spaces. All authors would like to express their gratitude to these institutions for their support and nice working environments.

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