Cantor sets with absolutely continuous harmonic measure
Journal de l’École polytechnique — Mathématiques, Volume 10 (2023), pp. 1277-1298.

We construct Ahlfors regular Cantor sets K of small dimension in the plane, such that the Hausdorff measure on K is equivalent to the harmonic measure associated to its complement. In particular Green’s function in 2 K satisfies G p (x)dist(x,K) δ whenever dist(x,K)1 and p is far from K.

Nous construisons des ensembles de Cantor K, Ahlfors-réguliers de petite dimension dans le plan, tels que la mesure de Hausdorff sur K est équivalente à la mesure harmonique associée à son complémentaire. En particulier, la fonction de Green dans 2 K satisfait G p (x)dist(x,K) δ lorsque dist(x,K)1 et p est loin de K.

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Accepted:
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DOI: 10.5802/jep.245
Classification: 31A15, 35J15
Keywords: Harmonic measure, Cantor set, Hausdorff measure
Mot clés : Mesure harmonique, Ensemble de Cantor, mesure de Hausdorff
Guy David 1; Cole Jeznach 2; Antoine Julia 1

1 Université Paris-Saclay, CNRS, Laboratoire de mathématiques d’Orsay 91405 Orsay, France
2 School of Mathematics, University of Minnesota Minneapolis, MN, 55455, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Guy David; Cole Jeznach; Antoine Julia. Cantor sets with absolutely continuous harmonic measure. Journal de l’École polytechnique — Mathématiques, Volume 10 (2023), pp. 1277-1298. doi : 10.5802/jep.245. https://jep.centre-mersenne.org/articles/10.5802/jep.245/

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