Cantor sets with absolutely continuous harmonic measure

David, Guy; Jeznach, Cole; Julia, Antoine

Journal de l’École polytechnique — Mathématiques, Tome 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 ${ℝ}^2\setminus K$ satisfies $G^p\left(x\right)\simeq dist{(x,K)}^{\delta }$ whenever $dist(x,K)\le 1$ 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 ${ℝ}^2\setminus K$ satisfait $G^p\left(x\right)\simeq dist{(x,K)}^{\delta }$ lorsque $dist(x,K)\le 1$ et $p$ est loin de $K$.

Classification : 31A15, 35J15
Keywords: Harmonic measure, Cantor set, Hausdorff measure
Mots-clés : Mesure harmonique, Ensemble de Cantor, mesure de Hausdorff

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