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arXiv · 2303.02055

Cantor sets with absolutely continuous harmonic measure

Guy David·Cole Jeznach·Antoine Julia
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Abstract

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 the Green function in $R^2 \backslash K$ satisfies $G^p (x) \simeq \mathrm{dist} (x, K)^\delta$ whenever $\mathrm{dist} (x, K) \le 1$ and $p$ is far from $K$.

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Guy David, Cole Jeznach, Antoine Julia. 2023-03-03. Cantor sets with absolutely continuous harmonic measure. https://arxiv.org/abs/2303.02055

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