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Cantor sets with absolutely continuous harmonic measure
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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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On the dimension drop for harmonic measure on uniformly non-flat Ahlfors-David regular boundaries
For uniformly non-flat Ahlfors-David regular boundaries of dimension between n−1−δ0 and n−1 in R^n (n≥3), harmonic measure is concentrated on a set of dimension strictly less than the boundary dimension.
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