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arXiv · 1511.09270

The weak-$A_\infty$ property of harmonic and $p$-harmonic measures implies uniform rectifiability

Steve Hofmann·Phi Le·José María Martell·Kaj Nyström
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Abstract

Let $E\subset \mathbb{R}^{n+1}$, $n\ge 2$, be an Ahlfors-David regular set of dimension $n$. We show that the weak-$A_\infty$ property of harmonic measure, for the open set $Ω:= \mathbb{R}^{n+1}\setminus E$, implies uniform rectifiability of $E$. More generally, we establish a similar result for the Riesz measure, $p$-harmonic measure, associated to the $p$-Laplace operator, $1<p<\infty$.

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Steve Hofmann, Phi Le, José María Martell, Kaj Nyström. 2015-11-30. The weak-$A_\infty$ property of harmonic and $p$-harmonic measures implies uniform rectifiability. https://doi.org/10.2140/apde.2017.10.513

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