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

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 $\Omega :={\mathbb{R}}^{n+1}\setminus E$, 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, $1<p<\infty$.

arXiv admin note: substantial text overlap with arXiv:1505.06499

References in corpus (2)

• Rectifiability of harmonic measure
• Uniform Rectifiability, Carleson measure estimates, and approximation of harmonic functions

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