Uniform Rectifiability and harmonic measure IV: Ahlfors regularity plus Poisson kernels in $L^p$ 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 $E$.

This is a preliminary version of our work on this topic, as presented by the first author at the Workshop on Harmonic Analysis and PDE held at ICMAT in Madrid, in January 2015. The final published version will be jointly authored with K. Nyström and P. Le, and in addition to the present results, will treat also the analogous theory for the $p$-Laplacian

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