Uniform rectifiability and harmonic measure III: Riesz transform bounds imply uniform rectifiability of boundaries of 1-sided nta domains

Abstract

Let $E\subset \ree$, $n\ge 2$, be a closed, Ahlfors-David regular set of dimension $n$ satisfying the ``Riesz Transform bound> $$\sup_{\eps>0}\int_E\left|\int_{\{y\in E:|x-y|>\eps\}}\frac{x-y}{|x-y|^{n+1}} \,f(y)\, dH^n(y)\right|^2 dH^n(x) \,\leq \,C \int_E|f|^2 dH^n\,.$$ Assume further that $E$ is the boundary of a domain $\Omega\subset\ree$ satisfying the Harnack Chain condition plus an interior (but not exterior) Corkscrew condition. Then $E$ is uniformly rectifiable.