Uniform Rectifiability and Harmonic Measure III: Riesz transform bounds imply uniform rectifiability of boundaries of 1-sided NTA domains (1207.1527v2)

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