Uniform rectifiability and harmonic measure. III: Riesz transform bounds imply uniform rectifiability of boundaries of 1-sided NTA domains (Q2929664)

Let \(E \subset \mathbb{R}^{n+1}\), \(n\geq 2\), be a closed Ahlfors-David regular set of dimension \(n\). It is shown that \(E\) is uniformly rectifiable provided that it is the boundary of a domain \(\Omega \subset \mathbb{R}^{n+1}\) satisfying the Harnack chain condition, an interior corkscrew condition, and a certain \(L^2\) Riesz transform bound.NEWLINENEWLINE For Part I and Part II see [Ann. Sci. Éc. Norm. Supér. (4) 47, No. 3, 577--654 (2014; Zbl 1302.31007); Duke Math. J. 163, No. 8, 1601--1654 (2014; Zbl 1323.31008)].