Uniform Rectifiability and Harmonic Measure III: Riesz Transform Bounds Imply Uniform Rectifiability of Boundaries of 1-sided NTA Domains

Let E ⊂ℝⁿ⁺¹, n= ≥ 2, be a closed, Ahlfors-David regular set of dimension n satisfying the "Riesz Transform bound" sup_ℰ>0∫E|∫_{{y∈E:|x-y|>e}}x - y/|x - y|n+1 f(y) dHⁿ(y)|²dHⁿ(x) ≤ C∫_E|f|² dHⁿ.Assume further that E is the boundary of a domain ω ⊂ℝⁿ⁺¹ satisfying the Harnack Chain condition plus an interior (but not exterior) corkscrew condition. Then E is uniformly rectifiable.