On the uniform rectifiability of AD-regular measures with bounded Riesz transform operator: the case of codimension 1

Abstract

We prove that if μ is a d-dimensional Ahlfors-David regular measure in ${\mathbb{R}}^{d+1}$ , then the boundedness of the d-dimensional Riesz transform in L²(μ) implies that the non-BAUP David–Semmes cells form a Carleson family. Combined with earlier results of David and Semmes, this yields the uniform rectifiability of μ.