Riesz transforms and the BAUPP and BWGL criteria for uniform rectifiability

Abstract

In this note it is shown that if $\mu$ is an $n$-Ahlfors regular measure in ${\mathbb{R}}^{n+1}$ such that the $n$-dimensional Riesz transform is bounded in $L^2(\mu )$ and the so-called BAUPP (bilateral approximation by unions of parallel planes) condition holds for $\mu$, then $\mu$ satisfies the BWGL (bilateral weak geometric lemma), and so $\mu$ is uniformly $n$-rectifiable. In this way, one can solve the David-Semmes problem in codimension one without relying on the BAUP (bilateral approximation by unions of planes) criterion of David and Semmes.