Riesz transforms and the BAUPP and BWGL criteria for uniform rectifiability
Xavier Tolsa
TL;DR
The note addresses the David–Semmes problem in codimension one by showing that an $n$-Ahlfors regular measure $\mu$ in ${\mathbb R}^{n+1}$ with bounded $n$-dimensional Riesz transform ${\mathcal{R}}_\mu$ and the BAUPP condition is automatic, leading to the bilateral weak geometric lemma (BWGL) and hence uniform $n$-rectifiability. The authors provide a direct argument that bypasses the original BAUP criterion, instead deriving BWGL from BAUPP and the Riesz transform via a dyadic David–Semmes lattice and Carleson-type estimates. A key step is a geometric-analytic lemma establishing a positive lower bound on the mean difference of ${\mathcal{R}}_\mu$ across cylinders formed by parallel planes, followed by a Lipschitz-approximation technique to convert these bounds into a global $L^2$-based Carleson estimate. Consequently, in concert with existing results (NToV), this yields that $L^2(\mu)$-boundedness of ${\mathcal{R}}_\mu$ implies uniform $n$-rectifiability in codimension one, without invoking the BAUP criterion.
Abstract
In this note it is shown that if $μ$ is an $n$-Ahlfors regular measure in $\mathbb R^{n+1}$ such that the $n$-dimensional Riesz transform is bounded in $L^2(μ)$ and the so-called BAUPP (bilateral approximation by unions of parallel planes) condition holds for $μ$, then $μ$ satisfies the BWGL (bilateral weak geometric lemma), and so $μ$ 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.