Variation for the Riesz transform and uniform rectifiability

Abstract

For $1\le n<d$ integers and $\rho >2$, we prove that an $n$-dimensional Ahlfors–David regular measure $\mu$ in ${\mathbb{R}}^d$ is uniformly $n$-rectifiable if and only if the $\rho$-variation for the Riesz transform with respect to $\mu$ is a bounded operator in $L^2(\mu )$. This result can be considered as a partial solution to a well known open problem posed by G. David and S. Semmes which relates the $L^2(\mu )$ boundedness of the Riesz transform to the uniform rectifiability of $\mu$.