Abstract

In this talk we will focus on variational estimates for singular integral operators defined on AD regular measures satisfying some geometric condition. In particular, I will present the following result, which is a joint work with Xavier Tolsa: let $0<n<d$ be integers and let $\mu$ be an $n$-dimensional AD regular measure in $\mathbb R^d$. Then, $\mu$ is uniformly $n$-rectifiable if and only if the variation for the Riesz transform with respect to $\mu$ is a bounded operator in $L^2(\mu)$. This result is related to an important open problem, posed by David and Semmes, about the equivalence between uniform rectifiability and $L^2$ boundedness of the Riesz transforms.