# Variation for the Riesz transform and uniform rectifiability

### Albert Mas

Universidad del Pais Vasco, Bilbao, Spain### Xavier Tolsa

Universitat Autónoma de Barcelona, Spain

## Abstract

For $1 ≤ n < d$ 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 $\mu$.

## Cite this article

Albert Mas, Xavier Tolsa, Variation for the Riesz transform and uniform rectifiability. J. Eur. Math. Soc. 16 (2014), no. 11, pp. 2267–2321

DOI 10.4171/JEMS/487