### Abstract

© European Mathematical Society 2014. For 1 ≤ n < d integers and ρ > 2, we prove that an n-dimensional Ahlfors-David regular measure μ in double-struck Rd is uniformly n-rectifiable if and only if the ρ-variation for the Riesz transform with respect to μ is a bounded operator in L2(μ). 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 L2(μ) boundedness of the Riesz transform to the uniform rectifiability of μ.

Original language | English |
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Pages (from-to) | 2267-2321 |

Journal | Journal of the European Mathematical Society |

Volume | 16 |

Issue number | 11 |

DOIs | |

Publication status | Published - 1 Jan 2014 |

### Keywords

- Calderón-Zygmund singular integrals
- Riesz transform
- Uniform rectifiability
- ρ-variation and oscillation

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## Cite this

Mas, A., & Tolsa, X. (2014). Variation for the Riesz transform and uniform rectifiability.

*Journal of the European Mathematical Society*,*16*(11), 2267-2321. https://doi.org/10.4171/JEMS/487