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