JournalsjemsVol. 16, No. 11pp. 2267–2321

Variation for the Riesz transform and uniform rectifiability

  • Albert Mas

    Universidad del Pais Vasco, Bilbao, Spain
  • Xavier Tolsa

    Universitat Autónoma de Barcelona, Spain
Variation for the Riesz transform and uniform rectifiability cover
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Abstract

For 1n<d1 ≤ n < d integers and ρ>2\rho >2, we prove that an nn-dimensional Ahlfors-David regular measure μ\mu in Rd\mathbb R^d is uniformly nn-rectifiable if and only if the ρ\rho-variation for the Riesz transform with respect to μ\mu is a bounded operator in L2(μ)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 L2(μ)L^2(\mu) boundedness of the Riesz transform to the uniform rectifiability of μ\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