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 integers and , we prove that an -dimensional Ahlfors–David regular measure in is uniformly -rectifiable if and only if the -variation for the Riesz transform with respect to is a bounded operator in . 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 boundedness of the Riesz transform to the uniform rectifiability of .

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