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. 2267–2321DOI 10.4171/JEMS/487