On the uniform rectifiability of AD-regular measures with bounded Riesz transform operator: the case of codimension 1

Fedor Nazarov Department of Mathematical Sciences, Kent State University Alexander Volberg Department of Mathematics, Michigan State University Xavier Tolsa ICREA/Universitat Autònoma de Barcelona, Bellaterra, Barcelona, Catalonia

Functional Analysis mathscidoc:1701.12006

Acta Mathematica, 213, (2), 237-321, 2012.12
We prove that if$μ$is a$d$-dimensional Ahlfors-David regular measure in $${\mathbb{R}^{d+1}}$$ , then the boundedness of the$d$-dimensional Riesz transform in$L$^{2}($μ$) implies that the non-BAUP David–Semmes cells form a Carleson family. Combined with earlier results of David and Semmes, this yields the uniform rectifiability of$μ$.
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@inproceedings{fedor2012on,
  title={On the uniform rectifiability of AD-regular measures with bounded Riesz transform operator: the case of codimension 1},
  author={Fedor Nazarov, Alexander Volberg, and Xavier Tolsa},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203403770854796},
  booktitle={Acta Mathematica},
  volume={213},
  number={2},
  pages={237-321},
  year={2012},
}
Fedor Nazarov, Alexander Volberg, and Xavier Tolsa. On the uniform rectifiability of AD-regular measures with bounded Riesz transform operator: the case of codimension 1. 2012. Vol. 213. In Acta Mathematica. pp.237-321. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203403770854796.
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