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

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Acta Mathematica

Abstract

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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Authors and Affiliations

  1. Department of Mathematical Sciences, Kent State University, Kent, OH, 44242, U.S.A

    Fedor Nazarov

  2. ICREA/Universitat Autònoma de Barcelona, 08193, Bellaterra, Barcelona, Catalonia

    Xavier Tolsa

  3. Department of Mathematics, Michigan State University, East Lansing, MI, 48823, U.S.A

    Alexander Volberg

Authors
  1. Fedor Nazarov
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  2. Alexander Volberg
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  3. Xavier Tolsa
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Correspondence to Xavier Tolsa.

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Nazarov, F., Volberg, A. & Tolsa, X. On the uniform rectifiability of AD-regular measures with bounded Riesz transform operator: the case of codimension 1. Acta Math 213, 237–321 (2014). https://doi.org/10.1007/s11511-014-0120-7

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  • DOI: https://doi.org/10.1007/s11511-014-0120-7

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