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Singular integrals unsuitable for the curvature method whose L2-boundedness still implies rectifiability

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Abstract

The well-known curvature method initiated in works of Melnikov and Verdera is now commonly used to relate the L2(μ)-boundedness of certain singular integral operators to the geometric properties of the support of measure μ, e.g., rectifiability. It can be applied, however, only if Menger curvature-like permutations directly associated with the kernel of the operator are non-negative. We give an example of an operator in the plane whose corresponding permutations change sign but the L2(μ)-boundedness of the operator still implies that the support of μ is rectifiable. To the best of our knowledge, it is the first example of this type. We also obtain several related results with Ahlfors–David regularity conditions.

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Acknowledgments

We are grateful to the referee for useful remarks on the first version of the paper.

Author information

Authors and Affiliations

  1. Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193, Bellaterra (Barcelona), Catalonia, Spain

    Petr Chunaev & Joan Mateu

  2. ICREA, Passeig Lluís Companys 23, 08010, Barcelona, Catalonia, Spain

    Xavier Tolsa

  3. Departament de Matemàtiques and BGSMath, Universitat Autònoma de Barcelona, 08193, Bellaterra (Barcelona), Catalonia, Spain

    Xavier Tolsa

Authors
  1. Petr Chunaev
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  2. Joan Mateu
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  3. Xavier Tolsa
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Corresponding author

Correspondence to Petr Chunaev.

Additional information

P. C. and X. T. were supported by the ERC grant 320501 (FP7/2007-2013).

J. M. was supported by MTM2013-44699-P and MTM2016-75390-P (Spain), 2017-SGR-395 (Catalonia), the Marie Curie ITN MAnET (FP7-607647) and MDM-2014-044 (MICINN, Spain).

X. T. was also supported byMTM2013-44304-P and MTM2016-77635-P (Spain), 2017-SGR-395 (Catalonia), the Marie Curie ITN MAnET (FP7-607647) and MDM-2014-044 (MICINN, Spain).

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Chunaev, P., Mateu, J. & Tolsa, X. Singular integrals unsuitable for the curvature method whose L2-boundedness still implies rectifiability. JAMA 138, 741–764 (2019). https://doi.org/10.1007/s11854-019-0043-5

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  • DOI: https://doi.org/10.1007/s11854-019-0043-5

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