Singular integrals unsuitable for the curvature method whose L 2-boundedness still implies rectifiability

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© 2019, The Hebrew University of Jerusalem. 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.
Original languageEnglish
Pages (from-to)741-764
JournalJournal d'Analyse Mathematique
Publication statusPublished - 1 Oct 2019


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