© 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.
Chunaev, P., Mateu, J., & Tolsa, X. (2019). Singular integrals unsuitable for the curvature method whose L 2-boundedness still implies rectifiability. Journal d'Analyse Mathematique, 138, 741-764. https://doi.org/10.1007/s11854-019-0043-5