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

Abstract

The well-known curvature method initiated in works of Melnikov and Verdera is now commonly used to relate the L 2 (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 L 2 (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. 2010 Mathematics Subject Classification. 42B20 (primary); 28A75 (secondary).How to relate the L 2 (µ)-boundedness of a certain CZO to the geometric properties of the support of µ is an old problem in harmonic analysis. It stems from Calderón's paper [1] where it is proved that the Cauchy transform, i.e. the CZO with K(z) = 1/z, is L 2 (H 1 ⌊E)-bounded if E is a Lipschitz graphs with small slope. Later on, Coifman, McIntosh and Meyer [5] removed the small Lipschitz constant assumption. In [6] David fully characterized rectifiable curves Γ, for which the Cauchy transform is L 2 (H 1 ⌊Γ)-bounded. These results led to further development of tools for understanding the above-mentioned problem.A new quantitative characterization of rectifiability in terms of the so-called βnumbers introduced by Jones [12] and the concept of uniform rectifiability proposed by David and Semmes [7,8] are among these tools. Several related definitions for the plane are in order. (We refer the reader to [7,8] for definitions and results in the multidimensional case). A Radon measure µ on C is called (1-dimensional ) Ahlfors-David regular (or AD-regular, for short) if it satisfies the inequalities C −1 r µ(B(z, r)) Cr, where z ∈ spt µ, 0 < r < diam (spt µ)

A New Family of Singular Integral Operators Whose $$L^2$$ L 2 -Boundedness Implies Rectifiability

“…This is indeed true. This result, among others with Ahlfors-David regularity condition, has appeared in [5].…”

Rectifiability; a survey

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Symmetrization of a family of Cauchy-Like kernels: Global instability

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