Calderón-Zygmund kernels and rectifiability in the plane

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Abstract

Let E⊂C be a Borel set with finite length, that is, 0<H1(E)<∞. By a theorem of David and Léger, the L2(H1⌊E)-boundedness of the singular integral associated to the Cauchy kernel (or even to one of its coordinate parts x/{pipe}z{pipe}2,y/{pipe}z{pipe}2,z=(x,y)∈C) implies that E is rectifiable. We extend this result to any kernel of the form x2n-1/{pipe}z{pipe}2n,z=(x,y)∈C,n∈N. We thus provide the first non-trivial examples of operators not directly related with the Cauchy transform whose L2-boundedness implies rectifiability. © 2012 Elsevier Ltd.

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Chousionis, V., Mateu, J., Prat, L., & Tolsa, X. (2012). Calderón-Zygmund kernels and rectifiability in the plane. Advances in Mathematics, 231(1), 535–568. https://doi.org/10.1016/j.aim.2012.04.025

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