Calderón-Zygmund kernels and rectifiability in the plane

Let E⊂C be a Borel set with finite length, that is, 0<H 1(E)<∞. By a theorem of David and Léger, the L 2(H 1⌊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 x 2n-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 L 2-boundedness implies rectifiability. © 2012 Elsevier Ltd.