Calderón–Zygmund kernels and rectifiability in the plane

AbstractLet 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/|z|2,y/|z|2,z=(x,y)∈C) implies that E is rectifiable. We extend this result to any kernel of the form x2n−1/|z|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