%0 Journal Article
%T Calder¨®n-Zygmund kernels and rectifiability in the plane
%A Vasilis Chousionis
%A Joan Mateu
%A Laura Prat
%A Xavier Tolsa
%J Mathematics
%D 2011
%I arXiv
%X Let $E \subset \C$ be a Borel set with finite length, that is, $0<\mathcal{H}^1 (E)<\infty$. By a theorem of David and L\'eger, the $L^2 (\mathcal{H}^1 \lfloor E)$-boundedness of the singular integral associated to the Cauchy kernel (or even to one of its coordinate parts $x / |z|^2,y / |z|^2,z=(x,y) \in \C$) implies that $E$ is rectifiable. We extend this result to any kernel of the form $x^{2n-1} /|z|^{2n}, z=(x,y) \in \C,n \in \mathbb{N}$. We thus provide the first non-trivial examples of operators not directly related with the Cauchy transform whose $L^2$-boundedness implies rectifiability.
%U http://arxiv.org/abs/1110.1302v3