Calder\'on-Zygmund kernels and rectifiability in the plane

Abstract

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.