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

TL;DR

This paper extends the known connection between the boundedness of certain singular integral operators and the rectifiability of sets in the plane, introducing new examples beyond the classical Cauchy kernel.

Contribution

It generalizes the result linking $L^2$-boundedness of singular integrals to rectifiability for a broader class of kernels of the form $x^{2n-1}/|z|^{2n}$, not directly related to the Cauchy transform.

Findings

$L^2$-boundedness of new kernels implies rectifiability

First examples of non-Cauchy kernels with this property

Extension of David and Léger's theorem to a wider class of kernels

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.