A new family of singular integral operators whose $L^2$-boundedness implies rectifiability

Abstract

Let $E\subset \mathbb{C}$ be a Borel set such that $0<{\mathcal{H}}^1(E)<\infty$. David and Léger proved that the Cauchy kernel $1/z$ (and even its coordinate parts $\text{Re}z/|z{|}^2$ and $\text{Im}z/|z{|}^2$, $z\in \mathbb{C}\setminus \{0\}$) has the following property $(\ast )$: the $L^2({\mathcal{H}}^1\lfloor E)$-boundedness of the corresponding singular integral operator implies the rectifiability of $E$. Recently Chousionis, Mateu, Prat and Tolsa extended this result to any kernel of the form $(\text{Re}z{)}^{2n-1}/|z{|}^{2n}$, $n\in \mathbb{N}$. In this paper, we prove that the property $(\ast )$ is valid for operators associated to the much wider class of kernels $(\text{Re}z{)}^{2N-1}/|z{|}^{2N}+t\cdot (\text{Re}z{)}^{2n-1}/|z{|}^{2n}$, where $n,N$ are positive integer numbers such that $N\ge n$, and $t\in \mathbb{R}\setminus (t_1,t_2)$ with $t_1,t_2$ depending only on $n$ and $N$.

5 figures