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

Let $E \subset \mathbb{C}$ be a Borel set such that $0<\mathcal{H}^1(E)<\infty$. David and L\'eger proved that the Cauchy kernel $1/z$ (and even its coordinate parts $\textrm{Re}\, z/|z|^2$ and $\textrm{Im}\, z/|z|^2$, $z\in \mathbb{C}\setminus\{0\}$) has the following property $(*)$: 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 $(\textrm{Re}\, z)^{2n-1}/|z|^{2n}$, $n\in \mathbb{N}$. In this paper, we prove that the property $(*)$ is valid for operators associated to the much wider class of kernels $(\textrm{Re}\, z)^{2N-1}/|z|^{2N}+t\cdot(\textrm{Re}\, z)^{2n-1}/|z|^{2n}$, 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 $N$.