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 $\mathrm{Re\,}z/|z|^2$ and $\mathrm{Im\,}z/|z|^2, z\in {\mathbb {C}}{\setminus }\{0\}$) has the following property: the $L^2({\mathcal {H}}^1\lfloor E)$-boundedness of the corresponding singular integral operator implies that E is rectifiable. Recently Chousionis, Mateu, Prat and Tolsa extended this result to any kernel of the form $(\mathrm{Re\,}z)^{2n-1}/|z|^{2n}, n\in {\mathbb {N}}$. In this paper, we prove that the above-mentioned property holds for operators associated with the much wider class of the kernels $(\mathrm{Re\,}z)^{2N-1}/|z|^{2N}+t\cdot (\mathrm{Re\,}z)^{2n-1}/|z|^{2n}$, where n and N are positive integer numbers such that $N\geqslant n$, and $t\in {\mathbb {R}}{\setminus } (t_1,t_2)$ with $t_1,t_2$ depending only on n and N.