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.