A family of singular integral operators which control the Cauchy transform

Abstract

We study the behaviour of singular integral operators $T_{k_t}$ of convolution type on ${\mathbb {C}}$ associated with the parametric kernels

$$\begin{aligned} k_t(z):=\frac{({\textsf {Re}\,}z)^{3}}{|z|^{4}}+t\cdot \frac{{\textsf {Re}\,}z}{|z|^{2}}, \quad t\in {\mathbb {R}},\qquad k_\infty (z):=\frac{{\textsf {Re}\,}z}{|z|^{2}}\equiv {\textsf {Re}\,}\frac{1}{z},\quad z\in {\mathbb {C}}{\setminus }\{0\}. \end{aligned}$$

It is shown that for any positive locally finite Borel measure with linear growth the corresponding $L^2$-norm of $T_{k_0}$ controls the $L^2$-norm of $T_{k_\infty }$ and thus of the Cauchy transform. As a corollary, we prove that the $L^2({\mathcal {H}}^1\lfloor E)$-boundedness of $T_{k_t}$ with a fixed $t\in (-t_0,0)$, where $t_0>0$ is an absolute constant, implies that E is rectifiable. This is so in spite of the fact that the usual curvature method fails to be applicable in this case. Moreover, as a corollary of our techniques, we provide an alternative and simpler proof of the bi-Lipschitz invariance of the $L^2$-boundedness of the Cauchy transform, which is the key ingredient for the bilipschitz invariance of analytic capacity.