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.