A family of singular integral operators which control the Cauchy transform

Abstract

We study the behaviour of singular integral operators 𝑇𝑘𝑡 of convolution type on ℂ associated with the parametric kernels 𝑘𝑡(𝑧):=(𝖱𝖾𝑧)3|𝑧|4+𝑡⋅𝖱𝖾𝑧|𝑧|2,𝑡∈ℝ,𝑘∞(𝑧):=𝖱𝖾𝑧|𝑧|2≡𝖱𝖾1𝑧,𝑧∈ℂ∖{0}. It is shown that for any positive locally finite Borel measure with linear growth the corresponding 𝐿2 norm of 𝑇𝑘0 controls the 𝐿2-norm of 𝑇𝑘∞ and thus of the Cauchy transform. As a corollary, we prove that the 𝐿2(1⌊𝐸)-boundedness of 𝑇𝑘𝑡 with a fixed 𝑡∈(−𝑡0,0), where 𝑡0>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 𝐿2-boundedness of the Cauchy transform, which is the key ingredient for the bilipschitz invariance of analytic capacity.