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(H1⌊ E) -boundedness of Tkt with a fixed t∈ (- t, 0) , where t> 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.