TY - JOUR

T1 - A family of singular integral operators which control the Cauchy transform

AU - Chunaev, Petr

AU - Mateu, Joan

AU - Tolsa, Xavier

PY - 2019/1/1

Y1 - 2019/1/1

N2 - © 2019, Springer-Verlag GmbH Germany, part of Springer Nature. 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 L 2 -norm of Tk0 controls the L 2 -norm of Tk∞ and thus of the Cauchy transform. As a corollary, we prove that the L 2 (H 1 ⌊ 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 L 2 -boundedness of the Cauchy transform, which is the key ingredient for the bilipschitz invariance of analytic capacity.

AB - © 2019, Springer-Verlag GmbH Germany, part of Springer Nature. 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 L 2 -norm of Tk0 controls the L 2 -norm of Tk∞ and thus of the Cauchy transform. As a corollary, we prove that the L 2 (H 1 ⌊ 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 L 2 -boundedness of the Cauchy transform, which is the key ingredient for the bilipschitz invariance of analytic capacity.

KW - Cauchy transform

KW - Corona type decomposition

KW - Rectifiability

KW - Singular integral operator

U2 - 10.1007/s00209-019-02332-7

DO - 10.1007/s00209-019-02332-7

M3 - Article

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

SN - 0025-5874

ER -