TY - JOUR
T1 - A family of singular integral operators which control the Cauchy transform
AU - Chunaev, Petr
AU - Mateu, Joan
AU - Tolsa, Xavier
PY - 2020/4/1
Y1 - 2020/4/1
N2 - 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.
AB - 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.
KW - Cauchy transform
KW - Corona type decomposition
KW - Rectifiability
KW - Singular integral operator
UR - http://www.scopus.com/inward/record.url?scp=85066036572&partnerID=8YFLogxK
UR - http://www.mendeley.com/research/family-singular-integral-operators-control-cauchy-transform
U2 - 10.1007/s00209-019-02332-7
DO - 10.1007/s00209-019-02332-7
M3 - Article
SN - 0025-5874
VL - 294
SP - 1283
EP - 1340
JO - Mathematische Zeitschrift
JF - Mathematische Zeitschrift
IS - 3-4
ER -