2018/03/07 by Petr Chunaev, Chunaev, Petr, Joan Mateu +3
Mathematics · #42B20 #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.1803.02854
openalex publication_date 2018/03/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the behaviour of singular integral operators Tkt of\nconvolution type on \ℂ associated with the parametric kernels \nkt(z):=
frac(
Re z)3|z|4+t
cdot
frac
Re z|z|2,
quad t
in\n
mathbbR,
qquad k_
infty(z):=
frac
Re z|z|2
equiv
Re\n
frac1z,
quad z
in
mathbbC
setminus
0
. It is shown that for any\npositive locally finite Borel measure with linear growth the corresponding\nL2-norm of Tk0 controls the L2-norm of Tk_\∞ and thus of\nthe Cauchy transform. As a corollary, we prove that the\nL2(\H1 lfloor E)-boundedness of Tkt with a fixed t\∈\n(-t0,0), where t0>0 is an absolute constant, implies that E is\nrectifiable. This is so in spite of the fact that the usual curvature method\nfails to be applicable in this case. Moreover, as a corollary of our\ntechniques, we provide an alternative and simpler proof of the bi-Lipschitz\ninvariance of the L2-boundedness of the Cauchy transform, which is the key\ningredient for the bi-Lipschitz invariance of analytic capacity.\n