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A note on sharp one-sided bounds for the Hilbert transform


Author: Michał Strzelecki
Journal: Proc. Amer. Math. Soc. 144 (2016), 1171-1181
MSC (2010): Primary 31A05, 60G44; Secondary 42A50, 42A61
DOI: https://doi.org/10.1090/proc/12773
Published electronically: July 1, 2015
MathSciNet review: 3447670
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Abstract: Let $ \mathcal {H}^{\mathbb{T}}$ denote the Hilbert transform on the circle. The paper contains the proofs of the sharp estimates

$\displaystyle \frac {1}{2\pi }\vert\{ \xi \in \mathbb{T} : \mathcal {H}^{\mathb... ...frac {\pi }{2}\Vert f\Vert _1\right )\right ) -1, \quad f\in L^{1}(\mathbb{T}),$    

and

$\displaystyle \frac {1}{2\pi }\vert\{ \xi \in \mathbb{T} : \mathcal {H}^{\mathb... ...q \frac {\Vert f\Vert _2^2}{1+\Vert f\Vert _2^2}, \quad f\in L^{2}(\mathbb{T}).$    

Related estimates for orthogonal martingales satisfying a subordination condition are also established.

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Additional Information

Michał Strzelecki
Affiliation: Department of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland
Email: m.strzelecki@mimuw.edu.pl

DOI: https://doi.org/10.1090/proc/12773
Keywords: Hilbert transform, martingale, differential subordination, weak-type inequality, best constants
Received by editor(s): November 21, 2014
Received by editor(s) in revised form: March 2, 2015
Published electronically: July 1, 2015
Communicated by: Mark M. Meerschaert
Article copyright: © Copyright 2015 American Mathematical Society