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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A sharp one-sided bound for the Hilbert transform


Author: Adam Osȩkowski
Journal: Proc. Amer. Math. Soc. 141 (2013), 873-882
MSC (2010): Primary 31B05, 60G44; Secondary 42A50, 42A61
Published electronically: November 14, 2012
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Abstract: Let $ \mathcal {H}^{\mathbb{T}}$, $ \mathcal {H}^{\mathbb{R}}$ denote the Hilbert transforms on the circle and real line, respectively. The paper contains the proofs of the sharp estimates

$\displaystyle \vert\{\zeta \in \mathbb{T}:\mathcal {H}^{\mathbb{T}}f(\zeta )\geq 1\}\vert\leq 2\pi \vert\vert f\vert\vert _1, \qquad f\in L^1(\mathbb{T}),$

and

$\displaystyle \vert\{x\in \mathbb{R}:\mathcal {H}^{\mathbb{R}}f(x)\geq 1\}\vert\leq \vert\vert f\vert\vert _1, \quad \qquad f\in L^1(\mathbb{R}).$

A related estimate for orthogonal martingales is also established.

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Additional Information

Adam Osȩkowski
Affiliation: Department of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland
Email: ados@mimuw.edu.pl

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11994-X
PII: S 0002-9939(2012)11994-X
Keywords: Hilbert transform, martingale, differential subordination, weak type inequality, best constants
Received by editor(s): June 27, 2011
Published electronically: November 14, 2012
Additional Notes: This research was partially supported by MNiSW Grant N N201 397437.
Communicated by: Mark M. Meerschaert
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.