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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
MathSciNet review: 3003680
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Abstract | References | Similar Articles | Additional Information

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

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


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

A related estimate for orthogonal martingales is also established.

References [Enhancements On Off] (What's this?)

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Additional Information

Adam Osȩkowski
Affiliation: Department of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland

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.

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