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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the best constants in the weak type inequalities for re-expansion operator and Hilbert transform


Author: Adam Osȩkowski
Journal: Trans. Amer. Math. Soc. 364 (2012), 4303-4322
MSC (2010): Primary 42B10, 60G44; Secondary 46E30
Published electronically: March 22, 2012
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Abstract: We study the weak type inequalities for the operator $ I-\mathcal {F}_s\mathcal {F}_c$, where $ \mathcal {F}_c$ and $ \mathcal {F}_s$ are the cosine and sine Fourier transforms on the positive half line, respectively, and $ I$ is the identity operator. We also derive sharp constants in related weak type estimates for $ I-\mathcal {H}^{\mathbb{T}}$, $ I-\mathcal {H}^{\mathbb{R}}$ and $ I-\mathcal {H}^{\mathbb{R}_+}$, where $ \mathcal {H}^\mathbb{T}$, $ \mathcal {H}^{\mathbb{R}}$ and $ \mathcal {H}^{\mathbb{R}_+}$ denote the Hilbert transforms on the circle, on the real line and the positive half-line, respectively. Our main tool is the weak type inequality for orthogonal martingales, which is of independent interest.


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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/S0002-9947-2012-05640-6
PII: S 0002-9947(2012)05640-6
Keywords: Martingale, re-expansion operator, Fourier transform
Received by editor(s): February 22, 2011
Published electronically: March 22, 2012
Additional Notes: The author was partially supported by MNiSW Grant N N201 364436.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.