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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On the best constants in the weak type inequalities for re-expansion operator and Hilbert transform
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by Adam Osȩkowski PDF
Trans. Amer. Math. Soc. 364 (2012), 4303-4322 Request permission

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
  • ORCID: 0000-0002-8905-2418
  • Email: ados@mimuw.edu.pl
  • 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.
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 364 (2012), 4303-4322
  • MSC (2010): Primary 42B10, 60G44; Secondary 46E30
  • DOI: https://doi.org/10.1090/S0002-9947-2012-05640-6
  • MathSciNet review: 2912456