A note on sharp one-sided bounds for the Hilbert transform
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- by Michał Strzelecki PDF
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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 \begin{equation*} \frac {1}{2\pi }|\{ \xi \in \mathbb {T} : \mathcal {H}^{\mathbb {T}}f(\xi ) \geq 1 \}| \leq \frac {4}{\pi }\arctan \left (\exp \left (\frac {\pi }{2}\|f\|_1\right )\right ) -1, \quad f\in L^{1}(\mathbb {T}), \end{equation*} and \begin{equation*} \frac {1}{2\pi }|\{ \xi \in \mathbb {T} : \mathcal {H}^{\mathbb {T}}f(\xi ) \geq 1 \}| \leq \frac {\|f\|_2^2}{1+\|f\|_2^2}, \quad f\in L^{2}(\mathbb {T}). \end{equation*} Related estimates for orthogonal martingales satisfying a subordination condition are also established.References
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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
- 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
- © Copyright 2015 American Mathematical Society
- 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
- MathSciNet review: 3447670