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

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**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.## References

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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