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

MathSciNet review:
2912456

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Abstract | References | Similar Articles | Additional Information

Abstract: We study the weak type inequalities for the operator , where and are the cosine and sine Fourier transforms on the positive half line, respectively, and is the identity operator. We also derive sharp constants in related weak type estimates for , and , where , and 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:
https://doi.org/10.1090/S0002-9947-2012-05640-6

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.