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On the generalization of the Riesz–Zygmund theorem for the ergodic Hilbert transform

Published online by Cambridge University Press:  14 November 2006

LASHA EPHREMIDZE  and
RYOTARO SATO
LASHA EPHREMIDZE
Affiliation:
A. Razmadze Mathematical Institute, Georgian Academy of Sciences, Tbilisi 0193, Georgia (e-mail: [email protected])
RYOTARO SATO
Affiliation:
Department of Mathematics, Okayama University, Okayama 700-8530, Japan (e-mail: [email protected]) 19-18 Higashi-hongo 2-chome, Midori-ku, Yokohama, 226-0002, Japan
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Abstract

A reverse weak type inequality for the conjugate function is generalized to the ergodic case. The inequality allows us to prove the Riesz–Zygmund theorem (if $f\geq 0$ and $f,\widetilde{f}\in L$, then $f\in L\log L$) for the ergodic Hilbert transform.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 27 , Issue 1 , February 2007 , pp. 113 - 122
Copyright
2006 Cambridge University Press

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