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On the Ergodic Averages and the Ergodic Hilbert Transform

Published online by Cambridge University Press:  20 November 2018

L. M. Fernández-Cabrera ,
F. J. Martín-Reyes  and
J. L. Torrea
L. M. Fernández-Cabrera
Affiliation:
Escuela Universitaria de Estadística Universidad Complutense Madrid Spain e–mail: luzfer@mat.ucm.es
F. J. Martín-Reyes
Affiliation:
Análisis Maternático Facultad de Ciencias Universidad de Málaga 29071–Málaga Spain e–mail: martin.reyes@ccuma.uma.es
J. L. Torrea
Affiliation:
Departamento de Matemáticas Facultad de Ciencias Universidad Autónoma de Madrid 28049 Madrid Spain e–mail: torrea@vml.sdi.uam.es
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Abstract

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Let T be an invertible measure-preserving transformation on a σ-finite measure space (X, μ) and let 1 < p < ∞. This paper uses an abstract method developed by José Luis Rubio de Francia which allows us to give a unified approach to the problems of characterizing the positive measurable functions v such that the limit of the ergodic averages or the ergodic Hilbert transform exist for all f ∈ Lp(νdμ). As a corollary, we obtain that both problems are equivalent, extending to this setting some results of R. Jajte, I. Berkson, J. Bourgain and A. Gillespie. We do not assume the boundedness of the operator Tf(x) = f(Tx) on Lp(νdμ). However, the method of Rubio de Francia shows that the problems of convergence are equivalent to the existence of some measurable positive function u such that the ergodic maximal operator and the ergodic Hilbert transform are bounded from LP(νdμ) into LP(udμ). We also study and solve the dual problem.

Keywords

28D0542B25Ergodic Hilbert transformergodic maximal operatormeasure-preserving transformationsvector valued inequalitiesweights
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 47 , Issue 2 , 01 April 1995 , pp. 330 - 343
Copyright
Copyright © Canadian Mathematical Society 1995

References

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