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Weighted inequalities for Cesàro maximal operators in Orlicz spaces

Published online by Cambridge University Press:  12 July 2007

Pedro Ortega Salvador
Affiliation:
Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spain (ortega@anamat.cie.uma.es; ramirez@anamat.cie.uma.es)
Consuelo Ramírez Torreblanca
Affiliation:
Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spain (ortega@anamat.cie.uma.es; ramirez@anamat.cie.uma.es)

Abstract

Let 0 < α ≤ 1 and let M+α be the Cesàro maximal operator of order α defined by In this work we characterize the pairs of measurable, positive and locally integrable functions (u, v) for which there exists a constant C > 0 such that the inequality holds for all λ > 0 and every f in the Orlicz space LΦ(v). We also characterize the measurable, positive and locally integrable functions w such that the integral inequality holds for every fLΦ(w). The discrete versions of this results allow us, by techniques of transference, to prove weighted inequalities for the Cesàro maximal ergodic operator associated with an invertible measurable transformation, T, which preserves the measure.

Finally, we give sufficient conditions on w for the convergence of the sequence of Cesàro-α ergodic averages for all functions in the weighted Orlicz space LΦ(w).

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2005

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