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On weighted inequalities for ergodic operators. (English) Zbl 0549.28018

We show that the method given by A. P. Calderón in Proc. Natl. Acad. Sci. USA 59, 349-353 (1968; Zbl 0185.218), can be also applied to problems involving weights. The special case of the ergodic maximal operator defined by a Vitali family of neighborhoods of zero, in a locally compact abelian group G, follows from the characterization of weights for which the Hardy-Littlewood maximal operator on spaces of homogeneous type is bounded in \(L^ p\) [see R. A. Macías, and C. Segovia ”A well-behaved quasi-distance for spaces of homogeneous type”. Trabajos de Matemática \(N\circ 32\), IAM, Buenos Aires (1981) and the author and R. A. Macías, Proc. Am. Math. Soc. 91, 213-216 (1984)]. When \(G={\mathbb{Z}}\) we obtain the weighted \(L^ p\) boundedness of the distance ergodic maximal operator given by E. Atencia and A. de la Torre in Stud. Math. 74, 35-47 (1982; Zbl 0442.28021).

MSC:

28D05 Measure-preserving transformations
28A25 Integration with respect to measures and other set functions
42B25 Maximal functions, Littlewood-Paley theory
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