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Transactions of the American Mathematical Society

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Inequalities for the ergodic maximal function and convergence of the averages in weighted $ L\sp p$-spaces


Author: F. J. Martín-Reyes
Journal: Trans. Amer. Math. Soc. 296 (1986), 61-82
MSC: Primary 28D05; Secondary 47A35
DOI: https://doi.org/10.1090/S0002-9947-1986-0837798-1
MathSciNet review: 837798
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Abstract: This paper is concerned with the characterization of those positive functions $ w$ such that Hopf's averages associated to an invertible measure preserving transformation $ T$ and a positive function $ g$ converge almost everywhere for every $ f \in {L^p}(w\,d\mu)$. We also study mean convergence when $ g$ satisfies a "doubling condition" over orbits. In order to do this, we first characterize the pairs of positive functions $ (u,w)$ such that the ergodic maximal operator associated to $ T$ and $ g$ is of weak or strong type with respect to the measures $ w\,d\mu $ and $ u\,d\mu $.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1986-0837798-1
Keywords: Ergodic maximal function, mean convergence, almost everywhere convergence, weighted inequalities, Hopf's averages
Article copyright: © Copyright 1986 American Mathematical Society

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