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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Author: F. J. Martín-Reyes
Journal: Trans. Amer. Math. Soc. 296 (1986), 61-82
MSC: Primary 28D05; Secondary 47A35
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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Keywords: Ergodic maximal function, mean convergence, almost everywhere convergence, weighted inequalities, Hopf’s averages
Article copyright: © Copyright 1986 American Mathematical Society