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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
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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  • [1] E. Atencia and F. J. Martín-Reyes, Weak type inequalities for the maximal ergodic function and the maximal ergodic Hilbert transform in weighted spaces, Studia Math. 78 (1984), 231-244. MR 782660 (86f:28020)
  • [2] E. Atencia and A. de la Torre, A dominated ergodic estimate for $ {L_p}$ spaces with weights, Studia Math. 74 (1982), 35-47. MR 675431 (84f:47005)
  • [3] R. Coifman, Peter W. Jones and José L. Rubio de Francia, On a constructive decomposition of $ BMO$ functions and factorizations of $ {A_p}$ weights, Proc. Amer. Math. Soc. 87 (1983), 675-676. MR 687639 (84c:42031)
  • [4] N. Dunford and J. T. Schwartz, Linear operators, Part I, Interscience, New York, 1957.
  • [5] J. E. Gilbert, Nikishin-Stein theory and factorization with applications, Proc. Sympos. Pure Math., vol. 35, Amer. Math. Soc., Providence, R. I., 1979, pp. 233-267. MR 545313 (81g:42023)
  • [6] R. Hunt, D. Kurtz and C. Neugebauer, A note on the equivalence of $ {A_p}$ and Sawyer's condition for equal weights, Conference on Harmonic Analysis in honor of Antoni Zygmund, Vols. I, II (Chicago, Ill., 1981), Wadsworth Math. Ser., Wadsworth, Belmont, Calif., 1983, pp. 156-158. MR 730066 (85f:42032)
  • [7] F. J. Martíin-Reyes and A. de la Torre, A dominated ergodic theorem for invertible $ {L_p}$-isometries in weighted $ {L_p}$-spaces (preprint).
  • [8] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207-226. MR 0293384 (45:2461)
  • [9] D. Revuz, Markov chains, North-Holland Mathematical Library, 1975. MR 758799 (86a:60097)
  • [10] J. L. Rubio de Francia, Boundedness of maximal functions and singular integrals in weighted $ {L^p}$ spaces, Proc. Amer. Math. Soc. 83 (1981), 673-679. MR 630035 (83a:42018)
  • [11] E. T. Sawyer, A characterization of a two-weight norm inequality for maximal operators, Studia Math. 75 (1982), 1-11. MR 676801 (84i:42032)
  • [12] -, Two weight norm inequalities for certain maximal and integral operators, Harmonic Analysis, Proceedings, Minneapolis, 1981, Springer-Verlag, Berlin and New York, 1982, pp. 102-127. MR 654182 (83k:42020b)
  • [13] J. Woś, The filling scheme and the ergodic theorems of Kesten and Tanny, Colloq. Math. (to appear). MR 893542 (88k:28023)

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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

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