Subsequence ergodic theorems for 𝐿^{𝑝} contractions

Jones, Roger L.; Olsen, James; Wierdl, Máté

doi:10.1090/S0002-9947-1992-1043860-6

Subsequence ergodic theorems for $L^ p$ contractions
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by Roger L. Jones, James Olsen and Máté Wierdl PDF

Trans. Amer. Math. Soc. 331 (1992), 837-850 Request permission

Abstract:

In this paper certain subsequence ergodic theorems which have previously been known in the case of measure preserving point transformations, or Dunford Schwartz operators, are extended to operators which are positive contractions on ${L^p}$ for $p$ fixed.

References

M. A. Akcoglu and L. Sucheston, Dilations of positive contractions on $L_{p}$ spaces, Canad. Math. Bull. 20 (1977), no. 3, 285–292. MR 458230, DOI 10.4153/CMB-1977-044-4
J. R. Baxter and J. H. Olsen, Weighted and subsequential ergodic theorems, Canadian J. Math. 35 (1983), no. 1, 145–166. MR 685822, DOI 10.4153/CJM-1983-010-7
A. Bellow and V. Losert, On sequences of density zero in ergodic theory, Conference in modern analysis and probability (New Haven, Conn., 1982) Contemp. Math., vol. 26, Amer. Math. Soc., Providence, RI, 1984, pp. 49–60. MR 737387, DOI 10.1090/conm/026/737387
A. Bellow and V. Losert, The weighted pointwise ergodic theorem and the individual ergodic theorem along subsequences, Trans. Amer. Math. Soc. 288 (1985), no. 1, 307–345. MR 773063, DOI 10.1090/S0002-9947-1985-0773063-8
J. Bourgain, On the maximal ergodic theorem for certain subsets of the integers, Israel J. Math. 61 (1988), no. 1, 39–72. MR 937581, DOI 10.1007/BF02776301
J. Bourgain, An approach to pointwise ergodic theorems, Geometric aspects of functional analysis (1986/87), Lecture Notes in Math., vol. 1317, Springer, Berlin, 1988, pp. 204–223. MR 950982, DOI 10.1007/BFb0081742
J. Bourgain, On the maximal ergodic theorem for certain subsets of the integers, Israel J. Math. 61 (1988), no. 1, 39–72. MR 937581, DOI 10.1007/BF02776301

On the pointwise ergodic theorem for arithmetic sets

A.-P. Calderón, Ergodic theory and translation-invariant operators, Proc. Nat. Acad. Sci. U.S.A. 59 (1968), 349–353. MR 227354, DOI 10.1073/pnas.59.2.349
Alberto de la Torre, A simple proof of the maximal ergodic theorem, Canadian J. Math. 28 (1976), no. 5, 1073–1075. MR 417819, DOI 10.4153/CJM-1976-106-8
Roger L. Jones, Necessary and sufficient conditions for a maximal ergodic theorem along subsequences, Ergodic Theory Dynam. Systems 7 (1987), no. 2, 203–210. MR 896793, DOI 10.1017/S0143385700003953
Roger L. Jones and James Olsen, Subsequence pointwise ergodic theorems for operators in $L^p$, Israel J. Math. 77 (1992), no. 1-2, 33–54. MR 1194784, DOI 10.1007/BF02808009
Charn Huen Kan, Ergodic properties of Lamperti operators, Canadian J. Math. 30 (1978), no. 6, 1206–1214. MR 511557, DOI 10.4153/CJM-1978-100-x
John Lamperti, On the isometries of certain function-spaces, Pacific J. Math. 8 (1958), 459–466. MR 105017
James Olsen, Dominated estimates of convex combinations of commuting isometries, Israel J. Math. 11 (1972), 1–13. MR 302862, DOI 10.1007/BF02761444
James H. Olsen, The individual ergodic theorem for Lamperti contractions, C. R. Math. Rep. Acad. Sci. Canada 3 (1981), no. 2, 113–118. MR 612399
Máté Wierdl, Pointwise ergodic theorem along the prime numbers, Israel J. Math. 64 (1988), no. 3, 315–336 (1989). MR 995574, DOI 10.1007/BF02882425