Discrepancy of behavior of perturbed sequences in 𝐿^{𝑝} spaces

Reinhold-Larsson, Karin

doi:10.1090/S0002-9939-1994-1169889-2

Discrepancy of behavior of perturbed sequences in $L^ p$ spaces
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by Karin Reinhold-Larsson PDF

Proc. Amer. Math. Soc. 120 (1994), 865-874 Request permission

Abstract:

Given $p \in [1,\infty )$, examples of sequences ${\{ {n_k}\} _{k \subset \mathbb {N}}}$ such that for any ergodic dynamical system $(X,\beta ,m,T)$ the averages \[ {A_N}f(x) = \frac {1} {N}\sum \limits _{k = 1}^N {f({T^{{n_k}}}x)} \] converge almost everywhere in all ${L^q}(X), q > p$, but fail to have a finite limit for some function in ${L^p}(X)$ are shown. Also, sequences such that for all ergodic dynamical systems the averages ${A_N}f(x)$ do not converge for some function $f \in {L^p}(X)$ for all $1 \leqslant p < \infty$ but do converge for all functions in ${L^\infty }(X)$ are shown.

References

Alexandra Bellow, Perturbation of a sequence, Adv. Math. 78 (1989), no. 2, 131–139. MR 1029097, DOI 10.1016/0001-8708(89)90030-3
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
Alexandra Bellow, Roger Jones, and Joseph Rosenblatt, Convergence for moving averages, Ergodic Theory Dynam. Systems 10 (1990), no. 1, 43–62. MR 1053798, DOI 10.1017/S0143385700005381
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, 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
Jean-Pierre Conze, Convergence des moyennes ergodiques pour des sous-suites, Contributions au calcul des probabilités, Bull. Soc. Math. France, Mém. No. 35, Soc. Math. France, Paris, 1973, pp. 7–15 (French). MR 0453975, DOI 10.24033/msmf.113
William R. Emerson, The pointwise ergodic theorem for amenable groups, Amer. J. Math. 96 (1974), 472–487. MR 354926, DOI 10.2307/2373555
Adriano M. Garsia, Topics in almost everywhere convergence, Lectures in Advanced Mathematics, No. 4, Markham Publishing Co., Chicago, Ill., 1970. MR 0261253
Joseph Rosenblatt, Universally bad sequences in ergodic theory, Almost everywhere convergence, II (Evanston, IL, 1989) Academic Press, Boston, MA, 1991, pp. 227–245. MR 1131794
S. Sawyer, Maximal inequalities of weak type, Ann. of Math. (2) 84 (1966), 157–174. MR 209867, DOI 10.2307/1970516