Hostname: page-component-848d4c4894-wzw2p Total loading time: 0 Render date: 2024-06-09T03:31:29.761Z Has data issue: false hasContentIssue false
  • English
  • Français

The Best Constants Associated with some Weak Maximal Inequalities in Ergodic Theory

Published online by Cambridge University Press:  20 November 2018

Ciprian Demeter
Ciprian Demeter*
Affiliation:
Department of Mathematics, University of Illinois at Urbana, Urbana, Illinois 61801, USA e-mail: demeter@math.uiuc.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We introduce a new device of measuring the degree of the failure of convergence in the ergodic theorem along subsequences of integers. Relations with other types of bad behavior in ergodic theory and applications to weighted averages are also discussed.

Keywords

37A05
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 56 , Issue 3 , 01 June 2004 , pp. 449 - 471
Copyright
Copyright © Canadian Mathematical Society 2004

References

[1] Akcoglu, M., Bellow, A., Jones, R. L., Losert, V., Reinhold-Larsson, K. and Wierdl, M., The strong sweeping out property for lacunary sequences, Riemann sums, convolution powers, and related matters. Ergodic Theory Dynamical Systems 16(1996), 207–253.Google Scholar
[2] Akcoglu, M., Jones, R. L. and Rosenblatt, J., The Worst sums in ergodic theory. Michigan Math. J. 47(2000), 265–285.Google Scholar
[3] Baxter, J. R. and Olsen, J. H., Weighted and subsequential ergodic theorems. Canad. J. Math. 35(1983), 145–166.Google Scholar
[4] Bellow, A., On “bad universal” sequences in ergodic theory II. In: Measure theory and its applications, Sherbrooke, QC, 1982, Lecture Notes in Math. 1033, Springer, Berlin, 1983, pp. 74–78.Google Scholar
[5] Bellow, A., Perturbation of a sequence. Adv. in Math. (2) 78(1989), 131–139.Google Scholar
[6] Bergh, J. and Löfström, J., Interpolation Spaces. Springer-Verlag, New York 1976.Google Scholar
[7] Birkhoff, G. D., Proof of the ergodic theorem. Proc. Nat. Acad. Sci. U.S.A. 17(1931), 656–660.Google Scholar
[8] Bourgain, J., On the maximal ergodic theorem for certain subsets of integers. Israel J. Math. 61(1988), 39–72.Google Scholar
[9] Bourgain, J., On the pointwise ergodic theorem on Lp for arithmetic sets. Israel J. Math. 61(1988), 73–84.Google Scholar
[10] Bourgain, J., Pointwise ergodic theorems for arithmetic sets. Inst. Hautes Études Sci. Publ. Math. 69(1989), 5–45.Google Scholar
[11] Bourgain, J., Furstenberg, H., Katznelson, Y. and Orstein, D., Return times of dynamical system. (Appendix to Bourgain [10]). Inst. Hautes Études Sci. Publ. Math. 69(1989), 47–50.Google Scholar
[12] Furstenberg, H., Poincaré recurrence and number theory. Bull. Amer. Math. Soc. (3) 5(1981), 211–234.Google Scholar
[13] Hardy, G. H. and Wright, E. M., An introduction to the theory of numbers. Oxford University Press, Oxford, 5th edition, 1979.Google Scholar
[14] Jones, R. L., Lacey, M. and Wierdl, M., Integer sequences with big gaps and the pointwise ergodic theorem. Ergodic Theory Dynamical Systems 19(1999), 1295–1308.Google Scholar
[15] Jones, R. L. and Wierdl, M., Convergence and divergence of ergodic averages. Ergodic Theory Dynamical Systems 14(1994), 515–535.Google Scholar
[16] Lin, M., Olsen, J. and Tempelman, A., On modulated ergodic theorems for Dunford-Schwartz operators. Illinois J. Math 43(1999), 542–567.Google Scholar
[17] Nair, R., On polynomials in primes and J. Bourgain's circle method approach to ergodic theory. II. Studia Math. 105(1993), 207–233.Google Scholar
[18] Petrov, V. V., Sums of Independent Random Variables. Springer-Verlag, New York, Heidelberg, Berlin, 1975.Google Scholar
[19] Reinhold-Larsson, K., Discrepancy of behavior of perturbated sequences in Lp spaces. Proc. Amer. Math. Soc. (3) 120(1994), 865–874.Google Scholar
[20] Rosenblatt, J., Universally bad sequences in ergodic theory. Almost everywhere convergence (II) (eds., Bellow, A., Jones, R.), Academic Press 1991, 227–245.Google Scholar
[21] Sawyer, S., Maximal inequalities of weak type. Ann.Math. 84(1966), 157–174.Google Scholar
[22] Wierdl, M., Almost everywhere convergence and recurrence along subsequences in ergodic theory. Ph.D. Thesis, The Ohio State University, 1989.Google Scholar
[23] Wierdl, M., Pointwise ergodic theorem along the prime numbers. Israel J. Math. 64(1988), 315–356.Google Scholar