Skip to main content
SpringerLink
Log in

Oscillation in ergodic theory: Higher dimensional results

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this paper we continue our investigations of square function inequalities. The results in [9] are primarily one dimensional, and here we extend all the results to multi-dimensional averages. Our basic tool is still a comparison of the ergodic averages with various dyadic (reversed) martingales, but the Fourier transform arguments are replaced by more geometric almost orthogonality arguments.

The results imply the pointwise ergodic theorem for the action of commuting measure preserving transformations, and give additional information such as control of the number of upcrossings of the ergodic averages. Related differentiation results are also discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. I. Assani, K. Petersen and H. White,Some connections between ergodic theory and harmonic analysis, inAlmost Everywhere Convergence (A. Bellow and R. L. Jones, eds.), Academic Press, San Diego, 1991, pp. 17–40.

    Google Scholar 

  2. E. Bishop,Foundations of Constructive Analysis, McGraw-Hill, New York, 1967.

    MATH  Google Scholar 

  3. J. Bourgain,On the maximal ergodic theorem for certain subsets of the integers, Israel Journal of Mathematics61 (1988), 39–72.

    MATH  MathSciNet  Google Scholar 

  4. J. Bourgain,Pointwise ergodic theorems for arithmetic sets (Appendix: The return time theorem), Publications Mathématiques de l’Institut des Hautes Études Scientifiques69 (1989), 5–45.

    Article  MATH  MathSciNet  Google Scholar 

  5. A. P. Calderón,Ergodic theory and translation invariant operators, Proceedings of the National Academy of Sciences of the United States of America59 (1968), 349–353.

    Article  MATH  Google Scholar 

  6. A. P. Calderón and A. Zygmund,On the existence of certain singular integrals, Acta Mathematica88 (1952), 85–139.

    Article  MATH  MathSciNet  Google Scholar 

  7. V. F. Gaposhkin,A theorem on the convergence almost everywhere of a sequence of measurable functions, and its applications to sequences of stochastic integrals, Mathematics of the USSR-Sbornik33 (1977), 1–17.

    Article  MATH  Google Scholar 

  8. V. F. Gaposhkin,Individual ergodic theorems for normal operators on L 2, Functional Analysis and its Applications15 (1981), 14–18.

    Article  MATH  MathSciNet  Google Scholar 

  9. R. Jones, R. Kaufman, J. Rosenblatt and M. Wierdl,Oscillation in ergodic theory, Ergodic Theory and Dynamical Systems18 (1998), 889–935.

    Article  MATH  MathSciNet  Google Scholar 

  10. R. Jones, J. Rosenblatt and M. Wierdl,Oscillation inequalities for rectangles, Proceedings of the American Mathematical Society129 (2001), 1349–1358.

    Article  MATH  MathSciNet  Google Scholar 

  11. B. Weiss and S. Kalikow,Fluctuations of ergodic averages, Illinois Journal of Mathematics43 (1999), 480–488.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, DePaul University, 2219 N. Kenmore, 60614, Chicago, IL, USA

    Roger L. Jones

  2. Department of Mathematics, University of Illinois at Urbana, 61801, Urbana, IL, USA

    Joseph M. Rosenblatt

  3. Department of Mathematical Sciences, University of Memphis, 38152, Memphis, TN, USA

    Máté Wierdl

Authors
  1. Roger L. Jones
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Joseph M. Rosenblatt
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Máté Wierdl
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Roger L. Jones.

Additional information

R. Jones is partially supported by NSF Grant DMS-9531526.

J. Rosenblatt is partially supported by NSF Grant DMS-9705228.

M. Wierdl is partially supported by NSF Grant DMS-9801602.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jones, R.L., Rosenblatt, J.M. & Wierdl, M. Oscillation in ergodic theory: Higher dimensional results. Isr. J. Math. 135, 1–27 (2003). https://doi.org/10.1007/BF02776048

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02776048

Keywords

Search

Navigation