Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-30T23:49:27.486Z Has data issue: false hasContentIssue false
  • English
  • Français

Square functions in ergodic theory

Published online by Cambridge University Press:  19 September 2008

Roger L. Jones ,
Iosif V. Ostrovskii  and
Joseph M. Rosenblatt
Roger L. Jones
Affiliation:
Department of Mathematics, DePaul University, Chicago, IL 60614, USA
Iosif V. Ostrovskii
Affiliation:
Institute for Low Temperature, Physics and Engineering, 47 Lenin Avenue, Kharkov 310164, Ukraine
Joseph M. Rosenblatt
Affiliation:
Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Given the usual averages in ergodic theory, let n1 ≤ n2 ≤ … and . There is a strong inequality ‖Sf‖2 ≤ 25‖f‖2 and there is a weak inequality m{Sf > λ} ≤ (7000/λ)‖f‖1. Related results and questions for other variants of this square function are also discussed.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 16 , Issue 2 , April 1996 , pp. 267 - 305
Copyright
Copyright © Cambridge University Press 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Assani, I., Petersen, K. and White, H.. Some connections between ergodic theory and harmonic analysis. Almost Everywhere Convergence II, Proc. Conf. on A.E. Convergence and Probability Theory, (Northwestern University, 1989). Academic Press, New York, 1991, pp. 17–40.Google Scholar
[2]Bellow, A., Jones, R. and Rosenblatt, J.. Almost everywhere convergence of weighted averages. Math. Annalen 293 (1992), 399–426.CrossRefGoogle Scholar
[3]Bellow, A., Jones, R. and Rosenblatt, J.. Almost everywhere convergence of convolution powers. Ergod. Th. & Dynam. Sys. 14 (1994), 415–432.CrossRefGoogle Scholar
[4]Benedek, A., Calderón, A. P. and Panzone, R.. Convolution operators on Banach space valued functions. Proc. National Academy of Sciences USA 48 (1962), 356–365.CrossRefGoogle ScholarPubMed
[5]Bourgain, J.. On the maximal ergodic theorem for certain subsets of the integers. Israel J. Math. 61 (1988), 39–72.CrossRefGoogle Scholar
[6]Bourgain, J.. Pointwise ergodic theorems for arithmetic sets. Publications Mathematiques, Institut des Hautes Etudes Scientifiques 69 (1989), 5–45.CrossRefGoogle Scholar
[7]Bradley, R. E.. Pseudospectral operators and the pointwise ergodic theorem. Proc. Amer. Math. Soc. 112 (1991), 863–870.CrossRefGoogle Scholar
[8]Burkholder, D. L.. Martingale transforms. Ann. Math. Statist. 37 (1966), 1494–1504.CrossRefGoogle Scholar
[9]Burkholder, D. L., Davis, B. J. and Gundy, R. F.. Integral inequalities for convex functions of operators on martingales. Proc. 6th Berkeley Symp. on Mathematical Statistics and Probability 2. University of California Press, Berkeley, CA, 1972, pp. 223–240.Google Scholar
[10]Calderón, A. P.. Ergodic theory and translation invariant operators. Proc. National Academy of Sciences USA 59 (1968) 349–353.CrossRefGoogle ScholarPubMed
[11]Duoandikoetxea, J. and de Francia, J. Rubio. Maximal and singular integral operators via Fourier transform estimates. Inventiones mathematicae 84 (1986), 541–561.CrossRefGoogle Scholar
[12]Gaposhkin, V. F.. A theorem on the convergence almost everywhere of a sequence of measurable functions, and its applications to sequences of stochastic integrals. Math USSR Sbornik 33 (1977), 1–17.CrossRefGoogle Scholar
[13]Gaposhkin, V. F.. Individual ergodic theorem for normal operators on L2. Functional Analysis and Applications 15 (1981), 14–18.CrossRefGoogle Scholar
[14]Gaposhkin, V. F. and Rosenblatt, J.. Almost everywhere convergence of sequences of powers. Proc. 5th Conf. on Probability and Statistics, (Vilnius, 1989), Vol I. Mokslas, Utrecht, 1990, pp. 391–400.Google Scholar
[15]Jones, R.. Inequalities for the ergodic maximal function. Studia Math 40 (1977), 111–129.CrossRefGoogle Scholar
[16]Rosenblatt, J. and Wierdl, M.. A new maximal inequality and its applications. Ergod. Th. & Dynam. Sys. 12 (1992), 509–558.CrossRefGoogle Scholar
[17]Rosenblatt, J. and Wierdl, M.. Pointwise ergodic theorems via harmonic analysis. Proc. Conf. on Ergodic Theory (Alexandria, Egypt, 1993). Cambridge University Press, 1994, pp. 3–151.Google Scholar
[18]de Francia, J. Rubio, Ruiz, F. and Torrea, J.. Calderón-Zygmund theory for operator-valued kernels. Adv. Math. 62 (1986), 7–48.CrossRefGoogle Scholar
[19]Sz-Nagy, B. and Foias, C.. Analyse harmonique des operateurs de l'espace de Hilbert. Akad. Kiado, Budapest. Mason, Paris, 1967.Google Scholar
[20]White, H.. The pointwise ergodic theorem and related analytic inequalities. Master's Thesis. University of North Carolina, Chapel Hill, NC, 1989.Google Scholar
[21]Wierdl, M.. Almost everywhere convergence and recurrence along subsequences in ergodic theory. Ph.D. Thesis. The Ohio State University, 1989.Google Scholar
[22]Wierdl, M.. Pointwise ergodic theorem along the prime numbers. Israel J. Math. 64 (1988), 315–336.CrossRefGoogle Scholar