Characterizing strong estimates
HTML articles powered by AMS MathViewer
- by Chaoyuan Liu and Joseph Rosenblatt PDF
- Proc. Amer. Math. Soc. 136 (2008), 557-567 Request permission
Abstract:
We describe necessary and sufficient conditions for square functions to map $L^\infty$ to $L^\infty$ for ergodic averages and Lebesgue derivatives.References
- Yuan Shih Chow and Henry Teicher, Probability theory, 2nd ed., Springer Texts in Statistics, Springer-Verlag, New York, 1988. Independence, interchangeability, martingales. MR 953964, DOI 10.1007/978-1-4684-0504-0
- John B. Conway, A course in functional analysis, 2nd ed., Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1990. MR 1070713
- Richard Durrett, Probability: theory and examples, 2nd ed., Duxbury Press, Belmont, CA, 1996. MR 1609153
- Adriano M. Garsia, Martingale inequalities: Seminar notes on recent progress, Mathematics Lecture Note Series, W. A. Benjamin, Inc., Reading, Mass.-London-Amsterdam, 1973. MR 0448538
- Roger L. Jones, Iosif V. Ostrovskii, and Joseph M. Rosenblatt, Square functions in ergodic theory, Ergodic Theory Dynam. Systems 16 (1996), no. 2, 267–305. MR 1389625, DOI 10.1017/S0143385700008816
- Roger L. Jones, Robert Kaufman, Joseph M. Rosenblatt, and Máté Wierdl, Oscillation in ergodic theory, Ergodic Theory Dynam. Systems 18 (1998), no. 4, 889–935. MR 1645330, DOI 10.1017/S0143385798108349
- Roger L. Jones, Joseph M. Rosenblatt, and Máté Wierdl, Oscillation in ergodic theory: higher dimensional results, Israel J. Math. 135 (2003), 1–27. MR 1996394, DOI 10.1007/BF02776048
- Shizuo Kakutani, Induced measure preserving transformations, Proc. Imp. Acad. Tokyo 19 (1943), 635–641. MR 14222
- C. Liu, The convergence of Lebesgue derivatives and ergodic averages, Thesis, University of Illinois at Urbana-Champaign, 2005.
- Karl Petersen, Ergodic theory, Cambridge Studies in Advanced Mathematics, vol. 2, Cambridge University Press, Cambridge, 1983. MR 833286, DOI 10.1017/CBO9780511608728
- V. A. Rokhlin, A ’general’ measure-preserving transformation is not mixing, Dokl. Akad. Nauk. SSSR, 60, 51-349, 1948.
- A. de la Torre and J. L. Torrea, One-sided discrete square function, Studia Math. 156 (2003), no. 3, 243–260. MR 1978442, DOI 10.4064/sm156-3-3
Additional Information
- Chaoyuan Liu
- Affiliation: Department of Mathematics and Statistics, Wallace 313, Eastern Kentucky University, Richmond, Kentucky 40475
- Email: mary.liu@eku.edu
- Joseph Rosenblatt
- Affiliation: Department of Mathematics, University of Illinois at Champaign-Urbana, 1409 W. Green Street, Urbana, Illinois 61801-2975
- MR Author ID: 150595
- Email: jrsnbltt@math.uiuc.edu
- Received by editor(s): October 5, 2006
- Published electronically: October 24, 2007
- Additional Notes: The second author recognizes the support of the NSF during the preparation of this article.
- Communicated by: Jane M. Hawkins
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 557-567
- MSC (2000): Primary 37A05, 37A50, 26A45; Secondary 28D05
- DOI: https://doi.org/10.1090/S0002-9939-07-09054-5
- MathSciNet review: 2358496