Oscillation inequalities for rectangles
HTML articles powered by AMS MathViewer
- by Roger L. Jones, Joseph M. Rosenblatt and Máté Wierdl PDF
- Proc. Amer. Math. Soc. 129 (2001), 1349-1358 Request permission
Abstract:
In this paper we extend previously obtained results on $L^p$ norm inequalities $(1<p<\infty )$ for square functions, oscillation and variation operators, with $\mathbb Z$ actions, to the case of $\mathbb Z^d$ actions. The technique involves the use of a result about vector valued maximal functions, due to Fefferman and Stein, to reduce the problem to a situation where we can apply our previous results.References
- Bellow, A., Transfer principles in ergodic theory, in Harmonic Analysis and Partial Differential Equations, Chicago Lectures in Mathematics, M. Christ, C. Kenig and C. Sadosky ed., University of Chicago Press, Chicago 1999, pages 27-39.
- A.-P. Calderón, Ergodic theory and translation-invariant operators, Proc. Nat. Acad. Sci. U.S.A. 59 (1968), 349–353. MR 227354, DOI 10.1073/pnas.59.2.349
- C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107–115. MR 284802, DOI 10.2307/2373450
- 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
- Jones, R.L., Rosenblatt, J. and Wierdl, M., Oscillation inequalities, the higher dimensional case, preprint.
- Roger L. Jones, Joseph M. Rosenblatt, and Máté Wierdl, Counting in ergodic theory, Canad. J. Math. 51 (1999), no. 5, 996–1019. MR 1718664, DOI 10.4153/CJM-1999-044-2
- Kalikow, S. and Weiss, B., Fluctuations of ergodic averages, Il. J. Math., 43 (1999) 480-488.
- Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
- A. Zygmund, Trigonometric series: Vols. I, II, Cambridge University Press, London-New York, 1968. Second edition, reprinted with corrections and some additions. MR 0236587
Additional Information
- Roger L. Jones
- Affiliation: Department of Mathematics, DePaul University, 2320 N. Kenmore, Chicago, Illinois 60614
- Email: rjones@condor.depaul.edu
- Joseph M. Rosenblatt
- Affiliation: Department of Mathematics, University of Illinois at Urbana, Urbana, Illinois 61801
- MR Author ID: 150595
- Email: jrsnbltt@symcom.math.uiuc.edu
- Máté Wierdl
- Affiliation: Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee 38152
- Email: wierdlm@mathsci.msci.memphis.edu
- Received by editor(s): July 15, 1999
- Published electronically: November 30, 2000
- Additional Notes: The first author was partially supported by NSF Grant DMS—9531526
The second author was partially supported by NSF Grant DMS—9705228
The third author was partially supported by NSF Grant DMS—9500577 - Communicated by: Michael Handel
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 1349-1358
- MSC (2000): Primary 42B25, 28D05; Secondary 40A30
- DOI: https://doi.org/10.1090/S0002-9939-00-06032-9
- MathSciNet review: 1814160