Oscillation inequalities for rectangles

Authors:
Roger L. Jones, Joseph M. Rosenblatt and Máté Wierdl

Journal:
Proc. Amer. Math. Soc. **129** (2001), 1349-1358

MSC (2000):
Primary 42B25, 28D05; Secondary 40A30

Published electronically:
November 30, 2000

MathSciNet review:
1814160

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we extend previously obtained results on norm inequalities for square functions, oscillation and variation operators, with actions, to the case of 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.

**1.**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.**2.**A.-P. Calderón,*Ergodic theory and translation-invariant operators*, Proc. Nat. Acad. Sci. U.S.A.**59**(1968), 349–353. MR**0227354****3.**C. Fefferman and E. M. Stein,*Some maximal inequalities*, Amer. J. Math.**93**(1971), 107–115. MR**0284802****4.**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**, 10.1017/S0143385700008816**5.**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**, 10.1017/S0143385798108349**6.**Jones, R.L., Rosenblatt, J. and Wierdl, M.,*Oscillation inequalities, the higher dimensional case*, preprint.**7.**Roger L. Jones, Joseph M. Rosenblatt, and Máté Wierdl,*Counting in ergodic theory*, Canad. J. Math.**51**(1999), no. 5, 996–1019. MR**1718664**, 10.4153/CJM-1999-044-2**8.**Kalikow, S. and Weiss, B.,*Fluctuations of ergodic averages*, Il. J. Math.,**43**(1999) 480-488. CMP**99:17****9.**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****10.**A. Zygmund,*Trigonometric series: Vols. I, II*, Second edition, reprinted with corrections and some additions, Cambridge University Press, London-New York, 1968. MR**0236587**

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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

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

DOI:
http://dx.doi.org/10.1090/S0002-9939-00-06032-9

Keywords:
Convergence of ergodic averages,
square functions,
variation,
oscillation,
upcrossing inequalities,
jump inequalities

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

Article copyright:
© Copyright 2000
American Mathematical Society