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Strong bounded operators for ergodic averages and Lebesgue derivatives

Published online by Cambridge University Press:  03 August 2009

CHAOYUAN LIU
CHAOYUAN LIU*
Affiliation:
Department of Mathematics and Statistics, Eastern Kentucky University, Richmond, KY 40475, USA (email: mary.liu@eku.edu)
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Abstract

The goal of this article is to give a new proof of there being strong type (p,p) bounds, for any p with 1>p>∞, for the class of square functions, oscillation operators and variation operators that arise when considering multiparameter differentiation and multiparameter ergodic averages. The method of proof is to first show that there is a strong estimate from L∞ to BMO for these operators in the case of multiparameter differentiation; then, by using known weak (1, 1) estimates and interpolation, one obtains strong bounds for these operators in the case of multiparameter differentiation. It follows from applying the Calderón transfer principle that these operators are of strong type (p,p), for all 1>p>∞, for the corresponding multiparameter ergodic averages.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 2 , April 2010 , pp. 547 - 564
Copyright
Copyright © Cambridge University Press 2009

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