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A new maximal inequality and its applications

Published online by Cambridge University Press:  19 September 2008

Joseph M. Rosenblatt  and
Mate Wierdl
Joseph M. Rosenblatt
Affiliation:
Department of Mathematics, The Ohio State University, 231 W. W. 18th Avenue, Columbus, OH. 43210, USA
Mate Wierdl
Affiliation:
Department of Mathematics, The Ohio State University, 231 W. W. 18th Avenue, Columbus, OH. 43210, USA
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Abstract

There is a maximal inequality on the integers which implies not only the classical ergodic maximal inequality and certain maximal inequalities for moving averages and differentiation theory, but it also has the following consequence: let P1 ≤ P2 ≤ … ≤ Pk+1 be positive integers. For a σ-finite measure-preserving system (Ω, β, μ, T) and an a.e. finite β-measurable f denote

Then for any λ > 0 and f ∈ L1(Ω)

We show how the multi-parametric and superadditive versions of the previous equation can be obtained from the corresponding inequality for reversed supermartingales. The possibility of similar theorems for martingales and other sequences is also discussed.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 12 , Issue 3 , September 1992 , pp. 509 - 558
Copyright
Copyright © Cambridge University Press 1992

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