Oscillation in ergodic theory: Higher dimensional results
type: Item
instance of: scholarly article
title: Oscillation in ergodic theory: Higher dimensional results
zbMATH Open document ID: 1052.28010
DOI: 10.1007/BF02776048
author: Roger Jones
author: Máté Wierdl
author: J. Rosenblatt
published in: Israel Journal of Mathematics
publication date: 2004-03-08T00:00:00Z
review text: This paper extends the studies of [\textit{R. L. Jones, R. Kaufmann, J. M. Rosenblatt}, and \textit{M. Wierdl}, Ergodic Theory Dyn. Syst. 18, No.~4, 889--935 (1998; Zbl 0924.28009)] of square function inequalities to multi-dimensional averages. The basic tool is a comparison of the ergodic averages with various dyadic reversed martingales, but in contrast to the 1-dimensional case, the Fourier transform arguments are now replaced by more geometric almost orthogonality arguments. Let \(X = \mathbb{Z}^d\), and let \(\sigma_n\) be the \(n\)-th dyadic \(\tau\)-algebra, generated by the dyadic cubes \([u,v)\), for which all coordinates of \(u\) and \(v\) are divisible by \((2^n)\). For \(A\subset X\) let \(M_A f(x)\) denote the average of all \(f(x + y)\) with \(y\in A\). Let \(H_n = [-2^{(n+1)}, 2^{(n+1)})^d\), and let \({\mathcal A}_n\) contain all cubes with side length between \(2^{(n-1)}\) and \(2^n\), contained in \(H_n\). If \({\mathcal E}_n\) is the conditional expectation with respect to \(\sigma_n\), it is shown that \(\|(\sum_n\sup_{(A\in{\mathcal A}_n)}| M_Af -\varepsilon_nf|^2)^\frac12 \;\| p\leq c_p\| f\|_p\) \((1 < p \leq 2)\) (the norms are in \(l_p\).) There is also a similar \(l_1\)-inequality. These estimates have several interesting consequences: Hardy-Littlewood inequalities, square function inequalities, oscillation inequalities, jump inequalities, and variation inequalities. The paper also discusses other averages than those over cubes, and one section deals with different results.
Mathematics Subject Classification ID: 28D05
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42B25
zbMATH DE Number: 2053055
zbMATH Keywords: ergodic inequalities
zbMATH Keywords: ergodic oscillation
zbMATH Keywords: Hardy-Littlewood inequalities
zbMATH Keywords: oscillation inequalities
reviewed by: Ulrich Krengel
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2053055