Dimension-free estimates for discrete Hardy-Littlewood averaging operators over the cubes in ${\mathbb{Z}}^d$

Abstract

Dimension-free bounds will be provided in maximal and $r$-variational inequalities on ${\ell }^p({\mathbb{Z}}^d)$ corresponding to the discrete Hardy-Littlewood averaging operators defined over the cubes in ${\mathbb{Z}}^d$. We will also construct an example of a symmetric convex body in ${\mathbb{Z}}^d$ for which maximal dimension-free bounds fail on ${\ell }^p({\mathbb{Z}}^d)$ for all $p\in (1,\infty )$. Finally, some applications in ergodic theory will be discussed.

arXiv comment28 pages, no figures. Referees suggestions implemented. Minor changes. Accepted for publication in the American Journal of Mathematics

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