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

TL;DR

This paper establishes dimension-free bounds for discrete Hardy-Littlewood averaging operators over cubes in integer lattices, explores limitations with symmetric convex bodies, and discusses applications in ergodic theory.

Contribution

It provides the first dimension-free bounds for these operators and constructs a counterexample showing such bounds do not hold universally.

Findings

Dimension-free bounds are achieved for cube averages in $ extbf{Z}^d$.

A symmetric convex body example shows maximal bounds fail universally.

Applications to ergodic theory are discussed.

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.