Abstract

abstract:

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