On discrete Hardy-Littlewood maximal functions over the balls in ${\mathbb{Z}}^d$: dimension-free estimates

We show that the discrete Hardy-Littlewood maximal functions associated with the Euclidean balls in ${\mathbb{Z}}^d$ with dyadic radii have bounds independent of the dimension on ${\mathrm{\ell }}^p({\mathbb{Z}}^d)$ for $p\in [2,\mathrm{\infty }]$.