Remark on dimension-free estimates for discrete maximal functions over ${\ell }^q$ balls: small dyadic scales

Abstract

We give a dimension-free bound on ${\ell }^p({\mathbb{Z}}^d)$, $p\in [2,\infty ]$ for the discrete Hardy-Littlewood maximal operator over the ${\ell }^q$ balls in ${\mathbb{Z}}^d$ with small dyadic radii. Our result combined with the work of Kosz, Mirek, Plewa, Wróbel gives dimension-free estimates on ${\ell }^p({\mathbb{Z}}^d)$, $p\in [2,\infty ]$ for the discrete dyadic Hardy-Littlewood maximal operator over ${\ell }^q$ balls for $q\ge 2$.

17 pages, no figures. New version with extended result, improved exposition and slightly changed title