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, Wr\'obel 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 $q \geq 2$.