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

We give a dimension-free bound on ${\mathrm{\ell }}^p({\mathbb{Z}}^d)$, $p\in [2,\mathrm{\infty }]$ for the discrete Hardy-Littlewood maximal operator over the ${\mathrm{\ell }}^q$ 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 ${\mathrm{\ell }}^p({\mathbb{Z}}^d)$, $p\in [2,\mathrm{\infty }]$ for the discrete dyadic Hardy-Littlewood maximal operator over ${\mathrm{\ell }}^q$ balls for $q\ge 2$.