Dimension-free estimates on $l^2({\mathbb{Z}}^d)$ for discrete dyadic maximal function over $l^1$ balls: small scales

We give a dimension-free bound on $l^p({\mathbb{Z}}^d)$ for discrete Hardy-Littlewood operator over $l^1$ balls in ${\mathbb{Z}}^d$ with small dyadic radii, where $p\in [2,\mathrm{\infty }]$.