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

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