Some remarks on dimension-free estimates for the discrete Hardy-Littlewood maximal functions

Dependencies of the optimal constants in strong and weak type bounds will be studied between maximal functions corresponding to the Hardy--Littlewood averaging operators over convex symmetric bodies acting on $\mathbb R^d$ and $\mathbb Z^d$. Firstly, we show, in the full range of $p\in[1,\infty]$, that these optimal constants in $L^p(\mathbb R^d)$ are always not larger than their discrete analogues in $\ell^p(\mathbb Z^d)$; and we also show that the equality holds for the cubes in the case of $p=1$. This in particular implies that the best constant in the weak type $(1,1)$ inequality for the discrete Hardy--Littlewood maximal function associated with centered cubes in $\mathbb Z^d$ grows to infinity as $d\to\infty$, and if $d=1$ it is equal to the largest root of the quadratic equation $12C^2-22C+5=0$. Secondly, we prove dimension-free estimates for the $\ell^p(\mathbb Z^d)$ norms, $p\in(1,\infty]$, of the discrete Hardy--Littlewood maximal operators with the restricted range of scales $t\geq C_q d$ corresponding to $q$-balls, $q\in[2,\infty)$. Finally, we extend the latter result on $\ell^2(\mathbb Z^d)$ for the maximal operators restricted to dyadic scales $2^n\ge C_q d^{1/q}$.