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

Abstract

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 $t\ge C_{q}d$ 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 $2^n\ge C_q{d}^{1/q}$.

21 pages, no figures. Referee report implemented and minor typos removed. To appear in the Israel Journal of Mathematics