Dimension-free estimates for discrete maximal functions and lattice points in high-dimensional spheres and balls with small radii

Abstract

We prove that the discrete Hardy-Littlewood maximal function associated with Euclidean spheres with small radii has dimension-free estimates on ${\ell }^p({\mathbb{Z}}^d)$ for $p\in [2,\infty ).$ This implies an analogous result for the Euclidean balls, thus making progress on a question of E.M. Stein from the mid 1990s. Our work provides the first dimension-free estimates for full discrete maximal functions related to spheres and balls without relying on comparisons with their continuous counterparts. An important part of our argument is a uniform (dimension-free) count of lattice points in high-dimensional spheres and balls with small radii. We also established a dimension-free estimate for a multi-parameter maximal function of a combinatorial nature, which is a new phenomenon and may be useful for studying similar problems in the future.

63 pages. Final version incorporating the referee's suggestions. Accepted for publication in Journal de Mathématiques Pures et Appliquées