Dimension-free estimates for discrete maximal functions and lattice points in high-dimensional spheres and balls with small radii
Jakub Niksiński, Błażej Wróbel
TL;DR
This work advances the discrete analogue of Stein's dimension-free maximal inequalities by proving dimension-free $\ell^p$ bounds for discrete maximal operators associated with Euclidean spheres and balls in high dimensions, in the small-scale regime. The authors develop a novel multi-parameter maximal operator $\mathcal{D}_{\overline{n}}$ and prove a dimension-free bound on $\ell^p(\mathbb{Z}^d)$ for $p\in[2,\infty]$, reducing the sphere problem to a combinatorial/multiplier framework built on Krawtchouk polynomials and diffusion semigroups. A cornerstone is a uniform saddle-point analysis yielding dimension-free lattice-point counts for $|\sqrt{n}B|$ and $|\sqrt{n}S|$ when $1\le n\le c d$, with explicit exponential-type asymptotics and initial coefficients $b_1=-\tfrac12$, $b_2=-\tfrac16$, $b_3=\tfrac{1}{24}$. By combining these lattice-counting results with precise multiplier difference estimates, the paper achieves dimension-free bounds for the full discrete maximal function over spheres (and, by a simple consequence, balls) on $\ell^p$ for $p\ge 2$ in the small-scale regime, providing a significant discrete analog to Stein's results and suggesting robust techniques for high-dimensional discrete harmonic analysis.
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.