Dimension-free estimates for the full discrete Euclidean ball maximal function

Abstract

Let $M_t$ denote the normalized average over the lattice points in the Euclidean ball of radius $t$ in ${\mathbb{Z}}^d$. We prove that the full maximal operator $f\mapsto {\sup }_{t\ge 0}|M_{t}f|$ is bounded on ${\ell }^p({\mathbb{Z}}^d)$, for every $1<p\le \infty$, with a constant independent of the dimension. In particular, this resolves a question of E.M. Stein from the mid 1990s. The principal ingredient in our proof is that, when $t\lesssim d$ with $t$ sufficiently large, the associated multiplier ${\mathfrak{m}}_{\sqrt{\lfloor t^2\rfloor }}(\xi )$ admits an asymptotic expansion of arbitrary prescribed order, uniform in $\xi$, whose resulting maximal operators can be controlled by the discrete normalized Gaussian maximal function studied by Mirek--Szarek--Wróbel \cite{MSW25}.

27 pages; Note that an independent proof of the main result obtained by Kaiwen Jin and Qingtang Su, has been posted on arXiv:2609.08433 on 8 Sep 2026