Dimension-free estimates for full discrete maximal functions associated with Euclidean balls and spheres
Abstract
We prove that the full discrete Hardy-Littlewood maximal operator associated with Euclidean balls satisfies dimension-free bounds on ℓp(ℤd) for every 1<p<∞. We also establish analogous dimension-free bounds for the full discrete spherical maximal operator when d≥ 5 and 2≤ p<∞. The main new idea is to approximate the relevant Fourier multipliers by finite linear combinations of normalized discrete Gaussian multipliers and their translates. We obtain these approximations uniformly in frequency and with uniformly bounded coefficients through a refined saddle-point analysis. The ball result resolves a question of E.M. Stein, while the spherical result gives, in the range p≥ 2, a dimension-free strengthening of the theorem of Magyar, Stein, and Wainger.
Related
- Dimension-free estimates for the full discrete Euclidean ball maximal function
- Dimension-free estimates for discrete maximal functions over cubes in ℤd
- Full-radius dimension-free maximal inequalities for discrete Euclidean balls
- Optimal dimension-dependent ℓp and ℓ1,∞ estimates of the second-order discrete Riesz transforms
- Endpoint estimates for commutators of singular integral operators associated with admissible functions
BibTeX
@misc{hormozi2026dimensionfree,
title = {{Dimension-free estimates for full discrete maximal functions associated with Euclidean balls and spheres}},
author = {Mahdi Hormozi and Jakub Niksiński and Błażej Wróbel},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10763},
eprint = {2609.10763},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10763},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Dimension-free estimates for full discrete maximal functions associated with Euclidean balls and spheres
AU - Mahdi Hormozi
AU - Jakub Niksiński
AU - Błażej Wróbel
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10763
UR - https://arxiv.org/abs/2609.10763
AB - We prove that the full discrete Hardy-Littlewood maximal operator associated with Euclidean balls satisfies dimension-free bounds on ℓ^p(ℤ^d) for every 1<p<∞. We also establish analogous dimension-free bounds for the full discrete spherical maximal operator when d≥ 5 and 2≤ p<∞. The main new idea is to approximate the relevant Fourier multipliers by finite linear combinations of normalized discrete Gaussian multipliers and their translates. We obtain these approximations uniformly in frequency and with uniformly bounded coefficients through a refined saddle-point analysis. The ball result resolves a question of E.M. Stein, while the spherical result gives, in the range p≥ 2, a dimension-free strengthening of the theorem of Magyar, Stein, and Wainger.
ER -