PreprintCC BY-NC-ND 4.0arXiv
Dimension-free estimates for the full discrete Euclidean ball maximal function
Abstract
Let Mt denote the normalized average over the lattice points in the Euclidean ball of radius t in ℤd. We prove that the full maximal operator f↦supt≥0lvert Mt frvert is bounded on ℓp(ℤd), for every 1<p≤∞, 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≲ d with t sufficiently large, the associated multiplier m√(⌊ t2⌋)(ξ) admits an asymptotic expansion of arbitrary prescribed order, uniform in ξ, whose resulting maximal operators can be controlled by the discrete normalized Gaussian maximal function studied by Mirek--Szarek--Wróbel \cite{MSW25}.
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BibTeX
@misc{mao2026dimensionfree,
title = {{Dimension-free estimates for the full discrete Euclidean ball maximal function}},
author = {Sheng-Chen Mao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10083},
eprint = {2609.10083},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10083},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Dimension-free estimates for the full discrete Euclidean ball maximal function
AU - Sheng-Chen Mao
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.10083
UR - https://arxiv.org/abs/2609.10083
AB - Let M_t denote the normalized average over the lattice points in the Euclidean ball of radius t in ℤ^d. We prove that the full maximal operator f↦sup_t≥0lvert M_t frvert is bounded on ℓ^p(ℤ^d), for every 1<p≤∞, 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≲ d with t sufficiently large, the associated multiplier m_√(⌊ t^2⌋)(ξ) admits an asymptotic expansion of arbitrary prescribed order, uniform in ξ, whose resulting maximal operators can be controlled by the discrete normalized Gaussian maximal function studied by Mirek--Szarek--Wróbel \cite{MSW25}.
ER -