PreprintarXiv licencearXiv
Full-radius dimension-free maximal inequalities for discrete Euclidean balls
Abstract
For every 1<p≤∞, we prove dimension-free maximal inequalities over all radii for normalized averages over Euclidean balls in ℤd. In particular, this settles the ℓ2 question attributed to Stein. The proof uses a two-saddle expansion at integer squared radii to compare ball multipliers with normalized discrete Gaussians at the zero and parity frequencies. First-order estimates give the full-radius ℓ2 bound. For 1<p<2, we combine higher-order residual estimates with dimension-uniform ℓ1 bounds for the residuals and levelwise interpolation to obtain the full range.
Related
- Dimension-free estimates for full discrete maximal functions associated with Euclidean balls and spheres
- Dimension-free estimates for the full discrete Euclidean ball maximal function
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- The complexity of computing the covering radius of a Euclidean lattice
- L1-Poincar\'e inequalities for differential forms on Euclidean spaces and Heisenberg groups
BibTeX
@misc{jin2026fullradius,
title = {{Full-radius dimension-free maximal inequalities for discrete Euclidean balls}},
author = {Kaiwen Jin and Qingtang Su},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08433},
eprint = {2609.08433},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08433},
}
RIS
TY - UNPB
TI - Full-radius dimension-free maximal inequalities for discrete Euclidean balls
AU - Kaiwen Jin
AU - Qingtang Su
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.08433
UR - https://arxiv.org/abs/2609.08433
AB - For every 1<p≤∞, we prove dimension-free maximal inequalities over all radii for normalized averages over Euclidean balls in ℤ^d. In particular, this settles the ℓ^2 question attributed to Stein. The proof uses a two-saddle expansion at integer squared radii to compare ball multipliers with normalized discrete Gaussians at the zero and parity frequencies. First-order estimates give the full-radius ℓ^2 bound. For 1<p<2, we combine higher-order residual estimates with dimension-uniform ℓ^1 bounds for the residuals and levelwise interpolation to obtain the full range.
ER -