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A sharp covering theorem and Solyanik estimates for Euclidean balls
Abstract
For every finite family of Euclidean balls in ℝn, n≥2, and every 0<δ<1/2, we select a subfamily whose (1+δ)-dilations cover the original union and whose undilated balls have multiplicity at most Cnδ−(n−1)/2. This proves the covering estimate conjectured by Han and Lu \cite{HL}. As an application, we determine the optimal Solyanik asymptotic 𝒞n(α)−1asympn(1−α)2/(n+1) (α↑1) for the uncentered Hardy--Littlewood maximal operator over Euclidean balls, which is Conjecture~1(b) of Hagelstein and Parissis \cite{HP14}. For the modified uncentered maximal operators, we determine the optimal weak (1,1) growth rate (k−1)−(n−1)/2 as k↓1, uniformly over Radon measures. We also obtain the corresponding Lp and Fefferman--Stein bounds.
Related
- Sharp Solyanik Estimates for Discrete and Ergodic Hardy--Littlewood Maximal Operators
- Solyanik Estimates for Ergodic Flows
- The complexity of computing the covering radius of a Euclidean lattice
- Besicovitch's covering theorem in the parabolic metric
- A Frankel type theorem in Euclidean and hyperbolic spaces
BibTeX
@misc{mukherjee2026sharp,
title = {{A sharp covering theorem and Solyanik estimates for Euclidean balls}},
author = {Mayukh Mukherjee},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07880},
eprint = {2609.07880},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07880},
}
RIS
TY - UNPB
TI - A sharp covering theorem and Solyanik estimates for Euclidean balls
AU - Mayukh Mukherjee
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07880
UR - https://arxiv.org/abs/2609.07880
AB - For every finite family of Euclidean balls in ℝ^n, n≥2, and every 0<δ<1/2, we select a subfamily whose (1+δ)-dilations cover the original union and whose undilated balls have multiplicity at most C_nδ^−(n−1)/2. This proves the covering estimate conjectured by Han and Lu \cite{HL}. As an application, we determine the optimal Solyanik asymptotic 𝒞_n(α)−1asymp_n(1−α)^2/(n+1) (α↑1) for the uncentered Hardy--Littlewood maximal operator over Euclidean balls, which is Conjecture~1(b) of Hagelstein and Parissis \cite{HP14}. For the modified uncentered maximal operators, we determine the optimal weak (1,1) growth rate (k−1)^−(n−1)/2 as k↓1, uniformly over Radon measures. We also obtain the corresponding L^p and Fefferman--Stein bounds.
ER -