On the endpoint estimate for discrete spherical average over sparse sequences
Abstract
Let d≥5, and let Aλ denote the normalized discrete spherical average on ℤd with squared radius λ. For a strictly increasing sequence (μk) of positive integers, set λk=μk! and consider the lacunary maximal operator A⋆ f:=supk|Aλkf|. For each fixed γ>0, we prove that the estimate #{x:A⋆ f(x)>α} ≤ K∑x|f(x)|/α (1+log+|f(x)|/α)γ holds with some finite K for all α>0 and finitely supported f if and only if supN≥2(#{k:μk≤ N})/((1+log N)γ)<∞. The summand is interpreted as zero when f(x)=0. The case γ=1 answers the ℓlogℓ endpoint question of Kesler, Lacey and Mena in the formulation stated above. For μk=22k, these estimates hold for every fixed γ>0, but A⋆ is not of weak type (1,1). For factorial blocks formed from arbitrary subsets of dyadic grids at widely separated scales, we characterize restricted weak type (p,p), 1<p≤ d/(d−1), by a uniform covering condition. In particular, for each 1<p*≤ d/(d−1), we construct a set of squared radii whose maximal operator is of restricted weak type (p*,p*), of strong type (p,p) for p>p*, and unbounded on ℓp for 1≤ p<p*.
Related
- Endpoint weighted Fourier estimate with vanishing spherical trace
- The strong type endpoint estimate for the local spherical maximal function
- On Entropy and Orlicz endpoint estimates and the dual Muckenhoupt-Wheeden estimate
- The Endpoint Fractional Riesz Estimate on the Hamming Cube
- A geometry-independent endpoint estimate for the Littlewood--Paley g-function generated by the Poisson semigroup
BibTeX
@misc{lee2026endpoint,
title = {{On the endpoint estimate for discrete spherical average over sparse sequences}},
author = {Sanghyuk Lee and Ji Li and Chong-Wei Liang and Chun-Yen Shen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.03004},
eprint = {2608.03004},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.03004},
}
RIS
TY - UNPB
TI - On the endpoint estimate for discrete spherical average over sparse sequences
AU - Sanghyuk Lee
AU - Ji Li
AU - Chong-Wei Liang
AU - Chun-Yen Shen
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2608.03004
UR - https://arxiv.org/abs/2608.03004
AB - Let d≥5, and let A_λ denote the normalized discrete spherical average on ℤ^d with squared radius λ. For a strictly increasing sequence (μ_k) of positive integers, set λ_k=μ_k! and consider the lacunary maximal operator A_⋆ f:=sup_k|A_λ_kf|. For each fixed γ>0, we prove that the estimate #{x:A_⋆ f(x)>α} ≤ K∑_x|f(x)|/α (1+log^+|f(x)|/α)^γ holds with some finite K for all α>0 and finitely supported f if and only if sup_N≥2(#{k:μ_k≤ N})/((1+log N)^γ)<∞. The summand is interpreted as zero when f(x)=0. The case γ=1 answers the ℓlogℓ endpoint question of Kesler, Lacey and Mena in the formulation stated above. For μ_k=2^2^k, these estimates hold for every fixed γ>0, but A_⋆ is not of weak type (1,1). For factorial blocks formed from arbitrary subsets of dyadic grids at widely separated scales, we characterize restricted weak type (p,p), 1<p≤ d/(d−1), by a uniform covering condition. In particular, for each 1<p_*≤ d/(d−1), we construct a set of squared radii whose maximal operator is of restricted weak type (p_*,p_*), of strong type (p,p) for p>p_*, and unbounded on ℓ^p for 1≤ p<p_*.
ER -