PreprintCC BY 4.0arXiv
Dimension-free estimates for discrete maximal functions over cubes in ℤd
Abstract
In this short note, we establish dimension-free ℓp(ℤd) bounds, for all p∈(1,∞], for the discrete Hardy--Littlewood maximal functions associated with cubes in ℤd, answering a question that had been open for a while. The key idea is to prove dimension-free bounds for the ℓp(ℤd) norms of the differences of the corresponding averages. This follows from an ad hoc interpretation of the associated discrete multipliers as a special continuous family of multipliers to which basic fractional integration and complex interpolation can be applied. The same method also yields an elementary proof of Bourgain's dimension-free Lp(ℝd) bounds for the Hardy--Littlewood maximal function associated with cubes in ℝd.
Related
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BibTeX
@misc{mirek2026dimensionfree,
title = {{Dimension-free estimates for discrete maximal functions over cubes in $\mathbb Z^d$}},
author = {Mariusz Mirek and Tomasz Z. Szarek and Błażej Wróbel},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24030},
eprint = {2609.24030},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24030},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Dimension-free estimates for discrete maximal functions over cubes in ℤ^d
AU - Mariusz Mirek
AU - Tomasz Z. Szarek
AU - Błażej Wróbel
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24030
UR - https://arxiv.org/abs/2609.24030
AB - In this short note, we establish dimension-free ℓ^p(ℤ^d) bounds, for all p∈(1,∞], for the discrete Hardy--Littlewood maximal functions associated with cubes in ℤ^d, answering a question that had been open for a while. The key idea is to prove dimension-free bounds for the ℓ^p(ℤ^d) norms of the differences of the corresponding averages. This follows from an ad hoc interpretation of the associated discrete multipliers as a special continuous family of multipliers to which basic fractional integration and complex interpolation can be applied. The same method also yields an elementary proof of Bourgain's dimension-free L^p(ℝ^d) bounds for the Hardy--Littlewood maximal function associated with cubes in ℝ^d.
ER -