Dimension-free estimates for discrete maximal functions and lattice points in high-dimensional spheres and balls with small radii
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866917964563873792 |
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| author | Niksiński, Jakub Wróbel, Błażej |
| author_facet | Niksiński, Jakub Wróbel, Błażej |
| contents | We prove that the discrete Hardy-Littlewood maximal function associated with Euclidean spheres with small radii has dimension-free estimates on $\ell^p(\mathbb{Z}^d)$ for $p\in[2,\infty).$ This implies an analogous result for the Euclidean balls, thus making progress on a question of E.M. Stein from the mid 1990s. Our work provides the first dimension-free estimates for full discrete maximal functions related to spheres and balls without relying on comparisons with their continuous counterparts. An important part of our argument is a uniform (dimension-free) count of lattice points in high-dimensional spheres and balls with small radii. We also established a dimension-free estimate for a multi-parameter maximal function of a combinatorial nature, which is a new phenomenon and may be useful for studying similar problems in the future. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_16952 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dimension-free estimates for discrete maximal functions and lattice points in high-dimensional spheres and balls with small radii Niksiński, Jakub Wróbel, Błażej Classical Analysis and ODEs Functional Analysis 42B25, 42B15, 05A10, 11P21 We prove that the discrete Hardy-Littlewood maximal function associated with Euclidean spheres with small radii has dimension-free estimates on $\ell^p(\mathbb{Z}^d)$ for $p\in[2,\infty).$ This implies an analogous result for the Euclidean balls, thus making progress on a question of E.M. Stein from the mid 1990s. Our work provides the first dimension-free estimates for full discrete maximal functions related to spheres and balls without relying on comparisons with their continuous counterparts. An important part of our argument is a uniform (dimension-free) count of lattice points in high-dimensional spheres and balls with small radii. We also established a dimension-free estimate for a multi-parameter maximal function of a combinatorial nature, which is a new phenomenon and may be useful for studying similar problems in the future. |
| title | Dimension-free estimates for discrete maximal functions and lattice points in high-dimensional spheres and balls with small radii |
| topic | Classical Analysis and ODEs Functional Analysis 42B25, 42B15, 05A10, 11P21 |
| url | https://arxiv.org/abs/2503.16952 |