高维离散1-对称凸体与极大函数的无维估计
High-dimensional discrete 1-symmetric convex bodies and dimension-free estimates for maximal functions
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中文总结 AI 辅助
该研究针对高维离散1-对称凸体，引入各向同性常数的离散类似物，证明离散二进极大算子等的无维ℓ^p界，为相关领域提供了推广性框架。
中文摘要 AI 辅助
我们研究1-对称凸体中的格点行为，这类凸体在坐标置换和符号变化下均保持不变。在此背景下，我们引入各向同性常数的离散类似物，并建立格点的质量集中性质，其与各向同性位置下凸体的经典质量集中结果相平行。这些几何估计被用于证明离散二进极大算子和小尺度区域内极大函数的无维ℓ^p界（p≥2）。此外，我们通过构造各向同性位置下的1-无条件凸体，证明坐标置换不变性是必要条件，此时上述无维估计失效。我们的结果提供了一个框架，推广了近期关于离散极大函数无维估计的若干研究，包括Bourgain、Mirek、Stein和Wróbel的工作。
英文摘要
We investigate the behavior of lattice points in 1-symmetric convex bodies--those invariant under both coordinate permutations and sign changes. In this setting we introduce a discrete analogue of the isotropic constant and establish concentration of mass properties for lattice points that parallel classical mass concentration results for convex bodies in isotropic position. These geometric estimates are applied to prove dimension-free $\ell^p$ bounds ($p \ge 2$) for discrete dyadic maximal operators and for maximal functions in the small-scale regime. Furthermore, we demonstrate that coordinate permutation invariance is a necessary condition by constructing 1-unconditional convex bodies in isotropic position for which these dimension-free estimates fail. Our results provide a framework that generalizes several recent works on dimension-free estimates for discrete maximal functions, including those by Bourgain, Mirek, Stein, and Wróbel.
