与欧氏球和球面相关的全离散极大函数的无维数估计
Dimension-free estimates for full discrete maximal functions associated with Euclidean balls and spheres
AI总结：
本文证明欧氏球全离散 Hardy-Littlewood 极大算子对所有 $p>1$ 及球面极大算子对 $d\geq5, p\geq2$ 满足无维数界，通过离散高斯乘子逼近解决 Stein 问题并加强 MSW 定理。
AI中文摘要：
我们证明了与欧氏球相关的全离散 Hardy-Littlewood 极大算子在每个 $1<p<\infty$ 下满足 $\u2113^p(\mathbb Z^d)$ 上的无维数界。我们还建立了当 $d\geq 5$ 且 $2\leq p<\infty$ 时全离散球面极大算子的类似无维数界。主要的新思想是通过归一化离散高斯乘子及其平移的有限线性组合来逼近相关的傅里叶乘子。我们通过精细的鞍点分析在频率上一致地获得这些逼近，且系数一致有界。球的结果解决了 E.M. Stein 的一个问题，而球面的结果在 $p\geq 2$ 范围内给出了 Magyar、Stein 和 Wainger 定理的无维数加强。
英文摘要：
We prove that the full discrete Hardy-Littlewood maximal operator associated with Euclidean balls satisfies dimension-free bounds on $\ell^p(\mathbb Z^d)$ for every $1<p<\infty$. We also establish analogous dimension-free bounds for the full discrete spherical maximal operator when $d\geq 5$ and $2\leq p<\infty$. The main new idea is to approximate the relevant Fourier multipliers by finite linear combinations of normalized discrete Gaussian multipliers and their translates. We obtain these approximations uniformly in frequency and with uniformly bounded coefficients through a refined saddle-point analysis. The ball result resolves a question of E.M. Stein, while the spherical result gives, in the range $p\geq 2$, a dimension-free strengthening of the theorem of Magyar, Stein, and Wainger.
