AI 中文总结
本文证明了离散欧几里得球上所有半径的归一化平均值的无维数极大不等式，解决了Stein提出的$\ell^2$问题，并通过双鞍点展开和插值方法覆盖$1<p\le\infty$全范围。
AI 中文摘要
对于每个 $1<p\le\infty$，我们证明了在 $\mathbb Z^d$ 中欧几里得球上归一化平均值的所有半径的无维数极大不等式。特别地，这解决了归因于 Stein 的 $\ell^2$ 问题。证明使用整数平方半径处的双鞍点展开，将球乘子与零频率和奇偶频率处的归一化离散高斯进行比较。一阶估计给出满半径 $\ell^2$ 界。对于 $1<p<2$，我们将高阶残差估计与残差的维数一致 $\ell^1$ 界以及逐层插值相结合，以获得完整范围。
英文摘要
For every $1<p\le\infty$, we prove dimension-free maximal inequalities over all radii for normalized averages over Euclidean balls in $\mathbb Z^d$. In particular, this settles the $\ell^2$ question attributed to Stein. The proof uses a two-saddle expansion at integer squared radii to compare ball multipliers with normalized discrete Gaussians at the zero and parity frequencies. First-order estimates give the full-radius $\ell^2$ bound. For $1<p<2$, we combine higher-order residual estimates with dimension-uniform $\ell^1$ bounds for the residuals and levelwise interpolation to obtain the full range.
Comments51 pages, no figures. This version is not intended for journal submission
