发表机构
Sichuan University（四川大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文证明了度量$X_p$不等式常数的最优上界为$O(p/\log p)$，匹配已知下界，通过结合Rosenthal比较与逆梯度估计，解决了Naor提出的开放问题。
AI 中文摘要
Naor证明了$L_p$满足具有尖锐缩放参数的度量$X_p$不等式，并询问其常数对$p$的最优依赖关系。先前已知的最佳上界为$O(p^4/\log p)$，而一次混沌给出阶为$p/\log p$的下界。我们证明了匹配的上界。更一般地，若$h$是Hamming立方体上的零均值标量函数，$p\geq2$，且$S$均匀分布于$[n]$的$k$-子集，则\\[ \bigl(\E_S\\|\E_{[n]\setminus S}h\\|_p^p\bigr)^{1/p} \lesssim \frac p{\log p} \left[\frac{k}{n}\sum_{j=1}^n\\|\partial_jh\\|_p^p+ \left(\frac{k}{n}\right)^{p/2}\\|h\\|_p^p\right]^{1/p}. \\] 证明结合了尖锐的固定基数Rosenthal比较与精确重构$h=\sum_jD_j\Delta^{-1}h$。通过积分Walsh热半群的逐点逆Poincaré不等式，得到无维数的逆梯度平方函数估计。Naor的转移论证随后产生最优度量界$O(p/\log p)$。
英文摘要
Naor proved that $L_p$ satisfies the metric $X_p$ inequality with the sharp scaling parameter and asked for the optimal dependence of its constant on $p$. The best previously known upper bound was $O(p^4/\log p)$, while first-degree chaos gives a lower bound of order $p/\log p$. We prove the matching upper bound. More generally, if $h$ is a mean-zero scalar function on the Hamming cube, $p\geq2$, and $S$ is uniformly distributed over the $k$-subsets of $[n]$, then \[ \bigl(\E_S\|\E_{[n]\setminus S}h\|_p^p\bigr)^{1/p} \lesssim \frac p{\log p} \left[\frac{k}{n}\sum_{j=1}^n\|\partial_jh\|_p^p+ \left(\frac{k}{n}\right)^{p/2}\|h\|_p^p\right]^{1/p}. \] The proof combines a sharp fixed-cardinality Rosenthal comparison with the exact reconstruction $h=\sum_jD_jΔ^{-1}h$. A dimension-free inverse-gradient square-function estimate follows by integrating a pointwise reverse Poincaré inequality for the Walsh heat semigroup. Naor's transference argument then yields the optimal metric bound $O(p/\log p)$.
