Sharp $p/\log p$ Bounds for Metric $X_p$ Inequalities and Arbitrary Rademacher Chaos

Abstract

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, $p\ge 2$, 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 $h={\sum }_j{D}_j{\Delta }^{-1}h$. 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 $O(p/\log p)$.