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(p⁴/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≥2, and S is uniformly distributed over the k-subsets of [n], then (E_S|E_{[n]∖ S}h|_p^p)^1/p ≲ p/(log p) [k/n∑_j=1ⁿ|∂_jh|_p^p+ (k/n)^p/2|h|_p^p]^1/p. The proof combines a sharp fixed-cardinality Rosenthal comparison with the exact reconstruction h=∑_jD_jΔ⁻¹h. 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).

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BibTeX RIS

BibTeX

@misc{zhang2026sharp,
  title = {{Sharp $p/\log p$ Bounds for Metric $X_p$ Inequalities and Arbitrary Rademacher Chaos}},
  author = {Yutong Zhang and Yaoran Yang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.26287},
  eprint = {2609.26287},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.26287},
  note = {CC BY-NC-ND 4.0},
}

RIS

TY  - UNPB
TI  - Sharp p/log p Bounds for Metric X_p Inequalities and Arbitrary Rademacher Chaos
AU  - Yutong Zhang
AU  - Yaoran Yang
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.26287
UR  - https://arxiv.org/abs/2609.26287
AB  - 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≥2, and S is uniformly distributed over the k-subsets of [n], then (E_S|E_[n]∖ Sh|_p^p)^1/p ≲ p/(log p) [k/n∑_j=1^n|∂_jh|_p^p+ (k/n)^p/2|h|_p^p]^1/p. The proof combines a sharp fixed-cardinality Rosenthal comparison with the exact reconstruction h=∑_jD_jΔ^−1h. 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).
ER  -

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