{
  "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, $p\\geq2$, 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 $O(p/\\log p)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26287",
  "author_total": 2,
  "authors": [
    {
      "name": "Yutong Zhang"
    },
    {
      "name": "Yaoran Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9ykbva8tr1pxx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26287",
  "fulltext_url": null,
  "id": "9ykbva8tr1pxx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26287",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7214081141",
  "pdf_url": "https://arxiv.org/pdf/2609.26287",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9ykbva8tr1pxx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26287",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26287"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp $p/\\log p$ Bounds for Metric $X_p$ Inequalities and Arbitrary Rademacher Chaos",
  "title_text": "Sharp p/log p Bounds for Metric X_p Inequalities and Arbitrary Rademacher Chaos",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/9ykbva8tr1pxx",
  "venue": "arXiv",
  "versions": []
}