{
  "abstract": "In this paper, we establish the metric cotype $2$ inequality for the Schatten trace class $S_{1}$ at the optimal order of the scaling parameter. More precisely, for every integer $n\\geq 1$, every even integer $m\\geq 4$, and every mapping $f:\\mathbb{Z}_{m}^{n}\\to S_{1}$, the following holds \\begin{align*} &\\left(\\sum_{j=1}^{n}\\mathbb{E}_{x}\\left\\|f\\left(x+\\frac{m}{2}e_{j}\\right)-f(x)\\right\\|_{1}^{2}\\right)^{1/2}\\\\ \\leq&\\left(3\\sqrt{70}\\,m+6\\sqrt{n}\\right)\\left(\\mathbb{E}_{x,\\varepsilon}\\left\\|f(x+\\varepsilon)-f(x)\\right\\|_{1}^{2}\\right)^{1/2}, \\end{align*} where $x$ and $\\varepsilon$ are independently uniform on $\\mathbb{Z}_{m}^{n}$ and $\\{-1,0,1\\}^{n}$, respectively. The order $\\sqrt{n}$ is necessary for any fixed metric cotype constant. This settles the optimal scaling problem for the Schatten trace class posed by Mendel and Naor \\cite[Section 8]{MN} (see also \\cite[Question 3]{GMN}).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.21445",
  "author_total": 3,
  "authors": [
    {
      "name": "Yong Jiao"
    },
    {
      "name": "Sijie Luo"
    },
    {
      "name": "Lian Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ad3fssv48fgr4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.21445",
  "fulltext_url": null,
  "id": "ad3fssv48fgr4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.21445",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7222607405",
  "pdf_url": "https://arxiv.org/pdf/2610.21445",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ad3fssv48fgr4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.21445",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.21445"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The Sharp Scale Inequality for the Schatten Trace Class",
  "title_text": "The Sharp Scale Inequality for the Schatten Trace Class",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/ad3fssv48fgr4",
  "venue": "arXiv",
  "versions": []
}