{
  "abstract": "We prove two dimension-free estimates in Gaussian space. The first is an optimal Meyer-type inequality for the Gaussian divergence: for $p\\geq2$, \\[ \\|δV\\|_{L^p(γ_d)} \\leq \\sqrt p\\,|\\mathbb E V| +Cp\\,\\|DV\\|_{L^p(γ_d;\\mathrm{HS}_d)}\\, . \\] Here $\\mathrm{HS}_d$ is the space of $d\\times d$ matrices with its Hilbert-Schmidt norm. Its main ingredient is a dimension-free weak-type $(1,1)$ bound for the second-order transform $D^2\\mathcal N^{-1}$, where $\\mathcal N := -Δ+x\\cdot D$ is the Ornstein-Uhlenbeck operator. We also include a direct change-of-variables proof of the weaker bounded derivative divergence inequality in the appendix. The second result is a Gaussian Bernstein-Markov inequality. If $P$ is a polynomial, $p\\geq2$, and $m_p := (\\mathbb E|g|^p)^{1/p}$ for a standard Gaussian $g$, then \\[ \\|DP\\|_{L^p(γ_d;\\mathbb R^d)} \\leq\\frac{2\\sqrt{2e\\, \\mathrm{deg}(P)}}{m_p}\\,\\|P\\|_{L^p(γ_d)}\\, . \\] The factor $2$ is unnecessary when $p$ is even, or $P$ is even or odd. Thus, the constant is of order $\\sqrt{\\mathrm{deg}(P)/p}$ throughout the range $p\\geq2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32851",
  "author_total": 4,
  "authors": [
    {
      "name": "Fan Chen"
    },
    {
      "name": "Sinho Chewi"
    },
    {
      "name": "Jianfeng Lu"
    },
    {
      "name": "Matthew S. Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bdf25rw8gvymt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32851",
  "fulltext_url": null,
  "id": "bdf25rw8gvymt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32851",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7214737885",
  "pdf_url": "https://arxiv.org/pdf/2609.32851",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bdf25rw8gvymt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.32851",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32851"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Sharp Gaussian divergence and Bernstein-Markov inequalities",
  "title_text": "Sharp Gaussian divergence and Bernstein-Markov inequalities",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/bdf25rw8gvymt",
  "venue": "arXiv",
  "versions": []
}