{
  "abstract": "We study the dependence on the dimension of stability constants for the Caffarelli--Kohn--Nirenberg inequalities, using the weighted energy distance to the manifold of optimizers. In the subcritical non-degenerate case, i.e., $b_{FS}(a)<b<a+1$ with $a<0$ and $a<b<a+1$ with $a\\ge0$, we prove \\[ \\frac{β_{a,b}}d\\le c_{d,a,b} \\le \\frac{4(1+a-b)}{d+4(1+a-b)} \\] with a positive constant depending only on the weights, uniformly over all admissible dimensions. The upper bound shows that this dimensional order is optimal. The critical case $a=b>0$ exhibits a different dimensional scale, \\[ \\frac{\\min\\{β_{\\mathrm{Sob}},4a\\}}{25d^2} \\le c_{d,a,a}\\le\\frac{16a}{d^2}+O_a(d^{-3}), \\] where $β_{\\mathrm{Sob}}>0$ is a universal Sobolev stability constant. In the degenerate case on the Felli--Schneider curve, for fixed $a<0$, we prove \\[ \\frac{β_a^{FS}}{d^3} \\le\\widetilde c_{d,a,b_{FS}(a)} \\le\\frac{32|a|}{d^3}+O_a(d^{-4}), \\] where $\\widetilde c$ is the quartic stability constant given in [Frank-Peteranderl, CV\\&PDE, 2024] and $β_a^{FS}>0$ is independent of $d$. The three cases thus have optimal dimensional orders $d^{-1}$, $d^{-2}$, and $d^{-3}$, respectively.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.41474",
  "author_total": 2,
  "authors": [
    {
      "name": "Houwang Li"
    },
    {
      "name": "Juncheng Wei"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9tay8r7h6wqp5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.41474",
  "fulltext_url": null,
  "id": "9tay8r7h6wqp5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.41474",
  "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.AP"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.41474",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9tay8r7h6wqp5.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.41474",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.41474"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A lower bound for the stability constant in the Caffarelli--Kohn--Nirenberg inequality",
  "title_text": "A lower bound for the stability constant in the Caffarelli--Kohn--Nirenberg inequality",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/9tay8r7h6wqp5",
  "venue": "arXiv",
  "versions": []
}