{
  "abstract": "We establish a sharp Sobolev-type inequality on $\\mathbb{R}^n$, $n\\geq 5$, that interpolates between the second-order energy and a weighted first-order term. Specifically, we determine the sharp constant $S_{4,2}$ in the inequality \\begin{equation*} \\int_{\\mathbb R^n} (Δu)^2\\,dx \\ge S_{4,2} \\left(\\int_{\\mathbb{R}^n}|\\nabla u|^{2}|u|^{\\frac{4}{n-4}}\\,dx\\right)^{\\frac{n-4}{n-2}}, \\end{equation*} for any $u\\in\\dot H^2(\\mathbb{R}^n)$, and we classify the cases of equality. This solves the problem of minimizing the total $Q$-curvature of among metrics on the sphere, conformal to the standard metric, with fixed total scalar curvature.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39659",
  "author_total": 4,
  "authors": [
    {
      "name": "Rupert L. Frank"
    },
    {
      "name": "Yuxin Ge"
    },
    {
      "name": "Guofang Wang"
    },
    {
      "name": "Wei Wei"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2t8d8j2qayzfm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39659",
  "fulltext_url": null,
  "id": "2t8d8j2qayzfm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39659",
  "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.AP"
    ],
    [
      "arxiv",
      "math.DG"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.39659",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2t8d8j2qayzfm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.39659",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39659"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A new sharp Sobolev inequality between the $Q$-curvature and the scalar curvature on Euclidean space",
  "title_text": "A new sharp Sobolev inequality between the Q-curvature and the scalar curvature on Euclidean space",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/2t8d8j2qayzfm",
  "venue": "arXiv",
  "versions": []
}