{
  "abstract": "For $α\\in(0,n)$, $p\\in(1,\\frac{n}α)$, $q:=\\frac{np}{n-αp}$, and $W\\in\\mathscr A_{p,q}$, we obtain the following matrix-weighted boundedness of the fractional integral $I_α$: for any $\\vec{f}\\in L^p(W^p)$, \\begin{align*} \\left\\|I_α\\vec f\\right\\|_{L^q(W^q)} \\lesssim[W]_{\\mathscr A_{p,q}}^{ (1-\\fracα{n})\\max\\{1,\\frac{p'}{q}\\}} \\left\\|\\vec f\\right\\|_{L^p(W^p)}, \\end{align*} where the implicit positive constant is independent of $W$ and $\\vec{f}$. For integer $n\\geq2$, $p\\in[1,n)$, $q:=\\frac{np}{n-p}$, and $W\\in\\mathscr A_{p,q}$, we establish the following matrix-weighted Sobolev inequality: for any smooth $\\mathbb{C}^d$-valued function $\\vec{f}$ with compact support, \\begin{align*} \\left\\|\\vec{f}\\right\\|_{L^{q}(W^q)} \\lesssim[W]_{\\mathscr A_{p,q}}^{\\frac{n-1}{n}} \\left\\|WD\\vec{f}\\right\\|_{L^p(\\mathbb R^n,\\mathbb C^{d\\times n})}, \\end{align*} where the implicit positive constant is independent of $W$ and $\\vec{f}$ and where $D\\vec{f}$ is the Jacobian matrix of $\\vec{f}$. Surprisingly, in both bounds above the exponents of $[W]_{\\mathscr A_{p,q}}$ coincide with the corresponding known optimal scalar exponents, and hence are optimal, which is different from the Calderón--Zygmund operator case. A key idea to obtain the above matrix-weighted boundedness of the fractional integrals is to establish the off-diagonal chain-packing principle of dyadic cubes, which is inspired by a recent work of A. K. Lerner.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.34403",
  "author_total": 3,
  "authors": [
    {
      "name": "Dachun Yang"
    },
    {
      "name": "Wen Yuan"
    },
    {
      "name": "Mingdong Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b4024qxv5rthq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.34403",
  "fulltext_url": null,
  "id": "b4024qxv5rthq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.34403",
  "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"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7214769280",
  "pdf_url": "https://arxiv.org/pdf/2609.34403",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b4024qxv5rthq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2609.34403",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.34403"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp Quantitative Matrix-Weighted Bounds for Fractional Integrals and Sobolev Inequalities",
  "title_text": "Sharp Quantitative Matrix-Weighted Bounds for Fractional Integrals and Sobolev Inequalities",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/b4024qxv5rthq",
  "venue": "arXiv",
  "versions": []
}