{
  "abstract": "The spaces $\\mathcal Q_α(\\mathbb R^n)$ for the critical index range $α\\in(0,\\min\\{1,n/2\\})$ form a scale-invariant family lying strictly between the space of constant functions and $\\mathrm{BMO}(\\mathbb R^n)$. A long-standing open problem, posed by Essén, Janson, Peng and Xiao in 2000, asks to characterize the quasisymmetric mappings $φ:\\ \\mathbb R^n\\to\\mathbb R^n$ for which the composition operator $\\mathcal C_φ(f)=f\\circφ^{-1}$ is bounded on $\\mathcal Q_α(\\mathbb R^n)$. This paper establishes the full intrinsic characterization by introducing a new geometric quantity, the capacity gauge $\\|φ\\|_{\\mathscr G_β}$, which measures the distortion of $φ$ through pullback volume ratios along the dyadic tree. We prove that for any quasisymmetric mapping $φ$ (with a mild Muckenhoupt \\(A_\\infty(\\mathbb R)\\) assumption on the Jacobian determinants of \\(φ\\) and \\(φ^{-1}\\) when \\(n=1\\)), the composition operator $\\mathcal C_φ$ is bounded on $\\mathcal Q_α(\\mathbb R^n)$ if and only if $\\|φ\\|_{\\mathscr G_{1-\\frac{2α}{n}}}<\\infty,$ with quantitative equivalence $$ \\|\\mathcal C_φ\\|_{\\mathcal Q_α(\\mathbb R^n)\\to\\mathcal Q_α(\\mathbb R^n)}^2 \\simeq \\|φ\\|_{\\mathscr G_{1-\\frac{2α}{n}}}. $$ This sharp, necessary and sufficient characterization refines earlier sufficient criteria of Koskela, Xiao, Zhang and Zhou in 2017, which were formulated in terms of local or global self-similar Minkowski dimension of the exceptional sets of the Jacobian. As applications, we obtain the composition stability of the finite capacity gauge, the invariance of $\\mathcal Q_α$-removability under quasisymmetric mappings with finite capacity gauge, and the propagation of $\\mathcal Q_α(\\mathbb R^n)$-regularity and initial-data stability for transport equations driven by quasisymmetric flows.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.27915",
  "author_total": 2,
  "authors": [
    {
      "name": "Liu Liguang"
    },
    {
      "name": "Xiao Jie"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/at3eqc1wer7cr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.27915",
  "fulltext_url": null,
  "id": "at3eqc1wer7cr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.27915",
  "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"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7204852499",
  "pdf_url": "https://arxiv.org/pdf/2608.27915",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/at3eqc1wer7cr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:28Z",
      "native_id": "2608.27915",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.27915"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A Sharp Capacity Gauge Solution to the Open Problem of Quasisymmetric Composition on $\\mathcal{Q}_α(\\mathbb R^n)$",
  "title_text": "A Sharp Capacity Gauge Solution to the Open Problem of Quasisymmetric Composition on 𝒬_α(ℝ^n)",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:28Z",
  "url": "https://research.draconitic.com/paper/at3eqc1wer7cr",
  "venue": "arXiv",
  "versions": []
}