{
  "abstract": "We investigate the optimal Bianchi-Egnell-type quantitative stability constant for the critical nonlocal Sobolev inequality on the Heisenberg group $\\mathbb{H}^{n}$, \\begin{equation}\\label{nS} S_{HL}(Q,μ)\\left(\\int_{\\mathbb{H}^{n}}\\int_{\\mathbb{H}^{n}} \\frac{|u(ξ)|^{Q^{\\ast}_μ}|u(η)|^{Q^{\\ast}_μ}} {|η^{-1}ξ|^μ}\\,dξdη\\right)^{\\frac{1}{Q^{\\ast}_μ}} \\leq \\int_{\\mathbb{H}^{n}}|\\nabla_{\\mathbb{H}}u|^{2}dξ, \\qquad u\\in S^{1,2}(\\mathbb{H}^{n}), \\end{equation} where $Q=2n+2$, $n\\geq1$, $0<μ<Q$, and $Q^{\\ast}_μ=(2Q-μ)/(Q-2)$. Let $\\mathfrak{M}$ denote the manifold of Jerison-Lee bubbles, and let $H_{NS}$ be the infimum of the quotient between the deficit in \\eqref{nS} and $\\mathrm{dist}(u,\\mathfrak{M})^{2}$. The central issue is that the Euclidean nonlocal problem and the local Folland-Stein-Sobolev problem each possess their own spectral and compactness structures, whereas the present problem couples the HLS interaction with the noncommutative conformal geometry of $\\mathbb{H}^{n}$. Under the stated conditions on $(n,μ)$, these strict thresholds, together with a Heisenberg-group profile decomposition and a nonlocal Br{e}zis-Lieb splitting, imply that $H_{NS}$ is attained. The minimizer then yields the strict comparison $H_{NS}>H_{BE}$ with the optimal stability constant for the local Folland-Stein-Sobolev inequality. We further prove that the sharp universal upper constant for the deficit-to-distance comparison is $1$ and characterize equality. Finally, for the associated Euler-Lagrange equation, we formulate the corresponding residual quotient and derive a strict single-bubble upper bound; this critical-point statement requires a separate expansion and does not follow from attainment of $H_{NS}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.09034",
  "author_total": 2,
  "authors": [
    {
      "name": "Wenjing Chen"
    },
    {
      "name": "Zexi Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/898w7cg7vdza2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.09034",
  "fulltext_url": null,
  "id": "898w7cg7vdza2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.09034",
  "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",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7202198474",
  "pdf_url": "https://arxiv.org/pdf/2608.09034",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/898w7cg7vdza2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2608.09034",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.09034"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group",
  "title_text": "Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/898w7cg7vdza2",
  "venue": "arXiv",
  "versions": []
}