{
  "abstract": "Let $Q_0:=(0,1)^n$. We define $\\mathcal{B}_q(Q_0)$ by replacing the local $L^1$ mean oscillation in the Bourgain--Brezis--Mironescu space $\\mathcal{B}(Q_0)$ with its $L^q$ counterpart, which coincides with $\\mathrm{BMO}(Q_0)$ when $n=1$. For $n\\geq2$, our results reveal different endpoint behaviors for general functions, characteristic functions and the exact perimeter formula. (i) For general functions, $\\mathcal{B}_q(Q_0)=\\mathcal{B}(Q_0)$ precisely when $q\\in(0,\\frac{n}{n-1})$. We also establish the corresponding sharp self-improvement and duality results for Hardy-type preduals. (ii) For characteristic functions, the endpoint is attained: the equivalence $\\|\\mathbf{1}_E\\|_{\\mathcal{B}_q(Q_0)}\\sim\\|\\mathbf{1}_E\\|_{\\mathcal{B}(Q_0)}\\sim\\min\\{1,P(E,Q_0)\\}$ holds uniformly over measurable $E\\subset Q_0$ precisely when $q\\in(0,\\frac{n}{n-1}]$, where $P(E,Q_0)$ denotes relative perimeter. (iii) The isotropic functional $I_{\\varepsilon,q}^{*}(E)$, defined using arbitrarily oriented cubes, converges to $\\frac{1}{2}\\min\\{1,P(E,Q_0)\\}$ as $\\varepsilon\\to0^+$ throughout $q\\in(0,\\frac{n}{n-1}]$. In dimension two, the formula extends to $q\\in(0,q_*]$ for some universal $q_*>\\frac{19}{9}$, whereas it fails in every dimension for every $q>q_0$, where $q_0>2$ is defined variationally and numerical computation suggests that $q_0\\approx2.15391$. The proofs of the three sharp self-improvements use sparse median domination, atomic block decompositions, and relative isoperimetry, respectively. For the perimeter formula, a uniform De Giorgi blow-up reduces the problem to a geometric cube-section inequality; geometric averaging and an explicit packing yield the planar extension and the obstruction, respectively.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.04849",
  "author_total": 2,
  "authors": [
    {
      "name": "Hongchao Jia"
    },
    {
      "name": "Xianjie Yan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/aws3vr702qpc8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.04849",
  "fulltext_url": null,
  "id": "aws3vr702qpc8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.04849",
  "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": "W7220684978",
  "pdf_url": "https://arxiv.org/pdf/2610.04849",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/aws3vr702qpc8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.04849",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.04849"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "From Sharp Self-Improvement to Exact Perimeter Formulas in Bourgain--Brezis--Mironescu Spaces",
  "title_text": "From Sharp Self-Improvement to Exact Perimeter Formulas in Bourgain--Brezis--Mironescu Spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/aws3vr702qpc8",
  "venue": "arXiv",
  "versions": []
}