{
  "abstract": "Let $p\\in[1,\\infty)$, and let $(X,ρ,μ)$ be a complete doubling $q$-Poincaré space for some $q\\in[1,p]$ with a lower dimension $d$ and an upper dimension $D$. In this article, we establish Poincaré inequalities for two families of nonlocal nonconvex functionals defined using distances or volumes. More precisely, for any ball $B$ with its radius $r_B\\in(0,\\operatorname{diam} X)$, any $λ\\in(0,\\infty)$, and any measurable function $f:X\\to\\mathbb R$, \\begin{align*} &\\frac{1}{μ(B)^2}\\int_B\\int_B|f(x)-f(y)|^p\\,dμ(x)\\,dμ(y)\\\\ &\\quad\\lesssimλ^pr_B^pϕ(B)^{γp}+\\frac{λ^pr_B^p}{μ(B)}\\mathop{\\int_{ΛB}\\int_{ΛB}}\\limits_{\\genfrac{}{}{}{2}{|f(x)-f(y)|}{ρ(x,y)ϕ(x,y)^γ}>λ}\\frac{ϕ(x,y)^{γp}}{V(x,y)}\\,dμ(x)\\,dμ(y), \\end{align*} where $V(x,y):=μ(B(x,ρ(x,y)))$ and either $ϕ=ρ$ with $ϕ(B)=r_B$ and $γ\\in(-\\infty,-\\frac{\\min\\{q,D\\}}{p}]$ or $ϕ=V$ with $ϕ(B)=μ(B)$ and $γ\\in(-\\infty,-\\frac{q}{p\\max\\{q,d\\}}]$. Here both ranges of $γ$ are sharp over the class of $q$-Poincaré spaces with the prescribed lower and upper dimensions, and the constant $Λ\\in[1,\\infty)$ and the implicit positive constant depend only on $p$, $γ$, and the geometric constants of $X$. We also obtain lower bounds for Sobolev energy and total variation in terms of the limit inferior of these functionals, for $γ$ in either the above negative ranges or $(0,\\infty)$. Our proofs use the curve characterization of Poincaré spaces and an adaptive dyadic refinement argument based on medians.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.40743",
  "author_total": 5,
  "authors": [
    {
      "name": "Feng Dai"
    },
    {
      "name": "Tuomas P. Hytönen"
    },
    {
      "name": "Dachun Yang"
    },
    {
      "name": "Wen Yuan"
    },
    {
      "name": "Yirui Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/024qh7vd2z6xz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.40743",
  "fulltext_url": null,
  "id": "024qh7vd2z6xz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.40743",
  "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.CA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.40743",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/024qh7vd2z6xz.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.40743",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.40743"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp Nonlocal Poincaré Inequalities and Liminf Lower Bounds for Sobolev and BV Energies on Metric Measure Spaces",
  "title_text": "Sharp Nonlocal Poincaré Inequalities and Liminf Lower Bounds for Sobolev and BV Energies on Metric Measure Spaces",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/024qh7vd2z6xz",
  "venue": "arXiv",
  "versions": []
}