{
  "abstract": "Let $N\\in\\mathbb N$ and $p\\in[1,\\infty)$. In this article, we consider a family $\\{Φ_{λ,p}^γ\\}_{λ\\in(0,\\infty)}$ of nonlocal nonconvex functionals with a real parameter $γ$ and establish two truncated difference estimates bounded by these functionals with the sharp range of $γ$; that is, for any $γ\\in(-\\infty,-1]$, any $λ\\in(0,\\infty)$, any cube $Q\\subset\\mathbb R^N$, and any measurable function $g:Q\\to\\mathbb R$, \\begin{align*} &\\int_Q\\int_Q\\left[\\frac{|g(x)-g(y)|-λ|x-y|^{1+γ/p}}{\\ell(Q)^{1+γ/p}}\\right]_+^p\\,dx\\,dy\\\\ &\\quad\\lesssim λ^p\\mathop{\\int_Q\\int_Q}\\limits_{\\frac{|g(x)-g(y)|}{|x-y|^{1+γ/p}}>λ}\\left[\\frac{|x-y|}{\\ell(Q)}\\right]^{γ-N}\\,dx\\,dy=:\\frac{Φ_{λ,p}^γ(g;Q)}{\\ell(Q)^{γ-N}}, \\end{align*} where the implicit positive constant depends only on $N$, $p$, and $γ$. For $γ\\in(0,\\infty)$, the above estimate holds with the left-most integrand replaced by $\\frac{\\min\\{|g(x)-g(y)|^p,λ^p|x-y|^{p+γ}\\}}{\\ell(Q)^{p+γ}}$. As applications, these estimates yield related nonlocal Poincaré inequalities with the sharp range of $γ$, and also recover the known $Γ$-liminf lower bounds and characterizations of Sobolev and BV spaces in terms of the liminf of the functionals $\\{Φ^γ_{λ,p}\\}_{λ\\in(0,\\infty)}$ for all $γ\\in\\mathbb{R}\\setminus(-1,0]$. We investigate further applications to compactness and variational problems as well as a new VMO criterion. Inspired by a recent work of R. L. Frank, we develop an elementary approach based on dyadic iteration of truncated differences along line segments, which allows us to treat the aforementioned problems in a unified way.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.04259",
  "author_total": 4,
  "authors": [
    {
      "name": "Feng Dai"
    },
    {
      "name": "Dachun Yang"
    },
    {
      "name": "Wen Yuan"
    },
    {
      "name": "Yirui Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0s0vds2fcyzfy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.04259",
  "fulltext_url": null,
  "id": "0s0vds2fcyzfy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.04259",
  "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"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7220695150",
  "pdf_url": "https://arxiv.org/pdf/2610.04259",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0s0vds2fcyzfy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.04259",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.04259"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp Truncated Difference Estimates for Nonlocal Nonconvex Functionals Defined by Superlevel Sets",
  "title_text": "Sharp Truncated Difference Estimates for Nonlocal Nonconvex Functionals Defined by Superlevel Sets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/0s0vds2fcyzfy",
  "venue": "arXiv",
  "versions": []
}