{
  "abstract": "Let $Ω\\subset\\mathbb{R}^n$ be a bounded Lipschitz domain and let $p:Ω\\to[1,\\infty)$ be a measurable, essentially bounded exponent; no regularity of $p$ -- in particular, no log-Hölder continuity -- is assumed. For a general non-negative measurable kernel $\\mathcal{K}(x,y)$ admitting a radial bounding function $κ(x,r)$ in the second variable, we establish the fractional modular Poincaré--Wirtinger-type inequality $$ \\iint_{Ω\\timesΩ}\\mathcal{K}(x,y)\\,|u(x)-u(y)|^{p(x)} d y\\, d x \\leqslant C\\int_Ωc(x)\\left(\\int_Ω \\frac{|\\nabla u(z)|^{p(x)}}{|z-x|^{n-1+θ}}\\, d z\\right) d x, $$ valid for every $u\\in W^{1,1}(Ω)$, in particular every $u\\in W^{1,p(\\cdot)}(Ω)$, and every $θ\\in[0,1)$. Here $$ c(x)=\\int_0^{D}r^{n+p(x)-2+θ}κ(x,r)\\,d r, $$ $D$ is the diameter of any fixed ball containing $\\overlineΩ$, and the constant $C$ depends only on $n,p^+,θ$ and the fixed domain $Ω$. For the fractional kernel $\\mathcal{K}(x,y)=|x-y|^{-n-λp(x)}$ we obtain a variable-exponent Gagliardo-modular estimate. The proof replaces the starshaped chain decompositions used there by a combination of a Morrey-type estimate on starshaped domains, a positive-kernel proof of weighted Poincaré inequalities, and one reflection extension for the weights $|z-x|^{-(n-1+θ)}$. The constants are uniform down to exponent $1$, and the inequalities are understood in the extended nonnegative sense when either side is infinite.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.36293",
  "author_total": 3,
  "authors": [
    {
      "name": "Giovanni Di Fratta"
    },
    {
      "name": "Alberto Fiorenza"
    },
    {
      "name": "Leon Happ"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f4n32twc6n0r8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36293",
  "fulltext_url": null,
  "id": "f4n32twc6n0r8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.36293",
  "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": "W7224091701",
  "pdf_url": "https://arxiv.org/pdf/2610.36293",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f4n32twc6n0r8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.36293",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36293"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A fractional modular Poincaré--Wirtinger-type inequality in Sobolev spaces with variable exponents",
  "title_text": "A fractional modular Poincaré--Wirtinger-type inequality in Sobolev spaces with variable exponents",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/f4n32twc6n0r8",
  "venue": "arXiv",
  "versions": []
}