{
  "abstract": "We prove sharp Hardy inequalities on $\\mathbb{Q}_p^n$ for the quadratic form of the Vladimirov--Taibleson operator, the $L^q$ norm of its fractional derivative, and a nonlinear difference energy. Under the self-dual Fourier normalization, the sharp quadratic-form constant for $0<α<n$ is $[(1-p^{-(n-α)/2})/(1-p^{-(n+α)/2})]^2$, with a remainder given by the ground state representation. For $1<q<\\infty$, $q'=q/(q-1)$, and $0<s<n/q$, a weighted Schur estimate determines the exact $L^q$ norm of the weighted Riesz potential $\\|\\boldsymbol{x}\\|_p^{-s}I_s$, giving the sharp linear Hardy constant $[Γ_{p}^{(n)}(n/q'+s)/Γ_{p}^{(n)}(n/q')]^q$. The two-point Picone inequality and shell decomposition give the sharp constant $2(1-p^{-n})\\sum_{m\\ge1}p^{-sqm}(1-p^{-(n/q-s)m})^q$ for the Gagliardo-type energy with kernel $\\|\\boldsymbol{x}-\\boldsymbol{y}\\|_p^{-n-sq}$. Annular truncations of the critical powers give explicit minimizing sequences with error estimates. Equality is attained only by the zero function in each stated class. For $q=2$, both $L^q$ inequalities reduce to the quadratic-form inequality. Uniform local constancy makes the difference energy finite for every Bruhat--Schwartz function and every $s>0$. The Hardy inequalities on the full test-function space therefore hold precisely when $sq<n$, without the restriction $s<1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.04463",
  "author_total": 1,
  "authors": [
    {
      "name": "Yaojia Sun"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/afxds01v6syk1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.04463",
  "fulltext_url": null,
  "id": "afxds01v6syk1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.04463",
  "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.FA"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7220734106",
  "pdf_url": "https://arxiv.org/pdf/2610.04463",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/afxds01v6syk1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.04463",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.04463"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp Hardy inequalities for the Vladimirov--Taibleson operator on $\\mathbb{Q}_p^n$",
  "title_text": "Sharp Hardy inequalities for the Vladimirov--Taibleson operator on ℚ_p^n",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/afxds01v6syk1",
  "venue": "arXiv",
  "versions": []
}