{
  "abstract": "For positive powers of the discrete Dirichlet Laplacian $(-Δ_{\\mathrm{D}})^σ$, non-trivial Hardy weights exist if and only if $0<σ<3/2$. For $0<σ\\leq 1$, an explicit optimal Hardy weight for $(-Δ_{\\mathrm{D}})^σ$ was obtained by Das and de la Fuente-Fernández. Their ground state argument does not apply when $1<σ<3/2$, because the off-diagonal matrix entries of $(-Δ_{\\mathrm{D}})^σ$ are no longer all non-positive. Nevertheless, we show that their result is true for the entire subcriticality range $0<σ<3/2$. For $1<σ<3/2$, our argument combines the factorisation $$(-Δ_{\\mathrm{D}})^σ=\\nabla_{\\mathrm{D}}^{*}(-Δ_{\\mathrm{N}})^{σ-1}\\nabla_{\\mathrm{D}},$$ where $\\nabla_{\\mathrm{D}}$ and $-Δ_{\\mathrm{N}}$ are the discrete Dirichlet gradient and the discrete Neumann Laplacian, respectively, with a suitable two-term ground state representation. In addition, we use these arguments to derive another explicit family of optimal Hardy weights for any $σ\\in(0,3/2)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.31171",
  "author_total": 2,
  "authors": [
    {
      "name": "František Štampach"
    },
    {
      "name": "Jakub Waclawek"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6n7g3sa0pks6n.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.31171",
  "fulltext_url": null,
  "id": "6n7g3sa0pks6n",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.31171",
  "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.SP"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7223399551",
  "pdf_url": "https://arxiv.org/pdf/2610.31171",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6n7g3sa0pks6n.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.31171",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.31171"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Optimal Hardy weights for the fractional discrete Dirichlet Laplacian on the half-line",
  "title_text": "Optimal Hardy weights for the fractional discrete Dirichlet Laplacian on the half-line",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/6n7g3sa0pks6n",
  "venue": "arXiv",
  "versions": []
}