{
  "abstract": "We study the boundary regularity of solutions for the tempered fractional Laplacian \\[ (-Δ)^s_λu=f \\quad \\text{in }Ω, \\qquad u=0 \\quad \\text{in }\\mathbb{R}^n\\setminusΩ, \\] where \\(0<s<1\\), \\(λ>0\\), and \\(Ω\\subset\\mathbb{R}^n\\) is a bounded Lipschitz domain satisfying the exterior ball condition. The tempered operator is defined by \\[ (-Δ)^s_λu(x) = c_{n,s}\\operatorname{P.V.} \\int_{\\mathbb{R}^n} \\frac{u(x)-u(y)}{|x-y|^{n+2s}} e^{-λ|x-y|}\\,dy, \\] which was originally proposed in [Koponen, Phys. Rev. E (1995), 1197-1199]. % Exponential tempering preserves the leading singularity of % the kernel but destroys its homogeneity, preventing a direct % application of classical boundary regularity estimates. % Extending these estimates [Ros-Oton and Serra, J. Math. Pures Appl. (2014), 275--302] to the tempered setting therefore % requires overcoming the difficulties caused by the loss % of scale invariance. By decomposing the operator into the classical fractional Laplacian and the correction operator \\(\\mathcal Q_λ^s\\), and combining approximation, localization, and iterative arguments with interpolation inequalities, we establish the global estimate \\[ \\|u\\|_{C^s(\\mathbb{R}^n)} \\le C\\|f\\|_{L^\\infty(Ω)}. \\] For more regular data, we derive interior and boundary-weighted Schauder estimates. In particular, under suitable regularity assumptions on \\(f\\), these estimates yield \\[ |D^νu(x)| \\le C\\,δ(x)^{s-|ν|}, \\quad |ν| > 0, \\] where \\(ν\\) is a multi-index and \\(δ(x)=\\operatorname{dist}(x,\\partialΩ)\\). These bounds quantify the possible growth of higher-order derivatives near the boundary and provide a regularity foundation for numerical error analysis of the tempered problem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.16437",
  "author_total": 2,
  "authors": [
    {
      "name": "Jie Zhao"
    },
    {
      "name": "Minghua Chen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2e3ht49bbymx1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.16437",
  "fulltext_url": null,
  "id": "2e3ht49bbymx1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.16437",
  "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.FA"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7221763312",
  "pdf_url": "https://arxiv.org/pdf/2610.16437",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2e3ht49bbymx1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.16437",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.16437"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Tempered fractional Laplacian: Boundary regularity",
  "title_text": "Tempered fractional Laplacian: Boundary regularity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/2e3ht49bbymx1",
  "venue": "arXiv",
  "versions": []
}