{
  "abstract": "Let $n\\geq2$, $s\\in(0,1)$, and let $Ω\\subset\\mathbb{R}^n$ be a bounded Lipschitz domain. In this paper, we establish optimal global Sobolev estimates for the fractional Dirichlet problem \\begin{equation*} \\left\\{\\begin{aligned} (-Δ)^su&=f & & \\text{in}\\ \\ Ω, u&=0 & & \\text{in}\\ \\ \\mathbb{R}^n\\setminusΩ. \\end{aligned}\\right. \\end{equation*} More precisely, for any given $t\\in[s,\\min\\{2s,\\,s+1/2\\})$, we prove that there exists a small positive constant $η=η(n,s,t,Ω)$ such that the following holds. If $n=2$ and $s\\in(1/2,1)$, then, for any $q\\in(1,\\frac{4}{1+2s}+η)$ and $p\\in(1,\\frac{nq}{n-(2s-t)q}]$, \\begin{equation*} \\|u\\|_{W^{t,p}(\\mathbb{R}^n)}\\le C\\|f\\|_{L^q(Ω)}, \\end{equation*} and, if either $n=2$ and $s\\in(0,1/2]$ or $n\\ge 3$ and $s\\in(0,1)$, then, for any $q\\in(1,\\frac{n(2s+1)}{nt+(2s-t)(2s+1)}+η)$ and $p\\in(1,\\frac{nq}{n-(2s-t)q}]$, \\begin{equation*} \\|u\\|_{W^{t,p}(\\mathbb{R}^n)}\\le C\\|f\\|_{L^q(Ω)}. \\end{equation*} Here the positive constant $C$ depends only on $n$, $s$, $t$, $p$, $q$, and $Ω$. The stated universal baselines are optimal in Dahlberg's sense: for any larger target exponent $p$ there is a bounded Lipschitz domain whose solution with right-hand side identically $1$ is not in $W^{t,p}(\\mathbb{R}^n)$. The counterexamples are constructed using regular Cantor sets on the boundary and, in the planar exceptional case, homogeneous solutions in sectors.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.05753",
  "author_total": 2,
  "authors": [
    {
      "name": "Wenxian Ma"
    },
    {
      "name": "Sibei Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0xfg07h84rptp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.05753",
  "fulltext_url": null,
  "id": "0xfg07h84rptp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.05753",
  "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"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7220694318",
  "pdf_url": "https://arxiv.org/pdf/2610.05753",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0xfg07h84rptp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.05753",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.05753"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Optimal Global Sobolev Estimates for Fractional Dirichlet Problems in Bounded Lipschitz Domains",
  "title_text": "Optimal Global Sobolev Estimates for Fractional Dirichlet Problems in Bounded Lipschitz Domains",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/0xfg07h84rptp",
  "venue": "arXiv",
  "versions": []
}