{
  "abstract": "We construct a counterexample to integral Aleksandrov-Bakelman-Pucci estimates for nonlocal inequalities of the form \\begin{equation*} L_au(x):=\\int_{\\mathbb R^n}δ_yu(x) \\frac{a(x,y)}{|y|^{n+σ}}\\,dy \\ge-f(x)\\quad\\text{in }B_1, \\end{equation*} with $u\\le0$ in $\\mathbb R^n\\setminus B_1$, where $δ_yu(x)=u(x+y)+u(x-y)-2u(x)$, and the measurable coefficient satisfies \\begin{equation}\\tag{C} λ\\le a(x,y)\\leΛ,\\qquad a(x,y)=a(x,-y),\\qquad x\\in\\mathbb R^n,\\quad y\\ne0. \\end{equation} More precisely, for every $n\\ge1$, $0<σ<2$, and $0<λ<Λ$, there exist a fixed coefficient $a(x,y)$ satisfying \\textup{(C)}, a constant $d>0$, and nonnegative functions $u_j,f_j\\in C_c^\\infty(B_1)$ such that $L_au_j\\ge-f_j$ in $B_1$, $0\\le f_j\\le1$, and $|\\operatorname{supp}f_j|\\to0$ as $j\\to\\infty$, while $u_j\\ge d$ on $B_{1/4}$ for every $j$. Consequently, this fixed operator admits no $L^p$ maximum estimate for any $1\\le p<\\infty$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.16162",
  "author_total": 2,
  "authors": [
    {
      "name": "Yongsheng Jiang"
    },
    {
      "name": "Lei Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/96p8fnpymx3s4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.16162",
  "fulltext_url": null,
  "id": "96p8fnpymx3s4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.16162",
  "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": "W7221767163",
  "pdf_url": "https://arxiv.org/pdf/2610.16162",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/96p8fnpymx3s4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.16162",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.16162"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A counterexample to integral Aleksandrov-Bakelman-Pucci estimates for a nonlocal operator",
  "title_text": "A counterexample to integral Aleksandrov-Bakelman-Pucci estimates for a nonlocal operator",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/96p8fnpymx3s4",
  "venue": "arXiv",
  "versions": []
}