{
  "abstract": "We consider nonlocal divergence-form operators of the type $$PV\\int_{\\mathbb{R}^n}(u(x)-u(y))K(x, y) dy=0\\quad \\text{in} \\; B_2,$$ where the kernel $K$ is symmetric ($K(x, y) = K(y, x)$). We first show that there exists a kernel satisfying the conditions of \\cite[Open Question 2.1]{FRRO24} for which this problem admits a bounded discontinuous weak solution, answering that question negatively. We then impose the stronger one-sided lower bound suggested in \\cite{CS20,IS20}, together with an $L^p$ bound of $K$ over annuli, and prove a coercivity estimate and interior Hölder regularity of weak solutions. In particular, this establishes the coercivity conjecture of \\cite{CS20,IS20} under the additional $L^p$ hypothesis for $p>1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.06408",
  "author_total": 1,
  "authors": [
    {
      "name": "Anup Biswas"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e6vhkf3ygwjh0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.06408",
  "fulltext_url": null,
  "id": "e6vhkf3ygwjh0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.06408",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7220691120",
  "pdf_url": "https://arxiv.org/pdf/2610.06408",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e6vhkf3ygwjh0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.06408",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.06408"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Coercivity and regularity for nonlocal divergence-form operators with measurable kernels",
  "title_text": "Coercivity and regularity for nonlocal divergence-form operators with measurable kernels",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/e6vhkf3ygwjh0",
  "venue": "arXiv",
  "versions": []
}