{
  "abstract": "We establish the Kato square root property for fractional elliptic operators $L = -\\operatorname{div}^α(μ\\nabla^α\\cdot)$ of order $2α\\in (0,2)$ on $\\mathbb{R}^d$, built from the Riesz fractional gradient and divergence, with bounded, measurable and uniformly elliptic complex coefficients $μ$. More precisely, we show that the domain of $L^{1/2}$ coincides with the distributional fractional Sobolev space $\\mathrm{S}^{α,2}(\\mathbb{R}^d)$ and that $\\|L^{1/2} u\\|_{\\mathrm{L}^2} \\simeq \\|\\nabla^αu\\|_{\\mathrm{L}^2}$, with constants that are locally uniform in $α$. As a consequence, we recover the Kato square root property for elliptic operators $-\\operatorname{div}(μ\\nabla\\cdot)$ in the limit $α\\to 1$, and we obtain continuity of the square root with respect to the parameter $α$. The proof follows the second-order approach to the Kato problem. Its two new ingredients are nonlocal off-diagonal estimates of polynomial order for the resolvents of $L$, and a construction of $T(b)$-test functions that circumvents the nonlocality of $\\nabla^α$ by factorizing it through Riesz potentials.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.06391",
  "author_total": 1,
  "authors": [
    {
      "name": "Luca Haardt"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5aej1t3tygj6p.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.06391",
  "fulltext_url": null,
  "id": "5aej1t3tygj6p",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.06391",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.FA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.06391",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5aej1t3tygj6p.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.06391",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.06391"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The Kato square root property for fractional elliptic operators in divergence form",
  "title_text": "The Kato square root property for fractional elliptic operators in divergence form",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:30Z",
  "url": "https://research.draconitic.com/paper/5aej1t3tygj6p",
  "venue": "arXiv",
  "versions": []
}