{
  "abstract": "Let $0<s<1$ and $A$ be a constant symmetric positive definite matrix. Given the quadratic form $$ Q_A(u)=\\int_{\\R^N} A\\nabla^s u(x)\\cdot \\nabla^s u(x)\\,dx, $$ where $\\nabla^s$ denotes the Riesz fractional gradient, we establish necessary and sufficient conditions such that $Q_A$ defines a Dirichlet form. In strike contrast with the local case $s=1$, positive definiteness of a constant matrix is not sufficient for the Markov property when $0<s<1$. We also describe the associated symmetric Markov semigroup and its $L^p$ realizations, prove the existence of a smooth nonnegative heat kernel and establish a bounded $H^\\infty$ functional calculus.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15809",
  "author_total": 1,
  "authors": [
    {
      "name": "Alessandro Carbotti"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ebjbwfdzjr3vp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15809",
  "fulltext_url": null,
  "id": "ebjbwfdzjr3vp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15809",
  "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": "W7213254693",
  "pdf_url": "https://arxiv.org/pdf/2609.15809",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ebjbwfdzjr3vp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.15809",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15809"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The semigroup generated by linear fractional divergence form operators in $L^p(\\mathbb{R}^N)$: the constant coefficient case",
  "title_text": "The semigroup generated by linear fractional divergence form operators in L^p(ℝ^N): the constant coefficient case",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/ebjbwfdzjr3vp",
  "venue": "arXiv",
  "versions": []
}