{
  "abstract": "For the Baouendi-Grushin operator $\\mathscr L_γ= Δ_x + |x|^{2γ}Δ_y$ in $\\R^{m+k}$, with $γ>0$, we prove that the time integral of the heat kernel with pole at the origin equals $C_{γ,Q}\\, N_γ^{2-Q}$, where $N_γ$ is the gauge and $Q$ the homogeneous dimension. When $γ\\not=1$ no closed form of the heat kernel is known. After a partial Fourier transform in $y$, the time integral of the heat kernel becomes the Green function of an anharmonic oscillator, which solves a generalized modified Bessel equation, and the gauge appears when the inverse Fourier transform is computed as a Hankel transform. In particular, the heat flow provides a new proof that $C_{γ,Q}\\, N_γ^{2-Q}$ is a fundamental solution of $-\\mathscr L_γ$ with pole at the origin.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01336",
  "author_total": 1,
  "authors": [
    {
      "name": "Nicola Garofalo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7z0bp7sbckhrz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01336",
  "fulltext_url": null,
  "id": "7z0bp7sbckhrz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01336",
  "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"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.01336",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7z0bp7sbckhrz.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.01336",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01336"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Heat flow and the fundamental solution of the Baouendi-Grushin operator",
  "title_text": "Heat flow and the fundamental solution of the Baouendi-Grushin operator",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/7z0bp7sbckhrz",
  "venue": "arXiv",
  "versions": []
}