{
  "abstract": "We construct quaternionic Möbius ($\\mathcal{QM}$ briefly) transformations on the quaternionic unit ball, which are used to define $\\mathcal{QM}$-invariant Laplacian operator $\\triangle$. A function annihilated by $\\triangle$ is called $\\mathcal{QM}$-harmonic. We prove that $\\mathcal{QM}$-harmonic functions can be expanded in terms of quaternionic spherical harmonics multiplied by hypergeometric functions as radial parts. By establishing a Green formula associated to $\\triangle$ and constructing the $\\mathcal{QM}$-Poisson kernel, we solve the Dirichlet problem for $\\mathcal{QM}$-invariant Laplace equation, which is degenerate elliptic. We also give a Fatou type theorem about non-tangential convergence of $\\mathcal{QM}$-Poisson integrals. Compared to the real and complex cases, the main difficulties come from the noncommutativity of the quaternionic algebra and the complexity of the quaternionic unitary group ${\\rm Sp}(n){\\rm Sp}(1)$ and its modules. However, they can be overcome by using the embedding of the quaternionic space to the complex matrix space and using more complicated algebraic tools.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.02153",
  "author_total": 1,
  "authors": [
    {
      "name": "Ruiwen Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8sqjdh6py1q2d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.02153",
  "fulltext_url": null,
  "id": "8sqjdh6py1q2d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.02153",
  "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.CV"
    ],
    [
      "openalex",
      "2604|Algebraic and Geometric Analysis"
    ]
  ],
  "openalex": "W7172435342",
  "pdf_url": "https://arxiv.org/pdf/2608.02153",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8sqjdh6py1q2d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:57Z",
      "native_id": "2608.02153",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.02153"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Quaternionic Möbius invariant Laplacian and quaternionic Möbius harmonic functions on the unit ball",
  "title_text": "Quaternionic Möbius invariant Laplacian and quaternionic Möbius harmonic functions on the unit ball",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:57Z",
  "url": "https://research.draconitic.com/paper/8sqjdh6py1q2d",
  "venue": "arXiv",
  "versions": []
}