{
  "abstract": "Hermitian Clifford analysis is a refinement of Euclidean Clifford analysis obtained by imposing a complex structure on $\\R^{2n}$ and considering the simultaneous null solutions (the $h$-monogenic functions) of two mutually adjoint Hermitian Dirac operators. This survey collects, in a unified notation and without proofs, the main results obtained along this line of research between 2008 and 2026. We review the matrix Borel--Pompeiu, Koppelman and Cauchy formulae, first for domains with Liapunov or Ahlfors--David regular boundary and then for Jordan domains whose boundary is fractal in the sense of Mandelbrot; the associated Cauchy transform, Hilbert transform, Hardy projections and their mapping properties, including Hölder norm estimates in terms of the $d$-sum of a Whitney decomposition; the jump decomposition, Dirichlet and Riemann boundary value problems in each setting, including the linear Riemann problems with variable circulant coefficients and their nonlinear counterparts. We also review a duality theory for the module of germs of $H$-monogenic matrix functions, together with its Euclidean model (the annihilators of harmonic differential forms) and its algebraic prerequisites (a Poincaré lemma and a Mittag--Leffler theorem for circulant matrices); the function theory associated with a Hermitian version of the Helmholtz equation, the corresponding $\\bar\\partial$-problem and its application to a formal two-dimensional Maxwell-type system; and finally, the parallel development in the quaternionic Hermitian setting, where four Dirac operators and circulant $(4\\times 4)$ matrices take over.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39087",
  "author_total": 3,
  "authors": [
    {
      "name": "Ricardo Abreu-Blaya"
    },
    {
      "name": "Juan Bory-Reyes"
    },
    {
      "name": "Carlos Daniel Tamayo-Castro"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6xgcaxkbhhbbf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39087",
  "fulltext_url": null,
  "id": "6xgcaxkbhhbbf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39087",
  "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.CV"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.39087",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6xgcaxkbhhbbf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.39087",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39087"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Hermitian Clifford Analysis on Euclidean Domains with Smooth and Fractal Boundaries: A Survey of Matrix Cauchy and Hilbert Transforms, Boundary Value Problems, Helmholtz-Type Dirac Operators and Duality",
  "title_text": "Hermitian Clifford Analysis on Euclidean Domains with Smooth and Fractal Boundaries: A Survey of Matrix Cauchy and Hilbert Transforms, Boundary Value Problems, Helmholtz-Type Dirac Operators and Duality",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/6xgcaxkbhhbbf",
  "venue": "arXiv",
  "versions": []
}