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PreprintCC BY 4.0arXiv

Quaternionic Möbius invariant Laplacian and quaternionic Möbius harmonic functions on the unit ball

  • Ruiwen Wang
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Abstract

We construct quaternionic Möbius (𝒬ℳ briefly) transformations on the quaternionic unit ball, which are used to define 𝒬ℳ-invariant Laplacian operator △. A function annihilated by △ is called 𝒬ℳ-harmonic. We prove that 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 △ and constructing the 𝒬ℳ-Poisson kernel, we solve the Dirichlet problem for 𝒬ℳ-invariant Laplace equation, which is degenerate elliptic. We also give a Fatou type theorem about non-tangential convergence of 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 Sp(n)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.

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Cite

BibTeXRIS
BibTeX
@misc{wang2026quaternionic,
  title = {{Quaternionic Möbius invariant Laplacian and quaternionic Möbius harmonic functions on the unit ball}},
  author = {Ruiwen Wang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.02153},
  eprint = {2608.02153},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.02153},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Quaternionic Möbius invariant Laplacian and quaternionic Möbius harmonic functions on the unit ball
AU  - Ruiwen Wang
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2608.02153
UR  - https://arxiv.org/abs/2608.02153
AB  - We construct quaternionic Möbius (𝒬ℳ briefly) transformations on the quaternionic unit ball, which are used to define 𝒬ℳ-invariant Laplacian operator △. A function annihilated by △ is called 𝒬ℳ-harmonic. We prove that 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 △ and constructing the 𝒬ℳ-Poisson kernel, we solve the Dirichlet problem for 𝒬ℳ-invariant Laplace equation, which is degenerate elliptic. We also give a Fatou type theorem about non-tangential convergence of 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 Sp(n)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.
ER  -