四元数单位球上的四元数默比乌斯不变拉普拉斯算子与四元数默比乌斯调和函数
Quaternionic Möbius invariant Laplacian and quaternionic Möbius harmonic functions on the unit ball
- School of Mathematical Sciences, Xiamen University（厦门大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究在四元数单位球上构造QM变换与QM不变拉普拉斯算子，求解其狄利克雷问题并证明法图型定理，克服四元数代数非交换性等困难，拓展了调和分析在四元数域的相关理论。
AI中文摘要：
我们在四元数单位球上构造了四元数默比乌斯（简称$\boldsymbol{\text{QM}}$）变换，将其用于定义$\boldsymbol{\text{QM}}$不变拉普拉斯算子$\triangle$。被$\triangle$零化的函数称为$\boldsymbol{\text{QM}}$调和函数。我们证明$\boldsymbol{\text{QM}}$调和函数可展开为四元数球调和函数乘以超几何函数作为径向部分的形式。通过建立与$\triangle$相关的格林公式并构造$\boldsymbol{\text{QM}}$泊松核，我们求解了退化椭圆型的$\boldsymbol{\text{QM}}$不变拉普拉斯方程的狄利克雷问题。我们还给出了关于$\boldsymbol{\text{QM}}$泊松积分非切向收敛的法图型定理。与实、复情形相比，主要困难来自四元数代数的非交换性以及四元数酉群$\boldsymbol{\text{Sp}(n)\text{Sp}(1)}$及其模的复杂性，但可通过将四元数空间嵌入复矩阵空间并使用更复杂的代数工具克服这些困难。
英文摘要：
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.
