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

Abstract

We construct quaternionic Möbius ($\mathcal{Q}\mathcal{M}$ briefly) transformations on the quaternionic unit ball, which are used to define $\mathcal{Q}\mathcal{M}$-invariant Laplacian operator $△$. A function annihilated by $△$ is called $\mathcal{Q}\mathcal{M}$-harmonic. We prove that $\mathcal{Q}\mathcal{M}$-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 $\mathcal{Q}\mathcal{M}$-Poisson kernel, we solve the Dirichlet problem for $\mathcal{Q}\mathcal{M}$-invariant Laplace equation, which is degenerate elliptic. We also give a Fatou type theorem about non-tangential convergence of $\mathcal{Q}\mathcal{M}$-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 $\mathrm{Sp}(n)\mathrm{Sp}(1)$ 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.