Quaternionic M\"{o}bius invariant Laplacian and quaternionic M\"{o}bius harmonic functions on the unit ball

Abstract

We construct quaternionic M\"{o}bius ($\mathcal{QM}$ 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.