A fundamental solution for the Laplace operator on the quaternionic Heisenberg group
Let \({\mathbb H}\) denote the quaternions, that is \({\mathbb R}\times {\mathbb R}^{3}\) with an associative noncommutative multiplication as follows: \[ (\lambda, {\mathbf u})(\mu, {\mathbf v})=(\lambda \mu - {\mathbf uv}, \lambda{\mathbf v} + \mu{\mathbf u} + {\mathbf u}\wedge {\mathbf v}). \] If \({\mathbf q}=(\lambda, {\mathbf u}) \in {\mathbf H}\), we denote \({\mathbf u}=\text{Im}{\mathbf q}\) and \(\overline{{\mathbf q}} =(\lambda, -{\mathbf u})\). We denote the quaternionic Heisenberg group of dimension \(n\) by \({\mathbb H}{\mathbb H}^{n}\). It is a topological group whose underlying manifold is \({\mathbf H}^{n} \times {\mathbb R}^{3}\) with coordinates \(({\mathbf u}, {\mathbf v})=({\mathbf u}_{1}; \cdots ; {\mathbf u}_{n}; v_{1}, v_{2}, v_{3})\) where \({\mathbf u}_{p}=(u_{p0}, u_{p1}, u_{p2}, u_{p3}) \in {\mathbf H}\). The group law for \({\mathbf H}{\mathbb H}^{n}\) is defined as: \[ ({\mathbf u}, {\mathbf v})({\mathbf w}, {\mathbf t})=({\mathbf u}+{\mathbf w}, {\mathbf v}+{\mathbf t} + 2\sum_{p=1}^{n}\text{Im}({\mathbf w}_{p}\overline{{\mathbf u}}_{p})). \] The aim of this paper is to generalize the results of \textit{G. B. Folland} [Bull. Am. Math. Soc. 79, 373-376 (1973; Zbl 0256.35020)] to the quaternionic Heisenberg group. The author constructs a fundamental solution for it and uses this solution to prove the \(L^{p}\)-boundedness and the weak \((1, 1)\)-boundedness of certain singular convolution operators on the quaternionic Heisenberg group.
scientific article; zbMATH DE number 1824207