The fundamental solution of a class of ultra-hyperbolic operators on Pseudo $H$-type groups

Pseudo $H$-type Lie groups $G_{r,s}$ of signature $(r,s)$ are defined via a module action of the Clifford algebra $C\ell_{r,s}$ on a vector space $V \cong \mathbb{R}^{2n}$. They form a subclass of all 2-step nilpotent Lie groups and based on their algebraic structure they can be equipped with a left-invariant pseudo-Riemannian metric. Let $\mathcal{N}_{r,s}$ denote the Lie algebra corresponding to $G_{r,s}$. A choice of left-invariant vector fields $[X_1, \ldots, X_{2n}]$ which generate a complement of the center of $\mathcal{N}_{r,s}$ gives rise to a second order operator \begin{equation*} \Delta_{r,s}:= \big{(}X_1^2+ \ldots + X_n^2\big{)}- \big{(}X_{n+1}^2+ \ldots + X_{2n}^2 \big{)}, \end{equation*} which we call ultra-hyperbolic. In terms of classical special functions we present families of fundamental solutions of $\Delta_{r,s}$ in the case $r=0$, $s>0$ and study their properties. In the case of $r>0$ we prove that $\Delta_{r,s}$ admits no fundamental solution in the space of tempered distributions. Finally we discuss the local solvability of $\Delta_{r,s}$ and the existence of a fundamental solution in the space of Schwartz distributions.