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Pseudo $H$-type Lie groups $G_{r,s}$ of signature $(r,s)$ are defined via amodule 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 andbased on their algebraic structure they can be equipped with a left-invariantpseudo-Riemannian metric. Let $\mathcal{N}_{r,s}$ denote the Lie algebracorresponding 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 ofclassical 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 caseof $r>0$ we prove that $\Delta_{r,s}$ admits no fundamental solution in thespace of tempered distributions. Finally we discuss the local solvability of$\Delta_{r,s}$ and the existence of a fundamental solution in the space ofSchwartz distributions.

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