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

Abstract

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.