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.

中文翻译：

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一类超双曲算子在伪H型群上的基本解

签名 $(r,s)$ 的伪 $H$ 型李群 $G_{r,s}$ 是通过 Clifford 代数 $C\ell_{r,s}$ 在向量空间 $ 上的模作用定义的V \cong \mathbb{R}^{2n}$。它们构成了所有 2 步幂零李群的子类，并且基于它们的代数结构，它们可以配备左不变的伪黎曼度量。令 $\mathcal{N}_{r,s}$ 表示对应于 $G_{r,s}$ 的李代数。左不变向量场 $[X_1, \ldots, X_{2n}]$ 的选择产生了 $\mathcal{N}_{r,s}$ 的中心的补集产生了二阶算子 \开始{方程*} \Delta_{r,s}:= \big{(}X_1^2+ \ldots + X_n^2\big{)}- \big{(}X_{n+1}^2+ \ ldots + X_{2n}^2 \big{)}, \end{equation*} 我们称之为超双曲线。在经典的特殊函数方面，我们提出了 $\Delta_{r 的基本解系列，s}$ 在 $r=0$, $s>0$ 的情况下，研究它们的性质。在 $r>0$ 的情况下，我们证明 $\Delta_{r,s}$ 在调和分布空间中没有基本解。最后，我们讨论了 $\Delta_{r,s}$ 的局部可解性和 Schwartz 分布空间中基本解的存在性。