Schwartz kernels on the Heisenberg group

Veneruso, Alessandro

Bollettino della Unione matematica italiana, Série 8, 6B (2003) no. 3, pp. 657-666

Résumé

Let $H_{n}$ be the Heisenberg group of dimension $2n+1$. Let $\mathcal{L}_{1},\ldots, \mathcal{L}_{n}$ be the partial sub-Laplacians on $H_{n}$ and $T$ the central element of the Lie algebra of $H_{n}$. We prove that the kernel of the operator $m(\mathcal{L}_{1},\ldots, \mathcal{L}_{n},-iT)$ is in the Schwartz space $S(H_{n})$ if $m\in S(\mathbb{R}^{n+1} )$. We prove also that the kernel of the operator $h(\mathcal{L}_{1},\ldots, \mathcal{L}_{n})$ is in $S(H_{n})$ if $h\in S(\mathbb{R}^{n})$ and that the kernel of the operator $g(\mathcal{L}, -iT)$ is in $S(H_{n})$ if $g\in S(\mathbb{R}^{2})$. Here $\mathcal{L}= \mathcal{L}_{1}+ \ldots+\mathcal{L}_{n}$ is the Kohn-Laplacian on $H_{n}$.