Geodesic
Schwartz kernels on the Heisenberg group
Bollettino della Unione matematica italiana, Série 8, 6B (2003) no. 3, pp. 657-666.

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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 $H_{n}$.
Sia $H_{n}$ il gruppo di Heisenberg di dimensione $2n+1$. Siano $\mathcal{L}_{1},\ldots, \mathcal{L}_{n}$ i sub-Laplaciani parziali su $H_{n}$ e $T$ l'elemento centrale dell'algebra di Lie di $H_{n}$. In questo lavoro dimostriamo che, data una funzione $m$ appartenente allo spazio di Schwartz $S(\mathbb{R}^{n+1})$, il nucleo dell'operatore $m(\mathcal{L}_{1},\ldots, \mathcal{L}_{n},-iT)$ è una funzione in $S(H_{n})$. Inoltre dimostriamo che, date altre due funzioni $h\in S(\mathbb{R}^{n} )$ e $g\in S(\mathbb{R}^{2})$, i nuclei degli operatori $h(\mathcal{L}_{1},\ldots, \mathcal{L}_{n})$ e $g(\mathcal{L}, -iT)$ stanno in $S(H_{n})$. Qui $\mathcal{L}= \mathcal{L}_{1}+ \ldots+\mathcal{L}_{n}$ è il sub-Laplaciano su $H_{n}$. 
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     author = {Veneruso, Alessandro},
     title = {Schwartz kernels on the {Heisenberg} group},
     journal = {Bollettino della Unione matematica italiana},
     pages = {657--666},
     publisher = {mathdoc},
     volume = {Ser. 8, 6B},
     number = {3},
     year = {2003},
     zbl = {1178.43007},
     mrnumber = {1612717},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BUMI_2003_8_6B_3_a10/}
}
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Veneruso, Alessandro. Schwartz kernels on the Heisenberg group. Bollettino della Unione matematica italiana, Série 8, 6B (2003) no. 3, pp. 657-666. http://geodesic.mathdoc.fr/item/BUMI_2003_8_6B_3_a10/

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