Schwartz kernels on the Heisenberg group

Veneruso, Alessandro

Bollettino dell'Unione Matematica Italiana, Tome 6-A (2003), p. 657-666 / Harvested from Biblioteca Digitale Italiana di Matematica

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Abstract
Résumé

Let $H_{n}$ be the Heisenberg group of dimension $2 n + 1$ . Let $L_{1}, \dots, L_{n}$ 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 $m (L_{1}, \dots, L_{n}, - i T)$ is in the Schwartz space $S (H_{n})$ if $m \in S (R^{n + 1})$ . We prove also that the kernel of the operator $h (L_{1}, \dots, L_{n})$ is in $S (H_{n})$ if $h \in S (R^{n})$ and that the kernel of the operator $g (L, - i T)$ is in $S (H_{n})$ if $g \in S (R^{2})$ . Here $L = L_{1} + \dots + L_{n}$ is the Kohn-Laplacian on $H_{n}$ .

Sia $H_{n}$ il gruppo di Heisenberg di dimensione $2 n + 1$ . Siano $L_{1}, \dots, L_{n}$ 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 $S (R^{n + 1})$ , il nucleo dell'operatore $m (L_{1}, \dots, L_{n}, - i T)$ è una funzione in $S (H_{n})$ . Inoltre dimostriamo che, date altre due funzioni $h \in S (R^{n})$ e $g \in S (R^{2})$ , i nuclei degli operatori $h (L_{1}, \dots, L_{n})$ e $g (L, - i T)$ stanno in $S (H_{n})$ . Qui $L = L_{1} + \dots + L_{n}$ è il sub-Laplaciano su $H_{n}$ .

Publié le : 2003-10-01

MR 2014825 | Zbl 1178.43007

@article{BUMI_2003_8_6B_3_657_0,
     author = {Alessandro Veneruso},
     title = {Schwartz kernels on the Heisenberg group},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {6-A},
     year = {2003},
     pages = {657-666},
     zbl = {1178.43007},
     mrnumber = {2014825},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2003_8_6B_3_657_0}
}

Veneruso, Alessandro. Schwartz kernels on the Heisenberg group. Bollettino dell'Unione Matematica Italiana, Tome 6-A (2003) pp. 657-666. http://gdmltest.u-ga.fr/item/BUMI_2003_8_6B_3_657_0/

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