A spectral Paley-Wiener theorem for the Heisenberg group and a support theorem for the twisted spherical means on Cn
نویسنده
چکیده
— We prove a spectral Paley-Wiener theorem for the Heisenberg group by means of a support theorem for the twisted spherical means on Cn. If f(z)e 1 4 |z| 2 is a Schwartz class function we show that f is supported in a ball of radius B in Cn if and only if f×μr(z) = 0 for r > B+ |z| for all z ∈ Cn. This is an analogue of Helgason’s support theorem on Euclidean and hyperbolic spaces. When n = 1 we show that the two conditions f × μr(z) = μr × f(z) = 0 for r > B + |z| imply a support theorem for a large class of functions with exponential growth. Surprisingly enough,this latter result does not generalize to higher dimensions. Résumé. — Nous prouvons un théorème de Paley-Wiener spectral pour le groupe d’Heisenberg en utilisant un théorème du support pour les moyennes sphériques tordues sur Cn. Si f(z)e 1 4 |z| 2 est une fonction dans la classe de Schwartz nous montrons que f a un support dans une boule de Cn de rayon B si et seulement si f×μr(z) = 0 pour r > B+ |z| et pour tout z ∈ Cn. C’est un analogue du théorème du support prouvé dans les contextes euclidiens et hyperboliques par Helgason. Lorsque n = 1 nous montrons que les deux conditions f × μr(z) = μr × f(z) = 0 pour r > B + |z| impliquent un théorème du support pour une grande classe de fonctions à croissance exponentielle. Il est surprenant de constater que ce dernier résultat ne se généralise pas aux dimensions supérieures.
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