A spectral Paley-Wiener theorem for the Heisenberg group and a support theorem for the twisted spherical means on ${ℂ}^n$

Narayanan, E. K.; Thangavelu, S.

Annales de l'Institut Fourier, Tome 56 (2006) no. 2, pp. 459-473.

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 ${ℂ}^n.$ Si $f\left(z\right)e^{\frac{1}{4}{\left|z\right|}^2}$ est une fonction dans la classe de Schwartz nous montrons que $f$ a un support dans une boule de ${ℂ}^n$ de rayon $B$ si et seulement si $f\times {\mu }_r\left(z\right)=0$ pour $r>B+\left|z\right|$ et pour tout $z\in {ℂ}^n.$ 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 $f\times {\mu }_r\left(z\right)={\mu }_r\times f\left(z\right)=0$ pour $r>B+\left|z\right|$ 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.

We prove a spectral Paley-Wiener theorem for the Heisenberg group by means of a support theorem for the twisted spherical means on ${ℂ}^n.$ If $f\left(z\right)e^{\frac{1}{4}{\left|z\right|}^2}$ is a Schwartz class function we show that $f$ is supported in a ball of radius $B$ in ${ℂ}^n$ if and only if $f\times {\mu }_r\left(z\right)=0$ for $r>B+\left|z\right|$ for all $z\in {ℂ}^n.$ This is an analogue of Helgason’s support theorem on Euclidean and hyperbolic spaces. When $n=1$ we show that the two conditions $f\times {\mu }_r\left(z\right)={\mu }_r\times f\left(z\right)=0$ for $r>B+\left|z\right|$ imply a support theorem for a large class of functions with exponential growth. Surprisingly enough,this latter result does not generalize to higher dimensions.

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