A spectral Paley-Wiener theorem for the Heisenberg group and a support theorem for the twisted spherical means on C n (Un théorème de Paley-Wiener spectral pour le groupe d'Heisenberg et un théorème de support pour les moyennes shériques tordues sur C n)

Abstract

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(z)e 1 4|z| 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×µ r (z)=0 for r>B+|z| for all z∈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×µ r (z)=µ r ×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.