Hardy's uncertainty principle on certain Lie groups

A theorem due to Hardy states that, if f is a function on R such that \(f) over cap (x)\ less than or equal to C e(-alpha \x\2) for all x in R and \f( xi)\ less than or equal to C e(-beta\xi \2) for all xi in R, where alpha > 0, beta > 0, and alpha beta > 1/4, then f = 0. A version of this celebrated theorem is proved for two classes of Lie groups: two-step nilpotent Lie gr oups and harmonic NA groups, the latter being a generalisation of noncompac t rank-1 symmetric spaces. In the first case the group Fourier transformati on is considered; in the second case an analogue of the Helgason-Fourier tr ansformation for symmetric spaces is considered.