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(x)\ ≤ C e-α\x\2 for all x in R and |f̂(ξ)| ≤ C e-β|ξl2 for all ξ, in R, where α > 0, β > 0, and αβ > 1/4, then f = 0. A version of this celebrated theorem is proved for two classes of Lie groups : two-step nilpotent Lie groups and harmonic NA groups, the latter being a generalisation of noncompact rank-1 symmetric spaces. In the first case the group Fourier transformation is considered; in the second case an analogue of the Helgason-Fourier transformation for symmetric spaces is considered