HARDY'S UNCERTAINTY PRINCIPLE ON CERTAIN LIE GROUPS

Abstract

A theorem due to Hardy states that, if f is a function on R such that |f(x)| [les ] C e^−α|x|2 for all x in R and |fˆ(ξ)| [les ] C e^−β|ξ|2 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.