Hardy's uncertainty principle on certain Lie groups

摘要

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 x2) for all x in R and (xi) less than or equal to C e(-betaxi 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 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. [References: 19]