Hardy’s uncertainty principle on semisimple groups

A theorem of 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|ξ|2 for all ξ in R, where α> 0, β> 0, and αβ> 1/4, then f = 0. Sitaram and Sundari generalised this theorem to semisimple groups with one conjugacy class of Cartan subgroups and to the K-invariant case for general semisimple groups. We ex-tend the theorem to all semisimple groups. 1. Introduction. The Uncertainty Principle states, roughly speaking, that a nonzero func-tion f and its Fourier transform f ̂ cannot both be sharply localised. This fact may be manifested in different ways. The version of this phenomenon described in the abstract is due to Hardy [3]; we call it Hardy’s Uncertaint