Hardy's uncertainty principle on semisimple groups

Abstract

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 |f(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 extend the theorem to all semisimple groups.