Local uncertainty inequalities for locally compact groups

Let G be a locally compact unimodular group equipped with Haar measure m, G^ its unitary dual and μ the Plancherel measure (or something closely akin to it) on G^. When G is a euclidean motion group, a noncompact semisimple Lie group or one of the Heisenberg groups we prove local uncertainty inequalities of the following type: given θ∈[O,½) there exists a constant Kθ such that for all ƒ in a certain class of functions on G and all measurable E ⊆ G^, (∫ETr(π(ƒ)∗π(ƒ))dμ(π)½ ≤ Kθμ(E)θ||Φθƒ||2 where Φθ is a certain weight function on G (for which an explicit formula is given). When G=Rk the inequality has been established with Φθ(x)=|x|kθ