Local uncertainty inequalities for locally compact groups

Abstract

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 K_θ 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 ≤ K_θμ(E)^θ||Φ_θƒ||₂ where Φ_θ is a certain weight function on G (for which an explicit formula is given). When G=R^k the inequality has been established with Φ_θ(x)=|x|^kθ.