Weak Paley-Wiener property for completely solvable Lie groups

We prove a weak Paley–Wiener property for completely solvable Lie groups, i.e. if the group Fourier transform of a measurable, bounded and compactly supported function van-ishes on a set of positive Plancherel measure then the function itself vanishes almost everywhere on the group. 1. Introduction. Let G be a connected, simply connected, and completely solvable Lie group, with the Lie algebra g. Let g ∗ be the dual of g. The equivalence classes of irreducible unitary representations G ̂ of G is parametrized by the coadjoint orbits g∗/G via the Kirillov-Bernat bijective map K: G ̂ → g∗/G. If ρ ∈ G