A Paley-Wiener Theorem for All Two- and Three-Step Nilpotent Lie Groups

A Paley-Wiener theorem for all connected, simply-connected two- and three-step nilpotent Lie groups is proved. If f ∈ L∞c (G), where G is a connected, simply-connected two- or three-step nilpotent Lie group such that the operator-valued Fourier transform φ̂(π) = 0 for all π in E, a subset of Ĝ of positive Plancherel measure, then it is shown that f = 0 a.e. on G. The proof uses representation-theoretic methods from Kirillov theory for nilpotent Lie groups and an integral formula for the operator-valued Fourier transform φ̂(π).