Integrability properties of quasi-regular representations of N A groups

Let G = N ⋉ A, where N is a graded Lie group and A = R+ acts on N via homogeneous dilations. The quasi-regular representation π = indGA(1) of G can be realised to act on L2(N). It is shown that for a class of analysing vectors the associated wavelet transform defines an isometry from L2(N) into L2(G) and that the integral kernel of the corresponding orthogonal projector has polynomial off-diagonal decay. The obtained reproducing formula is instrumental for obtaining decomposition theorems for function spaces on nilpotent groups.