Integrability properties of quasi-regular representations of $NA$ groups

van Velthoven, Jordy Timo

Comptes Rendus. Mathématique, Tome 360 (2022) no. G10, pp. 1125-1134.

Let $G=N⋊A$, where $N$ is a graded Lie group and $A={ℝ}^{+}$ acts on $N$ via homogeneous dilations. The quasi-regular representation $\pi ={\mathrm{ind}}_A^G\left(1\right)$ of $G$ can be realised to act on $L^2\left(N\right)$. It is shown that for a class of analysing vectors the associated wavelet transform defines an isometry from $L^2\left(N\right)$ into $L^2\left(G\right)$ 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.

Classification : 22E15, 22E27, 43A80, 44A35

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