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

TL;DR

This paper studies the integrability and decay properties of wavelet transforms derived from quasi-regular representations of certain semi-direct product groups, aiding in the analysis of function spaces on nilpotent Lie groups.

Contribution

It establishes isometric wavelet transforms and polynomial off-diagonal decay of kernels for quasi-regular representations of $NA$ groups, advancing harmonic analysis on these groups.

Findings

Wavelet transform acts as an isometry on $L^2(N)$

Orthogonal projector kernel exhibits polynomial decay

Facilitates decomposition theorems for function spaces

Abstract

Let $G = N \rtimes A$, where $N$ is a graded Lie group and $A = \mathbb{R}^+$ acts on $N$ via homogeneous dilations. The quasi-regular representation $\pi = \mathrm{ind}_A^G (1)$ of $G$ can be realised to act on $L^2 (N)$. It is shown that for a class of analysing vectors the associated wavelet transform defines an isometry from $L^2 (N)$ into $L^2 (G)$ and that the integral kernel of the corresponding orthogonal projector has polynomial off-diagonal decay. The obtained reproducing formula is instrumental for proving decomposition theorems for function spaces on nilpotent Lie groups.