An uncertainty principle for spectral projections on rank one symmetric spaces of noncompact type

TL;DR

This paper establishes an Ingham-type uncertainty principle for spectral projections on rank one noncompact symmetric spaces, extending classical harmonic analysis results to these geometric settings.

Contribution

It introduces a new uncertainty principle for spectral projections on rank one symmetric spaces and extends similar results to Dunkl Laplacian cases.

Findings

Proves an uncertainty principle for spectral projections on symmetric spaces.

Extends results to Dunkl Laplacian spectral projections.

Provides a framework for harmonic analysis on noncompact symmetric spaces.

Abstract

Let $G $ be a noncompact semisimple Lie group with finite centre. Let $X=G/K$ be the associated Riemannian symmetric space and assume that $X$ is of rank one. The spectral projections associated to the Laplace-Beltrami operator are given by $P_{\lambda}f =f\ast \Phi_{\lambda}$, where $\Phi_{\lambda}$ are the elementary spherical functions on $X$. In this paper, we prove an Ingham type uncertainty principle for $P_{\lambda}f$. Moreover, similar results are obtained in the case of spectral projections associated to Dunkl Laplacian.