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

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.