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. $$X=G/K$$ the associated Riemannian symmetric space and assume that X is of rank one. The generalized spectral projections to Laplace-Beltrami operator are given by $$P_{\lambda }f =f*\Phi _{\lambda }$$ , where $$\Phi elementary spherical functions on X. In this paper, we prove an Ingham type uncertainty principle for }f$$ . Moreover, similar results obtained in case Dunkl Laplacian.