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

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 generalized spectral projections associated to the Laplace-Beltrami operator are given by \(P_{\lambda }f =f*\Phi _{\lambda }\), 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 generalized spectral projections associated to Dunkl Laplacian.