Asymptotics of Harish-Chandra expansions, bounded hypergeometric functions associated with root systems, and applications

Abstract

A series expansion for Heckman-Opdam hypergeometric functions ${\phi }_{\lambda }$ is obtained for all $\lambda \in {\mathfrak{a}}_{\mathbb{C}}^{\ast }.$ As a consequence, estimates for ${\phi }_{\lambda }$ away from the walls of a Weyl chamber are established. We also characterize the bounded hypergeometric functions and thus prove an analogue of the celebrated theorem of Helgason and Johnson on the bounded spherical functions on a Riemannian symmetric space of the noncompact type. The $L^p$-theory for the hypergeometric Fourier transform is developed for $0<p<2$. In particular, an inversion formula is proved when $1\le p<2$.