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

A series expansion for Heckman-Opdam hypergeometric functions $φ_λ$ is obtained for all $λ\in \mathfrak a^*_{\mathbb C}.$ As a consequence, estimates for $φ_λ$ 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 $1\leq p <2$.

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