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

A series expansion for Heckman–Opdam hypergeometric functions φλφλ is obtained for all λ∈aC⁎. As a consequence, estimates for φλφλ 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 LpLp-theory for the hypergeometric Fourier transform is developed for 0