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

TL;DR

This paper develops asymptotic expansions and estimates for Heckman-Opdam hypergeometric functions related to root systems, characterizes bounded functions, and advances the $L^p$-theory for hypergeometric Fourier transforms, extending classical harmonic analysis results.

Contribution

It provides a comprehensive asymptotic analysis of hypergeometric functions, characterizes bounded cases, and develops the $L^p$-theory, including inversion formulas, for these functions.

Findings

Series expansion for all complex spectral parameters.

Estimates for hypergeometric functions away from Weyl chamber walls.

Characterization and proof of bounded hypergeometric functions.

Abstract

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