Contributions to the Hypergeometric Function Theory of Heckman and Opdam: Sharp Estimates, Schwartz Space, Heat Kernel

Abstract.

Under the assumption of positive multiplicity, we obtain basic estimates of the hypergeometric functions F _λ and G _λ of Heckman and Opdam, and sharp estimates of the particular functions F ₀ and G ₀. Next we prove the Paley–Wiener theorem for the Schwartz class, solve the heat equation and estimate the heat kernel.