Generalized Helgason-Fourier transforms associated to variants of the Laplace-Beltrami operators on the unit ball in $\mathbb{R}^{n}$

Abstract

In this paper we develop a harmonic analysis associated to the differential operators in a parallel way to that on real hyperbolic space. We make a detailed study of the generalized Helgason-Fourier transform and the 9-spherical transform associated to these differential operators. In particular, we obtain the inversion formula and the Plancherel theorem for them. As an application, we solve the relevant heat equation.