IUMJ

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

Authors: Congwen Liu and Lizhong Peng

Issue: Volume 58 (2009), Issue 3, 1457-1492

Abstract:

In this paper we develop a harmonic analysis associated to the differential operators \begin{equation*} \Delta_{\ind} \coloneqq \frac {1 - |x|^2}4 \bigg\{ (1 - |x|^2) \sum_{j=1}^n\frac {\partial^2} {\partial x_j^2} - 2\ind \sum_{j=1}^n x_j\frac {\partial} {\partial x_j} + \ind(2 - n - \ind) \bigg\}\end{equation*} in a parallel way to that on real hyperbolic space. We make a detailed study of the generalized Helgason-Fourier transform and the $\ind$-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.