Generalized Helgason-Fourier transforms associated to variants of the Laplace-Beltrami operators on the unit ball in $\mathbb{R}^{n}$ Congwen LiuLizhong Peng 43A8542B10generalized Helgason-Fourier transformsinversion formulaWeinstein operatorreal hyperbolic spacePoisson transformPlancherel theoremheat kernel 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. Indiana University Mathematics Journal 2009 text pdf 10.1512/iumj.2009.58.3588 10.1512/iumj.2009.58.3588 en Indiana Univ. Math. J. 58 (2009) 1457 - 1492 state-of-the-art mathematics http://iumj.org/access/