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
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10.1512/iumj.2009.58.3588
10.1512/iumj.2009.58.3588
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Indiana Univ. Math. J. 58 (2009) 1457 - 1492
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