Pointwise Fourier Inversion on Rank One Symmetric Spaces and Related Topics

This paper develops necessary and sufficient conditions for pointwise inversion of Fourier transforms on rank one symmetric spaces of non-compact type for functions in the piecewise smooth category. This extends results of Pinsky for isotropic Riemannian manifolds of constant curvature. Methodologically, a similar result on the Jacobi transform and a transplantation scheme from even dimensional spaces to certain odd dimensional real hyperbolic spaces naturally enter the picture.