On the inversion of the Laplace and Abel transforms for causal symmetric spaces
In this article we prove new growth estimates for the spherical functions on non-compactly causal symmetric spaces M. These are used to find Paley-Wiener type estimates for the spherical Laplace transform L defined on M. We then obtain a new inversion formula for L when the root multiplicities are even and proceed to establish a Paley-Wiener theorem for L in this special case. In analogy with the Riemannian case with even root multiplicities, we also prove that the inverse of the Abel transform defined on M is given by a differential operator which is a shift operator associated with the underlying root system.
Publication Source (Journal or Book title)
Andersen, N., Ólafsson, G., & Schlichtkrull, H. (2003). On the inversion of the Laplace and Abel transforms for causal symmetric spaces. Forum Mathematicum, 15 (5), 701-725. https://doi.org/10.1515/form.2003.038