On the inversion of the Laplace and Abel transforms for causal symmetric spaces

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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.

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Forum Mathematicum

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