Invariant Radon transforms on a symmetric space

Abstract

Injectivity and support theorems are proved for a class of Radon transforms, R μ {R_\mu } , for μ \mu a smooth family of measures defined on a certain space of affine planes in X 0 {\mathbb {X}_0} , where X 0 {\mathbb {X}_0} is the tangent space, of a Riemannian symmetric space of rank one. The transforms are defined by integrating against μ \mu over these planes. We show that if R μ f {R_\mu }f is supported inside a ball of radius R R then so is f f . This is true for f ∈ L c 2 ( X 0 ) f \in L_c^2({\mathbb {X}_0}) or f ∈ E ′ ( X 0 ) f \in \mathcal {E}’({\mathbb {X}_0}) . Furthermore, R μ {R_\mu } is invertible on either of these domains. The main technique is to use facts about spherical harmonics to reduce the problem to a one-dimensional integral equation.