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Transactions of the American Mathematical Society

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Invariant Radon transforms on a symmetric space

Author: Jeremy Orloff
Journal: Trans. Amer. Math. Soc. 318 (1990), 581-600
MSC: Primary 44A15; Secondary 43A85
MathSciNet review: 958898
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Abstract: Injectivity and support theorems are proved for a class of Radon transforms, $ {R_\mu }$, for $ \mu $ a smooth family of measures defined on a certain space of affine planes in $ {\mathbb{X}_0}$, where $ {\mathbb{X}_0}$ is the tangent space, of a Riemannian symmetric space of rank one. The transforms are defined by integrating against $ \mu $ over these planes. We show that if $ {R_\mu }f$ is supported inside a ball of radius $ R$ then so is $ f$. This is true for $ f \in L_c^2({\mathbb{X}_0})$ or $ f \in \mathcal{E}'({\mathbb{X}_0})$. Furthermore, $ {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.

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Article copyright: © Copyright 1990 American Mathematical Society