Radon inversion formulas over local fields

Abstract

Let F be a local field and n ≥ 2 an integer. We study the Radon transform as an operator M : C+ → C− from the space of smooth K-finite functions on Fn {0} with bounded support to the space of smooth K-finite functions on Fn \{0} supported away from a neighborhood of 0. These spaces naturally arise in the theory of automorphic forms. We prove that M is an isomorphism and provide formulas for M−1. In the real case, we show that when K-finiteness is dropped from the definitions, the analog of M is not surjective.