Radon inversion formulas over local fields

Let $F$ be a local field and $n \geq 2$ an integer. We study the Radon transform as an operator $M : \mathcal{C}_{+} \to \mathcal{C}_{-}$ from the space of smooth $K$-finite functions on $F^n \setminus \{ 0 \}$ with bounded support to the space of smooth $K$-finite functions on $F^n \setminus \{ 0 \}$ 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.

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Accepted 17 June 2015

Published 6 June 2016