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# Mathematical Research Letters

## Volume 23 (2016)

### Number 2

### Radon inversion formulas over local fields

Pages: 565 – 591

DOI: http://dx.doi.org/10.4310/MRL.2016.v23.n2.a13

#### Author

#### Abstract

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.