Radon inversion formulas over local fields

Let $F$ be a local field and $n\ge 2$ 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.