Elliptic pseudodifferential equations and Sobolev spaces overp-adic fields

We study the solutions of equations of type f ( D, α)u = v, where f ( D, α) is a p-adic pseudodifferential operator.If v is a Bruhat-Schwartz function, there exists a distribution E α , a fundamental solution, such that u = E α * v is a solution.However, it is unknown to which function space E α * v belongs.We show that if f ( D, α) is an elliptic operator, then u = E α * v belongs to a certain Sobolev space, and we give conditions for the continuity and uniqueness of u.By modifying the Sobolev norm, we establish that f ( D, α) gives an isomorphism between certain Sobolev spaces.