ELLIPTIC PSEUDO-DIFFERENTIAL EQUATIONS AND SOBOLEV SPACES OVER p-ADIC FIELDS

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