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 $E_α$, a fundamental solution, such that $u=E_α\ast v$ is a solution. However, it is unknown to which function space $E_α\ast v$ belongs. In this paper, we show that if $f(D,α)$ is an elliptic operator, then $u=E_α\ast v$ 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.