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We study the solutions of equations of type $f(D,\alpha)u=v$, where$f(D,\alpha)$ is a $p$-adic pseudo-differential operator. If $v$ is aBruhat-Schwartz function, then there exists a distribution $E_{\alpha}$, afundamental solution, such that $u=E_{\alpha}\ast v$ is a solution. However, itis unknown to which function space $E_{\alpha}\ast v$ belongs. In this paper,we show that if $f(D,\alpha)$ is an elliptic operator, then $u=E_{\alpha}\astv$ belongs to a certain Sobolev space. Furthermore, we give conditions for thecontinuity and uniqueness of $u$. By modifying the Sobolev norm, we canestablish that $f(D,\alpha)$ gives an isomorphism between certain Sobolevspaces.

Elliptic Pseudo-Differential Equations and Sobolev Spaces over p-adic Fields