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

Abstract

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 a Bruhat-Schwartz function, then there exists a distribution $E_{\alpha}$, a fundamental solution, such that $u=E_{\alpha}\ast v$ is a solution. However, it is 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 $u=E_{\alpha}\ast v$ 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 $f(D,\alpha)$ gives an isomorphism between certain Sobolev spaces.