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

TL;DR

This paper investigates solutions to p-adic pseudo-differential equations, establishing their membership in Sobolev spaces, and provides conditions for their continuity and uniqueness, thus advancing the understanding of p-adic analysis.

Contribution

It demonstrates that elliptic p-adic pseudo-differential operators map between specific Sobolev spaces and establishes isomorphism properties.

Findings

Solutions belong to certain Sobolev spaces

Conditions for continuity and uniqueness are provided

Operators act as isomorphisms between Sobolev spaces

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.