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arXiv · 0907.5545

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

J. J. Rodriguez-Vega·W. A. Zuniga-Galindo
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Abstract

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.

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J. J. Rodriguez-Vega, W. A. Zuniga-Galindo. 2009-07-31. Elliptic Pseudo-Differential Equations and Sobolev Spaces over p-adic Fields. https://arxiv.org/abs/0907.5545

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