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

2009/07/31 by J. J. Rodríguez-Vega, J. J. Rodriguez-Vega, Rodriguez-Vega, J. J. +3
Mathematics · Physics and Astronomy · #46S10 #47S10 #FOS: Physical sciences #Mathematical Physics (math-ph) #advanced mathematical theories #math-ph #math.MP #msc:46S10 #msc:47S10

paper · pdf · doi:10.48550/arxiv.0907.5545

13 pages. Accepted in the Pac. J. Math

arxiv created 2009/07/31 · openalex publication_date 2009/07/31 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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