p-Adic Fractional Pseudodifferential Equations and Sobolev Type Spaces overp-Adic Fields

Abstract

In this paper, we study the solutions of the pseudodifferential equations of typeDα u=voverp-adic fieldℚp, whereDαis ap-adic fractional pseudodifferential operator. Ifvis a Bruhat-Schwartz function, then there exists a distributionEα, a fundamental solution, such thatu=Eα*vis a solution. We also show that the solutionubelongs to a certain Sobolev space. Furthermore, we give conditions for the continuity and uniqueness ofu.