Elliptic pseudodifferential equations and Sobolev spaces over p-adic fields

J. J. Rodríguez-Vega and W. A. Zúñiga-Galindo

Vol. 246 (2010), No. 2, 407–420

DOI: 10.2140/pjm.2010.246.407

Abstract

We study the solutions of equations of type f(D,α)u = v, where f(D,α) is a p-adic pseudodifferential operator. If v is a Bruhat–Schwartz function, 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. We show that if f(D,α) is an elliptic operator, then u = E_α ∗ v belongs to a certain Sobolev space, and we give conditions for the continuity and uniqueness of u. By modifying the Sobolev norm, we establish that f(D,α) gives an isomorphism between certain Sobolev spaces.

Keywords

p-adic fields, p-adic pseudodifferential operators, fundamental solutions, p-adic Sobolev spaces

Mathematical Subject Classification 2000

Primary: 46S10, 47S10

Secondary: 35S05, 11S80

Milestones

Received: 13 March 2009

Revised: 27 July 2009

Accepted: 30 July 2009

Published: 1 June 2010