An Analysis Upon Surjectivity of Partial Differential Operators: Fundamental Solution to the Division Problem

Surjectivity of Partial Differential Operators in Periodic and Tempered Distributions

Abstract

We give a sufficient condition for the surjectivity of partial differentialoperators with constant coefficients on a class of distributions on  (herewe think of there being n spacedirections and one time direction), that are periodic in the spatial directionsand tempered in the time direction. By proving a topological Paley-Wiener Theorem for Riemannian symmetricspaces of non-compact type, we show that a non-zero invariant differentialoperator is a homeomorphism from the space of test functions onto its image andhence surjective when extended to the space of distributions.