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.