An Analysis Upon Surjectivity of Partial Differential Operators: Fundamental Solution to the Division Problem Surjectivity of Partial Differential Operators in Periodic and Tempered Distributions
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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.
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