On an interpretation of Hörmander's condition of local solvability

AbstractThe goal of this paper is to discuss a relation between microlocal solvability and an existence of microlocally complete set of eigenfunctions. Any differential operator with constant coefficients is locally solvable and this property can be derived from the existence of a complete set of eigenfunctions ei〈x,ξ〉. In this paper we prove that Hörmander's solvability condition {Rep, Imp} < 0 implies the existence of microlocally complete system of eigenfunctions and from this follows solvability. When a complete system of eigenfunctions is constructed, an operator with variable coefficients resembles very much the one with constant coefficients; for example one can introduce an analog of Fourier Transform and construct a parametrix exactly in the same way as it is done for operators with constant coefficients