A general formula for fundamental solutions of linear partial differential equations with constant coefficients

Abstract:

In this note we present a formula which furnishes particular fundamental solutions of linear partial differential equations with constant coefficients. Our construction extends an explicit formula of König (to appear) after the procedure of Malgrange (1955-1956). The crucial point is that he works with equations rather than with estimations as in the classical proof of the Malgrange-Ehrenpreis theorem. Following his ideas, we obtain fundamental solutions which are regular in the sense of Hörmander (1983); they are of basic importance. Our formula is as explicit as the zeros of a polynomial in one variable are explicit as functions of the coefficients.