A survey on explicit representation formulae for fundamental solutions of linear partial differential operators
The first general existence proofs for fundamental solutions of linear partial differential operators with constant coefficients were presented in 1953 by B. Malgrange and, independently, by L. Ehrenpreis. Their method of proof was based on the Hahn-Banach theorem but the question for explicit representation of the fundamental solution remained open. Thus the goal of the authors is to give a survey on such explicit representation formulas. In Chapter 2 they use a method which goes back to Hörmander for the construction of fundamental solutions with at most exponential growth as a finite sum of \(C^\infty\)-functions multiplied by the inverse Fourier transforms of \(L^\infty\)-functions. In Chapter 3 they study regular fundamental solutions by using temperate weight functions \(k=k (\xi)\) and the well-known \(B^{p,k}\)-spaces. The authors do not treat the existence of temperate fundamental solutions. This problem, construction of temperate fundamental solutions for general differential polynomials (Treves 1967), seems to be unsolved up to now.
scientific article; zbMATH DE number 1014488