Malgrange–Ehrenpreis theorem

In mathematics, the Malgrange–Ehrenpreis theorem states that every non-zero linear differential operator with constant coefficients has a Green's function. It was first proved independently by Leon Ehrenpreis (1954, 1955) and Bernard Malgrange (1955–1956).

This means that the differential equation

where is a polynomial in several variables and is the Dirac delta function, has a distributional solution . It can be used to show that

has a solution for any compactly supported distribution . The solution is not unique in general.

The analogue for differential operators whose coefficients are polynomials (rather than constants) is false: see Lewy's example.

Proofs

The original proofs of Malgrange and Ehrenpreis were non-constructive as they used the Hahn–Banach theorem. Since then several constructive proofs have been found.

There is a very short proof using the Fourier transform and the Bernstein–Sato polynomial, as follows. By taking Fourier transforms the Malgrange–Ehrenpreis theorem is equivalent to the fact that every non-zero polynomial has a distributional inverse. By replacing by the product with its complex conjugate, one can also assume that is non-negative. For non-negative polynomials the existence of a distributional inverse follows from the existence of the Bernstein–Sato polynomial, which implies that can be analytically continued as a meromorphic distribution-valued function of the complex variable ; the constant term of the Laurent expansion of at is then a distributional inverse of .

Other proofs, often giving better bounds on the growth of a solution, are given in (Hörmander 1983a, Theorem 7.3.10), (Reed & Simon 1975, Theorem IX.23, p. 48) and (Rosay 1991). (Hörmander 1983b, chapter 10) gives a detailed discussion of the regularity properties of the fundamental solutions.

A short constructive proof was presented in (Wagner 2009, Proposition 1, p. 458):

is a fundamental solution of , i.e., , if is the principal part of , with , the real numbers are pairwise different, and