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 the Dirac delta function, has a distributional solution
It can be used to show that has a solution for any compactly supported distribution
The analogue for differential operators whose coefficients are polynomials (rather than constants) is false: see Lewy's example.
The original proofs of Malgrange and Ehrenpreis were non-constructive as they used the Hahn–Banach theorem.
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
by the product with its complex conjugate, one can also assume that
can be analytically continued as a meromorphic distribution-valued function of the complex variable
; the constant term of the Laurent expansion 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