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.

The original proofs of Malgrange and Ehrenpreis did not use explicit constructions 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

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