### - Art Gallery -

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

$${\displaystyle P\left({\frac {\partial }{\partial x_{1}}},\ldots ,{\frac {\partial }{\partial x_{\ell }}}\right)u(\mathbf {x} )=\delta (\mathbf {x} ),}$$

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

$${\displaystyle P\left({\frac {\partial }{\partial x_{1}}},\ldots ,{\frac {\partial }{\partial x_{\ell }}}\right)u(\mathbf {x} )=f(\mathbf {x} )}$$

has a solution for any compactly supported distribution f. 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 P has a distributional inverse. By replacing P by the product with its complex conjugate, one can also assume that P is non-negative. For non-negative polynomials P the existence of a distributional inverse follows from the existence of the Bernstein–Sato polynomial, which implies that Ps can be analytically continued as a meromorphic distribution-valued function of the complex variable s; the constant term of the Laurent expansion of Ps at s = −1 is then a distributional inverse of P.

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):

$$E=\frac{1}{\overline{P_m(2\eta)}} \sum_{j=0}^m a_j e^{\lambda_j\eta x} \mathcal{F}^{-1}_{\xi}\left(\frac{\overline{P(i\xi+\lambda_j\eta)}}{P(i \xi + \lambda_j \eta)}\right)$$

is a fundamental solution of P(∂), i.e., P(∂)E = δ, if Pm is the principal part of P, η ∈ Rn with Pm(η) ≠ 0, the real numbers λ0, ..., λm are pairwise different, and

$$a_j=\prod_{k=0,k\neq j}^m(\lambda_j-\lambda_k)^{-1}.$$

References
Ehrenpreis, Leon (1954), "Solution of some problems of division. I. Division by a polynomial of derivation.", Amer. J. Math., 76 (4): 883–903, doi:10.2307/2372662, JSTOR 2372662, MR 0068123
Ehrenpreis, Leon (1955), "Solution of some problems of division. II. Division by a punctual distribution", Amer. J. Math., 77 (2): 286–292, doi:10.2307/2372532, JSTOR 2372532, MR 0070048
Hörmander, L. (1983a), The analysis of linear partial differential operators I, Grundl. Math. Wissenschaft., 256, Springer, doi:10.1007/978-3-642-96750-4, ISBN 978-3-540-12104-6, MR 0717035
Hörmander, L. (1983b), The analysis of linear partial differential operators II, Grundl. Math. Wissenschaft., 257, Springer, doi:10.1007/978-3-642-96750-4, ISBN 978-3-540-12139-8, MR 0705278
Malgrange, Bernard (1955–1956), "Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution", Annales de l'Institut Fourier, 6: 271–355, doi:10.5802/aif.65, MR 0086990
Reed, Michael; Simon, Barry (1975), Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, New York-London: Academic Press Harcourt Brace Jovanovich, Publishers, pp. xv+361, ISBN 978-0-12-585002-5, MR 0493420
Rosay, Jean-Pierre (1991), "A very elementary proof of the Malgrange-Ehrenpreis theorem", Amer. Math. Monthly, 98 (6): 518–523, doi:10.2307/2324871, JSTOR 2324871, MR 1109574
Rosay, Jean-Pierre (2001) [1994], "Malgrange–Ehrenpreis theorem", Encyclopedia of Mathematics, EMS Presss
Wagner, Peter (2009), "A new constructive proof of the Malgrange-Ehrenpreis theorem", Amer. Math. Monthly, 116 (5): 457–462, CiteSeerX 10.1.1.488.6651, doi:10.4169/193009709X470362, MR 2510844