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

[math]\displaystyle{ P\left(\frac{\partial}{\partial x_1}, \ldots, \frac{\partial}{\partial x_\ell} \right) u(\mathbf{x}) = \delta(\mathbf{x}), }[/math]

where [math]\displaystyle{ P }[/math] is a polynomial in several variables and [math]\displaystyle{ \delta }[/math] is the Dirac delta function, has a distributional solution [math]\displaystyle{ u }[/math]. It can be used to show that

[math]\displaystyle{ P\left(\frac{\partial}{\partial x_1}, \ldots, \frac{\partial}{\partial x_\ell} \right) u(\mathbf{x}) = f(\mathbf{x}) }[/math]

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

Other proofs, often giving better bounds on the growth of a solution, are given in (Hörmander 1983a), (Reed Simon) and (Rosay 1991). (Hörmander 1983b) gives a detailed discussion of the regularity properties of the fundamental solutions.

A short constructive proof was presented in (Wagner 2009):

[math]\displaystyle{ 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) }[/math]

is a fundamental solution of [math]\displaystyle{ P(\partial) }[/math], i.e., [math]\displaystyle{ P(\partial)E=\delta }[/math], if [math]\displaystyle{ P_m }[/math] is the principal part of [math]\displaystyle{ P }[/math], [math]\displaystyle{ \eta\in\mathbb{R}^n }[/math] with [math]\displaystyle{ P_m(\eta)\neq 0 }[/math], the real numbers [math]\displaystyle{ \lambda_0,\ldots,\lambda_m }[/math] are pairwise different, and

[math]\displaystyle{ a_j=\prod_{k=0,k\neq j}^m(\lambda_j-\lambda_k)^{-1}. }[/math]

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 
• 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 
• 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 
• 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 
• 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, http://aif.cedram.org/aif-bin/item?id=AIF_1956__6__271_0 
• 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 
• Rosay, Jean-Pierre (1991), "A very elementary proof of the Malgrange-Ehrenpreis theorem", Amer. Math. Monthly 98 (6): 518–523, doi:10.2307/2324871 
• Hazewinkel, Michiel, ed. (2001), "Malgrange–Ehrenpreis theorem", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=M/m120090 
• Wagner, Peter (2009), "A new constructive proof of the Malgrange-Ehrenpreis theorem", Amer. Math. Monthly 116 (5): 457–462, doi:10.4169/193009709X470362