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

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

where ${\displaystyle P}$ is a polynomial in several variables and ${\displaystyle \delta }$ is the Dirac delta function, has a distributional solution ${\displaystyle u}$. 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 ${\displaystyle f}$. 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 ${\displaystyle P}$ has a distributional inverse. By replacing ${\displaystyle P}$ by the product with its complex conjugate, one can also assume that ${\displaystyle P}$ is non-negative. For non-negative polynomials ${\displaystyle P}$ the existence of a distributional inverse follows from the existence of the Bernstein–Sato polynomial, which implies that ${\displaystyle P^{s}}$ can be analytically continued as a meromorphic distribution-valued function of the complex variable ${\displaystyle s}$; the constant term of the Laurent expansion of ${\displaystyle P^{s}}$ at ${\displaystyle s=-1}$ is then a distributional inverse of ${\displaystyle P}$.

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):

${\displaystyle E={\frac {1}{\overline {P_{m}(2\eta )}}}\sum _{j=0}^{m}a_{j}e^{\lambda _{j}\eta x}{\mathcal {F}}_{\xi }^{-1}\left({\frac {\overline {P(i\xi +\lambda _{j}\eta )}}{P(i\xi +\lambda _{j}\eta )}}\right)}$

is a fundamental solution of ${\displaystyle P(\partial )}$, i.e., ${\displaystyle P(\partial )E=\delta }$, if ${\displaystyle P_{m}}$ is the principal part of ${\displaystyle P}$, ${\displaystyle \eta \in \mathbb {R} ^{n}}$ with ${\displaystyle P_{m}(\eta )\neq 0}$, the real numbers ${\displaystyle \lambda _{0},\ldots ,\lambda _{m}}$ are pairwise different, and

${\displaystyle a_{j}=\prod _{k=0,k\neq j}^{m}(\lambda _{j}-\lambda _{k})^{-1}.}$

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., vol. 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., vol. 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 Press
• 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