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

$P(\frac{\partial }{\partial x_1},\dots ,\frac{\partial }{\partial x_{\ell }})u(\mathbf{𝐱})=\delta (\mathbf{𝐱}),$

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

$P(\frac{\partial }{\partial x_1},\dots ,\frac{\partial }{\partial x_{\ell }})u(\mathbf{𝐱})=f(\mathbf{𝐱})$

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 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 $P$ 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 $P^s$ can be analytically continued as a meromorphic distribution-valued function of the complex variable $s$; the constant term of the Laurent expansion of $P^s$ 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 , and . gives a detailed discussion of the regularity properties of the fundamental solutions.

A short constructive proof was presented in :

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

is a fundamental solution of $P(\partial )$, i.e., $P(\partial )E=\delta$, if $P_m$ is the principal part of $P$, $\eta \in {\mathbb{ℝ}}^n$ with $P_m(\eta )\ne 0$, the real numbers ${\lambda }_0,\dots ,{\lambda }_m$ are pairwise different, and

$a_j=\prod _{k=0,k\ne j}^m({\lambda }_j-{\lambda }_k{)}^{-1}.$

References

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