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 Template:Harvs and Template:Harvs.

This means that the differential equation

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

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

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

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 $P^s$ 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 Template:Harv, Template:Harv and Template:Harv. Template:Harv gives a detailed discussion of the regularity properties of the fundamental solutions.

A short constructive proof was presented in Template:Harv:

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

is a fundamental solution of $P(\partial )$, i.e., $P(\partial )E=\delta$, if $P_m$ is the principal part of $P$, $\eta \in {ℝ}^n$ 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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