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 and .

This means that the differential equation


 * $$P\left(\frac{\partial}{\partial x_1}, \ldots, \frac{\partial}{\partial x_\ell} \right) u(\mathbf{x}) = \delta(\mathbf{x}),$$

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


 * $$P\left(\frac{\partial}{\partial x_1}, \ldots, \frac{\partial}{\partial x_\ell} \right) u(\mathbf{x}) = f(\mathbf{x})$$

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 $$P^s$$ 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 :


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

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


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