Malgrange–Ehrenpreis theorem

A key question in mathematics and physics is how to model empty space with a point source, like the effect of a point mass on the gravitational potential energy, or a point heat source on a plate. Such physical phenomena are modeled by partial differential equations, having the form, where is a linear differential operator and is a delta function representing the point source. A solution to this problem is called a Green's function.
This motivates the question: given a linear differential operator, can we always solve ? The Malgrange–Ehrenpreis theorem answers this in the affirmative. It 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
where is a polynomial in several variables and is the Dirac delta function, has a distributional solution. It can be used to show that
has a solution for any compactly supported distribution. The solution is not unique in general.
The analogue for differential operators whose coefficients are polynomials 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 has a distributional inverse. By replacing by the product with its complex conjugate, one can also assume that is non-negative. For non-negative polynomials the existence of a distributional inverse follows from the existence of the Bernstein–Sato polynomial, which implies that can be analytically continued as a meromorphic distribution-valued function of the complex variable ; the constant term of the Laurent expansion of at is then a distributional inverse of.
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 :
is a fundamental solution of, i.e.,, if is the principal part of,
with, the real numbers are pairwise different, and