Malgrange–Ehrenpreis theorem explained

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( (\partial)/(\partial x _{1}) , \ldots, (\partial)/(\partial x _{\ell}) \right) 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 thatP\left( (\partial)/(\partial x _{1}) , \ldots, (\partial)/(\partial x _{\ell}) \right) 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, and . gives a detailed discussion of the regularity properties of the fundamental solutions.

A short constructive proof was presented in :

E = (1)/(\overline{P _{m(2η)}) } \sum_^m a_j e^ \mathcal^_\left(\frac\right)

is a fundamental solution of

P(\partial) , i.e., P(\partial)E = \delta , if P _{m} is the principal part of P , η \in R ^{n} with P _{m(η) ≠} 0 , the real numbers λ _{0,\ldots,λ} _{m} are pairwise different, anda _{j=\prod} ^{m( λ} ^{j -λ} ^{-1} ^{k)} .