Malgrange–Ehrenpreis theorem

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 if P is a polynomial (in several variables) then the differential equation

P(\partial/\partial x_i)u(x)=\delta(x)

has a distributional solution u, where δ is the Dirac delta function. It can be used to show that

P(\partial/\partial x_i)u(x)=f(x)

has a solution for any 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 proof 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 Ps can be analytically continued as a meromorphic distribution-valued function of the complex variable s; the constant term of the Laurent expansion of Ps 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 (Hörmander 1983a, Theorem 7.3.10), (Reed & Simon 1975, Theorem IX.23, p. 48) and (Rosay 1991). (Hörmander 1983b, chapter 10) gives a detailed discussion of the regularity properties of the fundamental solutions.

A short constructive proof was presented in (Wagner 2009, Proposition 1, p. 458):

 E=\frac1{\,\overline{P_m(2\eta)}\,}\sum_{j=0}^m
a_j\,\text{e}^{\lambda_j\eta x}\mathcal{F}^{-1}_{\xi}\left(
\frac{\,\overline{P(\text {i}\xi+\lambda_j\eta)}\,}{P(\text{i}
\xi+\lambda_j\eta)}\right)

is a fundamental solution of P(\partial), i.e., P(\partial)E=\delta, if Pm is the principal part of P, \eta\in\R^n with P_m(\eta)\neq0, the real numbers \lambda_0,\dots,\lambda_m are pairwise different, and a_j=\prod_{k=0,k\neq j}^m(\lambda_j-\lambda_k)^{-1}.

References


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