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 the differential equation

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

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

P\left(\frac{\partial}{\partial x_1}, \cdots, \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 (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=\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(∂), i.e., P(∂)E = δ, if P_m is the principal part of P, η ∈ Rⁿ with P_m(η) ≠ 0, the real numbers λ₀, ..., λ_m are pairwise different, and

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

References

Ehrenpreis, Leon (1954), "Solution of some problems of division. I. Division by a polynomial of derivation.", Amer. J. Math., American Journal of Mathematics, Vol. 76, No. 4, 76 (4): 883–903, doi:10.2307/2372662, JSTOR 2372662, MR 0068123
Ehrenpreis, Leon (1955), "Solution of some problems of division. II. Division by a punctual distribution", Amer. J. Math., American Journal of Mathematics, Vol. 77, No. 2, 77 (2): 286–292, doi:10.2307/2372532, JSTOR 2372532, MR 0070048
Hörmander, L. (1983a), The analysis of linear partial differential operators I, Grundl. Math. Wissenschaft., 256, Springer, doi:10.1007/978-3-642-96750-4, ISBN 3-540-12104-8, MR 0717035
Hörmander, L. (1983b), The analysis of linear partial differential operators II, Grundl. Math. Wissenschaft., 257, Springer, doi:10.1007/978-3-642-96750-4, ISBN 3-540-12139-0, MR 0705278
Malgrange, Bernard (1955–1956), "Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution", Annales de l'Institut Fourier, 6: 271–355, doi:10.5802/aif.65, MR 0086990
Reed, Michael; Simon, Barry (1975), Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, New York-London: Academic Press Harcourt Brace Jovanovich, Publishers, pp. xv+361, ISBN 0-12-585002-6, MR 0493420
Rosay, Jean-Pierre (1991), "A very elementary proof of the Malgrange-Ehrenpreis theorem", Amer. Math. Monthly, The American Mathematical Monthly, Vol. 98, No. 6, 98 (6): 518–523, doi:10.2307/2324871, JSTOR 2324871, MR 1109574
Rosay, Jean-Pierre (2001) [1994], "Malgrange–Ehrenpreis theorem", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Wagner, Peter (2009), "A new constructive proof of the Malgrange-Ehrenpreis theorem", Amer. Math. Monthly, 116 (5): 457–462, doi:10.4169/193009709X470362, MR 2510844

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