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 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):

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

is a fundamental solution of $P(\partial)$, i.e., $P(\partial)E=\delta$, if P_m 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

• 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, MR0068123 
• 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, MR0070048 
• Hörmander, L. (1983a), The analysis of linear partial differential operators I, Grundl. Math. Wissenschaft., 256, Springer, ISBN 3-540-12104-8, MR0717035 
• Hörmander, L. (1983b), The analysis of linear partial differential operators II, Grundl. Math. Wissenschaft., 257, Springer, ISBN 3-540-12139-0, MR0705278 
• Malgrange, Bernard (1955–1956), "Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution", Ann. Inst. Fourier, Grenoble 6: 271–355, MR0086990, http://aif.cedram.org/aif-bin/item?id=AIF_1956__6__271_0 
• 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 0125850026, MR0493420 
• 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, MR1109574 
• Rosay, Jean-Pierre (2001), "Malgrange–Ehrenpreis theorem", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/M/m120090.htm 
• Wagner, Peter (2009), "A new constructive proof of the Malgrange-Ehrenpreis theorem", Amer. Math. Monthly 116 (5): 457–462, doi:10.4169/193009709X470362, MR2510844