Malgrange–Ehrenpreis theorem

Remarks

Formally, the Fourier transform turns $P E = \delta$ into multiplication of $\widehat{E}$ by a polynomial $p(\xi)$, so one wants $\widehat{E} = 1 / p$. The difficulty is that $p$ may vanish on the real space. One proof avoids its zeros by moving the Fourier integration into the complex domain. Another uses the Hahn–Banach theorem together with an estimate $\norm{\varphi} \le C \norm{P^{*} \varphi}$ in suitable norms. The theorem was proved independently by Bernard Malgrange and Leon Ehrenpreis. Example: for $P = \Delta$ on $\R^3$, the function $E(x) = -1 / (4 \pi \abs{x})$ is a fundamental solution, and $u = E * f$ solves the Poisson equation $\Delta u = f$. The hypothesis of constant coefficients is needed: Lewy's operator, which has polynomial coefficients, has no local solvability at all. Constant-coefficient operators are even solvable in the large, by a further classical theorem. That is, $P u = f$ has a smooth solution on $\R^n$ for every smooth $f$ on $\R^n$.

Going deeper

For a student

The theorem is the starting point of the general theory of constant-coefficient operators. For particular operators one writes the fundamental solution down explicitly, such as the Newtonian potential for the Laplacian or the heat kernel for the heat operator. The general theorem guarantees existence but gives little information on $E$; finer questions, such as whether $P$ is hypoelliptic, depend on properties of the fundamental solution. The theorem fails for variable coefficients, and the study of when local solvability holds then becomes a hard problem.