---
tag: 3173Y813V7
kind: statement
role: theorem
name: Malgrange–Ehrenpreis theorem
msc: ["35E05", 35A08, 35A01]
uses: [K2NK9A8M5F, 6DZETZ5T9G, HS65DW0HXH]
about: [K2NK9A8M5F]
sources:
  - url: https://en.wikipedia.org/wiki/Malgrange%E2%80%93Ehrenpreis_theorem
same_as:
  wikidata: Q2226750
  wikipedia: Malgrange–Ehrenpreis theorem
---

## Statement

Let $P = \sum_{\abs{\alpha} \le m} c_\alpha \partial^\alpha$ be a [[linear differential operator|6DZETZ5T9G]] on $\R^n$ with constant complex coefficients $c_\alpha$, not all zero.
Then $P$ has a [[fundamental solution|K2NK9A8M5F]]: there is a distribution $E$ on $\R^n$ with
$$P E = \delta,$$
where $\delta$ is the Dirac distribution at the origin.
Consequently, for every [[test function|HS65DW0HXH]] $f$ on $\R^n$, the function $u = E * f$ is smooth and satisfies $P u = f$.

## 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|P6E9RZNRBZ]] 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 $\R^n$.

## Explanations

### L1 human

The theorem says that every linear differential equation with constant coefficients can be solved for a point source.
Once the response $E$ to a point source is known, the response to any source $f$ is obtained by adding up shifted copies of $E$, that is, by convolution.
So every such equation can be solved, at least for sources that vanish outside a bounded set.

### L2 human

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|KD404MF042]] for the heat operator.
The general theorem guarantees existence but gives little information on $E$; finer questions, such as whether $P$ is [[hypoelliptic|K9DSR21DQC]], 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.