Let $D\ne 0$ be a linear differential operator with constant coefficients acting on either real or complex valued functions on $\mathbb{R}^n$.


**Is it true that the equation $$Du=f$$
is solvable in any open ball, when $f,u\in C^\infty$?**



Solvability of general linear PDE with constant coefficients
The answer is yes. More generally, a (non-zero) constant coefficient differential operator $P$ induces a surjective map $P:C^\infty(X)\to C^\infty(X)$ for an open set $X\subset \mathbb{R}^n$ if and only if $X$ is $P$-convex for supports. A reference for this result is Theorem 10.6.6 and Corollary 10.6.8 in Hörmander's "The Analysis of Linear PDOs", volume 2.


An open set $X$ is $P$-convex for supports if for every compact set $K\subset X$ there is another compact set $K'\subset X$ such that, if $\phi \in C^\infty\_c(X)$ is such that $\text{supp}\,P(-D)\phi \subset K$, then $\text{supp}\,\phi \subset K'$. The basic result (Theorem 10.6.2 in the aformentioned book) is that every convex open set $X\subset \mathbb{R}^n$ is $P$-convex for supports for every operator $P\not=0$. In particular, $P:C^\infty(X)\to C^\infty(X)$ is surjective for every convex open set $X\subset \mathbb{R}^n$.


