I am looking for a reference for the following proposition:
Let $D: \mathcal{V} \to \mathcal{W}$ be an elliptic operator between smooth vector bundles on a smooth manifold $M$. Then the operator $D$ is an epimorphism of sheaves (i.e. surjective on stalks).
I wrote this in my notes years ago but can't seem to find a reference for it. This is the proof that I wrote when I was more familiar with this material, but I don't remember how I came up with it:
"We want to show that $D$ is surjective on germs, i.e. $D: \mathcal{V}_x \to \mathcal{W}_x$ is surjective for every $x \in M$. In other words given a germ $v$ in $\mathcal{F}_x$, we want to solve $D u = v$. Given some germ in $\mathcal{W}_x$, pick some representative section $\tau$ defined on some open neighbourhood $U$ of $x$. Now pick some compact neighbourhood $K$ of $x$ contained in $U$, and solve $D\sigma=\tau$ on $K$. By elliptic regularity, there exists a smooth solution $\sigma$. Taking germs, we get $D \sigma_x = \tau_x$, as required."