Let $p$ be a polynomial such that $p(D)$ is a hypoelliptic operator, i.e. if $u$ is a distribution satisfying $p(D)u = 0$ then $u$ is smooth. Here, $D = -i\partial$.
Let $E$ be a fundamental solution for $p(D)$; that is, assume that $E$ is a distribution such that $p(D)E = \delta$ where $\delta$ is the Dirac distribution.
I have read in a few texts and resources that since $p(D)E = 0$ away from the origin, $E$ must be smooth away from $0$. However, in every reference I've found, this has been stated without any kind of proof or justification. Nonetheless, this is not at all clear to me.
To prove the statement more rigorously, I tried multiplying $E$ by cut-off functions but that idea doesn't seem to lead anywhere. How could I go about proving the result?
EDIT: I know by assumption that if $u$ satisfies $p(D)=0$ everywhere then it is smooth. If $E$ is a fundamental solution for $p(D)$ then $p(D)E = 0$ away from the origin. So if I can show that p(D) is "local" I would be done. By this, I meant that I would like to show the following;
If $p(D)u = 0$ in some neighbourhood, then $u$ is smooth in that neighbourhood.
When I tried to introduce a cut-off function $\zeta$ as in the comments, I was still only able to obtain $p(D)(E\zeta) = 0$ outside of some small set - but not everywhere. So I was not able to directly apply my hypothesis.