Let $P=\sum_\alpha c_\alpha(x)\partial^\alpha$ be a partial differential operator with smooth coefficients in an open set of $\mathbb{R}^n$ ($\partial^\alpha=\frac{\partial^{\alpha_1+\cdots+\alpha_n}}{\partial x_1^{\alpha_1}\cdots\partial x_n^{\alpha_n}}$, as usual).
It's a theorem that if its formal adjoint $P^\star$ is hypoelliptic, than $P$ is locally solvable, which means in particular that for any $x$ we can find a distribution $K_x$, defined in a neighborhood of $x$, such that \begin{equation}P(K_x)=\delta_x.\end{equation}
Is it possible to glue this distributions together to find a $K\in \mathcal{D}'(U\times U)$ ($U$ a small open set) such that \begin{equation}P_xK(x,y)=\delta(x-y),\end{equation} (here $P_y$ is $P$ acting on the $y$ variable).
This could be a very well-known fact, but I didn't find a reference.
