## Local fundamental solution for hypoelliptic operators

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 $$P(K_x)=\delta_x.$$

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 $$P_xK(x,y)=\delta(x-y),$$ (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.

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 I believe that all you need to check is that $K_x$ depends continuously on $x$. This is more or less the same as showing that $K_x$ depends continuously on the coefficients of $P$. If this is not explicitly discussed in any reference you've looked at, then you just need to check that continuity with respect to the parameter $x$ is preserved at every step in the construction of $K_x$. – Deane Yang Feb 7 2012 at 10:15