Hörmander's theorem on hypoelliptic PDEs - existence of solution? I've seen many authors state that Hörmander theory implies the existence of a $C^\infty$ solution. For example, on Wikipedia it says:

The great achievement of Hörmander's 1967 paper was to show that a smooth fundamental solution exists under a considerably weaker assumption: the parabolic version of the condition that now bears his name.

http://en.wikipedia.org/wiki/H%C3%B6rmander%27s_condition#Application_to_the_Cauchy_problem
It is popular to cite:
Hörmander, Lars (1967). "Hypoelliptic second order differential equations". Acta Math. 119: 147–171
But from what I can understand, the main theorem 1.1 (usually referred to as "Hörmander's Theorem") says (roughly) that if a second order differential operator $P$ satisfies some conditions then it is hypoelliptic. Which in turn means that if $Pu$ is smooth, then $u$ must be smooth.
But this does not tell me anything about the existence of a solution to the equation $Pu=f$. I understand that if there exist a solution to $Pu=f$ and $f$ is smooth, then $u$ must be smooth if $P$ is hypoelliptic.
Am I missing something essential?
Thank you in advance!
 A: If you take a look at Hörmander's paper, the paragraph immediately following the statement of Theorem 1.1 states:

It is a simple exercise to verify that for the equation (1.2), the condition in Theorem 1.1 is the same that we needed to construct a smooth fundamental solution by the method of Kolmogorov [8].

So, if $P$ satisfies the conditions of the theorem, it seems that Kolmogorov has an argument that can then be used to construct the desired smooth fundamental solution.  The paper of Kolmogorov that Hörmander references is
Kolmogorov, A.  "Zufallige Bewegungen (Zur Theorie der Brownschen Bewegung)".  Annals of Mathematics, Vol. 35, No. 1, January, 1934.
Unfortunately, this paper is in German, so I can't read it, but this might get you started in the right direction.
A: To expand on @user48944's answer, the argument is actually given in Hörmander's paper. On page 148, he writes

This means that the null space of A contains a non-trivial invariant subspace for B. If this is not the case we obtain by inverting the Fourier transformation a two-sided fundamental solution which is a $C^{\infty}$ function off the diagonal.

and then

Thus it follows that (1.2) is hypoelliptic unless the null space of A contains a non-trivial subspace which is invariant for B.

Hörmander's Theorem 1.1 gives a sufficient condition for hypoellipticity. By the remark after Theorem 1.1, this condition, when applied to equation (1.2), reduces to the condition that A does not contain a non-trivial subspace which is invariant for B (this is the simple exercise). Therefore, the condition that guarantees hypoellipticity for equation (1.2) also guarantees the existence of a fundamental solution (and a way to construct it due to Kolmogorov).
The reference to Kolmogorov's paper is provided only to give him credit because he first introduced the method described on page 148 for studying the simpler equation (1.1).
