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.
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!