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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!

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1 Answer

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.

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