# hypoelliptic

###### Definition.

Let $P$ be a partial differential operator defined in an open subset $U\subset{\mathbb{R}}^{n}$. If for every distribution (http://planetmath.org/Distribution4) $u$ defined in an open subset $V\subset U$ such that $Pu$ is $C^{\infty}$ (smooth), $u$ must also be $C^{\infty}$, then $P$ is called hypoelliptic.

Similarly, if the same assertion holds with $C^{\infty}$ replaced by real analytic, then $P$ is said to be analytically hypoelliptic.

Note that some authors use “hypoelliptic” to mean “analytically hypoelliptic.” Hence, if it is not clear from context, it is best to specify the regularity when using the term. For example, $C^{\infty}$-hypoelliptic instead of just hypoelliptic.

## References

• 1 J. Barros-Neto, Ralph A. Artino. , Lecture Notes in Pure and Applied Mathematics, 53. Marcel Dekker, Inc., New York, 1980. http://www.ams.org/mathscinet-getitem?mr=81k:35031MR 81k:35031
• 2 Bernard Helffer, Francis Nier. , Lecture Notes in Mathematics, 1862. Springer-Verlag, Berlin, 2005. http://www.ams.org/mathscinet-getitem?mr=2006a:58039MR 2006a:58039
• 3 Norio Shimakura. , Kinokuniya Company Ltd., Tokyo, Japan, 1978.
Title hypoelliptic Hypoelliptic 2013-03-22 16:01:13 2013-03-22 16:01:13 jirka (4157) jirka (4157) 7 jirka (4157) Definition msc 35H10 analytically hypoelliptic analytic hypoelliptic