hypoelliptic

Definition.

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

Similarly, if the same assertion holds with $C^{\mathrm{\infty }}$ replaced byreal 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, $C^{\mathrm{\infty }}$-hypoelliptic instead of just hypoelliptic.