$U\; subset\{mathbb\{R\}\}^n$

is called hypoelliptic if for every distribution $u$ defined on an open subset $V\; subset\; U$ such that $Pu$ is $C^infty$ (smooth), $u$ must also be $C^infty$.

If this assertion holds with $C^infty$ replaced by real analytic, then $P$ is said to be analytically hypoelliptic.

Every elliptic operator is hypoelliptic. In particular, the Laplacian is an example of a hypoelliptic operator (the Laplacian is also analytically hypoelliptic). The heat equation operator

$P(u)=u\_t\; -\; kDelta\; u,$

(where $k>0$) is hypoelliptic but not elliptic. The wave equation operator

$P(u)=u\_\{tt\}\; -\; c^2Delta\; u,$

(where $cne\; 0$) is not hypoelliptic.

References

• Shimakura, Norio Partial differential operators of elliptic type: translated by Norio Shimakura. American Mathematical Society, Providence, R.I.
• Egorov, Yu. V.; Schulze, Bert-Wolfgang Pseudo-differential operators, singularities, applications. Birkhäuser.
• Vladimirov, V. S. Methods of the theory of generalized functions. Taylor & Francis.