, more specifically in the theory of partial differential equations
, a partial differential operator
defined on an open subset
is called hypoelliptic if for every distribution defined on an open subset such that is (smooth), must also be .
If this assertion holds with replaced by real analytic, then 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
(where ) is hypoelliptic but not elliptic. The wave equation operator
(where ) is not hypoelliptic.
- 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.