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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 with coefficients 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 (1992). Partial differential operators of elliptic type: translated by Norio Shimakura. American Mathematical Society, Providence, R.I. ISBN0-8218-4556-X.