Hypoelliptic operator

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 with coefficients is hypoelliptic. In particular, the Laplacian is an example of a hypoelliptic operator (the Laplacian is also analytically hypoelliptic). In addition, the operator for the heat equation () (where ) is hypoelliptic but not elliptic. However, the operator for the wave equation () (where ) is not hypoelliptic.