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, 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. In addition, the operator for the heat equation
is hypoelliptic but not elliptic. However, the operator for the wave equation
is not hypoelliptic.