In the theory of partial differential equations, a partial differential operator defined on an open subset

is called hypoelliptic if for every distribution (smooth.

is said to be analytically hypoelliptic.

Every elliptic operator 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)

) is hypoelliptic but not elliptic. However, the operator for the wave equation)

) is not hypoelliptic.

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