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

If this assertion holds with

is said to be analytically hypoelliptic.

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 (

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