Hypoelliptic operator

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


 * $$U \subset{\mathbb{R}}^n$$

is called hypoelliptic if for every distribution $$u$$ defined on an open subset $$V \subset U$$ such that $$Pu$$ is $$C^\infty$$ (smooth), $$u$$ must also be $$C^\infty$$.

If this assertion holds with $$C^\infty$$ replaced by real-analytic, then $$P$$ is said to be analytically hypoelliptic.

Every elliptic operator with $$C^\infty$$ 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 ($$P(u)=u_t - k\,\Delta u\,$$)
 * $$P= \partial_t - k\,\Delta_x\,$$

(where $$k>0$$) is hypoelliptic but not elliptic. However, the operator for the wave equation ($$P(u)=u_{tt} - c^2\,\Delta u\,$$)
 * $$      P= \partial^2_t - c^2\,\Delta_x\,$$

(where $$c\ne 0$$) is not hypoelliptic.