Hypoelliptic operator

Summary

A hypoelliptic operator is a linear differential operator $P$ with smooth coefficients such that every distribution $u$ with $Pu$ smooth is itself smooth. So $P$ cannot create singularities in its solutions that are not already in the right-hand side. Elliptic operators with smooth coefficients are hypoelliptic, and so is the heat operator, which is not elliptic. Hörmander characterized hypoelliptic operators with constant coefficients and gave a wide class of hypoelliptic sums of squares of vector fields.