Hypoelliptic operator
In the theory of partial differential equations, a partial differential operator P {\displaystyle P} defined on an open subset U ⊂ R n {\displaystyle U\subset {\mathbb {R} }^{n}} is called hypoelliptic if for every distribution u {\displaystyle u} defined on an open subset V ⊂ U {\displaystyle V\subset U} such that P u {\displaystyle Pu} is C ∞ {\displaystyle C^{\infty }} (smooth), u {\displaystyle u} must also be C ∞ {\displaystyle C^{\infty }} . If this assertion holds with C ∞ {\displaystyle C^{\infty }} replaced by real-analytic, then P {\displaystyle P} is said to be analytically hypoelliptic.