In the theory of partial differential equations, a partial differential operator formula_1 defined on an open subset 
formula_2 
is called hypoelliptic if for every distribution formula_3 defined on an open subset formula_4 such that formula_5 is formula_6 (smooth), formula_3 must also be formula_6.
If this assertion holds with formula_6 replaced by real-analytic, then formula_1 is said to be "analytically hypoelliptic". 
Every elliptic operator with formula_6 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 (formula_12) 
formula_13 
(where formula_14) is hypoelliptic but not elliptic. However, the operator for the wave equation (formula_15) 
formula_16 
(where formula_17) is not hypoelliptic.
References.
"This article incorporates material from Hypoelliptic on PlanetMath, which is licensed under the ."