# Definition:Hypoelliptic Operator

## Definition

Let $T \in \map {\DD'} \R$ be a distribution.

Let $T_f$ and $T_g$ be distributions associated with real functions $f$ and $g$.

Let $D$ be a differential operator.

Suppose in the distributional sense it holds that:

$D T_f = T_g$

Suppose:

$g \in \map {\CC^\infty} \R \implies f \in \map {\CC^\infty} \R$

where $\map {\CC^\infty} \R$ denotes the space of smooth real functions.

Then $D$ is called the hypoelliptic operator.