\begin{definition}[Definition:Hypoelliptic Operator]
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'''.
\end{definition}