Definition:Hypoelliptic Operator

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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.


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