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:
- $g \in \map {C^\infty} \R \implies f \in \map {C^\infty} \R$
where $\map {C^\infty} \R$ denotes the space of smooth real functions.
Then $D$ is called the hypoelliptic operator.
Also see
- Results about hypoelliptic operators can be found here.
Sources
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis: Chapter $\S 6.4$: Multiplication by $C^\infty$ functions