Elliptic regularity

In the theory of partial differential equations, a partial differential operator ${\displaystyle P}$ defined on an open subset

is called hypoelliptic if for every distribution ${\displaystyle u}$ defined on an open subset ${\displaystyle V\subset U}$ such that ${\displaystyle Pu}$ is ${\displaystyle C^{\infty }}$ (smooth), ${\displaystyle u}$ must also be ${\displaystyle C^{\infty }}$.

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