# CR function

###### Definition.

Let $M\subset{\mathbb{C}}^{N}$ be a CR submanifold and let $f$ be a $C^{k}(M)$ ($k$ times continuously differentiable) function to ${\mathbb{C}}$ where $k\geq 1$. Then $f$ is a CR function if for every CR vector field $L$ on $M$ we have $Lf\equiv 0$. A distribution (http://planetmath.org/Distribution4) $f$ on $M$ is called a CR distribution if similarly every CR vector field annihilates $f$.

For example restrictions of holomorphic functions in ${\mathbb{C}}^{N}$ to $M$ are CR functions. The converse is not always true and is not easy to see. For example the following basic theorem is very useful when you have real analytic submanifolds.

###### Theorem.

Let $M\subset{\mathbb{C}}^{N}$ be a generic submanifold which is real analytic (the defining function is real analytic). And let $f\colon M\to{\mathbb{C}}$ be a real analytic function. Then $f$ is a CR function if and only if $f$ is a restriction to $M$ of a holomorphic function defined in an open neighbourhood of $M$ in ${\mathbb{C}}^{N}$.

## References

• 1 M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. , Princeton University Press, Princeton, New Jersey, 1999.
Title CR function CRFunction 2013-03-22 14:57:10 2013-03-22 14:57:10 jirka (4157) jirka (4157) 5 jirka (4157) Definition msc 32V10 CR distribution