# CR function

###### Definition.

Let $M\subset {\u2102}^{N}$ be a CR submanifold and let $f$ be a ${C}^{k}(M)$ ($k$ times continuously differentiable) function to $\u2102$ where $k\ge 1$. 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 ${\u2102}^{N}$ 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 $M\mathrm{\subset}{\mathrm{C}}^{N}$ be a generic submanifold which is real analytic (the defining function is real analytic). And let $f\mathrm{:}M\mathrm{\to}\mathrm{C}$ 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 ${\mathrm{C}}^{N}$.

## References

- 1 M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. , Princeton University Press, Princeton, New Jersey, 1999.

Title | CR function |
---|---|

Canonical name | CRFunction |

Date of creation | 2013-03-22 14:57:10 |

Last modified on | 2013-03-22 14:57:10 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 5 |

Author | jirka (4157) |

Entry type | Definition |

Classification | msc 32V10 |

Defines | CR distribution |