CR function

Definition.

Let $M\subset {ℂ}^N$ be a CR submanifold and let $f$ be a $C^k(M)$ ($k$ times continuously differentiable) function to $ℂ$ 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 ${ℂ}^N$ 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 $M\mathrm{\subset }{\mathrm{C}}^N$ 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 ${\mathrm{C}}^N$.

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