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 vectorfield $L$ on $M$ we have $Lf\equiv 0$. Adistribution (http://planetmath.org/Distribution4)$f$ on $M$ is called aCR 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 tosee. For example the following basic theorem is very useful when you havereal analytic submanifolds.

Theorem.

Let $M\mathrm{\subset }{\mathrm{C}}^N$ be a generic submanifold which is realanalytic (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 ifand only if $f$ is a restriction to $M$ of a holomorphic functiondefined in an open neighbourhood of $M$ in ${\mathrm{C}}^N$.