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CR function (Definition)
Definition 1   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 $ 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 1   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$.

Bibliography

1
M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. Real Submanifolds in Complex Space and Their Mappings, Princeton University Press, Princeton, New Jersey, 1999.



"CR function" is owned by jirka.
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Also defines:  CR distribution
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Cross-references: neighbourhood, open, real analytic, generic submanifold, real analytic submanifolds, easy to see, converse, holomorphic functions, restrictions, CR vector field, function, continuously differentiable, CR submanifold
There are 4 references to this entry.

This is version 2 of CR function, born on 2005-01-16, modified 2005-03-07.
Object id is 6647, canonical name is CRFunction.
Accessed 2321 times total.

Classification:
AMS MSC: 32V10 (Several complex variables and analytic spaces :: CR manifolds :: CR functions)
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