CR function


Let MN be a CR submanifold and let f be a Ck(M) (k times continuously differentiable) function to where k1. Then f is a CR function if for every CR vector field L on M we have Lf0. A distribution ( f on M is called a CR distribution if similarly every CR vector field annihilates f.

For example restrictionsPlanetmathPlanetmath of holomorphic functionsMathworldPlanetmath in N to M are CR functions. The converseMathworldPlanetmath 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.


Let MCN be a generic submanifold which is real analytic (the defining function is real analytic). And let f:MC 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 CN.


  • 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