---
tag: JFKYYK2CEB
kind: concept
name: CR function
msc: [32V10, 32V05]
sources:
  - url: https://en.wikipedia.org/wiki/CR_manifold
---

## Summary

A CR function on a [[CR manifold|ZC78JKZX9S]] is a function annihilated by the anti-holomorphic tangent vectors of the CR structure, see the [[definition|Z23FCT4WTZ]].
On a real hypersurface of $\C^n$, these are the solutions of the [[tangential Cauchy–Riemann equations|X7GP9BECD8]], and the restriction of any holomorphic function is one.
The [[Lewy extension theorem|V2YCGHD719]] shows that on a strongly pseudoconvex hypersurface, CR functions extend locally holomorphically to one side.
So they are locally boundary values of holomorphic functions.

## Explanations

### L1 human

On a surface in $\C^n$ one cannot ask a function to be holomorphic, since the surface is not a complex space.
One can still ask it to satisfy the Cauchy–Riemann equations in the directions that the surface allows.
These are the CR functions.
Restrictions of holomorphic functions are the main examples, and in good cases every CR function is of this kind.