# Lewy extension theorem

Let $M\subset {\mathbf{C}}^{n}$ be a smooth real hypersurface.
Let $\rho $ be a defining function for $M$ near $p.$ That is, for some neighbourhood
of $p,$ the submanifold^{} $M$ is defined by $\rho =0.$
For a neighbourhood $U\subset {\u2102}^{n},$ define the set
${U}_{+}$ to be the set $U\cap \{\rho >0\}.$ We will say that
$M$ has at least one negative eigenvalue if the Levi form defined by $\rho $ has at least one negative
eigenvalue. That is, if

$$ |

###### Theorem.

Let $f$ be a smooth CR function on $M\mathrm{.}$ Suppose that near $p\mathrm{\in}M$ the Levi form of $M$ has at least one positive eigenvalue at $p\mathrm{.}$ Then there exists a neighbourhood $U$ of $p\mathrm{,}$ such that for every smooth CR function $f$ on $M\mathrm{,}$ there exists a function $F$ holomorphic in ${U}_{\mathrm{+}}$ and ${C}^{\mathrm{1}}$ up to $M\mathrm{,}$ such that ${F\mathrm{|}}_{U\mathrm{\cap}M}\mathrm{=}{f\mathrm{|}}_{U\mathrm{\cap}M}\mathrm{.}$

By considering $-\rho $ instead of $\rho $ as a defining function, we get the corresponding result for at least one negative eigenvalue. If the Levi form of $M$ has both positive and negative eigenvalues at a point, then $f$ extends to both sides of $M$ and is then a restriction of a holomorphic function.

A point is the fact that $U$ is fixed and does not depend on $f.$ To see why this is necessary, imagine a Levi flat example. Let $M$ be defined in ${\u2102}^{2}$ in coordinates $(z,w)$ by $\mathrm{Im}w=0.$ The domains $$ for $\u03f5>0,$ are pseudoconvex and hence there exist functions holomorphic on ${\mathrm{\Omega}}_{\u03f5}$ (and hence CR on $M$) that do not extend past any point of the boundary. No neighbourhood of a point on $M$ fits in all ${U}_{\u03f5}.$ So at least one nonzero eigenvalue of the Levi form is needed.

The statement of this theorem is not exactly the theorem that Lewy formulated[4], but this is generally called the Lewy extension. There have been many results in this direction since Lewy’s original paper, but this is the most result.

## References

- 1 M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. , Princeton University Press, Princeton, New Jersey, 1999.
- 2 Albert Boggess. , CRC, 1991.
- 3 Lars Hörmander. , North-Holland Publishing Company, New York, New York, 1973.
- 4 Hans Lewy. Ann. of Math. (2) 64 (1956), 514–522.

Title | Lewy extension theorem |
---|---|

Canonical name | LewyExtensionTheorem |

Date of creation | 2013-03-22 17:39:44 |

Last modified on | 2013-03-22 17:39:44 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 4 |

Author | jirka (4157) |

Entry type | Theorem |

Classification | msc 32V25 |

Synonym | Lewy extension |