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 $U\subset{\mathbb{C}}^{n},$ 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

\sum_{j,k=1}^{n}\frac{\partial^{2}\rho(p)}{\partial z_{j}\partial\bar{z}_{k}}w% _{j}\bar{w}_{k}<0~{}\text{ for some }~{}w\in{\mathbb{C}}^{n}~{}\text{ such % that }~{}\sum_{j=1}^{n}w_{j}\frac{\partial\rho(p)}{\partial z_{j}}=0.

Theorem.

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

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 ${\mathbb{C}}^{2}$ in coordinates $(z,w)$ by $\operatorname{Im}w=0.$ The domains $U_{\epsilon}:=\{\lvert\operatorname{Im}w\rvert<\epsilon\},$ for $\epsilon>0,$ are pseudoconvex and hence there exist functions holomorphic on $\Omega_{\epsilon}$ (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 $U_{\epsilon}.$ 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