# 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, 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 LewyExtensionTheorem 2013-03-22 17:39:44 2013-03-22 17:39:44 jirka (4157) jirka (4157) 4 jirka (4157) Theorem msc 32V25 Lewy extension