Lewy extension theorem

Let $M\subset {𝐂}^n$ be a smooth real hypersurface.Let $\rho$ be a defining function for $M$ near $p.$ That is, for some neighbourhoodof $p,$ the submanifold $M$ is defined by $\rho =0.$For a neighbourhood $U\subset {ℂ}^n,$ 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 negativeeigenvalue. That is, if

Theorem.

Let $f$ be a smooth CR function on $M\mathrm{.}$ Suppose thatnear $p\mathrm{\in }M$ the Levi form of $M$ has at least one positive eigenvalue at $p\mathrm{.}$ Then there exists aneighbourhood $U$ of $p\mathrm{,}$ such that for every smooth CR function $f$ on $M\mathrm{,}$ there exists afunction $F$ 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 forat least one negative eigenvalue.If the Levi form of $M$ has both positive and negative eigenvalues at a point, then $f$ extends to both sidesof $M$ 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 ${ℂ}^2$ in coordinates $(z,w)$ by $\mathrm{Im}w=0.$ The domains for $ϵ>0,$ are pseudoconvex and hencethere exist functionsholomorphic on ${\mathrm{\Omega }}_{ϵ}$ (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 $U_{ϵ}.$ So at least one nonzero eigenvalue of the Levi formis 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 resultsin this direction since Lewy’s original paper, but this is the most result.