[proofplan]
We first prove the Levi-flat assertion in an explicit flat model. The hypersurface is the real hyperplane $\operatorname{Im} z_n=0$, whose CR directions are exactly the complex tangential directions in the first $n-1$ variables. We choose a smooth one-variable boundary germ that is not the smooth boundary value of a [holomorphic function](/page/Holomorphic%20Function) from either side, multiply it by a nonzero holomorphic factor in the leaf variables, and observe that any ambient extension would restrict to a forbidden one-variable extension. The final assertion is precisely Henkin's constructed degenerate mixed-Levi counterexample, which is used here as an external existence theorem rather than derived from the displayed Levi-sign conditions alone.
[/proofplan]
[step:Choose a smooth transverse germ with no one-sided holomorphic extension]
We use the following classical one-variable fact: there exists a function $a\in C^\infty((-1,1);\mathbb C)$ such that neither of the following alternatives holds near $0$:
there are $\varepsilon>0$ and $A_+\in\mathcal O(\{w\in\mathbb C: |w|<\varepsilon,\operatorname{Im}w>0\})$ with $A_+$ extending $C^\infty$ to $\{w\in\mathbb C: |w|<\varepsilon,\operatorname{Im}w\ge 0\}$ and satisfying
\begin{align*}
A_+(t)=a(t)
\end{align*}
for all $t\in(-\varepsilon,\varepsilon)$;
and there are $\varepsilon>0$ and $A_-\in\mathcal O(\{w\in\mathbb C: |w|<\varepsilon,\operatorname{Im}w<0\})$ with $A_-$ extending $C^\infty$ to $\{w\in\mathbb C: |w|<\varepsilon,\operatorname{Im}w\le 0\}$ and satisfying
\begin{align*}
A_-(t)=a(t)
\end{align*}
for all $t\in(-\varepsilon,\varepsilon)$.
This is the standard smooth non-extendable boundary germ in one complex variable; it may be obtained, for example, by prescribing incompatible positive and negative microlocal boundary spectrum at the origin.
[guided]
The transverse variable in the Levi-flat model will be a real coordinate $t$. To obstruct extension to either side, it is not enough to choose $a$ merely non-real-analytic: smooth boundary values of one-sided holomorphic functions need not be real-analytic. What is required is stronger. We choose a germ
\begin{align*}
a:(-1,1)\to\mathbb C
\end{align*}
with $a\in C^\infty((-1,1);\mathbb C)$ and with no smooth holomorphic filling from either the upper or lower half-plane.
Concretely, this means that there is no $\varepsilon>0$ and no holomorphic function
\begin{align*}
A_+:\{w\in\mathbb C: |w|<\varepsilon,\operatorname{Im}w>0\}\to\mathbb C
\end{align*}
which is $C^\infty$ up to the boundary interval and satisfies $A_+(t)=a(t)$ for all real $t$ with $|t|<\varepsilon$. The analogous lower half-plane condition is also excluded. This is a classical one-variable boundary-value construction: one prescribes a smooth germ whose microlocal boundary spectrum contains obstructions for both choices of side. This is the only one-variable input needed in the Levi-flat part.
[/guided]
[/step]
[step:Build the Levi-flat hypersurface and the CR function]
Write points of $\mathbb C^n$ as $z=(z',z_n)$, where $z'=(z_1,\dots,z_{n-1})\in\mathbb C^{n-1}$ and $z_n=s+it$ with $s,t\in\mathbb R$. Define
\begin{align*}
M:=\{(z',z_n)\in\mathbb C^{n-1}\times\mathbb C:\operatorname{Im}z_n=0\}.
\end{align*}
Let
\begin{align*}
p:=(0,\dots,0)\in M.
\end{align*}
The function
\begin{align*}
\rho:\mathbb C^n&\to\mathbb R
\end{align*}
\begin{align*}
(z',z_n)&\mapsto \operatorname{Im}z_n
\end{align*}
is a $C^\infty$ defining function for $M$, and $d\rho\ne0$ on $M$. Hence $M$ is a $C^\infty$ embedded real hypersurface. Its complex tangent bundle is spanned by the coordinate vector fields $\partial_{\bar z_1},\dots,\partial_{\bar z_{n-1}}$, and therefore its Levi form is identically zero. Thus $M$ is Levi-flat.
Define
\begin{align*}
u:M&\to\mathbb C
\end{align*}
\begin{align*}
(z',s)&\mapsto a(s).
\end{align*}
Since $a\in C^\infty((-1,1);\mathbb C)$, after multiplying by a smooth cutoff in the $s$ variable away from $0$ if desired, $u$ is a $C^\infty$ function on a neighbourhood of $p$ in $M$, and it extends to a global smooth function on $M$. For each $j\in\{1,\dots,n-1\}$,
\begin{align*}
\partial_{\bar z_j}u=0
\end{align*}
on $M$, because $u$ depends only on $s=\operatorname{Re}z_n$. Hence $u$ is CR on $M$.
[/step]
[step:Restrict any side extension to a complex normal line]
Suppose first that $u$ has a holomorphic extension to the side $\{\operatorname{Im}z_n>0\}$ near $p$. Then there are an open neighbourhood $U\subset\mathbb C^n$ of $p$ and a holomorphic function
\begin{align*}
F_+:U\cap\{\operatorname{Im}z_n>0\}\to\mathbb C
\end{align*}
with smooth boundary value $u|_{U\cap M}$. Choose $\varepsilon>0$ such that
\begin{align*}
\{(0,\dots,0,w)\in\mathbb C^n: |w|<\varepsilon\}\subset U.
\end{align*}
Define
\begin{align*}
A_+:\{w\in\mathbb C: |w|<\varepsilon,\operatorname{Im}w>0\}&\to\mathbb C
\end{align*}
\begin{align*}
w&\mapsto F_+(0,\dots,0,w).
\end{align*}
Since $F_+$ is holomorphic, $A_+$ is holomorphic. Since $F_+$ has smooth boundary value $u$ on $M$, $A_+$ has smooth boundary value
\begin{align*}
A_+(s)=u(0,\dots,0,s)=a(s)
\end{align*}
for all $s\in(-\varepsilon,\varepsilon)$. This contradicts the choice of $a$.
The same argument applies to the side $\{\operatorname{Im}z_n<0\}$. If
\begin{align*}
F_-:U\cap\{\operatorname{Im}z_n<0\}\to\mathbb C
\end{align*}
were a holomorphic function with smooth boundary value $u|_{U\cap M}$, then
\begin{align*}
A_-:\{w\in\mathbb C: |w|<\varepsilon,\operatorname{Im}w<0\}&\to\mathbb C
\end{align*}
\begin{align*}
w&\mapsto F_-(0,\dots,0,w)
\end{align*}
would be a forbidden lower half-plane holomorphic extension of $a$. Therefore $u$ has no local holomorphic extension to either side of $M$ at $p$.
[/step]
[step:Invoke Henkin's degenerate mixed-Levi construction]
For $n\ge3$, Henkin's degenerate mixed-Levi non-extension construction gives a $C^\infty$ embedded real hypersurface
\begin{align*}
M_H\subset\mathbb C^n,
\end{align*}
a point $p_H\in M_H$, and a function
\begin{align*}
u_H:M_H\to\mathbb C
\end{align*}
with $u_H\in C^\infty(M_H;\mathbb C)$ such that $u_H$ is CR on $M_H$, the Levi form at $p_H$ is degenerate, every neighbourhood of $p_H$ contains points where the Levi form has a positive eigenvalue and points where it has a negative eigenvalue, and $u_H$ has no local holomorphic extension to either side of $M_H$ at $p_H$.
This cited Henkin construction is a special model construction: the obstruction is arranged by the tangential Cauchy-Riemann cohomology and support properties of the datum, not merely by the existence of mixed signs in the Levi form. Hence it supplies exactly the second existential assertion of the theorem.
[/step]
[step:Combine the explicit flat model with Henkin's model]
The first three steps produce, for every $n\ge2$, a $C^\infty$ embedded Levi-flat real hypersurface $M\subset\mathbb C^n$, a point $p\in M$, and a smooth CR function $u\in C^\infty(M;\mathbb C)$ with no local holomorphic extension to either local side at $p$. The Henkin construction in the preceding step supplies, for every $n\ge3$, the asserted degenerate mixed-Levi model and its two-sided non-extension datum. These are precisely the two assertions in the theorem.
[/step]
