[proofplan]
The proof reduces boundary extension to a compact-support $\bar\partial$ problem in the interior. First one extends the CR boundary function smoothly into $\overline{\Omega}$ so that its $\bar\partial$ is a compactly supported $\bar\partial$-closed $(0,1)$-form in $\Omega$. Strict pseudoconvexity and the hypothesis $n\ge 2$ give compact-support solvability for this form. Subtracting the compactly supported solution produces a [holomorphic function](/page/Holomorphic%20Function) with the prescribed boundary value, and uniqueness follows from the maximum principle.
[/proofplan]
[step:Reduce the boundary extension problem to compactly supported $\bar\partial$ solvability]
By [citetheorem:9213], applied to the bounded smooth strictly pseudoconvex domain $\Omega\subset\mathbb C^n$ with $n\ge 2$ and to the CR function $u\in C^\infty(M;\mathbb C)$, there exists a smooth extension
\begin{align*}
\widetilde u:\overline{\Omega}\to \mathbb C
\end{align*}
with $\widetilde u|_M=u$ such that the $(0,1)$-form
\begin{align*}
f:=\bar\partial \widetilde u
\end{align*}
belongs to $C_c^\infty(\Omega;\Lambda^{0,1})$ and satisfies $\bar\partial f=0$ in $\Omega$. The compact support means that there is a compact set $K\subset\Omega$ such that $\operatorname{supp} f\subset K$.
[guided]
We need to convert a boundary-value problem into an interior equation. The input is a smooth CR function
$u\in C^\infty(M;\mathbb C)$, where $M=\partial\Omega$. The CR condition is exactly the compatibility condition that permits a smooth extension into the collar of the boundary whose failure to be holomorphic vanishes to infinite order at $M$. The reduction theorem [citetheorem:9213] packages this collar construction and the cutoff argument.
The hypotheses of [citetheorem:9213] are satisfied here: $\Omega\subset\mathbb C^n$ has smooth strictly pseudoconvex boundary, $n\ge 2$, and $u$ is a smooth CR function on $M$. Therefore there is a function
\begin{align*}
\widetilde u:\overline{\Omega}\to \mathbb C
\end{align*}
with $\widetilde u\in C^\infty(\overline{\Omega};\mathbb C)$ and $\widetilde u|_M=u$, such that
\begin{align*}
f:=\bar\partial\widetilde u
\end{align*}
is a smooth compactly supported $(0,1)$-form on $\Omega$. In symbols,
\begin{align*}
f\in C_c^\infty(\Omega;\Lambda^{0,1}).
\end{align*}
The same reduction also gives $\bar\partial f=0$. This last identity is the necessary compatibility condition for solving $\bar\partial v=f$, since $\bar\partial^2=0$ forces every right-hand side of a $\bar\partial$ equation to be $\bar\partial$-closed.
[/guided]
[/step]
[step:Solve the compactly supported $\bar\partial$ equation in the interior]
The form $f$ satisfies the hypotheses of [citetheorem:9214]: the domain $\Omega$ is bounded and smoothly strictly pseudoconvex, $n\ge 2$, $f\in C_c^\infty(\Omega;\Lambda^{0,1})$, and $\bar\partial f=0$. Hence there exists a function
\begin{align*}
v:\Omega\to\mathbb C
\end{align*}
with $v\in C_c^\infty(\Omega)$ such that
\begin{align*}
\bar\partial v=f.
\end{align*}
Because $v$ has compact support in $\Omega$, there is an open neighbourhood $U_M\subset\mathbb C^n$ of $M$ such that $v=0$ on $U_M\cap\Omega$.
[/step]
[step:Subtract the interior solution to obtain the holomorphic extension]
Define
\begin{align*}
F:\Omega \to \mathbb C,\qquad z \mapsto \widetilde u(z)-v(z)
\end{align*}
Since $\widetilde u\in C^\infty(\overline{\Omega};\mathbb C)$ and $v\in C_c^\infty(\Omega)$ vanishes on a neighbourhood of $M$, extending $v$ by $0$ near $M$ shows that $F\in C^\infty(\overline{\Omega};\mathbb C)$. Moreover,
\begin{align*}
\bar\partial F=\bar\partial\widetilde u-\bar\partial v=f-f=0
\end{align*}
in $\Omega$. Therefore $F\in\mathcal O(\Omega)$.
Since $v=0$ on $U_M\cap\Omega$, the trace of $v$ on $M$ is $0$. Hence
\begin{align*}
F|_M=\widetilde u|_M-v|_M=u-0=u.
\end{align*}
Thus $F\in\mathcal O(\Omega)\cap C^\infty(\overline{\Omega};\mathbb C)$ is the required extension.
[/step]
[step:Apply the maximum principle to prove uniqueness on connected domains]
Assume now that $\Omega$ is connected. Let
\begin{align*}
F_1,F_2:\Omega\to\mathbb C
\end{align*}
be two functions in $\mathcal O(\Omega)\cap C^\infty(\overline{\Omega};\mathbb C)$ satisfying $F_1|_M=F_2|_M=u$. Define
\begin{align*}
G:\Omega \to \mathbb C,\qquad z \mapsto F_1(z)-F_2(z)
\end{align*}
Then $G\in\mathcal O(\Omega)\cap C^\infty(\overline{\Omega};\mathbb C)$ and $G|_M=0$.
Since $\Omega$ is bounded, $\overline{\Omega}$ is compact in $\mathbb C^n$. The [continuous function](/page/Continuous%20Function) $|G|:\overline{\Omega}\to[0,\infty)$ attains its maximum on $\overline{\Omega}$. The maximum principle for holomorphic functions on bounded connected domains, applied to $G$ (citing a result not yet in the wiki: maximum principle for holomorphic functions), gives
\begin{align*}
\sup_{z\in\Omega}|G(z)|\le \sup_{\zeta\in M}|G(\zeta)|.
\end{align*}
Because $G|_M=0$, the right-hand side is $0$, and therefore $G=0$ on $\Omega$. Hence $F_1=F_2$, so the extension is unique when $\Omega$ is connected.
[/step]
