[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 $\operatorname{supp} f\subset K$.[/step]

[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: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]