Let $n\ge 2$, let $\Omega\subset\mathbb C^n$ be a bounded domain with $C^\infty$ strictly pseudoconvex boundary $M:=\partial\Omega$, and let $u\in C^\infty(M;\mathbb C)$ be a CR function on $M$. Then there exists a function $\widetilde u\in C^\infty(\overline\Omega;\mathbb C)$ such that $\widetilde u|_M=u$ and the $(0,1)$-form
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\begin{align*}
f:=\bar\partial \widetilde u
\end{align*}
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belongs to $C_c^\infty(\Omega;\Lambda^{0,1}T^*\Omega)$ and satisfies $\bar\partial f=0$.
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Assume moreover that every form $f\in C_c^\infty(\Omega;\Lambda^{0,1}T^*\Omega)$ with $\bar\partial f=0$ admits a solution $v\in C_c^\infty(\Omega;\mathbb C)$ such that
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\begin{align*}
\bar\partial v=f.
\end{align*}
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Then there exists a [holomorphic function](/page/Holomorphic%20Function) $F\in\mathcal O(\Omega)\cap C^\infty(\overline\Omega;\mathbb C)$ such that $F|_M=u$.