Let $n\ge 2$, let $M\subset \mathbb C^n$ be a $C^\infty$ embedded real hypersurface, let $p\in M$, and let $\Omega^+$ be a chosen local side of $M$ near $p$. Choose an open neighbourhood $U\subset\mathbb C^n$ of $p$ and a real-valued $C^\infty$ local defining function $\rho:U\to\mathbb R$ such that

\begin{align*} M\cap U=\{z\in U:\rho(z)=0\}. \end{align*}

Assume that $d\rho_q\ne 0$ for every $q\in M\cap U$ and that

\begin{align*} \Omega^+\cap U=\{z\in U:\rho(z)>0\}. \end{align*}

Let $\Omega^-\cap U:=\{z\in U:\rho(z)<0\}$ be the opposite local side. Let $T^{1,0}_pM:=T^{1,0}_p\mathbb C^n\cap (T_pM\otimes_{\mathbb R}\mathbb C)$ be the CR tangent space, and let

\begin{align*} \mathcal L_{\rho,p}:T^{1,0}_pM\times T^{1,0}_pM\to\mathbb C \end{align*}

be the Hermitian Levi form determined by $\rho$. Assume that $\mathcal L_{\rho,p}$ has at least one positive eigenvalue. Assume as analytic input Lewy's local one-sided [extension theorem](/theorems/59) in the following precise form: for a $C^\infty$ real hypersurface with a $C^\infty$ defining function $r$ whose Levi form has a positive eigenvalue at a point, every $C^\infty$ CR function near that point extends holomorphically, smoothly up to the edge, into the side $\{r<0\}$. Then every function $u\in C^\infty(M\cap U;\mathbb C)$ satisfying $\bar\partial_bu=0$ on $M\cap U$ admits a one-sided holomorphic extension to the side $\Omega^-$. More precisely, after possibly shrinking $U$, there exist an open neighbourhood $V\subset U$ of $p$ in $\mathbb C^n$, an [open set](/page/Open%20Set) $W\subset V\cap \Omega^-$ with $p\in\overline W$ and $M\cap V\subset \partial W$, and a function

\begin{align*} F:W\to\mathbb C \end{align*}

holomorphic on $W$ and smooth up to $M\cap V$, such that

\begin{align*} F|_{M\cap V}=u|_{M\cap V}. \end{align*}