Let $N\in\mathbb N$, let $0<\alpha<1$, and let $M\subset\mathbb C^N$ be a generic embedded CR submanifold of class $C^{2,\alpha}$, with CR tangent bundle $T^cM:=TM\cap iTM$. Let $U\subset M$ be open in the relative topology, and let

\begin{align*} \gamma:[0,1]\to U \end{align*}

be a continuous curve for which there is a finite partition $0=t_0<t_1<\cdots<t_m=1$ such that $\gamma|_{[t_{k-1},t_k]}$ is $C^1$ for each $1\le k\le m$ and, at every $t\in[0,1]$ where $\dot\gamma(t)$ exists,

\begin{align*} \dot\gamma(t)\in T^c_{\gamma(t)}M. \end{align*}

Let

\begin{align*} f:U\to\mathbb C \end{align*}

be a continuous CR function on $U$. A wedge with edge $M\cap V$ at $p\in M$ means an [open set](/page/Open%20Set) $W\subset V\subset\mathbb C^N$ which, in local $C^{2,\alpha}$ coordinates near $p$ and after choosing a real normal complement to $T_pM$, contains all sufficiently small points obtained from an edge neighbourhood in $M\cap V$ by adding normal vectors in a non-empty open cone; that cone is called the normal cone of the wedge. Suppose that, for $p_0:=\gamma(0)$, there are an open neighbourhood $V_0\subset\mathbb C^N$ of $p_0$, an open wedge $W_0\subset V_0$ with edge $M\cap V_0$ at $p_0$, and a [holomorphic function](/page/Holomorphic%20Function)

\begin{align*} F_0:W_0\to\mathbb C \end{align*}

such that $F_0$ has continuous boundary value $f$ on the edge portion $M\cap V_0\cap U$.

Then, for every $t\in[0,1]$, there are an open neighbourhood $V_t\subset\mathbb C^N$ of $\gamma(t)$, an open wedge $W_t\subset V_t$ with edge $M\cap V_t$ at $\gamma(t)$, and a holomorphic function

\begin{align*} F_t:W_t\to\mathbb C \end{align*}

such that $F_t$ has continuous boundary value $f$ on $M\cap V_t\cap U$. The neighbourhood $V_t$, the wedge $W_t$, and the normal cone of $W_t$ may depend on $t$ and may be smaller than the initial wedge data.