Each $f_j$ vanishes to order one on $N$ since $F^{-1}(z,w',w'')=(z,w',w''-h)=(z,w',t)$. $\blacksquare$

[**Proof of Corollary \[t3\].**]{} Let $M$ and $N$ be as in the hypothesis of theorem \[t3\]. After a linear of variables, we may assume that $p_0=0$ and that near the origin, $M$ is parametrized by $$M=\{z,u+iv(z,u):(z,u) \in \C^k \times \R^{p-k}\}.$$ By the implicit function theorem, we may assume that $N$ is given as a subset of $M$ by

$$\begin{cases}
u_{p-k-n}=\mu_1(z,u_1,...,u_{p-k-n-1}),...,
u_{p-k}=\mu_n(z,u_1,...,u_{p-k-n-1}),\\
\mu(0)=d\mu(0)=0.
\end{cases}$$ Denote by $s=(u_1,...,u_{p-k-n-1}) \in \R^m$ and $t=(u_{p-k-n},...,u_{p-k}) \in \R^n$. Setting $t'=t-\mu$, in the $(z,s,t')$ coordinates, we have $N$ given as a subset of $M$ by $t'=0$ and $$N=\{(z,w'(z,s),h(z,w'):(z,s)\in \C^k  \times \R^m\},$$ where $h$ is a CR map from $\n:=\{z,w'(z,s)\}$. We can now apply theorem \[t1\] to obtain the CR extension as the restriction of the holomorphic extension of $f$ to $\W \cap M$. The second part follows in the same manner. $\blacksquare$

The non Decomposable Case
-------------------------

[**Trépreau’s Example**]{}

We wish to note that on the particular class of manifolds of CR dimension $k$ containing through the origin $\C^k$, we do get a complex transversal extension result for any CR functions. Trépreau’s original example of a CR submanifold of $\C^3$ which admits non decomposable CR functions, 