$\CC^{\infty}$ functions so that

$$\label{e8}
\begin{cases}
b_j|_{\B_{\epsilon} \times B_{\epsilon}}=a_j|_{\B_{\epsilon} \times
B_{\epsilon}},\\
b_j=d~{\rm if}~s \in \R^{m} \setminus
B_{2\epsilon}
\end{cases}$$

Define the generic submanifold of $\C^k \times \C^m$ $\ti \n$ as follows

$$\label{e9}
\ti \n=\{(z,s+ib(z,s)):(z,s) \in \C^k \times \R^m\}.$$

[**Notation.**]{} We shall use the following convention, if $M$ is a manifold defined in a neighborhood of the origin, parametrized by $\C^k \times \R^{m}$, then $M|_{B_r}$ is defined by

$$M|_{B_r} =\{(Z(z,x)):x \in B_r\}.$$ With the above notation we then have $$\ti \n|_{ B_{\epsilon}}= \n|_{ B_{\epsilon}},$$

which we illustrate with the following picture.

p2.pstex\_t

\[p1\] For $\epsilon$ small enough, there exists $b_j$ as in \[e8\] so that $f$ (as in proposition \[Prop1\]) extends to a CR distribution on $\ti \n$ in a $\eta$ neighborhood of the origin.

\[coro1\] There is no loss of generality in assuming that the functions $a$ in \[e1\] verify \[e8\], i.e., one can replace $\n$ by $\ti \n$.

[**Proof of Proposition \[p1\].**]{} The main tool is the following lemma.

\[l1\] There exists $b_j $ so that for $\epsilon$ small enough we have

$$\left (\ti \n \cap V_{\eta}\right ) \subset \overline \W,$$ where $\W$ and $V_{\eta}$ are as in \[e4\].

[**Proof of Lemma \[l1\].**]{}

By \[e2\] we have $$\label{e10}
\|a_j\|_{L^{\infty}(\B_{3\epsilon} \times 