By lemma \[LL1\] $  L_j(S(g))$ vanishes on $\partial \Omega$, hence by theorem \[proba\] we have $  L_j(S(g))=0$ in $\Omega$. $\blacksquare$

Analytic Vector Theory
----------------------

In this section we present the results developed by Baouendi and Treves [@[Ba-Tr]] and later by Treves in [@[Tre]]. We have included this section for the sake of completeness and claim no originality whatsoever. We will state the main results needed here, sending the reader who wishes to read the details of the proofs to [@[Tre]].

Using the notations $R_{m+j}={{\partial}\over{\partial t_j}}$ and $R^{\alpha}=R_1^{\alpha_1}...R_{m+n}^{\alpha_{m+n}}$ for $\alpha \in
\NN^{m+n}$, we shall say that a continuous function $f$ in $\omega$ is an analytic vector of the system of vector fields $\{R_1,...,R_{n+m}\}$ if $R^{\alpha}f \in \CC^0 $ for any $\alpha \in
\NN^{n+m}$, and if to every compact set $K$ of $\omega$ there is a constant $\rho>0$ such that, in $K$, $$\label{2.2.1}
\sup_{\alpha \in
\NN^{n+m}} \left ( \rho^{|\alpha|} {{ |R^{\alpha}f|} \over {\alpha!}}
\right )<\infty.$$

Let $p
=(z,  w(z,s_0,t_0)) \in \Omega$ so that with our previous notation $(s_0,t_0) \in \omega$. We are using the following convention

$$(z,  w(z,s_0,t_0)) \in \Omega \Leftrightarrow (z,s_0,t_0) \in
\B_{3\epsilon}\times \omega.$$

\[p5\] (Lemma 4.1 in [@[Ba-Tr]]). Let $p
=(z,  w(z,s_0,t_0)) \in \Omega$, let $\U$ be an open neighborhood 