of (i). We thus determine the $R_j$ by solving for their coefficients in (i) (see for example [@[BER]] Lemma 8.7.13 page 234).

\[r1\] We note that the $R_j$’s verifies the following property,

if $j \leq m$ then $$R_j|_{0}= R_j|_{\C^k\times \left [\R^{m}\setminus
B_{2\epsilon}\right ]}={{\partial}\over {\partial s_j}}.$$

The equivalent of lemma \[l3\] for $\n\times\R^n$, is given by

\[l’3\] The $m+n$ vector fields given by $\{R_1,....,R_m,{{\partial}\over{\partial
t_1}},...,{{\partial}\over{\partial t_n}}\}$ verify the same properties as in lemma \[l3\], where we replace $  \n$ by $\n\times\R^n$.

Define $\Delta_{\n\times\R^n}$ by

$$\label{e1.1}
\Delta_{\n\times\R^n}=\sum_{j=1}^{m}R_j^2 +\sum_{j=1}^{n}
{{\partial^2}\over{{\partial
t_j}}^2} .$$

From \[r1\] and lemma \[l3\] (iii) we deduce immediately the following:

\[p2\] The operator $-\Delta_{\n\times\R^n}$ is strongly elliptic of degree two on $\n\times\R^n$ with smooth ($\CC^{\infty}$) coefficients and no constant terms.

$\Delta_{\n\times\R^n}$ is a differential operator in the variables $s$ and $t$ whose coefficient functions depend smoothly on the $z$ variable. We are going to study a Dirichlet problem on $\n\times\R^n$. To do this, we need $\Omega$ some open set in $\n\times\R^n$ with boundary $\partial \Omega$ parametrized by some closed submanifold of $\R^{n+m}$ of codimension one (recall that $\Delta_{\n\times\R^n}$ is a differential operator in the variables $s$ and $t$).

The reason we went to the trouble of redefining $  \n$ was 