p4.pstex\_t

Denote by $(D)$ the Dirichlet problem on $\Omega$, i.e. $${(D)}~~~~~~~~~~~~~~~~~
\begin{cases}
\Delta_{\n\times\R^n} (u)=0~~{\rm in}~\Omega,\\
u=g~~{\rm on}~\partial \Omega.
\end{cases}$$

We now have the following result, which seems well known but is not so easy to find in the literature.

\[proba\] For $g \in \D'(\partial \Omega)$, $(D)$ has a unique solution.

For a reference, we send the reader to [@[C-P]] theorem 5.2 and remark 5.3 page 263-265, as well as [@[Ho]] chapter 10.

Let $S$ be the solution operator for $(D)$, that is for $g \in \D'(\partial \Omega)$, we have

$$\label{ee2}
\begin{cases}
\begin{array}{l}
\Delta_{\n\times\R^n} (S(g))=0~~{\rm in}~~\Omega,\\
S(g)=g ~~{\rm on}~\partial \Omega.
\end{array}
\end{cases}$$

\[p3\] Let $  L_j \in  
\L$ and let $g$ be a CR distribution on $\partial {\Omega}$ then we have $$L_j(S(g))=0.$$

\[LL1\] If $f$ is a CR distribution on $\partial \Omega $ then for any $  L_j \in   \L$ we have $$L_j(S(f))|_{\partial \Omega}=0.$$

[**Proof of Lemma \[LL1\].**]{} The lemma holds since $\partial \Omega$ is a CR submanifold of $\n\times\R^n$ and $f$ is a CR distribution on $\partial \Omega$, so $ L_j(S(f))|_{\partial \Omega}=  L|_{\partial \Omega}(f|_{\partial \Omega})=0$.

[**Proof of Proposition \[p3\].**]{} Consider $  L_j(S(g))$. By lemma \[l3\] (iii) we have $$\Delta_{\n\times\R^n}   L_j(S(g))=  L_j\Delta_{\n\times\R^n}(
S(g)),$$ hence we have $$\label{e1.3}
\Delta_{\n\times\R^n} \left [  L_j(S(g))\right]=0.$$

