[@[Tu-2]] as a well as in a recent paper by Eastwood and Graham [@[Ea-Gr]]. For general background on CR geometry, we recommend the books by Baouendi, Ebenfelt, Rothschild [@[BER]], Boggess [@[Bo]] and Jacobowitz [@[Ja]] and the survey paper on holomorphic extension by Trépreau [@[Tr-2]].

Outline of the Paper
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The main part of this paper is the proof of theorem\[t1\], we then obtain theorem \[th\] as a corollary. We show that it suffices to consider the case where $N$ is given as a trivial CR graph, that is $N=\{(\n,0)\}$. We then solve a Dirichlet problem on some open subset $\Omega$ of $\n \times \R^n$, by Treves’s analytic vector theory, this yields holomorphy in the $w$ variables. To obtain holomorphy in $z$ we need to proceed with a deformation on $\n$ so that the “boundary” of $\Omega$ (boundary with respect to the variables of the elliptic operator) $\partial
\Omega$ is a CR submanifold of $\n \times \R^n$ of same CR dimension.

Acknowledgements
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The author would like to thank Jean-Pierre Rosay for some helpful remarks and fruitful discussions as well as Jean-Michel Bony for the references for the Dirichlet problem for elliptic operators and Joel Merker for the help with the 