Holomorphic extension of decomposable distributions from a CR submanifold of $\mathbb C^L$

Abstract

Given $N$ a non generic smooth CR submanifold of $\C^L$, $N=\{(\n,h(\n))\}$ where $\n$ is generic in $\C^{L-n}$ and $h$ is a CR map from $\n$ into $\C^n$. We prove, using only elementary tools, that if $h$ is decomposable at $p'\in \n$ then any decomposable CR distribution on $N$ at $p=(p',h(p'))$ extends holomorphically to a complex transversal wedge. This gives an elementary proof of the well known equivalent for totally real non generic submanifolds, i.e if $N$ is a smooth totally real submanifold of $\C^L$ any continuous function on $N$ admits a holomorphic extension to a complex transverse wedge