generic submanifolds. We wish to point out the differences between our results and the previously known results on holomorphic extension of CR functions. The most general result on holomorphic extension is Tumanov’s theorem which states that if $M$ is a generic submanifold of $\C^L$ that is minimal at some $p_0$ then there exists a wedge $\W$ with edge $M$ (near $p_0$), such that one gets a “forced” unique local holomorphic extension of CR functions defined in a neighborhood of $p_0$ in $M$.

One should note that the question of CR extension can be viewed as a Cauchy problem with Cauchy data on a characteristic set $N$. In [@[Ei]] we constructed an example of an abstract CR structure where there is no such CR extension propriety. It is very easy to construct an abstract CR structure where there is no CR function vanishing on $N$ to finite order (see the example at the end of this paper).

Off course the situation is greatly different if we impose holomorphic extension to a full neighborhood of $N$, as the next example shows.

[**Example**]{} Let $I_n~~n>0$ be a sequence of disjointed intervals (separated by some open sets) in $\R$ accumulating to the origin and 