Normal forms for generic manifolds and holomorphic extension of CR functions (Q1093775)

The following situation is considered: Let M be a generic CR manifold i.e. M is defined near the origin of \({\mathbb{C}}^{n+l}\) by Im w\(=\Phi (z,\bar z,Re w)\), with \(z\in {\mathbb{C}}^ n\), \(w\in {\mathbb{C}}^ l\) and \(\Phi\) a smooth \({\mathbb{R}}^ l\)-valued function defined near the origin of \({\mathbb{R}}^{2n+l}\) with \(\Phi (0)=0\) and \(d\Phi (0)=0\). A wedge of edge M is an open set of \({\mathbb{C}}^{n+l}\) of the form \[ W(U,C)= \{(z,w)\in U| \quad Im w - \Phi (z,\bar z,Re w)\in C\} \] where U is a neighbourhood of 0 in \({\mathbb{C}}^{n+l}\) and C is a convex open cone in \({\mathbb{R}}^ l.\) The main results of this paper give normal forms for coordinates on M, conditions for local holomorphic extendability to a wedge of edge M and decomposition of CR distributions as boundary values of holomorphic functions. In particular the class of semirigid M's is considered: semirigidity means essentially that \(\Phi\) can be chosen to depend, in appropriate local coordinates on \(Re w\) only of higher order; in particular all hypersurfaces are semirigid. Then we have: Theorem 1. If M is semirigid of finite type, then every CR function on M extends to be holomorphic in a wedge with edge M. Theorem 2. Let M be generic CR and let T be a CR distribution on M with hypoanalytic wavefront contained in a disjoint union of strictly convex closed cones in \({\mathbb{R}}^ l-\{0\}\) then T is a finite sum of boundary values of holomorphic functions in some wedges.