$CR$ Extension from manifolds of higher type

Abstract

In this paper, a generalization of the "sector property" theorem first pioneered by Baouendi, Rothschild and Treves is given. The main contribution consists in showing that if a submanifold of $\C^n$ with higher codimension is locally presented in a weighted normal form, similar to that described by Bloom and Graham, then a characterization is given as to when a vector in the tangent space belongs to the analytic wave front set for the set of locally defined CR functions on the submanifold. This characterization is described in terms of the sign of the inner product of this vector with the local graphing functions for the submanifold on sectors of suitable size along complex lines in its tangent space. Examples are given to show that under certain circumstances, the results are sharp. Previous results by Baouendi et. al. contained a semi-rigidity assumption which is not assumed in the present paper. The hypoanalytic wave front set determines the cone of directions in which CR functions extend analytically to the ambient space and thus provides an explicit description of the local hull of holomorphy of a submanifold of $\C^n$.