Extending Cr Functions from Manifolds with Boundaries

The extendibility of CR functions on smooth real manifolds in C has been extensively studied by many specialists (see, e.g., [Bo] for references). In this paper we generalize the results of [T1] and [T2] on the extendibility of CR functions to manifolds with boundaries and edges. Recall that a generic manifold M ⊂ C is said to be minimal at p ∈ M if there is no proper CR submanifold S ⊂ M passing through p with the same CR dimension as M . According to [T1], if M is minimal at p, then all CR functions on M extend to the same full dimensional wedge with edge M near p. Conversely, Baouendi and Rothschild [BR] show that if M is not minimal at p, then there are CR functions in a neighborhood of p on M that do not extend to any full dimensional wedge with edge M . Nevertheless, all CR functions on M generally still extend to a larger manifold [BP] [T2]. This manifold has the form of a wedge of dimension lower than the dimension of the ambient space. We call it a manifold with edge M , reserving the term wedge for full dimensional sets in C . Since the natural domains of extended CR functions are manifolds with edges, it is appropriate to consider CR functions initially defined on such sets. The simplest case of a manifold with edge is a manifold with boundary. Let M ⊂ C be a manifold with generic boundary M0. Suppose M is minimal at all interior points. Then by [T1] all CR functions on M extend over a wedge W with edge M . A natural question is the boundary behavior of this wedge at M0. We show that W approaches M0 as a wedge with edge M0 (Corollary 1.3). In the general case, we introduce the defect of a manifold M with edge M0 at a point p ∈ M0 and show that unless the defect is maximal, all CR