$\mathcal{C}^{\infty}$-hypoellipticity and extension of CR functions

Abstract

Let M be a CR submanifold of a complex manifold X. The main result of this article is to show that CR-hypoellipticity at p0 ϵ M is necessary and sufficient for holomorphic extension of all germs at p0 of CR functions on M to an ambient neighborhood of p0 in X. As an application, we obtain that CR-hypoellipticity implies the existence of global generic embeddings and prove holomorphic extension for a large class of CR manifolds satisfying a higher order Levi pseudoconcavity condition. We also obtain results on the relationship of holomorphic wedge-extension and the C∞-wave front set for CR distributions.