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

TL;DR

This paper establishes that CR-hypoellipticity at a point on a CR submanifold is both necessary and sufficient for the holomorphic extension of CR functions, with applications to embeddings and pseudoconcavity conditions.

Contribution

It proves the equivalence between CR-hypoellipticity and holomorphic extension for CR functions, and applies this to embedding and extension problems under pseudoconcavity.

Findings

CR-hypoellipticity is necessary and sufficient for holomorphic extension.

CR-hypoellipticity implies the existence of generic embeddings.

Holomorphic extension holds for CR manifolds with higher order Levi pseudoconcavity.

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 $p_0\in{M}$ is necessary and sufficient for holomorphic extension of all germs of $CR$ functions to an ambient neighborhood in $X$. As an application, we obtain that $CR$-hypoellipticity implies the existence of generic embeddings and prove holomorphic extension for a large class of $CR$ manifolds satisfying a higher order Levi pseudoconcavity condition.