Regularity in the local CR embedding problem

Abstract

We consider a formally integrable, strictly pseudoconvex CR manifold $M$ of hypersurface type, of dimension $2n-1\ge 7$. Local CR, i.e. holomorphic, embeddings of $M$ are known to exist from the works of Kuranishi and Akahori. We address the problem of regularity of the embedding in standard Hölder spaces $C^a(M)$, $a\in \mathbf{R}$. If the structure of $M$ is of class $C^m$, $m\in \mathbf{Z}$, $4\le m\le \infty$, we construct a local CR embedding near each point of $M$. This embedding is of class $C^a$, for every $a$, $0\le a<m+(1/2)$. Our method is based on Henkin's local homotopy formula for the embedded case, some very precise estimates for the solution operators in it, and a substantial modification of a previous Nash-Moser argument due to the second author.