Regularity in the local CR embedding problem

TL;DR

This paper proves that for certain CR manifolds, local holomorphic embeddings are regular in H"older spaces, with the regularity depending on the smoothness of the manifold, using advanced homotopy formulas and Nash-Moser techniques.

Contribution

It establishes the regularity of local CR embeddings in H"older spaces for manifolds of class C^{m}, extending previous existence results with precise regularity estimates.

Findings

Embeddings are of class C^{a} for all a < m + 1/2 when the manifold is C^{m}.

The method combines Henkin's homotopy formula with a modified Nash-Moser argument.

Provides regularity results for CR embeddings beyond existence, in standard function spaces.

Abstract

We consider a formally integrable, strictly pseudoconvex CR manifold $M$ of hypersurface type, of dimension $2n-1\geq7$. 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 H\"older spaces $C^{a}(M)$, $a\in\mathbf{R}$. If the structure of $M$ is of class $C^{m}$, $m\in\mathbf{Z}$, $4\leq m\leq\infty$, we construct a local CR embedding near each point of $M$. This embedding is of class $C^{a}$, for every $a$, $0\leq a < m+(1/2)$. 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.