Regularity in the local CR embedding problem

Details Regularity in the local CR embedding problem Regularity in the local CR embedding problem

2009-11-24

arxiv.org/abs/0911.4550v1

Mathematics

Complex Variables

Scientific paper

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.

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