Regularity in the local CR embedding problem

2009/11/24 by Xianghong Gong, Gong, Xianghong, S. M. Webster +1 · 1 citation
Mathematics · #32V30 #35N10 #Complex Variables (math.CV) #FOS: Mathematics #math.CV #msc:32V30 #msc:35N10

paper · pdf · doi:10.48550/arxiv.0911.4550

arxiv created 2009/11/24 · arxiv updated 2009/12/08

Abstract

We consider a formally integrable, strictly pseudoconvex CR manifold M of hypersurface type, of dimension 2n-1≥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\inR. If the structure of M is of class C^m, m\inZ, 4≤ m≤∞, we construct a local CR embedding near each point of M. This embedding is of class C^a, for every a, 0≤ 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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