Embedding of pseudoconvex CR manifolds of Levi-forms with one degenerate eigenvalue. (Q1879992)

The author extends the results of \textit{M. Kuranishi} [Ann. Math. (2) 115, No. 3, 451--500 (1982; Zbl 0505.32018)], \textit{T. Akahori} [Mem. Am. Math. Soc. 366 (1987; Zbl 0628.32025)], \textit{S. M. Webster} [Ann. Inst. H. Poincaré, Anal. Non Linéaire 6, No. 3, 183--207 (1989; Zbl 0679.32020)] and \textit{D. W. Catlin} [J. Geom. Anal. 4, No. 4, 467--538 (1994; Zbl 0841.32012)], that states that a strictly pseudoconvex abstract CR hypersurface \(M\) of real dimension \(\geq 7\) is locally embeddable into a complex manifold \(X\), to the case where the Levi form of \(M\) is allowed to have one zero eigenvalue. The proof follows Catlin's approach, first giving a one-side extension of the CR structure to a complex manifold with boundary. This requires establishing subelliptic estimates with weights for the \(D_q\)-Neumann problem for an almost-complex structure and the use of the Nash-Moser iteration process.