Regularity in the local CR embedding problem (Q692173)

Let \(M\) be an abstract strictly pseudoconvex CR manifold of hypersurface type of dimension \(2n - 1 \geq 7\). By ``CR manifold'' we mean that there exists a complex subbundle \(L \subset {\mathbb C} \otimes TM\), that is formally integrable, \([L, L] \subset L\), and almost Lagrangian, \(L \cap \overline{L} = \{ 0\}\). By ``hypersurface type'' we mean that \(L\) is of rank \(n\). If the Levi-form \(\frac{1}{2i} [ v, \bar{w} ] \bmod L \oplus \overline{L}\) is positive definite then \(M\) is said to be strictly pseudoconvex. Such manifolds for dimension 9 and higher are known to be locally integrable by the work of Kuranishi and in dimension 7 by the work of Akahori. By integrability we mean that there exist \(n\) independent solutions \(z_1,\ldots,z_n\) to \(L\), giving a local embedding as an embedded strictly pseudoconvex submanifold of \({\mathbb C}^n\). In this paper it is proved that if \(M\) is of class \(C^m\) for an integer \(m\), \(4 \leq m \leq \infty\), then there exists an embedding as above of class \(C^a\) for \(0 \leq a \leq m + \frac{1}{2}\).