On the smoothness of Levi-foliations (Q1114049)

Let S be a real hypersurface in \({\mathbb{C}}^ n\) of class \({\mathcal C}^ k\) with vanishing Levi form. The maximal complex subspace is an integral distribution of class \({\mathcal C}^{k-1}\) and real co-dimension 1 so that S admits a foliation of class \({\mathcal C}^{k-1}\) by complex hypersurfaces. The authors show that in fact this unique foliation is of class \({\mathcal C}^ k\).