Regularity for the CR vector bundle problem. I (Q541359)

Let \(\xi=\{E@>>>M\}\) be a locally trivial complex vector bundle over a \(CR\) manifold \(M\). The integrability problem for \(\xi\) is stated in terms of a connection \(D\) on \(\xi\). For a frame field \(e=(e_1,\hdots,e_r)\), setting \[ De=\omega\otimes{e},\quad\Omega=d\omega-\omega\wedge\omega, \] the integrability condition is expressed by the fact that the components \(\Omega^{i}_j\) of \(\Omega\) belong to the ideal \(\mathcal{J}(M)\) of \((1,0)\)-forms of \(M\). The question is to find a new frame \(\tilde{e}\) for which also the components \(\tilde{\omega}^i_j\) of the corresponding \(\tilde{\omega}\) belong to \(\mathcal{J}(M)\). If \(\tilde{e}=Ae\), this translates into the equations \[ \omega+A^{-1}\bar\partial_MA=0,\quad \bar\partial_M\omega=\omega\wedge\omega. \] The authors consider the case where \(M\) is a strongly pseudoconvex hypersurface in \(\mathbb{C}^n\), of class \(\mathcal{C}^{\ell}\) with \(\ell\geq{5}\), and \(\roman{dim}_{\mathbb{R}}M=2n-1\geq{7}\). They prove that, when \(\omega\) is of class \(\mathcal{C}^k\) with \(1\leq{k}\leq \ell-4\), then there is a local solution \(A\) of class \(\mathcal{C}^{k,\alpha}\) for \(0\leq\alpha\leq\frac{1}{2}\). In particular \(A\in\mathcal{C}^{\infty}\) when both \(M,\omega\) are of class \(\mathcal{C}^{\infty}\). For the special case where \(M\) is the Heisenberg group, they also obtain a more precise result in the Folland-Stein spaces. Namely, they show that, for \(\omega\) of class \(\mathcal{C}_{FS}^{k,\alpha}\), with \(1\leq{k}<\infty\) and \(0<\alpha<\frac{1}{2}\), then the local solution \(A\) can be taken in \(\mathcal{C}_{FS}^{k+1,\alpha}\). The proofs utilize the KAM rapid convergence argument, and avoid the more difficult Nash-Moser methods previously employed in the study of similar problems.