On the regularity of CR structures for almost CR vector bundles (Q1346325)

Let \(M\) be a \({\mathcal C}^ m\)-smooth strongly pseudoconvex real hypersurface in \(\mathbb{C}^ n\), \((E,D)\) a \({\mathcal C}^ m\)-smooth almost CR vector bundle over \(M\). It is proved that \((E,D)\) is a \({\mathcal C}^ k\)- smooth CR vector bundle, if \(m \geq k + 7\), \(k \geq 6\). This is an improvement of \textit{S. M. Webster}'s result in Proc. Symp. Pure Math. 52, Part 3, 355-368 (1991; Zbl 0744.32002), which requires \(m \geq 4n + 5k + 13\), \(k \geq 8\).