Complex manifolds in \(q\)-convex boundaries (Q2354339)

Let \(b\Omega\) be \(q\)-convex, let \(M \subset b\Omega\) be a CR submanifold which is complex tangential of finite bracket type, i.e., the subsequent brackets of \(\mathcal{C}^\infty\) vector fields with values in \(T^{\mathbb{C}}M\) generate the whole tangent bundle \(TM\). Let \(\mathcal{W}\) be the wedge complexification of \(M\). It is shown that \(\mathcal{W}\) is tangent to \(b\Omega\) of infinite order along \(M\). In addition, running the Kohn algorithm, the authors prove a subelliptic estimate for \(q\)-forms: assume that in a neighborhood of \(z_0, \;b\Omega\) is real analytic, \(q\)-convex, and contains no germ of a holomorphic manifold of dimension \(\geq q\). Then a subelliptic estimate in degree \(k\geq q\) for the \(\overline \partial\)-Neumann problem holds in a neighborhood \(U\) of \(z_0,\) that is, for some \(\epsilon >0\) one has \(\|u\|^2_\epsilon \lesssim Q(u,u)\) for any \(u\in {\text{dom}\,}\overline \partial^* \cap \mathcal{C}_c^\infty (\overline \Omega \cap U)^k\).