Subellipticity at higher degree of a boundary condition associated with construction of the versal family of strongly pseudo-convex domains (Q1092289)

Let N be a complex manifold of \(\dim_{{\mathbb{C}}}N=n\geq 4\), \(\Omega\) a relatively compact domain of N with a strongly pseudoconvex boundary \(\partial \Omega =M\) and \({}^ 0T''\) a CR structure on M induced from the complex structure on N. The purpose of this paper is to prove: (1) If \(2\leq q\leq n-2\), then there exist positive constants c and c' such that \[ c'\| \phi \|^ 2_{}\leq \| \phi \|^{'2} \leq c(\| {\bar \partial}\phi \|^ 2+ \| \vartheta \phi \|^ 2+ \| \phi \|^ 2) \] for any \(\phi\in \Gamma ({\bar \Omega},T'N\otimes \Lambda^ q(T''N)^*)\) satisfying \(\tau\phi\in \Gamma (M,Eq)\) and \(<\sigma (\vartheta,dr)\phi,y>=0\) on M for all \(y\in E_{q-1}.\) (2) If \(2\leq q\leq n-2\), then there exists a positive constant c such that \[ \| \phi \|^{'2}\leq c(\| {\bar \partial}_ b\phi \|^ 2+ \| (\vartheta_ b\phi)_{E_{q-1}}\|^ 2+ \| \phi \|^ 2) \] for any \(\phi\in \Gamma (M,Eq)\), where \((\vartheta_ b\phi)_{E_{q-1}}\) denotes the orthogonal projection onto \(E_{q-1}\) with respect to the Levi metric. (3) If \(1\leq q\leq n-3\), then there exists a positive constant c such that \[ \| \phi \|^{'2}\leq c(\| ({\bar \partial}_ b\phi)_{E^{\perp}_{p+1}}\|^ 2+\| \vartheta_ b\phi \|^ 2+ \| \phi \|^ 2) \] for any \(\phi \in \Gamma (M,E_ q^{\perp}).\) The proofs of the theorems (2), (3) are higher degree versions of the ones of degree \(q=2\) of \textit{T. Akahori}, Invent. Math. 68, 317-352 (1982; Zbl 0575.32021) and \(q=1\) in [\textit{T. Akahori}, Math. Ann. 264, 525-535 (1983; Zbl 0575.32022)], respectively.