The complex Green operator on CR-submanifolds of \(\mathbb{C}^{n}\) of hypersurface type: compactness (Q2841346)

A compact set \(K\subset \mathbb{C}^n\) satisfies property \((P_q)\) if for every \(A>0\) there is a \(\mathcal{C}^2\) function \(\lambda_A\) defined in some neighborhood \(U_A\) of \(K\) such that \(0\leq \lambda_A(z) \leq 1\), \(z\in U_A\) and NEWLINE\[NEWLINE\sum_{|J|=q-1}' \sum_{j,k=1}^n\frac{\partial^2 \lambda_A(z)}{\partial z_j \partial \overline z_k}\, w_{jJ} \overline {w_{kJ}} \geq A |w|^2 \, , \quad z\in U_A,\quad w\in \Lambda_z^{0,q},NEWLINE\]NEWLINENEWLINENEWLINE where \( \Lambda_z^{0,q}\) denotes the space of \((0,q)\)-forms at \(z.\) It is shown that for a smooth compact pseudoconvex CR-submanifold \(M\) of hypersurface type with property \((P_q)\) and property \((P_{m-1-q})\), \(1\leq q \leq m-2\), the following compactness estimate holds: for all \(\epsilon >0\), there exists a constant \(C_\epsilon\) such that NEWLINENEWLINENEWLINE\[NEWLINE\|u\|_{L^2_{0,q}(M)} \leq \epsilon \Big( \|\overline \partial_b u \|_{L^2_{0,q+1}(M)} + \|\overline \partial_b^* u \|_{L^2_{0,q-1}(M)} \Big) + C_\epsilon \|u\|_{W^{-1}_{(0,q)}(M)}NEWLINE\]NEWLINENEWLINENEWLINE for all \(u\in {\text{dom}}(\overline \partial_b) \cap {\text{dom}}(\overline \partial_b^*)\), here \(m-1\) is the CR-dimension of \(M.\) In addition, it is proved that \(M\) satisfies property \((P_q)\) on the nullspace of the Levi form if and only if \(M\) satisfies property \((P_q).\) The author points out that A. Raich developed microlocal methods to derive compactness estimates when \(M\) satisfies a CR-analogue of property \((P)\) and is also orientable [\textit{A. Raich}, Math. Ann. 348, No. 1, 81--117 (2010; Zbl 1238.32032)]. In this paper a new proof is given which does not require orientability.