A sufficient condition for compactness of the \(\overline{\partial}\)-Neumann operator
\textit{D. W. Catlin} has proved [Proc. Symp. Pure Math. 41, 39-49 (1984; Zbl 0578.32031)] that if a domain \(\Omega\) satisfies the so-called condition \((P_1)\) (i.e. forall \(M>0\) there exists \(\varphi\in C^1(\Omega)\) such that \[ |\xi |\leq 1\text{ on }\Omega,\tag{1} \] \[ \sum_{k,l} {\partial^2 \varphi \over\partial z_l\partial \overline z_k}(p)\xi_k \overline\xi_l\geq M\|\xi \|^2 \text{ for }p\in b\Omega\text{ and }\xi\in \mathbb{C}^n),\tag{2} \] then the \(\overline \partial\)-Neumann operator acting on \((0,1)\)-forms is compact. The author gives a weaker condition, \((\widetilde P_q)\), which, if satisfied by the domain \(\Omega\), ensures the compactness of the \(\overline\partial\)-Neumann operator \(N_q\) (condition \((\widetilde P_q)\) is: for every \(M>0\) there exists \(\varphi= \varphi_M\in C^\infty (\Omega)\) such that (1) \(|\partial \varphi |_{i\partial\overline\partial_f}\leq 1\), (2) The sum of any \(q\) eigenvalues of the matrix \(({\partial^2 \varphi\over \partial z_k\partial \overline z_l})\) is \(\geq M\) for all \(z\in b\Omega)\). Here \(|\partial \varphi|_{i \partial \overline\partial f}\) denotes the length of the 1-form \(\partial \varphi\) measured in the Kähler metric with fundamental form \(i\partial \overline \partial f\). If \[ \left|\sum_k{\partial f\over\partial z_k} (z)\xi_k\right |^2\leq c\sum_{k,l} {\partial^2 f\over\partial z_k\partial \overline z_l}(z) \xi_k\overline \xi_l\qquad\text{for all }\xi\in\mathbb{C}^n\text{ and }z\in \Omega, \] the author uses the notation \(|\partial f|_{i\partial \overline \partial f}\leq C\) and says that \(f\in C^2(\Omega) \cup PSH(\Omega)\) has self-bounded complex gradient. But as \(-e^{-{1\over c}f}\in PSH(\Omega)\) is equivalent with the above inequality, the self-boundedness condition can be interpretated also for non-smooth functions. The main result is: If \(\Omega\) is a smoothly bounded, pseudoconvex, relatively compact domain in \(\mathbb{C}^n\) and if \(\Omega\) satisfies condition \((\widetilde P_q)\) then \(N_q\) is compact. This follows from a slightly improved Hörmander inequality (Proposition 3.1 and Proposition 3.2), an easy but useful abstract lemma, which ensures the compacity of an operator \(T\) provided it satisfies a family of estimates, and a duality argument. The author introduces also the condition \((\widetilde P_q)\) in the weak sense (\(\forall M>0\) \(\exists \varphi =\varphi_M\in PSH(\Omega)\) and a neighbourhood \(U\) of \(b\Omega\) such that, in the sense of currents (i) \(\partial \varphi\wedge \overline\partial \varphi\leq i\partial\overline \partial \varphi\) on \(\Omega\), (ii) \(i\partial\overline \partial\geq Mi\partial\overline \partial |z|^2\) in \(U\cap\Omega)\). By Straube regularity techniques one gets that even \((\widetilde P_q)\) in the weak sense ensures the compactness of the \(\overline\partial\)-Neumann operator \(N_q\) for \(\Omega\) a bounded pseudoconvex domain in \(\mathbb{C}^n\). The author shows also, in the case of a complex ellipsoid in \(\mathbb{C}^2\), the simplification that the use of the \(\widetilde P_1\) condition, in showing the compactness of \(N_1\), brings in the Catlin construction.
scientific article; zbMATH DE number 1888346
scientific article