Global regularity for the \(\bar{\delta}\)-Neumann operator and bounded plurisubharmonic exhaustion functions (Q643445)

Let \(\Omega =\{ \rho <0 \} \subset \mathbb{C}^n\) be a smooth, bounded, pseudoconvex domain. Suppose that there are plurisubharmonic functions \(\lambda_\epsilon \) with self-bounded gradient and transverse \((1,0)\)-vector fields \(v_\epsilon\) satisfying \(|\arg d\rho (v_\epsilon)| < \epsilon\) such that \[ i\partial \overline \partial \lambda_\epsilon \geq i \, \frac{1}{\epsilon^2} \, \frac{(d\rho([\overline \partial, v_\epsilon])^-}{d\rho(v_\epsilon)} \wedge \frac{d\rho([\overline \partial, v_\epsilon])}{d\rho(v_\epsilon)} \] on the boundary of \(\Omega\) for all \(\epsilon >0.\) It is shown that this is a new sufficient condition for global regularity of the \(\overline \partial\)-Neumann operator that generalizes McNeal's property \((\tilde P )\), the approximately holomorphic vector fields of Boas and Straube, and a condition involving bounded plurisubharmonic exhaustion functions due to Kohn.