Abstract

By establishing a unified estimate of the twisted Kohn–Morrey–Hörmander estimate and the $q$-pseudoconvex Ahn–Zampieri estimate, we discuss variants of Property $\left(P_q\right)$ of Catlin and Property $\left(\widetilde{P_q}\right)$ of McNeal on the boundary of a smooth pseudoconvex domain in ${ℂ}^n$ for certain high level forms. These variant conditions on the one side, imply $L^2$-compactness of the $\bar{\partial }$-Neumann operator on the associated domain, on the other side, are different from the classical Property $\left(P_q\right)$ and Property $\left(\widetilde{P_q}\right)$. As an application of our result, we show that if the Hausdorff $\left(2n-2\right)$-dimensional measure of the weakly pseudoconvex points on the boundary of a smooth bounded pseudoconvex domain is zero, then the $\bar{\partial }$-Neumann operator $N_{n-1}$ is $L^2$-compact on $\left(0,n-1\right)$-level forms. This result generalizes Boas and Sibony’s results on $\left(0,1\right)$-level forms.

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