Sufficient conditions for compactness of the $\bar{\mathrm{\partial }}$-Neumann operator on high level forms

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