#### Vol. 313, No. 1, 2021

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

### Yue Zhang

Vol. 313 (2021), No. 1, 213–249
DOI: 10.2140/pjm.2021.313.213
##### 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 $\left(\stackrel{̃}{{P}_{q}}\right)$ 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 $\stackrel{̄}{\partial }$-Neumann operator on the associated domain, on the other side, are different from the classical Property $\left({P}_{q}\right)$ and Property $\left(\stackrel{̃}{{P}_{q}}\right)$. 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 $\stackrel{̄}{\partial }$-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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$\bar\partial$-Neumann operator, compactness estimates, pseudoconvex domains