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

By establishing a unified estimate of the twisted Kohn–Morrey–Hörmander estimate and the q-pseudoconvex Ahn–Zampieri estimate, we discuss variants of Property (Pq) of Catlin and Property (Pq ̃) of McNeal on the boundary of a smooth pseudoconvex domain in n for certain high level forms. These variant conditions on the one side, imply L2-compactness of the ̄-Neumann operator on the associated domain, on the other side, are different from the classical Property (Pq) and Property (Pq ̃). 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 ̄-Neumann operator Nn1 is L2-compact on (0,n1)-level forms. This result generalizes Boas and Sibony’s results on (0,1)-level forms.

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$\bar\partial$-Neumann operator, compactness estimates, pseudoconvex domains
Mathematical Subject Classification
Primary: 32W05, 35N15
Received: 3 August 2020
Revised: 22 February 2021
Accepted: 3 May 2021
Published: 17 September 2021
Yue Zhang
Department of Mathematics
Zhe Jiang Normal University
Jin Hua