A sufficient condition for compactness of the \(\overline{\partial}\)-Neumann operator
type: Item
instance of: scholarly article
title: A sufficient condition for compactness of the \(\overline{\partial}\)-Neumann operator
zbMATH Open document ID: 1023.32029
published in: Journal of Functional Analysis
publication date: 2003-03-26T00:00:00Z
review text: \textit{D. W. Catlin} has proved [Proc. Symp. Pure Math. 41, 39-49 (1984; Zbl 0578.32031)] that if a domain \(\Omega\) satisfies the so-called condition \((P_1)\) (i.e. forall \(M>0\) there exists \(\varphi\in C^1(\Omega)\) such that \[ |\xi |\leq 1\text{ on }\Omega,\tag{1} \] \[ \sum_{k,l} {\partial^2 \varphi \over\partial z_l\partial \overline z_k}(p)\xi_k \overline\xi_l\geq M\|\xi \|^2 \text{ for }p\in b\Omega\text{ and }\xi\in \mathbb{C}^n),\tag{2} \] then the \(\overline \partial\)-Neumann operator acting on \((0,1)\)-forms is compact. The author gives a weaker condition, \((\widetilde P_q)\), which, if satisfied by the domain \(\Omega\), ensures the compactness of the \(\overline\partial\)-Neumann operator \(N_q\) (condition \((\widetilde P_q)\) is: for every \(M>0\) there exists \(\varphi= \varphi_M\in C^\infty (\Omega)\) such that (1) \(|\partial \varphi |_{i\partial\overline\partial_f}\leq 1\), (2) The sum of any \(q\) eigenvalues of the matrix \(({\partial^2 \varphi\over \partial z_k\partial \overline z_l})\) is \(\geq M\) for all \(z\in b\Omega)\). Here \(|\partial \varphi|_{i \partial \overline\partial f}\) denotes the length of the 1-form \(\partial \varphi\) measured in the Kähler metric with fundamental form \(i\partial \overline \partial f\). If \[ \left|\sum_k{\partial f\over\partial z_k} (z)\xi_k\right |^2\leq c\sum_{k,l} {\partial^2 f\over\partial z_k\partial \overline z_l}(z) \xi_k\overline \xi_l\qquad\text{for all }\xi\in\mathbb{C}^n\text{ and }z\in \Omega, \] the author uses the notation \(|\partial f|_{i\partial \overline \partial f}\leq C\) and says that \(f\in C^2(\Omega) \cup PSH(\Omega)\) has self-bounded complex gradient. But as \(-e^{-{1\over c}f}\in PSH(\Omega)\) is equivalent with the above inequality, the self-boundedness condition can be interpretated also for non-smooth functions. The main result is: If \(\Omega\) is a smoothly bounded, pseudoconvex, relatively compact domain in \(\mathbb{C}^n\) and if \(\Omega\) satisfies condition…
Mathematics Subject Classification ID: 32W05
Mathematics Subject Classification ID: 32N15
zbMATH DE Number: 1888346
zbMATH Keywords: \(\overline{\partial}\)-Neumann problem
zbMATH Keywords: compactness
zbMATH Keywords: plurisubharmonic functions
author: J. D. McNeal
reviewed by: Gheorghe Gussi
MaRDI profile type: Publication
full work available at URL: jfan.2002.3958
OpenAlex ID: W1968804278
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DOI: 10.1006/JFAN.2002.3958
description: scientific article; zbMATH DE number 1888346