Compact and subelliptic estimates for the \(\overline\partial\)-Neumann operator on \(C^2\) pseudoconvex domains
type: Item
instance of: scholarly article
title: Compact and subelliptic estimates for the \(\overline\partial\)-Neumann operator on \(C^2\) pseudoconvex domains
zbMATH Open document ID: 1118.32027
author: Phillip S. Harrington
published in: Mathematische Annalen
publication date: 2007-01-26T00:00:00Z
review text: The following theorems are shown. Theorem 1.1 For \(k\geq 2\), let \(\Omega\) be a bounded \({\mathcal C}^k\) pseudoconvex domain in \(\mathbb{C}^n\) satisfying Catlin's property (P). Then the \(\overline\partial\)-Neumann operator is regular on \(W^{k-1}(\Omega)\). Theorem 1.2 For \(k\geq 2\), let \(\Omega\) be a bounded \({\mathcal C}^k\) pseudoconvex domain in \(\mathbb{C}^n\). Suppose that for some constants \(0< \varepsilon< {1\over 2}\) and \(c> 0\) we have a plurisubharmonic function \(\lambda\in {\mathcal C}^\infty(\overline\Omega)\) for every \(\widetilde\delta> 0\) such that \(0\leq\lambda\leq 1\) on \(\Omega\) and \(i\partial\overline\partial\lambda\geq ic\), \(\widetilde\delta^{-2\varepsilon}\partial\overline\partial|z|^2\) whenever \(\text{dist}(z, b\Omega)\leq\widetilde\delta\). Then we have the subelliptic estimates: \[ \| f\|^2_{k-1+\varepsilon}\precsim \|\overline\partial f\|^2_{k-1}+ \|\overline\partial^* f\|^2_{k-1} \] for all \(f\in\text{Dom}(\overline\partial^*)\cap \text{Dom}(\overline\partial)\) with \(\|\overline\partial f\|^2_{k-1} \|\overline\partial^* f\|^2_{k-1}< \infty\) and: \[ \| f\|_{k-1+\varepsilon}\precsim \|\overline\partial f\|_{k-1} \] for all \(f\in\ker(\overline\partial)^\perp\) with \(\|\overline\partial f\|_{k-1}< \infty\). In particular, the \(\overline\partial\)-Neumann operator is bounded from \(W^{k-1-\varepsilon}(\Omega)\) to \(W^{k-1+\varepsilon}(\Omega)\). The hypotheses of the theorem are redundant in the smooth case. The proof relies on the construction of 2 tangential differential operators \(T_1\) and \(T_2\) with the same principal part. Then one uses the Bergman projection in order to define a new operator \(T= (I- P)T_1+ PT_2\). The advantage is that \(T\) allows the computation of necessary estimates with only second derivatives of the defining function of the domain. This gives estimates that are one derivative better than previously obtained results.
reviewed by: Joachim Michel
Mathematics Subject Classification ID: 32W05
Mathematics Subject Classification ID: 35H20
Mathematics Subject Classification ID: 32T99
zbMATH DE Number: 5118865
zbMATH Keywords: subelliptic estimates
zbMATH Keywords: Neumann operator
zbMATH Keywords: Catlin's condition \((P)\)
MaRDI profile type: Publication
full work available at URL: s00208-006-0037-3
OpenAlex ID: W2009133765
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DOI: 10.1007/S00208-006-0037-3
description: scientific article; zbMATH DE number 5118865