\(L^ p\) estimates for local solutions of \({\bar \partial}_ b\) on strongly pseudo-convex CR-manifolds
type: Item
instance of: scholarly article
title: \(L^ p\) estimates for local solutions of \({\bar \partial}_ b\) on strongly pseudo-convex CR-manifolds
zbMATH Open document ID: 0687.32017
DOI: 10.1007/BF01444520
author: Mei-Chi Shaw
published in: Mathematische Annalen
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164722
review text: Let M be a strongly pseudo-convex hypersurface in \({\mathbb{C}}^ n\), \(n\geq 3\), and \(z_ 0\in M\). We prove that the tangential Cauchy-Riemann operators are locally solvable with \(L^ p\) estimates away from the top degree (the Lewy case). We show that there exists neighborhood base \(\omega\) of \(z_ 0\) such that for any \({\bar \partial}_ b\)-closed (0,q) form \(\alpha\) on \(\omega\), \(1\leq q\leq n-2\), with \(L^ p\)- coefficients, \(1<p<\infty\), there exists a (0,q-1) form u on \(\omega\) with \(L^ p\)-coefficients satisfying \({\bar \partial}_ bu=\alpha\) in the distribution sense and \[ \| u\|_{L^ p_{(0,q- 1)}(\omega)}\leq c_ p\| \alpha \|_{L^ p_{(0,q)}(\omega)} \] where \(c_ p\) is independent of \(\alpha\). In particular, the range of \({\bar \partial}_ b\) is closed in \(L^ p_{(0,q)}(\omega)\) when \(1\leq q\leq n-2\). This result is in sharp contrast to the top degree case (when \(q=n-1)\) where the range of \({\bar \partial}_ b\) is not closed in \(L^ p_{(0,n-1)}(\omega)\) for any neighborhood \(\omega\) of \(z_ 0\). The \(L^ p\)-estimates are sharp and cannot be improved.
Mathematics Subject Classification ID: 32W05
zbMATH DE Number: 4125789
zbMATH Keywords: strongly pseudoconvex CR-manifold
zbMATH Keywords: (partial d) bar sub b
zbMATH Keywords: equation
zbMATH Keywords: tangential Cauchy-Riemann operator
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4125789