\({\mathcal H}^1\)-estimates of Jacobians by subdeterminants
type: Item
instance of: scholarly article
title: \({\mathcal H}^1\)-estimates of Jacobians by subdeterminants
zbMATH Open document ID: 1055.42011
author: Tadeusz Iwaniec
author: Jani Onninen
published in: Mathematische Annalen
publication date: 2002-12-01T00:00:00Z
review text: Let \(\Omega\) be a nonempty open subset in \(\mathbb R^n\), \(n\geq 2\). A mapping \(f=(f^{1},\dots,f^{2}):\Omega\to\mathbb R^n\) in the Sobolev space \(W^{1,n-1}_{\text{loc}}(\Omega,\mathbb R^n)\) is studied. The differential matrix is \(Df=[\partial f^{i}/\partial x_{j}]\) and its cofactor matrix is \(D^{\sharp}f\). The cofactor matrix is assumed to satisfy that \(| D^{\sharp}f| \in L^{n/(n-1)}(\Omega)\). The main result of this paper is that the Jacobian determinant \(\det Df(x)\) belongs to the Hardy space \({\mathcal H}^{1}(\Omega)\) and that the following uniform estimate holds \[ \| \det Df\| _{{\mathcal H}^{1}(\Omega)}\leq C(n) \int_{\Omega}| D^{\sharp}f(x)| ^{n/(n-1)}\,dx. \] This theorem answers a question raised by \textit{S. Müller}, \textit{T. Qi} and \textit{B. S. Yan} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 11, No. 2, 217--243 (1994; Zbl 0863.49002)].
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 26B10
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 1837401
zbMATH Keywords: Hardy space
zbMATH Keywords: Jacobian determinant
zbMATH Keywords: differential matrix
zbMATH Keywords: cofactor matrix
MaRDI profile type: Publication
full work available at URL: s00208-002-0341-5
OpenAlex ID: W2312340508
DOI: 10.1007/S00208-002-0341-5
description: scientific article; zbMATH DE number 1837401