Sobolev Inequalities and the ∂¯-Neumann Operator

We study a complex-valued version of the Sobolev inequalities and its relationship to compactness of the ∂ ¯ -Neumann operator. For this purpose we use an abstract characterization of compactness derived from a general description of precompact subsets in L 2 -spaces. Finally we remark that the ∂ ¯...

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Vydané v:The Journal of geometric analysis Ročník 26; číslo 1; s. 287 - 293
Hlavný autor: Haslinger, Friedrich
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: New York Springer US 01.01.2016
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Abstract We study a complex-valued version of the Sobolev inequalities and its relationship to compactness of the ∂ ¯ -Neumann operator. For this purpose we use an abstract characterization of compactness derived from a general description of precompact subsets in L 2 -spaces. Finally we remark that the ∂ ¯ -Neumann operator can be continuously extended provided a subelliptic estimate holds.
AbstractList We study a complex-valued version of the Sobolev inequalities and its relationship to compactness of the ∂ ¯ -Neumann operator. For this purpose we use an abstract characterization of compactness derived from a general description of precompact subsets in L 2 -spaces. Finally we remark that the ∂ ¯ -Neumann operator can be continuously extended provided a subelliptic estimate holds.
Author Haslinger, Friedrich
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Keywords Primary 32W05
Neumann problem
35P10
Secondary 30H20
Compactness
Sobolev inequalities
Language English
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PublicationTitle The Journal of geometric analysis
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References AdamsRAFournierJJFSobolev Spaces, Pure and Applied Mathematics2006Waltham, MAAcademic Press
BonamiASibonyNSobolev embedding in Cn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{C}^n$$\end{document} and the ∂¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\partial }}$$\end{document}-equationJ. Geom. Anal.199113073270743.32015112934510.1007/BF02921308
KrantzSOptimal Lipschitz and Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document} estimates for the equation ∂¯u=f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\partial }} u=f$$\end{document} on strongly pseudoconvex domainsMath. Ann.19762192332600303.3505939702010.1007/BF01354286
BealsRGreinerPCStantonNKLp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document} and Lipschitz estimates for the ∂¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\partial }}$$\end{document}-equation the ∂¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\partial }}$$\end{document}-Neumann problemMath. Ann.19872771851960598.3507788641810.1007/BF01457358
Straube, E.: The L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-Sobolev theory of the ∂¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\partial }}$$\end{document}-Neumann problem. ESI Lectures in Mathematics and Physics, EMS (2010)
BrezisHAnalyse Fonctionnelle, Théorie et Applications1983ParisMasson0511.46001
CatlinDWNecessary conditions for subellipticity of the ∂¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\partial }$$\end{document}-Neumann problemAnn. Math.19831171471710552.3201768380510.2307/2006974
LiebIRangeRMIntegral representations and estimates in the theory of the ∂¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\partial }}$$\end{document}-Neumann problemAnn. Math.19861232653010589.3203483576310.2307/1971272
KimMInheritance of noncompactness of the ∂¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\partial }}$$\end{document}-Neumann problemJ. Math. Anal. Appl.20053024504561076.32033210784510.1016/j.jmaa.2004.05.010
CatlinDWBoundary invariants of pseudoconvex domainsAnn. Math.19841205295860583.3204876916310.2307/1971087
HaslingerFCompactness for the ∂¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\partial }}$$\end{document}- Neumann problem-a functional analysis approachCollectanea Math.2011621211291217.32014279251510.1007/s13348-010-0013-9
CatlinDWSubelliptic estimates for the ∂¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\partial }$$\end{document}-Neumann problem on pseudoconvex domainsAnn. Math.19871261311910627.3201389805410.2307/1971347
D’AngeloJPReal hypersurfaces, orders of contact, and applicationsAnn. Math.197911561563765724110.2307/2007015
D’AngeloJPFinite type conditions for real hypersurfacesJ. Differen. Geom.19791459660411.3200857787810.1111/j.1432-0436.1979.tb01012.x
References_xml – reference: HaslingerFCompactness for the ∂¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\partial }}$$\end{document}- Neumann problem-a functional analysis approachCollectanea Math.2011621211291217.32014279251510.1007/s13348-010-0013-9
– reference: AdamsRAFournierJJFSobolev Spaces, Pure and Applied Mathematics2006Waltham, MAAcademic Press
– reference: BealsRGreinerPCStantonNKLp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document} and Lipschitz estimates for the ∂¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\partial }}$$\end{document}-equation the ∂¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\partial }}$$\end{document}-Neumann problemMath. Ann.19872771851960598.3507788641810.1007/BF01457358
– reference: CatlinDWNecessary conditions for subellipticity of the ∂¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\partial }$$\end{document}-Neumann problemAnn. Math.19831171471710552.3201768380510.2307/2006974
– reference: BonamiASibonyNSobolev embedding in Cn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{C}^n$$\end{document} and the ∂¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\partial }}$$\end{document}-equationJ. Geom. Anal.199113073270743.32015112934510.1007/BF02921308
– reference: D’AngeloJPFinite type conditions for real hypersurfacesJ. Differen. Geom.19791459660411.3200857787810.1111/j.1432-0436.1979.tb01012.x
– reference: BrezisHAnalyse Fonctionnelle, Théorie et Applications1983ParisMasson0511.46001
– reference: CatlinDWSubelliptic estimates for the ∂¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\partial }$$\end{document}-Neumann problem on pseudoconvex domainsAnn. Math.19871261311910627.3201389805410.2307/1971347
– reference: KimMInheritance of noncompactness of the ∂¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\partial }}$$\end{document}-Neumann problemJ. Math. Anal. Appl.20053024504561076.32033210784510.1016/j.jmaa.2004.05.010
– reference: Straube, E.: The L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-Sobolev theory of the ∂¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\partial }}$$\end{document}-Neumann problem. ESI Lectures in Mathematics and Physics, EMS (2010)
– reference: LiebIRangeRMIntegral representations and estimates in the theory of the ∂¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\partial }}$$\end{document}-Neumann problemAnn. Math.19861232653010589.3203483576310.2307/1971272
– reference: CatlinDWBoundary invariants of pseudoconvex domainsAnn. Math.19841205295860583.3204876916310.2307/1971087
– reference: D’AngeloJPReal hypersurfaces, orders of contact, and applicationsAnn. Math.197911561563765724110.2307/2007015
– reference: KrantzSOptimal Lipschitz and Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document} estimates for the equation ∂¯u=f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\partial }} u=f$$\end{document} on strongly pseudoconvex domainsMath. Ann.19762192332600303.3505939702010.1007/BF01354286
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Snippet We study a complex-valued version of the Sobolev inequalities and its relationship to compactness of the ∂ ¯ -Neumann operator. For this purpose we use an...
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StartPage 287
SubjectTerms Abstract Harmonic Analysis
Convex and Discrete Geometry
Differential Geometry
Dynamical Systems and Ergodic Theory
Fourier Analysis
Global Analysis and Analysis on Manifolds
Mathematics
Mathematics and Statistics
Title Sobolev Inequalities and the ∂¯-Neumann Operator
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