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 |
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| Médium: | Journal Article |
| Jazyk: | English |
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New York
Springer US
01.01.2016
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| ISSN: | 1050-6926, 1559-002X |
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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 |
| Author_xml | – sequence: 1 givenname: Friedrich surname: Haslinger fullname: Haslinger, Friedrich email: friedrich.haslinger@univie.ac.at organization: Fakultät für Mathematik, Universität Wien |
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| DOI | 10.1007/s12220-014-9549-3 |
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| Keywords | Primary 32W05 Neumann problem 35P10 Secondary 30H20 Compactness Sobolev inequalities |
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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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∂
¯
-Neumann operator. For this purpose we use an... |
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| 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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