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Vector-valued Sobolev Spaces and Gagliardo-Nirenberg Inequalities

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Nonlinear Elliptic and Parabolic Problems

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 64))

Abstract

We prove Gagliardo-Nirenberg inequalities for vector-valued Lizorkin-Triebel spaces. Therefrom we derive the corresponding assertions for the vector-valued Sobolev spaces Wm p (Rn,E) . Here we do not assume the UMD property for E.

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References

  1. R.A. Adams, Sobolev spaces, Academic Press, New York, 1975.

    Google Scholar 

  2. H. Amann, Linear and quasilinear parabolic problems, Vol. 11, Birkäuser, Basel, 1995.

    Google Scholar 

  3. H. Amann, Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications, Math. Nachr. 186 (1997), 5–56.

    MATH  MathSciNet  Google Scholar 

  4. H. Amann, Vector-valued distributions and Fourier multipliers, Preprint, Zürich, 1997.

    Google Scholar 

  5. O.V. Besov, L.D. Kudrayavtsev, P.I. Lizorkin and S.M. Nikol’skij, Investigations in the theory of spaces of differentiable functions of several variables, Proc. Steklov Inst. 182 (1990), 73–139.

    Google Scholar 

  6. J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, Ark. Mat. 21 (1983), 163–168.

    MATH  MathSciNet  Google Scholar 

  7. H. Brezis and P. Mironescu, Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces, J. Evol. Equ. 1 (2001), 387–404.

    MathSciNet  Google Scholar 

  8. A.V. Bukhvalov, Sobolev spaces of vector-valued functions, J. Math. Sciences 71 (1994), 2173–2179.

    MathSciNet  Google Scholar 

  9. P. Grisvard, Commutativitdé deux foncteurs d’interpolation at applications, J. Math. Pures Appl. 45 (1966), 143–290.

    MATH  MathSciNet  Google Scholar 

  10. H. König, Eigenvalue distribution of compact operators, Birkhäuser, Basel, 1986.

    Google Scholar 

  11. P.I. Lizorkin, On multipliers of Fourier integrals in the spaces L pθ , Trudy Math. Inst. Steklov 89 (1967), 231–248.

    MATH  MathSciNet  Google Scholar 

  12. T.R. McConnell, On Fourier multiplier transforms of Banach-valued functions, Trans. Amer. Math. Soc. 285 (1984), 739–757.

    MATH  MathSciNet  Google Scholar 

  13. T. Muramatu, Besov spaces and Sobolev spaces of generalized functions defined on a general region, Publ. R.I.M.S. Kyoto Univ. 9 (1974), 325–396.

    MATH  MathSciNet  Google Scholar 

  14. J. Peetre, New thoughts on Besov spaces, Duke Univ. Press, Durham, 1976.

    Google Scholar 

  15. A. Pelczynski and M. Wojciechowski, Molecular decompositions and embedding theorems for vector-valued Sobolev spaces with gradient norm, Studia Math 107 (1993), 61–100.

    MathSciNet  Google Scholar 

  16. A. Pietsch, Eigenvalues and s-numbers. Cambridge Univ. Press, Cambridge, 1987.

    Google Scholar 

  17. A. Pietsch and J. Wenzel, Orthonormal systems and Banach space geometry. Cambridge Univ. Press, Cambridge, 1998.

    Google Scholar 

  18. G. Pisier, Type des espaces normés. Seminaire Maurey-Schwartz 1973–1974, Ecole Polytechnique, Palaiseau.

    Google Scholar 

  19. J. Prüss, Evolutionary integral equations and applications, Birkhäuser, Basel, 1993.

    Google Scholar 

  20. J.L. Rubio de Francia and J.L. Torrea, Some Banach techniques in vector-valued Fourier analysis, Coll. Math. 54 (1987), 273–284.

    MathSciNet  Google Scholar 

  21. H.-J. Schmeisser, Vector-valued Sobolev and Besov spaces, In: Seminar Analysis of the Karl-Weierstraß-Institute, 4–44, Teubner-Texte Math. 96, Teubner, Leipzig, 1987.

    Google Scholar 

  22. H.-J. Schmeißer and W. Sickel, Traces, Gagliardo-Nirenberg inequalities and Sobolev type embeddings for vector-valued function spaces, Jenaer Schriften zur Mathematik und Informatik Math/Inf/24/01, Jena, 2001.

    Google Scholar 

  23. G. Sparr, Interpolation of several Banach spaces, Ann. Mat. Pura Appl. 99 (1974), 247–316.

    MATH  MathSciNet  Google Scholar 

  24. E.M. Stein, Harmonic analysis. Real-variable methods, orthogonality, and oscillatory integrals. Princeton Univ. Press, Princeton, 1993.

    Google Scholar 

  25. H. Triebel, Theory of function spaces, Birkhuäuser, Basel, 1983.

    Google Scholar 

  26. H. Triebel, Fractals and spectra related to Fourier analysis and function spaces, Birkhäuser, Basel, 1997.

    Google Scholar 

  27. F. Zimmermann, On vector-valued Fourier multiplier theorems, Stud. Math. 93 (1989), 201–222.

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Mathematisches Institut, Friedrich-Schiller-Universität Jena, Ernst-Abbe-Platz 1-2, D-07743, Jena, Germany

    Hans-Jürgen Schmeißer & Winfried Sickel

Authors
  1. Hans-Jürgen Schmeißer
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  2. Winfried Sickel
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Editor information

Editors and Affiliations

  1. Universitá Pierre et Marie Curie, Paris

    Haim Brezis

  2. Rutgers University, New Brunswick, N.J.

    Haim Brezis

  3. Angewandte Mathematik, Universität Zurich, Winterthurerstr. 190, 8057, Zürich, Switzerland

    Michel Chipot

  4. Institute of Applied Mathematics, University of Hannover, Welfengarten 1, 30167, Hannover, Germany

    Joachim Escher

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© 2005 Birkhäuser Verlag Basel/Switzerland

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Schmeißer, HJ., Sickel, W. (2005). Vector-valued Sobolev Spaces and Gagliardo-Nirenberg Inequalities. In: Brezis, H., Chipot, M., Escher, J. (eds) Nonlinear Elliptic and Parabolic Problems. Progress in Nonlinear Differential Equations and Their Applications, vol 64. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7385-7_27

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