Czechoslovak Mathematical Journal

, Volume 61, Issue 4, pp 923–940

Compact embeddings of besov spaces involving only slowly varying smoothness

  • António Caetano
  • Amiran Gogatishvili
  • Bohumír Opic


We characterize compact embeddings of Besov spaces Bp,r0,b(ℝn) involving the zero classical smoothness and a slowly varying smoothness b into Lorentz-Karamata spaces \({L_{p,q;\overline b }}\)(Ω), where is a bounded domain in ℝn and \(\overline b \) is another slowly varying function.


Besov spaces with generalized smoothness Lorentz-Karamata spaces compact embeddings 

MSC 2010

46E35 46E30 


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  1. [1]
    C. Bennett, R. Sharpley: Interpolation of Operators. Academic Press, Boston, 1988.MATHGoogle Scholar
  2. [2]
    A. M. Caetano, W. Farkas: Local growth envelopes of Besov spaces of generalized smoothness. Z. Anal. Anwendungen 25 (2006), 265–298.MathSciNetMATHGoogle Scholar
  3. [3]
    A. M. Caetano, A. Gogatishvili, B. Opic: Sharp embeddings of Besov spaces involving only logarithmic smoothness. J. Approx. Theory 152 (2008), 188–214.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    A. M. Caetano, A. Gogatishvili, B. Opic: Embeddings and the growth envelope of Besov spaces involving only slowly varying smoothness. J. Approx. Theory 163 (2011), 1373–1399.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    A. M. Caetano, D. D. Haroske: Continuity envelopes of spaces of generalised smoothness: a limiting case; embeddings and approximation numbers. J. Function Spaces Appl. 3 (2005), 33–71.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    A. M. Caetano, S. D. Moura: Local growth envelopes of spaces of generalized smoothness: the sub-critical case. Math. Nachr. 273 (2004), 43–57.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    A. M. Caetano, S. D. Moura: Local growth envelopes of spaces of generalized smoothness: the critical case. Math. Ineq. & Appl. 7 (2004), 573–606.MathSciNetMATHGoogle Scholar
  8. [8]
    M. J. Carro, J. A. Raposo, J. Soria: Recent developments in the theory of Lorentz spaces and weighted inequalities. Mem. Amer. Math. Soc. 187 (2007).Google Scholar
  9. [9]
    M. J. Carro, J. Soria: Weighted Lorentz spaces and the Hardy operator. J. Funct. Anal. 112 (1993), 480–494.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    N. Dunford, J. T. Schwartz: Linear Operators, part I. Interscience, New York, 1957.Google Scholar
  11. [11]
    D. E. Edmunds, W. D. Evans: Hardy Operators, Functions Spaces and Embeddings. Springer, Berlin, Heidelberg, 2004.Google Scholar
  12. [12]
    D. E. Edmunds, P. Gurka, B. Opic: Compact and continuous embeddings of logarithmic Bessel potential spaces. Studia Math. 168 (2005), 229–250.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    D. E. Edmunds, R. Kerman, L. Pick: Optimal Sobolev imbeddings involving rearrangementinvariant quasinorms. J. Funct. Anal. 170 (2000), 307–355.MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    W. Farkas, H.-G. Leopold: Characterisations of function spaces of generalised smoothness. Ann. Mat. Pura Appl. 185 (2006), 1–62.MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    M. L. Gol’dman: Embeddings of Nikol’skij-Besov spaces into weighted Lorent spaces. Trudy Mat. Inst. Steklova 180 (1987), 93–95. (In Russian.)Google Scholar
  16. [16]
    M. L. Gol’dman: Rearrangement invariant envelopes of generalized Besov, Sobolev, and Calderon spaces. Burenkov, V. I. (ed.) et al., The interaction of analysis and geometry. International school-conference on analysis and geometry, Novosibirsk, Russia, August 23–September 3, 2004. Providence, RI: American Mathematical Society (AMS). Contemporary Mathematics 424 (2007), 53–81.Google Scholar
  17. [17]
    M. L. Gol’dman, R. Kerman: On optimal embedding of Calderón spaces and generalized Besov spaces. Tr. Mat. Inst. Steklova 243 (2003), 161–193 (In Russian.); English translation: Proc. Steklov Inst. Math. 243 (2003), 154–184.MathSciNetGoogle Scholar
  18. [18]
    P. Gurka, B. Opic: Sharp embeddings of Besov spaces with logarithmic smoothness. Rev. Mat. Complutense 18 (2005), 81–110.MathSciNetMATHGoogle Scholar
  19. [19]
    P. Gurka, B. Opic: Sharp embeddings of Besov-type spaces. J. Comput. Appl. Math. 208 (2007), 235–269.MathSciNetMATHCrossRefGoogle Scholar
  20. [20]
    D. D. Haroske, S. D. Moura: Continuity envelopes of spaces of generalized smoothness, entropy and approximation numbers. J. Approximation Theory 128 (2004), 151–174.MathSciNetMATHCrossRefGoogle Scholar
  21. [21]
    G. A. Kalyabin, P. I. Lizorkin: Spaces of functions of generalized smoothness. Math. Nachr. 133 (1987), 7–32.MathSciNetMATHCrossRefGoogle Scholar
  22. [22]
    Yu. Netrusov: Imbedding theorems of Besov spaces in Banach lattices. J. Soviet. Math. 47 (1989), 2871–2881; Translated from Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklova (LOMI) 159 (1987), 69–82.MathSciNetMATHCrossRefGoogle Scholar
  23. [23]
    H. Triebel: Theory of Function Spaces II. Birkhäuser, Basel, 1992.MATHCrossRefGoogle Scholar
  24. [24]
    H. Triebel: Theory of Function Spaces III. Birkhäuser, Basel, 2006.MATHGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2011

Authors and Affiliations

  • António Caetano
    • 1
  • Amiran Gogatishvili
    • 2
  • Bohumír Opic
    • 3
  1. 1.Departamento de MatemáticaUniversidade de AveiroAveiroPortugal
  2. 2.Institute of MathematicsAcademy of Sciences of the Czech RepublicPrague 1Czech Republic
  3. 3.Department of Mathematical Analysis, Faculty of Mathematics and PhysicsCharles UniversityPrague 8Czech Republic

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