Potential Analysis

, Volume 32, Issue 3, pp 201–228 | Cite as

Optimal Embeddings of Bessel-Potential-Type Spaces into Generalized Hölder Spaces Involving k-Modulus of Smoothness

  • Amiran Gogatishvili
  • Júlio S. Neves
  • Bohumír Opic


We establish necessary and sufficient conditions for embeddings of Bessel potential spaces H σ X(IR n ) with order of smoothness σ ∈ (0, n), modelled upon rearrangement invariant Banach function spaces X(IR n ), into generalized Hölder spaces (involving k-modulus of smoothness). We apply our results to the case when X(IR n ) is the Lorentz-Karamata space \(L_{p,q;b}({{\rm I\kern-.17em R}}^n)\). In particular, we are able to characterize optimal embeddings of Bessel potential spaces \(H^{\sigma}L_{p,q;b}({{\rm I\kern-.17em R}}^n)\) into generalized Hölder spaces. Applications cover both superlimiting and limiting cases. We also show that our results yield new and sharp embeddings of Sobolev-Orlicz spaces W k + 1 L n/k(logL) α (IR n ) and W k L n/k(logL) α (IR n ) into generalized Hölder spaces.


Slowly varying functions Lorentz-Karamata spaces Rearrangement-invariant Banach function spaces Bessel potentials (fractional) Sobolev-type spaces Hölder-type spaces Zygmund-type spaces Embedding theorems 

Mathematics Subject Classifications (2000)

46E35 46E30 26B35 26A12 26D15 26A15 26A16 26B35 47B38 26D10 


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Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • Amiran Gogatishvili
    • 1
  • Júlio S. Neves
    • 2
  • Bohumír Opic
    • 1
    • 3
  1. 1.Institute of MathematicsAcademy of Sciences of the Czech RepublicPrague 1Czech Republic
  2. 2.CMUC, Department of MathematicsUniversity of CoimbraCoimbraPortugal
  3. 3.Department of Mathematics and Didactics of MathematicsTechnical University of LiberecLiberecCzech Republic

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