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

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Abstract

We establish necessary and sufficient conditions for embeddings of Bessel potential spaces H σ X(IRn) with order of smoothness σ ∈ (0, n), modelled upon rearrangement invariant Banach function spaces X(IRn), into generalized Hölder spaces (involving k-modulus of smoothness). We apply our results to the case when X(IRn) 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(IRn) and W k L n/k(logL)α(IRn) into generalized Hölder spaces.

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Authors and Affiliations

  1. Institute of Mathematics, Academy of Sciences of the Czech Republic, Z̆itná 25, 11567, Prague 1, Czech Republic

    Amiran Gogatishvili & Bohumír Opic

  2. CMUC, Department of Mathematics, University of Coimbra, Apartado 3008, 3001-454, Coimbra, Portugal

    Júlio S. Neves

  3. Department of Mathematics and Didactics of Mathematics, Technical University of Liberec, Hálkova 6, 46117, Liberec, Czech Republic

    Bohumír Opic

Authors
  1. Amiran Gogatishvili
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  2. Júlio S. Neves
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  3. Bohumír Opic
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Correspondence to Júlio S. Neves.

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The work was partially supported by grants nos. 201/05/2033 and 201/08/0383 of the Grant Agency of the Czech Republic, by the Institutional Research Plan no. AV0Z10190503 of the Academy of Sciences of the Czech Republic (AS CR), by a joint project between AS CR and FCT (Fundação para a Ciência e a Tecnologia), and by Centre of Mathematics of the University of Coimbra.

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Gogatishvili, A., Neves, J.S. & Opic, B. Optimal Embeddings of Bessel-Potential-Type Spaces into Generalized Hölder Spaces Involving k-Modulus of Smoothness. Potential Anal 32, 201–228 (2010). https://doi.org/10.1007/s11118-009-9148-2

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