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Embeddings of Sobolev-type spaces into generalized Hölder spaces involving \(k\)-modulus of smoothness

  • Amiran Gogatishvili
  • Susana D. Moura
  • Júlio S. Neves
  • Bohumír Opic
Article

Abstract

We use an estimate of the \(k\)-modulus of smoothness of a function \(f\) such that the norm of its distributional gradient \(|\nabla ^kf|\) belongs locally to the Lorentz space \(L^{n/k, 1}({\mathbb {R}}^n),\,k \in {\mathbb {N}},\,k\le n\), and we prove its reverse form to establish necessary and sufficient conditions for continuous embeddings of Sobolev-type spaces. These spaces are modelled upon rearrangement-invariant Banach function spaces \(X({\mathbb {R}}^n)\). Target spaces of our embeddings are generalized Hölder spaces defined by means of the \(k\)-modulus of smoothness \((k\in {\mathbb {N}})\). General results are illustrated with examples.

Keywords

Rearrangement-invariant Banach function space Modulus of smoothness Distributional gradient Lorentz space Sobolev-type space Banach lattice Hölder-type space Embeddings 

Mathematics Subject Classification (2010)

26D15 26B35 26A15 26A16 46E30 46E35 46B42 

Notes

Acknowledgments

We thank the referees for their useful comments and suggestions.

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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Amiran Gogatishvili
    • 1
  • Susana D. Moura
    • 2
  • Júlio S. Neves
    • 2
  • Bohumír Opic
    • 3
  1. 1.Institute of MathematicsAcademy of Sciences of the Czech RepublicPrague 1Czech Republic
  2. 2.Department of Mathematics, CMUCUniversity of CoimbraCoimbraPortugal
  3. 3.Department of Mathematical Analysis, Faculty of Mathematics and PhysicsCharles UniversityPrague 8Czech Republic

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