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The Journal of Geometric Analysis

, Volume 26, Issue 4, pp 2930–2954 | Cite as

Optimal Rearrangement Invariant Sobolev Embeddings in Mixed Norm Spaces

  • Nadia Clavero
  • Javier Soria
Article

Abstract

We improve the Sobolev-type embeddings due to Gagliardo (Ric Mat 7:102–137, 1958) and Nirenberg (Ann Sc Norm Sup Pisa 13:115–162, 1959) in the setting of rearrangement invariant (r.i.) spaces. In particular, we concentrate on seeking the optimal domains and the optimal ranges for these embeddings between r.i. spaces and mixed norm spaces. As a consequence, we prove that the classical estimate for the standard Sobolev space \(W^{1}L^{p}\) by Poornima (Bull Sci Math 107(3):253–259,  1983), O’Neil (Duke Math J 30:129–142,  1963) and Peetre (Ann Inst Fourier 16(1):279–317,  1966) (\(1 \le p < n\)), and by Hansson (Math Scand 45(1):77–102,  1979, Brezis and Wainger (Commun Partial Differ Equ 5(7):773–789,  1980) and Maz’ya (Sobolev spaces,  1985) (\(p=n\)) can be further strengthened by considering mixed norms on the target spaces.

Keywords

Sobolev embeddings Rearrangement-invariant spaces Hardy operator Optimal range Optimal domain 

Mathematics Subject Classification

28A35 46E30 46E35 

Notes

Acknowledgments

We would like to thank the referee for his/her careful revision which has improved the final version of this work. Both authors have been partially supported by the Grants MTM2013-40985-P (Spanish Government) and 2014SGR289 (Catalan Autonomous Government).

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Copyright information

© Mathematica Josephina, Inc. 2015

Authors and Affiliations

  1. 1.Department of Applied Mathematics and AnalysisUniversity of BarcelonaBarcelonaSpain

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