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Journal of Fourier Analysis and Applications

, Volume 20, Issue 5, pp 1020–1049 | Cite as

Ulyanov-type Inequalities Between Lorentz–Zygmund Spaces

  • Amiran Gogatishvili
  • Bohumír Opic
  • Sergey Tikhonov
  • Walter Trebels
Article

Abstract

We establish inequalities of Ulyanov-type for moduli of smoothness relating the source Lorentz–Zygmund space \(\, L^{p,r}(\log L)^{\alpha -\gamma },\, \gamma >0,\) and the target space \(\, L^{p^*,s}(\log L)^\alpha \) over \(\, {\mathbb R}^n\) if \(\, 1<p<p^*<\infty \) and over \(\, \mathbb {T}^n\) if \(\, 1<p \le p^*<\infty .\) The stronger logarithmic integrability (corresponding to \(\, L^{p^*,s}(\log L)^\alpha \)) is balanced by an additional logarithmic smoothness.

Keywords

Moduli of smoothness \(K\)-functionals Lorentz–Zygmund spaces 

Mathematics Subject Classification

Primary 41A17 46E30 Secondary 42B15 46E35 

Notes

Acknowledgments

The authors thank the referees for useful comments and suggestions that led to the improvement of the paper. Part of this work was done while the authors were at the Centre de Recerca Matemàtica (Barcelona) in 2011. This research was partially supported by the MTM 2011-27637, 2014 SGR 289, RFFI 13-01-00043, RVO: 67985840, the Grant Agency of the Czech Republic, Grants Nos. 201/08/0383, and P 201 13-14743S. The research of A. Gogatishvili was partially supported by the research grant no. 31/48 of the Shota Rustaveli National Science Foundation.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Amiran Gogatishvili
    • 1
  • Bohumír Opic
    • 2
  • Sergey Tikhonov
    • 3
  • Walter Trebels
    • 4
  1. 1.Institute of Mathematics of the Academy of Sciences of the Czech RepublicPrague 1Czech Republic
  2. 2.Department of Mathematical Analysis, Faculty of Mathematics and PhysicsCharles UniversityPrague 8Czech Republic
  3. 3.ICREA and Centre de Recerca MatemàticaBarcelonaSpain
  4. 4.Fb. Mathematik, AG AlgebraTU DarmstadtDarmstadtGermany

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