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Ulyanov-type Inequalities Between Lorentz–Zygmund Spaces

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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.

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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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Authors and Affiliations

  1. Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, 115 67 , Prague 1, Czech Republic

    Amiran Gogatishvili

  2. Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 , Prague 8, Czech Republic

    Bohumír Opic

  3. ICREA and Centre de Recerca Matemàtica, Apartat 50, Bellaterra, 08193 , Barcelona, Spain

    Sergey Tikhonov

  4. Fb. Mathematik, AG Algebra, TU Darmstadt, 64289 , Darmstadt, Germany

    Walter Trebels

Authors
  1. Amiran Gogatishvili
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  2. Bohumír Opic
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  3. Sergey Tikhonov
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  4. Walter Trebels
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Correspondence to Sergey Tikhonov.

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Communicated by Hans G. Feichtinger.

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Gogatishvili, A., Opic, B., Tikhonov, S. et al. Ulyanov-type Inequalities Between Lorentz–Zygmund Spaces . J Fourier Anal Appl 20, 1020–1049 (2014). https://doi.org/10.1007/s00041-014-9343-4

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  • DOI: https://doi.org/10.1007/s00041-014-9343-4

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