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Sobolev, Hardy, Gagliardo–Nirenberg, and Caffarelli–Kohn–Nirenberg-type inequalities for some fractional derivatives

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Abstract

In this paper, we show different inequalities for fractional-order differential operators. In particular, the Sobolev, Hardy, Gagliardo–Nirenberg, and Caffarelli–Kohn–Nirenberg-type inequalities for the Caputo, Riemann–Liouville, and Hadamard derivatives are obtained. In addition, we show some applications of these inequalities.

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Acknowledgements

The authors were supported in part by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations. Michael Ruzhansky was supported in part by the EPSRC Grant EP/R003025/1 and by the Leverhulme Research Grant RPG-2017-151. Aidyn Kassymov was supported in part by the MESRK Grant AP08053051 of the Ministry of Education and Science of the Republic of Kazakhstan.

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Authors and Affiliations

  1. Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Ghent, Belgium

    Aidyn Kassymov, Michael Ruzhansky, Niyaz Tokmagambetov & Berikbol T. Torebek

  2. Al-Farabi Kazakh National University, Almaty, Kazakhstan

    Aidyn Kassymov

  3. Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan

    Aidyn Kassymov

  4. School of Mathematical Sciences, Queen Mary University of London, London, United Kingdom

    Michael Ruzhansky

  5. Al-Farabi Kazakh National University, Almaty, Kazakhstan

    Niyaz Tokmagambetov

  6. Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan

    Berikbol T. Torebek

Authors
  1. Aidyn Kassymov
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  2. Michael Ruzhansky
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  3. Niyaz Tokmagambetov
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  4. Berikbol T. Torebek
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Correspondence to Aidyn Kassymov.

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Communicated by Juan Seoane Sepślveda.

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Kassymov, A., Ruzhansky, M., Tokmagambetov, N. et al. Sobolev, Hardy, Gagliardo–Nirenberg, and Caffarelli–Kohn–Nirenberg-type inequalities for some fractional derivatives. Banach J. Math. Anal. 15, 6 (2021). https://doi.org/10.1007/s43037-020-00097-4

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  • DOI: https://doi.org/10.1007/s43037-020-00097-4

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