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Rellich, Gagliardo—Nirenberg, Trudinger and Caffarelli—Kohn—Nirenberg inequalities for Dunkl operators and applications

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Abstract

In this paper we introduce several extended versions of the classical Caffarelli—Kohn—Nirenberg inequalities. Moreover, we obtain weighted higher order Rellich, weighted Gagliardo—Nirenberg, Caffarelli—Kohn—Nirenberg, Trudinger inequalities and the uncertainty principle for Dunkl operators. Furthermore, we give an application of the Gagliardo—Nirenberg inequality to the Cauchy problem for the nonlinear damped wave equations for the Dunkl Laplacian.

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Acknowledgements

The authors wish to thank Michael Ruzhansky for introducing them to several of the problems studied in this paper and for useful advice, and also Sergey Tikhonov and Hatem Mejjaoli for pointing out missing references. The first author gratefully acknowledges financial support from EPSRC.

Author information

Authors and Affiliations

  1. Department of Mathematics, Imperial College London Huxley Building, 180 Queen’s Gate, London, SW7 2AZ, UK

    Andrei Velicu

  2. Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062, Toulouse, France

    Andrei Velicu

  3. Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, 281 Krijgslaan, Building S8, Ghent, Belgium

    Nurgissa Yessirkegenov

  4. Suleyman Demirel University, Kaskelen, Kazakhstan

    Nurgissa Yessirkegenov

  5. Institute of Mathematics and Mathematical Modeling, 125 Pushkin str., Almaty, 050010, Kazakhstan

    Nurgissa Yessirkegenov

Authors
  1. Andrei Velicu
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  2. Nurgissa Yessirkegenov
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Corresponding author

Correspondence to Andrei Velicu.

Additional information

The second author was supported by the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (Grant No. AP09058474) and by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations.

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Velicu, A., Yessirkegenov, N. Rellich, Gagliardo—Nirenberg, Trudinger and Caffarelli—Kohn—Nirenberg inequalities for Dunkl operators and applications. Isr. J. Math. 247, 741–782 (2022). https://doi.org/10.1007/s11856-021-2261-7

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  • DOI: https://doi.org/10.1007/s11856-021-2261-7

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