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Existence of Extremals of Dunkl-Type Sobolev Inequality and of Stein–Weiss Inequality for Dunkl Riesz Potential

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Abstract

In this paper, we prove the existence of an extremal for the Dunkl-type Sobolev inequality in the case of \(p=2\). Also we prove the existence of an extremal of the Stein–Weiss inequality for the Dunkl Riesz potential in the case of \(r=2\).

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Data Availability Statement

Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

References

  1. Abdelkefi, C., Rachdi, M.: Some properties of the Riesz potentials in Dunkl analysis. Ricerche mat. 64(4), 195–215 (2015)

    Article  MathSciNet  Google Scholar 

  2. Amri, B., Sifi, M.: Riesz transforms for Dunkl transform. Ann. Math. Blaise Pascal 19, 247–262 (2012)

    Article  MathSciNet  Google Scholar 

  3. Anker, J.P., Dziubański, J., Hejna, A.: Harmonic functions, conjugate harmonic functions and the hardy space H 1 in the rational Dunkl setting. J. Fourier Anal. Appl. (2019). https://doi.org/10.1007/s00041-019-09666-0

    Article  MathSciNet  MATH  Google Scholar 

  4. Beckner, W.: Weighted inequalities and Stein–Weiss potentials. Forum Math. 20, 587–606 (2008)

    Article  MathSciNet  Google Scholar 

  5. Beckner, W.: Inequalities in Fourier analysis. Ann. Math. 102, 159–182 (1975)

    Article  MathSciNet  Google Scholar 

  6. Ben Salem, N., Garna, A., Kallel, S.: Bessel and Flett Potentials associated with Dunkl operators on \({\mathbb{R}}^d\). Methods Appl. Anal. 15(4), 477–494 (2008)

    MathSciNet  MATH  Google Scholar 

  7. Chen, L., Lu, G.: Existence of extremal functions for the Stein-Weiss inequalities on the Heisenberg group. J. Funct. Anal. 277(4), 1112–1138 (2019)

    Article  MathSciNet  Google Scholar 

  8. Dunkl, C.F.: Differential-difference operators associated to reflection groups. Trans. Am. Math. Soc. 311(1), 167–183 (1989)

    Article  MathSciNet  Google Scholar 

  9. Frank, R. L.: Sobolev inequalities and uncertainty principles in mathematical physics, Part 1, pp. 1601–1610. http://www.math.caltech.edu/~rlfrank/sobweb1.pdf

  10. Gadjiev, A.D., Guliyev, V.S., Serbetci, A., Guliyev, E.V.: The Stein–Weiss type inequalities for the B-Riesz potentials. J. Math. Ineq. 5(1), 87–106 (2011)

    Article  MathSciNet  Google Scholar 

  11. Gorbachev, D.V., Ivanov, V.I., Yu, S.: Tikhonov, positive \(L^{p}-\)bounded Dunkl-type generalized translation operator and its applications. Constr. Approx. (2018). https://doi.org/10.1007/s00365-018-9435-5

    Article  Google Scholar 

  12. Gorbachev, D.V., Ivanov, V.I., Yu, S.: Tikhonov, Riesz potential and maximal function for Dunkl transform. Potential Anal. (2020). https://doi.org/10.1007/s11118-020-09867-z

    Article  Google Scholar 

  13. Han, X., Lu, G., Zhu, J.: Hardy–Littlewood–Sobolev and Stein–Weiss inequalities and integral systems on the Heisenberg group. Nonlinear Anal. 75, 4296–4314 (2012)

    Article  MathSciNet  Google Scholar 

  14. Han, X.: Existence of maximizers for Hardy–Littlewood–Sobolev inequalities on the Heisenberg group. Indiana Univ. Math. J. 62, 737–751 (2013)

    Article  MathSciNet  Google Scholar 

  15. Hassani, S., Mustapha, S., Sifi, M.: Riesz potentials and fractional maximal function for the Dunkl transform. J. Lie Theory. 19, 725–734 (2009)

    MathSciNet  MATH  Google Scholar 

  16. Kawazoe, T., Mejjaoli, H.: Generalized Besov spaces and their applications. Tokyo. J. Math. 35(2), 297–320 (2012)

    Article  MathSciNet  Google Scholar 

  17. Ledoux, M.: On improved Sobolev embedding theorems. Math. Res. Lett. 10, 659–669 (2003)

    Article  MathSciNet  Google Scholar 

  18. Lieb, E.H., Loss, M.: Analysis, Second edition, Graduate studies in Mathematics, vol. 14. American Mathematical Society, Providence (2001)

    Google Scholar 

  19. Lieb, E.: Sharp constants in the Hardy Littlewood–Sobolev and related inequalities. Ann. Math. 118, 349–374 (1983)

    Article  MathSciNet  Google Scholar 

  20. Maz’ya, V.G.: Sobolev spaces. Springer, Berlin (1985)

    Book  Google Scholar 

  21. Mejjaoli, H.: Littlewood-Paley decomposition associated with the Dunkl operators and paraproduct operators J. Inequal. Pure Appl. Math. 9, No. 4, Article 95 (2008)

  22. Mejjaoli, H.: Generalized homogeneous Besov spaces and their applications. Serdica. Math. J. 38, 575–614 (2012)

    MathSciNet  MATH  Google Scholar 

  23. Mejjaoli, H.: Dunkl heat semigroup and applications. Appl. Anal. Int. J. 92(9), 1980–2007 (2013)

    Article  MathSciNet  Google Scholar 

  24. Mejjaoli, H.: Generalized Lorentz spaces and applications. J. Funct. Spaces Appl., Art. ID 302941 (2013)

  25. Mejjaoli, H.: Generalized heat equation and applications. Integr. Transf. Spec. Funct. 25, 15–33 (2014)

    Article  MathSciNet  Google Scholar 

  26. Mustapha, S., Sifi, M.: Relative Nash-type and \(L^{2}\)-Sobolev inequalities for Dunkl operators and applications. ar**v:2009.03405

  27. De Napoli, P., Drelichman, I., Salort, A.: Weighted inequalities for the Fractional Laplacian and the existence of extremals. Commun. Contemp. Math. (2019). https://doi.org/10.1142/S0219199718500347

    Article  MathSciNet  MATH  Google Scholar 

  28. Rösler, M.: Generalized Hermite polynomials and the heat equation for Dunkl operators. Commun. Math. Phys. 192, 519–542 (1998)

    Article  MathSciNet  Google Scholar 

  29. Rösler, M.: A positive radial product formula for the Dunkl kernel. Trans. Am. Math. Soc. 355, 2413–2438 (2003)

    Article  MathSciNet  Google Scholar 

  30. Rösler, M.: Dunkl operators: theory and applications. In: Orthogonal Polynomials and Special Functions (Leuven, 2002), volume 1817 of Lecture Notes in Math., pp. 93–135. Springer, Berlin (2003)

  31. Luca, R.: Regularity criteria with angular integrability for the Navier–Stokes equation. Nonlinear Anal. 105, 24–40 (2014)

    Article  MathSciNet  Google Scholar 

  32. Saloff-Coste, L.: Aspects of Sobolev-type inequalities, Volume 289, London Mathematical Society, Lecture note, Cambridge University Press (2002)

  33. Stein, E., Weiss, G.: Fractional integrals on n-dimensional Euclidean space. J. Math. Mech. 7, 503–514 (1958)

    MathSciNet  MATH  Google Scholar 

  34. Talenti, G.: Best constant in Sobolev inequality. Ann. Mat. Pura. Appl. 110, 353–372 (1976)

    Article  MathSciNet  Google Scholar 

  35. Thangavelu, S., Xu, Y.: Convolution operator and maximal function for the Dunkl transform. J. Anal. Math. 97(1), 25–55 (2005)

    Article  MathSciNet  Google Scholar 

  36. Thangavelu, S., Xu, Y.: Riesz transform and Riesz potentials for Dunkl transform. J. Comput. Appl. Math. 199, 181–195 (2007)

    Article  MathSciNet  Google Scholar 

  37. Velicu, A.: Sobolev-type inequality for Dunkl operator. J. Funct. Anal. 279, 108695 (2020). https://doi.org/10.1016/j.jfa.2020.108695

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We thank Hatem Mejjaoli for pointing out the missing references and the anonymous referee for meticulously reading the manuscript and giving us valuable comments and suggestions. Most of the work was carried out when first two authors were in NISER Bhubaneswar and they acknowledge the financial support received from NISER.

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Author notes

  1. Saswata Adhikari

    Present address: Department of Mathematics, SRM University-AP, Amravati, Andhra Pradesh, 522502, India

Authors and Affiliations

  1. School of Mathematical Sciences, NISER Bhubaneswar, Jatni, Odisha, 752050, India

    Saswata Adhikari & Sanjay Parui

  2. Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai, 400094, India

    Saswata Adhikari & Sanjay Parui

  3. Department of Mathematics, Indian Institute of Science, Bangalore, 560012, India

    V. P. Anoop

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  1. Saswata Adhikari
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Correspondence to Saswata Adhikari.

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Communicated by Jussi Behrndt, Fabrizio Colombo and Sergey Naboko.

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This article is part of the topical collection “Spectral Theory and Operators in Mathematical Physics” edited by Jussi Behrndt, Fabrizio Colombo and Sergey Naboko.

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Adhikari, S., Anoop, V.P. & Parui, S. Existence of Extremals of Dunkl-Type Sobolev Inequality and of Stein–Weiss Inequality for Dunkl Riesz Potential . Complex Anal. Oper. Theory 15, 28 (2021). https://doi.org/10.1007/s11785-020-01068-1

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