Potential Analysis

, Volume 44, Issue 1, pp 109–136 | Cite as

Sharp Estimates for Potential Operators Associated with Laguerre and Dunkl-Laguerre Expansions

Open Access


We study potential operators associated with Laguerre function expansions of convolution and Hermite types, and with Dunkl-Laguerre expansions. We prove qualitatively sharp estimates of the corresponding potential kernels. Then we characterize those 1 ≤ p,q8, for which the potential operators are Lp - Lq bounded. These results are sharp analogues of the classical Hardy-Littlewood-Sobolev fractional integration theorem in the Laguerre and Dunkl-Laguerre settings.


Laguerre Expansion Dunkl-Laguerre Expansion Laguerre Operator Dunkl Harmonic Oscillator Negative Power Potential Operator Fractional Integral Potential Kernel 

Mathematics Subject Classification (2010)

Primary 42C10 47G40 Secondary 31C15 26A33 


  1. 1.
    Anker, J.-Ph.: Sharp estimates for some functions of the Laplacian on noncompact symmetric spaces. Duke Math. J. 65, 257–297 (1992)CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Auscher, P., Martell, J.M.: Weighted norm inequalities for fractional operators. Indiana Univ. Math. J. 57, 1845–1869 (2008)CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Bongioanni, B., Torrea, J.L.: Sobolev spaces associated to the harmonic oscillator. Proc. Indian Acad. Sci. Math. Sci. 116, 337–360 (2006)CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Ciaurri, Ó., Roncal, L.: Vector-valued extensions for fractional integrals of Laguerre expansions. arXiv:1212.4715
  5. 5.
    Lebedev, N.N.: Special functions and their applications. Dover, New York (1972)MATHGoogle Scholar
  6. 6.
    Nåsell, I.: Rational bounds for ratios of modified Bessel functions. SIAM J. Math. Anal. 9, 1–11 (1978)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Nowak, A., Roncal, L.: Potential operators associated with Jacobi and Fourier-Bessel expansions. J. Math. Anal. Appl. 422, 148–184 (2015)CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Nowak, A., Stempak, K.: Riesz transforms for multi-dimensional Laguerre function expansions. Adv. Math. 215, 642–678 (2007)CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Nowak, A., Stempak, K.: Negative powers of Laguerre operators. Can. J. Math. 64, 183–216 (2012)CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    Nowak, A., Stempak, K.: Sharp estimates of the potential kernel for the harmonic oscillator with applications. Nagoya Math. J. 212, 1–17 (2013)CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Nowak, A., Stempak, K.: Potential operators associated with Hankel and Hankel-Dunkl transforms. J. Anal. Math. arXiv:1402.3399
  12. 12.
    Nowak, A., Szarek, T.Z.: Calderón-Zygmund operators related to Laguerre function expansions of convolution type. J. Math. Anal. Appl. 388, 801–816 (2012)CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    Stempak, K.: Almost everywhere summability of Laguerre series. Studia Math. 100, 129–147 (1991)MathSciNetMATHGoogle Scholar
  14. 14.
    Thangavelu, S.: Lectures on Hermite and Laguerre expansions, vol. 42. Princeton University Press, Princeton (1993)MATHGoogle Scholar

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© The Author(s) 2015

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Instytut MatematycznyPolska Akademia NaukWarszawaPoland
  2. 2.Wydział MatematykiPolitechnika WrocławskaWrocławPoland

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