Morrey spaces for Schrödinger operators with certain nonnegative potentials, Littlewood–Paley and Lusin functions on the Heisenberg groups

Abstract

Let \({\mathcal {L}}=-\varDelta _{{\mathbb {H}}^n}+V\) be a Schrödinger operator on the Heisenberg group \({\mathbb {H}}^n\), where \(\varDelta _{{\mathbb {H}}^n}\) is the sublaplacian on \({\mathbb {H}}^n\) and the nonnegative potential V belongs to the reverse Hölder class \(RH_q\) with \(q\ge Q/2\). Here \(Q=2n+2\) is the homogeneous dimension of \({\mathbb {H}}^n\). In this paper the author first introduces a class of Morrey spaces associated with the Schrödinger operator \({\mathcal {L}}\) on \({\mathbb {H}}^n\). Then by using some pointwise estimates of the kernels related to the nonnegative potential V, the author establishes the boundedness properties of the Littlewood–Paley function \({\mathfrak {g}}_{{\mathcal {L}}}\) and the Lusin area integral \({\mathcal {S}}_{{\mathcal {L}}}\)(with respect to the heat semigroup \(\{e^{-s{\mathcal {L}}}\}_{s>0}\)) acting on the Morrey spaces. It can be shown that the same conclusions also hold for the operators \({\mathfrak {g}}_{\sqrt{{\mathcal {L}}}}\) and \({\mathcal {S}}_{\sqrt{{\mathcal {L}}}}\) with respect to the Poisson semigroup \(\{e^{-s\sqrt{{\mathcal {L}}}}\}_{s>0}\).

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References

  1. 1.

    Adams, D.R.: Morrey Spaces, Lecture Notes in Applied and Numerical Harmonic Analysis. Birkhäuser/Springer, Cham (2015)

    Google Scholar 

  2. 2.

    Adams, D.R., Xiao, J.: Morrey spaces in harmonic analysis. Ark. Math. 50, 201–230 (2012)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Adams, D.R., Xiao, J.: Nonlinear potential analysis on Morrey spaces and their capacities. Indiana Univ. Math. J. 53, 1629–1663 (2004)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Bongioanni, B., Harboure, E., Salinas, O.: Classes of weights related to Schrödinger operators. J. Math. Anal. Appl. 373, 563–579 (2011)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Bongioanni, B., Harboure, E., Salinas, O.: Weighted inequalities for commutators of Schrödinger–Riesz transforms. J. Math. Anal. Appl. 392, 6–22 (2012)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Bongioanni, B., Cabral, A., Harboure, E.: Extrapolation for classes of weights related to a family of operators and applications. Potential Anal. 38, 1207–1232 (2013)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Bongioanni, B., Cabral, A., Harboure, E.: Lerner’s inequality associated to a critical radius function and applications. J. Math. Anal. Appl. 407, 35–55 (2013)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Bui, T.A.: Weighted estimates for commutators of some singular integrals related to Schrödinger operators. Bull. Sci. Math. 138, 270–292 (2014)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Dziubański, J., Zienkiewicz, J.: Hardy spaces associated with some Schrödinger operators. Stud. Math. 126, 149–160 (1997)

    Article  Google Scholar 

  10. 10.

    Dziubański, J., Zienkiewicz, J.: Hardy space \(H^1\) associated to Schrödinger operator with potential satisfying reverse Hölder inequality. Rev. Mat. Iberoamericana 15, 279–296 (1999)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Dziubański, J., Zienkiewicz, J.: \(H^p\) spaces associated with Schrödinger operators with potentials from reverse Hölder classes. Colloq. Math. 98, 5–38 (2003)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Dziubański, J., Garrigós, G., Martínez, T., Torrea, J.L., Zienkiewicz, J.: \(BMO\) spaces related to Schrödinger operators with potentials satisfying a reverse Hölder inequality. Math. Z. 249, 329–356 (2005)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Di Fazio, G., Ragusa, M.A.: Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients. J. Funct. Anal 112, 241–256 (1993)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Di Fazio, G., Palagachev, D.K., Ragusa, M.A.: Global Morrey regularity of strong solutions to the Dirichlet problem for elliptic equations with discontinuous coefficients. J. Funct. Anal. 166, 179–196 (1999)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Folland, G.B.: Harmonic Analysis in Phase Space, Annals of Mathematics Studies. Princeton Univ. Press, Princeton (1989)

    Google Scholar 

  16. 16.

    Folland, G.B., Stein, E.M.: Estimates for the \(\bar{\partial }_b\) complex and analysis on the Heisenberg group. Comm. Pure Appl. Math. 27, 429–522 (1974)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Goldstein, J.A.: Semigroups of Linear Operators and Applications. Oxford Univ. Press, New York (1985)

    Google Scholar 

  18. 18.

    Guliyev, V.S., Eroglu, A., Mammadov, Y.Y.: Riesz potential in generalized Morrey spaces on the Heisenberg group. J. Math. Sci. (N.Y.) 189, 365–382 (2013)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Hofmann, S., Lu, G.Z., Mitrea, D., Mitrea, M., Yan, L.X.: Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates. Mem. Am. Math. Soc. 214, (2011)

  20. 20.

    Jiang, R.J., Jiang, X.J., Yang, D.C.: Maximal function characterizations of Hardy spaces associated with Schrödinger operators on nilpotent Lie groups. Rev. Mat. Complut. 24, 251–275 (2011)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Li, H.Q.: Estimations \(L^p\) des opérateurs de Schrödinger sur les groupes nilpotents. J. Funct. Anal. 161, 152–218 (1999)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Lin, C.C., Liu, H.P.: \(BMO_{L}({\mathbb{H}}^n)\) spaces and Carleson measures for Schrödinger operators. Adv. Math. 228, 1631–1688 (2011)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Lu, G.Z.: A Fefferman–Phong type inequality for degenerate vector fields and applications. Panam. Math. J. 6, 37–57 (1996)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Mizuhara, T.: Boundedness of Some Classical Operators on Generalized Morrey Spaces, Harmonic Analysis, ICM-90 Satellite Conference Proceedings, pp. 183–189. Springer, Tokyo (1991)

    Google Scholar 

  25. 25.

    Morrey, C.B.: On the solutions of quasi-linear elliptic partial differential equations. Trans. Am. Math. Soc. 43, 126–166 (1938)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Pan, G.X., Tang, L.: Boundedness for some Schrödinger type operators on weighted Morrey spaces, J. Funct. Spaces, Art. ID 878629, 10 pp (2014)

  27. 27.

    Polidoro, S., Ragusa, M.A.: Harnack inequality for hypoelliptic ultraparabolic equations with a singular lower order term. Rev. Mat. Iberoam. 24, 1011–1046 (2008)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Ragusa, M.A.: Necessary and sufficient condition for a \(VMO\) function. Appl. Math. Comput. 218, 11952–11958 (2012)

    MathSciNet  MATH  Google Scholar 

  29. 29.

    Shen, Z.W.: \(L^p\) estimates for Schrödinger operators with certain potentials. Ann. Inst. Fourier (Grenoble) 45, 513–546 (1995)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Song, L., Yan, L.X.: Riesz transforms associated to Schrödinger operators on weighted Hardy spaces. J. Funct. Anal. 259, 1466–1490 (2010)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Univ. Press, Princeton (1970)

    Google Scholar 

  32. 32.

    Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Univ. Press, Princeton (1993)

    Google Scholar 

  33. 33.

    Tang, L.: Weighted norm inequalities for Schrödinger type operators. Forum Math. 27, 2491–2532 (2015)

    MathSciNet  MATH  Google Scholar 

  34. 34.

    Taylor, M.E.: Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations. Comm. Partial Differ. Equ. 17, 1407–1456 (1992)

    MathSciNet  Article  Google Scholar 

  35. 35.

    Thangavelu, S.: Harmonic Analysis on the Heisenberg Group, Progress in Mathematics, vol. 159. Birkhäuser, Boston/Basel/Berlin (1998)

    Google Scholar 

  36. 36.

    Zhao, J.M.: Littlewood-Paley and Lusin functions on nilpotent Lie groups. Bull. Sci. Math. 132, 425–438 (2008)

    MathSciNet  Article  Google Scholar 

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Acknowledgements

The author would like to express his deep gratitude to the referee for his/her careful reading, valuable comments and suggestions which made this article more readable.

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Correspondence to Hua Wang.

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The author declares that he has no conflict of interest.

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Communicated by Maria Alessandra Ragusa.

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Wang, H. Morrey spaces for Schrödinger operators with certain nonnegative potentials, Littlewood–Paley and Lusin functions on the Heisenberg groups. Banach J. Math. Anal. (2020). https://doi.org/10.1007/s43037-020-00076-9

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Keywords

  • Schrödinger operator
  • Littlewood–Paley function
  • Lusin area integral
  • Heisenberg group
  • Morrey spaces
  • Reverse Hölder class

Mathematics Subject Classification

  • 42B20
  • 35J10
  • 22E25
  • 22E30