Revista Matemática Complutense

, Volume 24, Issue 1, pp 251–275 | Cite as

Maximal function characterizations of Hardy spaces associated with Schrödinger operators on nilpotent Lie groups

  • Renjin Jiang
  • Xiaojuan Jiang
  • Dachun Yang


Let G be a connected and simply connected nilpotent Lie group and L≡−Δ+W be the Schrödinger operator on L 2(G), where \(0\le W\in L^{1}_{\mathrm{loc}}(G)\). In this paper, the authors establish some equivalent characterizations of the Hardy space \(H^{p}_{L}(G)\) for p∈(0,1] in terms of the radial maximal functions and non-tangential maximal functions associated with \(\{e^{-t^{2}L}\}_{t>0}\) and \(\{e^{-t\sqrt{L}}\}_{t>0}\), respectively. The boundedness of the Riesz transform \(\nabla L^{-\frac{1}{2}}\) from \(H^{p}_{L}(G)\) to L p (G) with p∈(0,1] and from \(H^{p}_{L}(G)\) to H p (G) with p∈(D/(D+1),1] are also obtained, where D is the dimension at infinity of G.


Nilpotent Lie group Hardy space Schrödinger operator Maximal function Riesz transform 

Mathematics Subject Classification (2000)

42B25 42B20 42B30 22E25 43A15 


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  1. 1.
    Alexopoulos, G.: Spectral multipliers on Lie groups of polynomial growth. Proc. Am. Math. Soc. 120, 973–979 (1994) zbMATHMathSciNetGoogle Scholar
  2. 2.
    Auscher, P., Duong, X.T., McIntosh, A.: Boundedness of Banach space valued singular integral operators and Hardy spaces. Unpublished manuscript (2005) Google Scholar
  3. 3.
    Auscher, P., McIntosh, A., Russ, E.: Hardy spaces of differential forms on Riemannian manifolds. J. Geom. Anal. 18, 192–248 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Auscher, P., Russ, E.: Hardy spaces and divergence operators on strongly Lipschitz domains of ℝn. J. Funct. Anal. 201, 148–184 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Calderón, A.P., Torchinsky, A.: Parabolic maximal functions associated with a distribution. Adv. Math. 16, 1–64 (1975) zbMATHCrossRefGoogle Scholar
  6. 6.
    Christ, M., Geller, D.: Singular integral characterizations of Hardy spaces on homogeneous groups. Duke Math. J. 51, 547–598 (1984) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Coifman, R.R., Meyer, Y., Stein, E.M.: Some new function spaces and their applications to harmonic analysis. J. Funct. Anal. 62, 304–335 (1985) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Coifman, R.R., Weiss, G.: Analyse harmonique non-commutative sur certains espaces homogènes. Lecture Notes in Math., vol. 242. Springer, Berlin (1971) zbMATHGoogle Scholar
  9. 9.
    Coifman, R.R., Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83, 569–645 (1977) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Duong, X.T., Yan, L.: Duality of Hardy and BMO spaces associated with operators with heat kernel bounds. J. Am. Math. Soc. 18, 943–973 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Dziubańki, J., Zienkiewicz, J.: Hardy space H 1 associated to Schrödinger operator with potential satisfying reverse Hölder inequality. Rev. Mat. Iberoamericana 15, 279–296 (1999) MathSciNetGoogle Scholar
  12. 12.
    Dziubański, J., Zienkiewicz, J.: H p spaces for Schrödinger operators. In: Fourier Analysis and Related Topics. Banach Center Publ., vol. 56, pp. 45–53. Polish Acad. Sci., Warsaw (2002) Google Scholar
  13. 13.
    Fefferman, C., Stein, E.M.: H p spaces of several variables. Acta Math. 129, 137–193 (1972) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Folland, G.B., Stein, E.M.: Hardy Spaces on Homogeneous Groups. Princeton University Press/University of Tokyo Press, Princeton/Tokyo (1982) zbMATHGoogle Scholar
  15. 15.
    Grafakos, L., Liu, L., Yang, D.: Maximal function characterizations of Hardy spaces on RD-spaces and their applications. Sci. China Ser. A 51, 2253–2284 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Guivarc’h, Y.: Croissance polynômiale et périodes des fonctions harmoniques. Bull. Soc. Math. Fr. 101, 333–379 (1973) zbMATHMathSciNetGoogle Scholar
  17. 17.
    Han, Q., Lin, F.: Elliptic Partial Differential Equations. Courant Lecture Notes in Mathematics, vol. 1. American Mathematical Society, Providence (1997) New York University, Courant Institute of Mathematical Sciences, New York Google Scholar
  18. 18.
    Hofmann, S., Lu, G., Mitrea, D., Mitrea, M., Yan, L.: Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates. Mem. Am. Math. Soc. (to appear) Google Scholar
  19. 19.
    Hofmann, S., Mayboroda, S.: Hardy and BMO spaces associated to divergence form elliptic operators. Math. Ann. 344, 37–116 (2009) zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Hofmann, S., Mayboroda, S.: Correction to “Hardy and BMO spaces associated to divergence form elliptic operators”. Math. Ann. (to appear). arXiv:0907.0129
  21. 21.
    Hofmann, S., Mayboroda, S., McIntosh, A.: Second order elliptic operators with complex bounded measurable coefficients in L p, Sobolev and Hardy spaces. arXiv:1002.0792
  22. 22.
    Han, Y., Müller, D., Yang, D.: Littlewood-Paley characterizations for Hardy spaces on spaces of homogeneous type. Math. Nachr. 279, 1505–1537 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Han, Y., Müller, D., Yang, D.: A theory of Besov and Triebel-Lizorkin spaces on metric measure spaces modeled on Carnot-Carathéodory spaces. Abstr. Appl. Anal. 2008, Art ID 893409 (2008), 250 pp. CrossRefGoogle Scholar
  24. 24.
    Hu, G., Yang, D., Zhou, Y.: Boundedness of singular integrals in Hardy spaces on spaces of homogeneous type. Taiwan. J. Math. 13, 91–135 (2009) zbMATHMathSciNetGoogle Scholar
  25. 25.
    Jiang, R., Yang, D.: Orlicz-Hardy spaces associated with operators satisfying Davies-Gaffney estimates. arXiv:0903.4494
  26. 26.
    Lin, C., Liu, H., Liu, Y.: The Hardy space \(H_{L}^{1}\) associated with Schödinger operators on the Heisenberg group (submitted) Google Scholar
  27. 27.
    Li, H.: Estimations L p des opérateurs de Schrödinger sur les groupes nilpotents. J. Funct. Anal. 161, 152–218 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Nagel, A., Ricci, F., Stein, E.M.: Harmonic analysis and fundamental solutions on nilpotent Lie groups. In: Analysis and Partial Differential Equations. Lecture Notes in Pure and Appl. Math., vol. 122, pp. 249–275. Dekker, New York (1990) Google Scholar
  29. 29.
    Nagel, A., Ricci, F., Stein, E.M.: Fundamental solutions and harmonic analysis on nilpotent groups. Bull. Am. Math. Soc. (N.S.) 23, 139–144 (1990) zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Nagel, A., Stein, E.M.: The \(\overline{\partial}_{b}\)-complex on decoupled boundaries in ℂn. Ann. Math. (2) 164, 649–713 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Nagel, A., Stein, E.M., Wainger, S.: Balls and metrics defined by vector fields. Acta Math. 155, 103–147 (1985) zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Ouhabaz, E.M.: Analysis of Heat Equations on Domains. London Math. Soc. Monographs, vol. 31. Princeton University Press, Princeton (2004) Google Scholar
  33. 33.
    Saloff-Coste, L.: Fonctions maximales sur certains groupes de Lie. C. R. Acad. Sci. Paris Sér. I Math. 305, 457–459 (1987) zbMATHMathSciNetGoogle Scholar
  34. 34.
    Saloff-Coste, L.: Analyse rélle sur les groupes à croissance polynômiale. C. R. Acad. Sci. Paris Sér. I Math. 309, 149–151 (1989) zbMATHMathSciNetGoogle Scholar
  35. 35.
    Saloff-Coste, L.: Analyse sur les groupes de Lie à croissance polynômiale. Ark. Mat. 28, 315–331 (1990) zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Shen, Z.: Estimates in L p for magnetic Schrödinger operators. Indiana Univ. Math. J. 45, 817–841 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Stein, E.M.: Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993) zbMATHGoogle Scholar
  38. 38.
    Stein, E.M.: Singular integrals: The roles of Calderón and Zygmund. Not. Am. Math. Soc. 45, 1130–1140 (1998) zbMATHGoogle Scholar
  39. 39.
    Stein, E.M.: Some geometrical concepts arising in harmonic analysis. Geom. Funct. Anal. Special Volume (Part I), 434–453 (2000) Google Scholar
  40. 40.
    Varopoulos, N.Th.: Analysis on Lie groups. J. Funct. Anal. 76, 346–410 (1988) zbMATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Varopoulos, N.Th., Saloff-Coste, L., Coulhon, T.: Analysis and Geometry on Groups. Cambridge University Press, Cambridge (1992) Google Scholar
  42. 42.
    Yosida, K.: Functional Analysis. Springer, Berlin (1999) Google Scholar

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© Revista Matemática Complutense 2010

Authors and Affiliations

  1. 1.School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex SystemsMinistry of EducationBeijingPeople’s Republic of China

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