Potential Analysis

, Volume 38, Issue 3, pp 711–739 | Cite as

Variation Operators for Semigroups and Riesz Transforms on BMO in the Schrödinger Setting

  • Jorge J. Betancor
  • Juan C. Fariña
  • Eleonor Harboure
  • Lourdes Rodríguez-MesaEmail author


In this paper we prove that the variation operators of the heat semigroup and the truncations of Riesz transforms associated to the Schrödinger operator are bounded on a suitable BMO type space.


Schrödinger operators Variation Riesz transforms Heat semigroups 

Mathematics Subject Classifications (2000)

42B20 42B25 40A30 


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  1. 1.
    Bennet, C., DeVore, R.A., Sharpley, R.: Weak-L and BMO. Ann. Math. (2) 113(3), 601–611 (1981)CrossRefGoogle Scholar
  2. 2.
    Betancor, J.J., Fariña, J.C., Harboure, E., Rodríguez-Mesa, L.: L p-boundedness properties of variation and oscillation operators in the Schrödinger setting. Rev. Mat. Complut. (2012). doi: 10.1007/s13163-012-0094-y
  3. 3.
    Bongioanni, B., Harboure, E., Salinas, O.: Weighted inequalities for negative powers of Schrödinger operators. J. Math. Anal. Appl. 348(1), 12–27 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bongioanni, B., Harboure, E., Salinas, O.: Riesz transform related to Schrödinger operators acting on BMO type spaces. J. Math. Anal. Appl. 357, 115–131 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bongioanni, B., Harboure, E., Salinas, O.: Commutators of Riesz transform related to Schrödinger operators. J. Fourier Anal. Appl. 17(1), 115–134 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bourgain, J.: Pointwise ergodic theorems for arithmetic sets. Publ. Math. IHES 69, 5–45 (1989)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Campbell, J.T., Jones, R.L., Reinhold, K., Wierdl, M.: Oscillation and variation for the Hilbert transform. Duke Math. J. 105(1), 59–83 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Campbell, J.T., Jones, R.L., Reinhold, K., Wierdl, M.: Oscillation and variation for singular integrals in higher dimensions. Trans. Am. Math. Soc. 355(5), 2115–2137 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Crescimbeni, R., Macías, R.A., Menárguez, T., Torrea, J.L., Viviani, B.: The ρ-variation as an operator between maximal operators and singular integrals. J. Evol. Equ. 9(1), 81–102 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Crescimbeni, R., Martín-Reyes, F., de la Torre, A., Torrea, J.L.: The ρ-variation of the Hermitian Riesz transform. Acta Math. Sin. Engl. Ser. 26(10), 1827–1838 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Dong, J.-F., Liu, H.-P.: The BMO L space and Riesz transforms associated with Schrödinger operators. Acta Math. Sin. 26(4), 659–668 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Dziubański, J., Zienkiewicz, J.: Hardy space H 1 associated to Schrödinger operator with potential satisfying reverse Hölder inequality. Rev. Mat. Iberoam. 15(2), 279–296 (1999)zbMATHCrossRefGoogle Scholar
  13. 13.
    Dziubański, J., Zienkiewicz, J.: H p spaces for Schrödinger operators. In: Fourier Analysis and Related Topics, vol. 56, pp. 45–53. Banach Center Publications (2002)Google Scholar
  14. 14.
    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(2), 329–356 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Dziubański, J., Zienkiewicz, J.: H p spaces associated with Schrödinger operators with potentials from reverse Hölder classes. Colloq. Math. 98(1), 5–38 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Gehring, F.W.: The L p-integrability of the partial derivatives of a quasiconformal mapping. Acta Math. 130, 265–277 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Gillespie, T.A., Torrea, J.L.: Dimension free estimates for the oscillation of Riesz transforms. Isr. J. Math. 141, 125–144 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Guo, Z., Li, P., Peng, L.: L p boundedness of commutators of Riesz transforms associated to Schrödinger operator. J. Math. Anal. Appl. 341, 421–432 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Harboure, E., Macías, R., Menárguez, T., Torrea, J.L.: Oscillation and variation for the Gaussian Riesz transforms and Poisson integral. Proc. R. Soc. Edinb. Sect. A 135(1), 85–104 (2005)zbMATHCrossRefGoogle Scholar
  20. 20.
    Jones, R.L., Wang, G.: Variation inequalities for the Fejer and Poisson kernels. Trans. Am. Math. Soc. 356(11), 4493–4518 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Jones, R.L., Kaufman, R., Rosenblatt, J., Wierdl, M.: Oscillation in ergodic theory. Ergod. Theory Dyn. Syst. 18(4), 889–936 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Jones, R.L., Reinhold, K.: Oscillation and variation inequalities for convolution powers. Ergod. Theory Dyn. Syst. 21(6), 1809–1829 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Jones, R.L., Seeger, A., Wright, J.: Strong variational and jump inequalities in harmonic analysis. Trans. Am. Math. Soc. 360(12), 6711–6742 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Liu, Y., Ding, Y.: Some estimates of Schrödinger-type operators with certain nonnegative potentials. Int. J. Math. Math. Sci., article ID 214030, 8 pp. (2008). doi: 10.1155/2008/214030
  25. 25.
    Liu, Y., Dong, J.: Some estimates of higher order Riesz transform related to Schrödinger-type operators. Potential Anal. 32(1), 41–55 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Oberlin, R., Seeger, A., Tao, T., Thiele, C., Wright, J.: A variation norm Carleson theorem. J. Eur. Math. Soc. 14(2), 421–464 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Shen, Z.W.: L p estimates for Schrödinger operators with certain potentials. Ann. Inst. Fourier (Grenoble) 45(2), 513–546 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, vol. 43, xiv+695 (1993). With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, IIIGoogle Scholar
  29. 29.
    Sugano, S.: L p estimates for some Schrödinger operators and a Calderón-Zygmund operator of Schrödinger type. Tokyo J. Math. 30, 179–197 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Torchinsky, A.: Real-Variable Methods in Harmonic Analysis. Dover, Mineola (2004)zbMATHGoogle Scholar
  31. 31.
    Yang, D., Yang, D.: Characterizations of localized BMO\(({\mathbb R}^n)\) via commutators of localized Riesz transforms and fractional integrals associated to Schrödinger operators. Collect. Math. 61(1), 65–79 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Yang, D., Yang, D., Zhou, Y.: Endpoint properties of localized Riesz transforms and fractional integrals associated to Schrödinger operators. Potential Anal. 30, 271–300 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Zhong, J.: Harmonic analysis for some Schrödinger type operators. Ph.D. thesis, Princeton University (1993)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  • Jorge J. Betancor
    • 1
  • Juan C. Fariña
    • 1
  • Eleonor Harboure
    • 2
  • Lourdes Rodríguez-Mesa
    • 1
    Email author
  1. 1.Departamento de Análisis MatemáticoUniversidad de la LagunaLa Laguna (Sta. Cruz de Tenerife)Spain
  2. 2.Instituto de Matemática Aplicada del Litoral, IMALCONICET-Universidad Nacional del LitoralSanta FeArgentina

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