Journal of Fourier Analysis and Applications

, Volume 17, Issue 1, pp 115–134 | Cite as

Commutators of Riesz Transforms Related to Schrödinger Operators

  • B. BongioanniEmail author
  • E. Harboure
  • O. Salinas


In this work we obtain boundedness on L p , for 1<p<∞, of commutators T b f=bTfT(bf) where T is any of the Riesz transforms or their conjugates associated to the Schrödinger operator −Δ+V with V satisfying an appropriate reverse Hölder inequality. The class where b belongs is larger than the usual BMO. We also obtain a substitute result for p=∞, under a slightly stronger condition on b.


Schrödinger operator BMO Commutators Riesz transforms 

Mathematics Subject Classification (2000)

42B35 35J10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bongioanni, B., Harboure, E., Salinas, O.: Riesz transforms related to Schrödinger operators acting on BMO type spaces. J. Math. Anal. Appl. 357(1), 115–131 (2009) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Coifman, R.R., Rochberg, R., Weiss, G.: Factorization theorems for Hardy spaces in several variables. Ann. Math. (2) 103(3), 611–635 (1976) CrossRefMathSciNetGoogle Scholar
  3. 3.
    Dziubański, J., Zienkiewicz, J.: Hardy spaces H 1 associated to Schrödinger operators with potential satisfying reverse Hölder inequality. Rev. Mat. Iberoam. 15(2), 279–296 (1999) zbMATHGoogle Scholar
  4. 4.
    Dziubański, J., Garrigós, G., Martínez, T., Torrea, J., Zienkiewicz, J.: BMO spaces related to Schrödinger operators with potentials satisfying a reverse Hölder inequality. Math. Z. 249(2), 329–356 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Gehring, F.W.: The L p-integrability of the partial derivatives of a quasiconformal mapping. Acta Math. 130, 265–277 (1973) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Guo, Z., Li, P., Peng, L.: L p boundedness of commutators of Riesz transforms associated to Schrödinger operator. J. Math. Anal. Appl. 341(1), 421–432 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Harboure, E., Segovia, C., Torrea, J.L.: Boundedness of commutators of fractional and singular integrals for the extreme values of p. Ill. J. Math. 41(4), 676–700 (1997) MathSciNetGoogle Scholar
  8. 8.
    John, F., Nirenberg, L.: On functions of bounded mean oscillation. Commun. Pure Appl. Math. 14, 415–426 (1961) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Pérez, C.: Sharp estimates for commutators of singular integrals via iterations of the Hardy-Littlewood maximal function. J. Fourier Anal. Appl. 3(6), 743–756 (1997) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Pradolini, G., Salinas, O.: Commutators of singular integrals on spaces of homogeneous type. Czechoslov. Math. J. 57(1), 75–93 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Shen, Z.: L p estimates for Schrödinger operators with certain potentials. Ann. Inst. Fourier (Grenoble) 45(2), 513–546 (1995) zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Instituto de Matemática Aplicada del Litoral CONICET-UNLSanta FeArgentina

Personalised recommendations