Skip to main content
SpringerLink
Log in

Characterizations of localized BMO(ℝn) via commutators of localized Riesz transforms and fractional integrals associated to Schrödinger operators

  • Published:
Collectanea mathematica Aims and scope Submit manuscript

Abstract

Let\(\mathcal{L} \equiv - \Delta + V\) be the Schrödinger operator in ℝn, whereV is a nonnegative function satisfying the reverse Hölder inequality. Let ρ be an admissible function modeled on the known auxiliary function determined byV. In this paper, the authors establish several characterizations of the space BMOρ(ℝn) in terms of commutators of several different localized operators associated to ρ, respectively; these localized operators include localized Riesz transforms and their adjoint operators, the localized fractional integral and its adjoint operator, the localized fractional maximal operator and the localized Hardy-Littlewood-type maximal operator. These results are new even for the space\(\mathcal{L} \equiv - \Delta + V\) introduced by J. Dziubański, G. Garrigóset al.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. Chanillo, A note on commutators,Indiana Univ. Math. J. 31 (1982), 7–16.

    Article  MATH  MathSciNet  Google Scholar 

  2. R.R. Coifman, R. Rochberg, and G. Weiss, Factorization theorems for Hardy spaces in several variables,Ann. of Math. (2) 103 (1976), 611–635.

    Article  MathSciNet  Google Scholar 

  3. X.T. Duong and L. Yan, New function spaces of BMO type, the John-Nirenberg inequality, interpolation, and applications,Comm. Pure Appl. Math. 58 (2005), 1375–1420.

    Article  MATH  MathSciNet  Google Scholar 

  4. X.T. Duong and L. Yan, Duality of Hardy and BMO spaces associated with operators with heat kernel bounds,J. Amer. Math. Soc. 18 (2005), 943–973.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  6. J. Dziubański and J. Zienkiewicz, Hardy spaceH 1 associated to Schrödinger operator with potential satisfying reverse Hölder inequality,Rev. Mat. Iberoamericana 15 (1999), 279–296.

    MATH  MathSciNet  Google Scholar 

  7. C.L. Fefferman, The uncertainty principle,Bull. Amer. Math. Soc. (N.S.) 9 (1983), 129–206.

    Article  MATH  MathSciNet  Google Scholar 

  8. D. Goldberg, A local version of real Hardy spaces,Duke Math. J. 46 (1979), 27–42.

    Article  MATH  MathSciNet  Google Scholar 

  9. L. Grafakos,Classical and Modern Fourier Analysis, Pearson Education, Inc., Upper Saddle River N.J., 2004.

    MATH  Google Scholar 

  10. Z. Guo, P. Li, and L. Peng,L p boundedness of commutators of Riesz transforms associated to Schrödinger operator,J. Math. Anal. Appl. 341 (2008), 421–432.

    Article  MATH  MathSciNet  Google Scholar 

  11. S. Janson, Mean oscillation and commutators of singular integral operators,Ark. Mat. 16 (1978), 263–270.

    Article  MATH  MathSciNet  Google Scholar 

  12. F. John and L. Nirenberg, On functions of bounded mean oscillation,Comm. Pure Appl. Math. 14 (1961), 415–426.

    Article  MATH  MathSciNet  Google Scholar 

  13. C. Segovia and J.L. Torrea, Vector-valued commutators and applications,Indiana Univ. Math. J. 38 (1989), 959–971.

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  15. E.M. Stein,Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mahtematical Series43, Princeton University Press, Princeton, N.J., 1993.

    MATH  Google Scholar 

  16. Da. Yang, Do. Yang, and Y. Zhou, Localized BMO and BLO spaces on RD-spaces and applications to Schrödinger operators, arXiv: 0903.4536.

  17. Da. Yang, Do. Yang, and Y. Zhou, Endpoint properties of localized Riesz transforms and fractional integrals associated to Schrödinger operators,Potential Anal. 30 (2009), 271–300.

    Article  MATH  MathSciNet  Google Scholar 

  18. Da. Yang and Y. Zhou, Localized Hardy spacesH 1 related to admissible functions on RD-spaces and applications to Schrödinger operators, arXiv: 0903.4581.

Download references

Author information

Authors and Affiliations

  1. Laboratory of Mathematics and Complex systems, Ministry of Education, School of Mathematical Sciences, Beijing Normal University, 100875, Beijing, People’s Republic of China

    Dachun Yang & Dongyong Yang

Authors
  1. Dachun Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dongyong Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dachun Yang.

Additional information

The first author is supported by the National Natural Science Foundation (Grant No. 10871025) of China.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yang, D., Yang, D. Characterizations of localized BMO(ℝn) via commutators of localized Riesz transforms and fractional integrals associated to Schrödinger operators. Collect. Math. 61, 65–79 (2010). https://doi.org/10.1007/BF03191227

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03191227

Keywords

MSC2000

Search

Navigation