Weighted Norm Inequalities for Area Functions Related to Schrödinger Operators

  • L. TangEmail author
  • J. Wang
  • H. Zhu
Real and Complex Analysis


Let L = −Δ + V be a Schrödinger operator, where Δ is the Laplacian operator on ℝn, and V is a nonnegative potential belonging to certain reverse Hölder class. In this paper, we establish some weighted norm inequalities for area functions related to Schrödinger operators and their commutators.


Area function Schrödinger operator weighted norm inequality 

MSC2010 numbers

42B25 42B20 


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  1. 1.
    B. Bongioanni, E. Harboure and O. Salinas, “Commutators of Riesz transforms related to Schrödinger operators”, J. Fourier Ana Appl., 17, 115–134, 2011.CrossRefzbMATHGoogle Scholar
  2. 2.
    B. Bongioanni, E. Harboure and O. Salinas, “Class of weights related to Schrödinger operators”, J. Math. Anal. Appl., 373, 563–579, 2011.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    B. Bongioanni, E. Harboure and O. Salinas, “Weighted inequalities for commutators of Schrödinger Riesz transforms”, J.Math. Anal. Appl., 392, 6–22, 2012.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    B. Bongioanni, A. Cabral and E. Harboure and O. Salinas, “Lerner’s inequality associated to a critical radius function and applications”, J.Math. Anal. Appl., 407, 35–55, 2013.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    J. Dziubański and J. Zienkiewicz, “Hardy space H1 associated to Schrödinger operator with potential satisfying reverse Hölder inequality”, Rev.Math. Iber., 15, 279–296, 1999.CrossRefzbMATHGoogle Scholar
  6. 6.
    J. Dziubański, G. Garrigós, J. Torrea and J. Zienkiewicz, “BMO spaces related to Schrödinger operators with potentials satisfying a reverse Hölder inequality”, Math. Z., 249, 249–356, 2005.zbMATHGoogle Scholar
  7. 7.
    J. Garciá–Cuerva and J. Rubio de Francia, Weighted norm inequalities and related topics (North–Holland, Amsterdam, 1985).zbMATHGoogle Scholar
  8. 8.
    S. Hartzstein, O. Salinas, “Weighted BMO and Carleson measures on spaces of homogeneous”, J. Math. Anal. Appl., 342, 950–969, 2008.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    B. Muckenhoupt, “Weighted norm inequalities for the Hardy maximal functions”, Trans. Amer.Math. Soc., 165, 207–226, 1972.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    C. Pérez, “Endpoint estimates for commutators of singular integral operators”, J. Funct. Anal., 128, 163–185, 1995.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    M. M. Rao and Z. D. Ren, Theory of Orlicz spaces, Monogr. Textbooks Pure Appl. Math.146, (Marcel Dekker, New York, 1991).Google Scholar
  12. 12.
    Z. Shen, “Lp estimates for Schrödinger operators with certain potentials”, Ann. Inst. Fourier. Grenoble, 45, 513–546, 1995.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    L. Tang, “Weighted norm inequalities for Schrödinger type operators”, ForumMath., 27, 2491–2532, 2015.zbMATHGoogle Scholar
  14. 14.
    L. Tang, “Weighted norm inequalities for commutators of Littlewood–Paley functions related to Schrödinger operators”, arXiv:1109.0100.Google Scholar
  15. 15.
    J. Zhong, Harmonic analysis for some Schrödinger type operators, Ph.D. Thesis. (Princeton University, Princeton, 1993).Google Scholar

Copyright information

© Allerton Press, Inc. 2019

Authors and Affiliations

  1. 1.Peking UniversityBeijingChina
  2. 2.Tufts UniversityMedfordUSA
  3. 3.Beijing International Studies UniversityBeijingChina

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