Skip to main content
SpringerLink
Log in

On Boundedness Property of Singular Integral Operators Associated to a Schrödinger Operator in a Generalized Morrey Space and Applications

  • Published:
Acta Mathematica Scientia Aims and scope Submit manuscript

Abstract

In this paper, we provide the boundedness property of the Riesz transforms associated to the Schrödinger operator \({\cal L} = \Delta + {\bf{V}}\) in a new weighted Morrey space which is the generalized version of many previous Morrey type spaces. The additional potential V considered in this paper is a non-negative function satisfying the suitable reverse Hölder’s inequality. Our results are new and general in many cases of problems. As an application of the boundedness property of these singular integral operators, we obtain some regularity results of solutions to Schrödinger equations in the new Morrey space.

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.

Similar content being viewed by others

References

  1. Adams D, Xiao J. Nonlinear potential analysis on Morrey spaces and their capacities. Indiana Univ Math, 2004, 53: 1629–1663

    Article  MathSciNet  Google Scholar 

  2. Akbulut1 A, Guliyev V, Omarova M. Marcinkiewicz integrals associated with Schrödinger operator and their commutators on vanishing generalized Morrey spaces. Boundary Value Problems, 2017, 2017: Art 121

  3. Bui T A. The weighted norm inequalities for Riesz transforms of magnetic Schrödinger operators. Differential Integral Equations, 2010, 23: 811–826

    MathSciNet  MATH  Google Scholar 

  4. Bui T A. Weighted estimates for commutators of some singular integrals related to Schrödinger operators. Bulletin des Sciences Mathematiques, 2014, 138(2): 270–292

    Article  MathSciNet  Google Scholar 

  5. Bongioanni B, Harboure E, Salinas O. Riesz transform related to Schrödinger operators acting on BMO type spaces. J Math Anal Appl, 2009, 357(2): 115–131

    Article  MathSciNet  Google Scholar 

  6. Bongioanni B, Harboure E, Salinas O. Classes of weights related to Schrödinger operators. J Math Anal Appl, 2011, 373: 563–579

    Article  MathSciNet  Google Scholar 

  7. Coulhon T, Duong X T. Riesz transforms for 1 ≤ p ≤ 2. Trans Amer Math Soc, 1999, 351(3): 1151–1169

    Article  MathSciNet  Google Scholar 

  8. Duong X T, Xiao J, Yan L. Old and new Morrey spaces with heat kernel bounds. J Fourier Anal Appl, 2007, 13: 87–111

    Article  MathSciNet  Google Scholar 

  9. Dziubánski J, Garrigos G, Martinez T, Torrea J, Zienkiewicz J. BMO spaces related to Schrödinger operators with potentials satisfying reverse Hölder inequality. Mat Z, 2005, 49(2): 329–356

    Article  Google Scholar 

  10. Dziubánski J, Zienkiewicz J. Hp spaces for Schrödinger operators. Fourier Anal Related Top, 2002, 56: 45–53

    MATH  Google Scholar 

  11. Feuto J, Fofana I, Koua K. Espaces de fonctions moyenne fractionnaire intgrables sur les groupes localement compacts. Afrika Mat, 2003, 3(13): 73–91

    MATH  Google Scholar 

  12. Feuto J, Fofana I, Koua K. Integrable fractional mean functions on spaces of homogeneous type. Afr Diaspora J Math, 2010, 9(1): 8–30

    MathSciNet  MATH  Google Scholar 

  13. Feuto J. Norm inequalities in generalized morrey spaces. J Fourier Anal Appl, 2014, 20(4): 896–909

    Article  MathSciNet  Google Scholar 

  14. Fofana I. Étude d’une classe d’espaces de fonctions contenant les espaces de Lorentz. Afrika Mat, 1988, 1 (2): 29–50

    MathSciNet  MATH  Google Scholar 

  15. Grafakos L. Classical Fourier Analysis. Graduate Texts in Mathematics 250. 3rd ed. New York: Springer, 2014

    MATH  Google Scholar 

  16. Komori Y, Shirai S. Weighted Morrey spaces and a singular integral operator. Math Nachr, 2009, 282: 219–231

    Article  MathSciNet  Google Scholar 

  17. Kurata K, Sugano S. Estimate of the fundamental solution for magnetic Schröodinger operators and their applications. Tohoku Math J, 2000, 52: 367–382

    Article  MathSciNet  Google Scholar 

  18. Komori Y, Shirai S. Weighted Morrey spaces and a singular integral operator. Math Nachr, 2009, 282: 219–231

    Article  MathSciNet  Google Scholar 

  19. Li P, Wan X, Zhang C. Schröodinger type operators on generalized Morrey spaces. J Inequal Appl, 2015, 2015: Art 229

  20. Liu Y. The weighted estimates for the operators Vα(−ΔG+V)−β and Vα∇(−ΔG+V)−β on the stratified Lie group G. J Math Anal Appl, 2009, 349: 235–244

    Article  MathSciNet  Google Scholar 

  21. Liu Y, Wang L. Boundedness for Riesz transform associated with Schröodinger operators and its commutator on weighted Morrey spaces related to certain nonnegative potentials. J Inequal Appl, 2014, 2014: Art 194

  22. Ly F K. Second order Riesz transforms associated to the Schröodinger operator for p ≤ 1. J Math Anal Appl, 2014, 410(1): 391–402

    Article  MathSciNet  Google Scholar 

  23. Morrey C. On the solutions of quasi-linear elliptic partial differential equations. Trans Amer Math Soc, 1938, 43: 126–166

    Article  MathSciNet  Google Scholar 

  24. Muckenhoupt B, Wheeden R L. Weighted norm inequalities for fractional integrals. Trans Amer Math Soc, 1974, 192: 261–274

    Article  MathSciNet  Google Scholar 

  25. Peetre J. On the theory of Lp,λ spaces. J Funct Anal, 1969, 4: 71–87

    Article  Google Scholar 

  26. Pan G, Tang L. Solvability for Schrödinger equations with discontinuous coefficients. J Funct Anal, 2016, 270: 88–133

    Article  MathSciNet  Google Scholar 

  27. Ren B, Wang H. Boundedness of higher order Riesz transforms associated with Schröodinger type operator on generalized Morrey spaces. J Nonlinear Sci Appl, 2017, 10: 2757–2766

    Article  MathSciNet  Google Scholar 

  28. Samko N. Weighted Hardy and singular operators in Morrey spaces. J Math Anal Appl, 2009, 350: 56–72

    Article  MathSciNet  Google Scholar 

  29. Shen Z. Lp estimates for Schrödinger operators with certain potentials. Ann Inst Fourier, 1995, 45: 513–546

    Article  MathSciNet  Google Scholar 

  30. Shen Z. Estimates in Lp for magnetic Schrödinger operators. Indiana Univ Math J, 1996, 45: 817–841

    Article  MathSciNet  Google Scholar 

  31. Song L, Yang L. Riesz transforms associated to Schrödinger operators on weighted Hardy spaces. J Funct Anal, 2010, 259: 1466–1490

    Article  MathSciNet  Google Scholar 

  32. Sugano S. Estimates for the operators Vα(−Δ +V)−β and Vα∇(−Δ +V)−β with certain nonnegative potentials V. Tokyo J Math, 1988, 21: 441–452

    Article  MathSciNet  Google Scholar 

  33. Tang L, Dong J. Boundedness for some Schrödinger type operators on Morrey spaces related to certain nonnegative potentials. J Math Anal Appl, 2009, 355: 101–109

    Article  MathSciNet  Google Scholar 

  34. Xiao J. Homothetic variant of fractional Sobolev space with application to Navier-Stokes system. Dyn Partial Differ Equ, 2007, 4: 227–245

    Article  MathSciNet  Google Scholar 

  35. Zhong J. Harmonic Analysis for Some Schröodinger Type Operators [D]. Princeton University, 1993

Download references

Author information

Authors and Affiliations

  1. Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Xuan Truong Le

  2. Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Xuan Truong Le

  3. Department of Mathematics, Ho Chi Minh City University of Education, Ho Chi Minh City, Vietnam

    Thanh Nhan Nguyen

  4. Department of Primary Education, Ho Chi Minh City University of Education, Ho Chi Minh City, Vietnam

    Ngoc Trong Nguyen

Authors
  1. Xuan Truong Le
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Thanh Nhan Nguyen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ngoc Trong Nguyen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Xuan Truong Le, Thanh Nhan Nguyen or Ngoc Trong Nguyen.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Le, X.T., Nguyen, T.N. & Nguyen, N.T. On Boundedness Property of Singular Integral Operators Associated to a Schrödinger Operator in a Generalized Morrey Space and Applications. Acta Math Sci 40, 1171–1184 (2020). https://doi.org/10.1007/s10473-020-0501-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10473-020-0501-2

Key words

2010 MR Subject Classification

Search

Navigation