Skip to main content
SpringerLink
Log in

Lp-gradient estimates for the commutators of the Kato square roots of second-order elliptic operators on ℝn

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

Let L = –div(A∇) be a second-order divergent-form elliptic operator, where A is an accretive n × n matrix with bounded and measurable complex coefficients on ℝn: Herein, we prove that the commutator [b; \(\sqrt L \)] of the Kato square root \(\sqrt L \) and b with ∇b ∈ Ln(ℝn)(n > 2), is bounded from the homogenous Sobolev space \(\dot L_1^p(\mathbb{R}^n)\) to Lp(ℝn) (p-(L) < p < p+(L)).

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. Auscher P. On Lp estimates for square roots of second order elliptic operators on ℝn. Publ Mat, 2004, 48: 159–186

    Article  MathSciNet  Google Scholar 

  2. Auscher P. On Necessary and Sufficient Conditions for Lp-Estimates of Riesz Transforms Associated to Elliptic Operators on ℝn and Related Estimates. Memoirs of the American Mathematical Society, vol. 186. Providence: Amer Math Soc, 2007

    Google Scholar 

  3. Auscher P, Coulhon T, Tchamitchian P. Absence de principe du maximum pour certaines equations paraboliques complexes. Colloq Math, 1996, 71: 87–95

    Article  MathSciNet  Google Scholar 

  4. Auscher P, Hofmann S, Lacey M, et al. The solution of the Kato square root problem for second order elliptic operators on ℝn. Ann of Math (2), 2002, 156: 633–654

    Article  MathSciNet  Google Scholar 

  5. Auscher P, Hofmann S, Lewis J, et al. Extrapolation of Carleson measures and the analyticity of Kato’s square root operators. Acta Math, 2001, 187: 161–190

    Article  MathSciNet  Google Scholar 

  6. Auscher P, Martell J M. Weighted norm inequalities, off-diagonal estimates and elliptic operators Part I: General operator theory and weights. Adv Math, 2007, 212: 225–276

    Article  MathSciNet  Google Scholar 

  7. Auscher P, Martell J M. Weighted norm inequalities, off-diagonal estimates and elliptic operators Part II: Off-diagonal estimates on spaces of homogeneous type. J Evol Equ, 2007, 7: 265–316

    Article  MathSciNet  Google Scholar 

  8. Auscher P, Tchamitchian P. Square Root Problem for Divergence Operators and Related Topics. Astérisque, no. 249. Paris: Soc Math France, 1998

    MATH  Google Scholar 

  9. Bui T A. Commutators of BMO functions with spectral multiplier operators. J Math Soc Japan, 2012, 64: 885–902

    Article  MathSciNet  Google Scholar 

  10. Bui T A, Duong X T. Weighted norm inequalities for commutators of BMO functions and singular integral operators with non-smooth kernels. J Geom Anal, 2014, 24: 1368–1397

    Article  MathSciNet  Google Scholar 

  11. Calderón A P. Commutators of singular integral operators. Proc Natl Acad Sci USA, 1965, 53: 1092–1099

    Article  MathSciNet  Google Scholar 

  12. Calderón A P. Commutators, singular integrals on Lipschitz curves and application. Proc Int Congr Math, 1978, 1: 85–96

    Google Scholar 

  13. Cao J, Mayboroda S, Yang D. Maximal function characterizations of Hardy spaces associated to homogeneous higher order elliptic operators. Forum Math, 2016, 28: 823–856

    Article  MathSciNet  Google Scholar 

  14. Cao J, Mayboroda S, Yang D. Local Hardy spaces associated with inhomogeneous higher order elliptic operators. Anal Appl, 2017, 15: 137–224

    Article  MathSciNet  Google Scholar 

  15. Chen Y, Ding Y. Gradient estimates for commutators of square roots of elliptic operators with complex bounded measurable coefficient. J Geom Anal, 2017, 27: 466–491

    Article  MathSciNet  Google Scholar 

  16. Chen Y, Ding Y, Hofmann S. The commutator of the Kato square root for second order elliptic operators on ℝn. Acta Math Sin (Engl Ser), 2016, 32: 1121–1144

    Article  MathSciNet  Google Scholar 

  17. Coifman R, McIntosh A, Meyer Y. L’integrale de Cauchy définit un opérateur borné sur L2(ℝn) pour les courbes lipschitziennes. Ann of Math (2), 1982, 116: 361–387

    Article  MathSciNet  Google Scholar 

  18. Fabes E, Jerison D, Kenig C. Multilinear square functions and partial differential equations. Amer J Math, 1985, 107: 1325–1368

    Article  MathSciNet  Google Scholar 

  19. Geng J, Shen Z, Song L. Uniform W1,p estimates for systems of linear elasticity in a periodic medium. J Funct Anal, 2012, 262: 1742–1758

    Article  MathSciNet  Google Scholar 

  20. Hofmann S, Lacey M, McIntosh A. The solution of the Kato problem for divergence form elliptic operators with Gaussian heat kernel bounds. Ann of Math (2), 2002, 156: 623–631

    Article  MathSciNet  Google Scholar 

  21. Hofmann S, Martell J M. Lp bounds for Riesz tranforms and square roots associated to second order elliptic operators. Publ Mat, 2003, 47: 497–515

    Article  MathSciNet  Google Scholar 

  22. Jiang R, Yang D. New Orlicz-Hardy spaces associated with divergence form elliptic operators. J Funct Anal, 2010, 258: 1167–1224

    Article  MathSciNet  Google Scholar 

  23. John F, Nirenberg L. On functions of bounded mean oscillation. Comm Pure Appl Math, 1961, 14: 415–426

    Article  MathSciNet  Google Scholar 

  24. Kato T. Franctional powers of dissipative operators. J Math Soc Japan, 1961, 13: 246–274

    Article  MathSciNet  Google Scholar 

  25. Kato T. Perturbation Theory for Linear Operators. Berlin-Heidelberg: Springer, 1995

    Book  Google Scholar 

  26. Kenig C, Meyer Y. Kato’s square roots of accretive operators and Cauchy kernels on Lipschitz curves are the same. North-Holland Math Stud, 1985, 111: 123–143

    Article  MathSciNet  Google Scholar 

  27. Liu L, Chang D, Fu X, et al. Endpoint boundedness of commutators on spaces of homogeneous type. Appl Anal, 2017, 96: 2408–2433

    Article  MathSciNet  Google Scholar 

  28. McIntosh A. On representing closed accretive sesquilinear forms as (A1/2u;A*1/2v). In: Nonlinear Partial Differential Equations and their Applications. Collège de France Seminar, vol. 3. Research Notes in Mathematics, vol. 70. Boston-Mass-London: Pitman, 1982: 252–267

    Google Scholar 

  29. McIntosh A. Square roots of operators and applications to hyperbolic PDEs. In: Miniconference on Operator Theory and Partial Differential Equations. Proceedings of the Centre for Mathematics and its Applications, Australian National University, vol. 5. Canberra: Austral National University, 1984: 124–136

    Google Scholar 

  30. Meyer Y, Coifman R. Ondelettes et opérateurs. Paris: Hermann, 1990

    MATH  Google Scholar 

  31. Pérez C, Trujillo-Gonzá R. Sharp weighted estimates for multilinear commutators. J Lond Math Soc (2), 2002, 65: 672–692

    Article  MathSciNet  Google Scholar 

  32. Pradolini G, Ramos W. Characterization of Lipschitz functions via the commutators of singular and fractional integral operators in variable Lebesgue spaces. Potential Anal, 2017, 46: 499–525

    Article  MathSciNet  Google Scholar 

  33. Shen S, Song L. On Lp estimates in homogenization of elliptic equations of Maxwell’s type. Adv Math, 2014, 252: 7–21

    Article  MathSciNet  Google Scholar 

  34. Wu Q, Fu Z. Weighted p-adic Hardy operators and their commutators on p-adic central Morrey spaces. Bull Malays Math Sci Soc (2), 2017, 40: 635–654

    Article  MathSciNet  Google Scholar 

  35. Yang D, Zhang J. Weighted Lp estimates of Kato square roots associated to degenerate elliptic operators. Publ Mat, 2017, 61: 395–444

    Article  MathSciNet  Google Scholar 

  36. Zhang J, Cao J, Jiang R, et al. Non-tangential maximal function characterizations of Hardy spaces associated with degenerate elliptic operators. Canad J Math, 2015, 67: 1161–1200

    Article  MathSciNet  Google Scholar 

  37. Zhang J, Chang D, Yang D. Characterizations of Sobolov spaces associated to operators satisfying off-diagonal estimates on balls. Math Methods Appl Sci, 2017, 40: 2907–2929

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 11471033), Program for New Century Excellent Talents in University of China (Grant No. NCET-11-0574), the Fundamental Research Funds for the Central Universities (Grant No. FRF-BR-17-001B) and the Fundamental Research Funds for Doctoral Candidate of University of Science and Technology Beijing (Grant No. FRF-BR-17-018). The authors express their gratitude to the referees for their helpful comments.

Author information

Authors and Affiliations

  1. School of Mathematics and Physics, University of Science and Technology Beijing, Beijing, 100083, China

    Wenyu Tao, Yanping Chen & Liwei Wang

  2. Department of Mathematical Sciences, Ball State University, Muncie, IN, 47306, USA

    Yayuan Xiao

  3. School of Mathematics and Physics, Anhui Polytechnic University, Wuhu, 241000, China

    Liwei Wang

Authors
  1. Wenyu Tao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yanping Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yayuan Xiao
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Liwei Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yanping Chen.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tao, W., Chen, Y., Xiao, Y. et al. Lp-gradient estimates for the commutators of the Kato square roots of second-order elliptic operators on ℝn. Sci. China Math. 63, 575–594 (2020). https://doi.org/10.1007/s11425-017-9310-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-017-9310-0

Keywords

MSC(2010)

Search

Navigation