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

  • Wenyu Tao
  • Yanping ChenEmail author
  • Yayuan Xiao
  • Liwei Wang


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 ∇bLn(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)).


commutator Kato square root elliptic operators Sobolev space 


42B20 42B25 


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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.


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Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Wenyu Tao
    • 1
  • Yanping Chen
    • 1
    Email author
  • Yayuan Xiao
    • 2
  • Liwei Wang
    • 1
    • 3
  1. 1.School of Mathematics and PhysicsUniversity of Science and Technology BeijingBeijingChina
  2. 2.Department of Mathematical SciencesBall State UniversityMuncieUSA
  3. 3.School of Mathematics and PhysicsAnhui Polytechnic UniversityWuhuChina

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