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Gradient Estimates for Commutators of Square Roots of Elliptic Operators with Complex Bounded Measurable Coefficients

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Abstract

Let \(L=-\mathrm{div}(A\nabla )\) be a second order divergence form elliptic operator and A an accretive \(n\times n\) matrix with bounded measurable complex coefficients in \({\mathbb R}^n\). Let \(\nabla b\in L^n({\mathbb R}^n)\,(n>2)\). In this paper, we prove that the commutator generated by b and the square root of L, which is defined by \([b,\sqrt{L}]f(x)=b(x)\sqrt{L}f(x)-\sqrt{L}(bf)(x)\), is bounded from the homogenous Sobolev space \({\dot{L}}_1^2({\mathbb R}^n)\) to \(L^2({\mathbb R}^n)\).

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Acknowledgments

The authors would like to express their deep gratitude to the referee for giving many valuable suggestions. Y. Chen is supported by NSF of China (Grant: 11471033), NCET of China (Grant: NCET-11-0574) and the Fundamental Research Funds for the Central Universities (FRF-TP-12-006B). Y. Ding is supported by NSF of China (Grant: 11371057), SRFDP of China (Grant: 20130003110003) and the Fundamental Research Funds for the Central Universities (2014KJJCA10).

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Authors and Affiliations

  1. Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing, 100083, People’s Republic of China

    Yanping Chen

  2. Laboratory of Mathematics and Complex Systems (BNU), Ministry of Education, School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, People’s Republic of China

    Yong Ding

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  1. Yanping Chen
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Correspondence to Yanping Chen.

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Chen, Y., Ding, Y. Gradient Estimates for Commutators of Square Roots of Elliptic Operators with Complex Bounded Measurable Coefficients. J Geom Anal 27, 466–491 (2017). https://doi.org/10.1007/s12220-016-9687-x

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  • DOI: https://doi.org/10.1007/s12220-016-9687-x

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