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Gradient Estimates for the Commutator with Fractional Differentiation for Second Order Elliptic Perators

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Abstract

Let L = −div(A∇) be a second order divergence form elliptic operator, where A is an accretive, n×n matrix with bounded measurable complex coefficients on ℝn. Let \(L^{\frac{\alpha}{2}}(0<\alpha<1)\) denotes the fractional differential operator associated with L and \((-\Delta)^{\frac{\alpha}{2}}b\;\in\;L^{n/\alpha}(\mathbb{R}^n)\). In this article, we prove that the commutator [b, \(L^{\frac{\alpha}{2}}\)] is bounded from the homogenous Sobolev space \(\dot{L}_\alpha^2\) (ℝn) to L2(ℝn).

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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, China

    Wenyu Tao, Yanping Chen & Jili Li

Authors
  1. Wenyu Tao
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  2. Yanping Chen
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  3. Jili Li
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Corresponding author

Correspondence to Yanping Chen.

Additional information

The second author was supported by NSFC (11471033), NCET of China (NCET-11-0574) and the Fundamental Research Funds for the Central Universities (FRF-BR- 16-011A).

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Tao, W., Chen, Y. & Li, J. Gradient Estimates for the Commutator with Fractional Differentiation for Second Order Elliptic Perators. Acta Math Sci 39, 1255–1264 (2019). https://doi.org/10.1007/s10473-019-0505-y

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  • DOI: https://doi.org/10.1007/s10473-019-0505-y

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