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The Boundedness for Commutator of Fractional Integral Operator With Rough Variable Kernel

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Abstract

For b ∈ BMO(ℝn) and 0 < α ≤ 1/2, the commutator of the fractional integral operator T Ω,α with rough variable kernel is defined by

$$ [b, T_{\Omega, \alpha}]f(x)= \int_{\mathbb{R}^n} \frac{\Omega(x,x-y)}{|x-y|^{n-\alpha}}(b(x)-b(y))f(y)dy. $$

In this paper the authors prove that the commutator [b, T Ω,α ] is a bounded operator from \(L^{\frac{2n}{n+2\alpha}}(\mathbb{R}^n)\) to L 2(ℝn). The result obtained in this paper is substantial improvement and extension of some known results.

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Authors and Affiliations

  1. Department of Applied Mathematics, University of Science and Technology Beijing, Beijing, 100083, China

    Yanping Chen

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

    Yong Ding

  3. School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

    Ran Li

Authors
  1. Yanping Chen
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  2. Yong Ding
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  3. Ran Li
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Corresponding author

Correspondence to Yanping Chen.

Additional information

The research was supported by NSF of China (Grant: 10901017, 10931001), SRFDP of China (Grant: 20090003110018) and the Fundamental Research Funds for the Central Universities.

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Chen, Y., Ding, Y. & Li, R. The Boundedness for Commutator of Fractional Integral Operator With Rough Variable Kernel. Potential Anal 38, 119–142 (2013). https://doi.org/10.1007/s11118-011-9267-4

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  • DOI: https://doi.org/10.1007/s11118-011-9267-4

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