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Compact Commutators of Rough Singular Integral Operators

  • Jiecheng Chen,
    Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, P. R. China
  • Guoen Hu,
    Department of Applied Mathematics, Zhengzhou Information Science and Technology Institute, P. O. Box 1001-747, Zhengzhou 450002, P. R. China
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Abstract

Let $b\in \mathrm{BMO}(\mathbb{R}^n)$ and $T_{\Omega}$ be the singular integral operator with kernel $\frac{\Omega(x)}{|x|^n}$, where $\Omega$ is homogeneous of degree zero, integrable and has mean value zero on the unit sphere $S^{n-1}$. In this paper, by Fourier transform estimates and approximation to the operator $T_{\Omega}$ by integral operators with smooth kernels, it is proved that if $b\in \mathrm{CMO}(\mathbb{R}^n)$ and $\Omega$ satisfies a certain minimal size condition, then the commutator generated by $b$ and $T_{\Omega}$ is a compact operator on $L^p(\mathbb{R}^n)$ for appropriate index $p$. The associated maximal operator is also considered.
Keywords: commutator, singular integral operator, compact operator, maximal operator commutator, singular integral operator, compact operator, maximal operator
MSC Classifications: 42B20 show english descriptions Singular and oscillatory integrals (Calderon-Zygmund, etc.) 42B20 - Singular and oscillatory integrals (Calderon-Zygmund, etc.)
 

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