Compact Commutators of Rough Singular Integral Operators. Issue 1 (1st March 2015) - Chen, Jiecheng; Hu, Guoen
Abstract: Let $b\, \in \, \text{BMO}\left( {{\mathbb{R}}^{n}} \right)$ and ${T_{Ω }}$ be the singular integral operator with kernel $Ω \left( x \right)/{{\left| x \right|}^{n}}$, where $Ω$ is homogeneous of degree zero, integrable, and has mean value zero on the unit sphere ${S^{n-1}}$ . In this paper, using Fourier transform estimates and approximation to the operator ${T_{Ω }}$ by integral operators with smooth kernels, it is proved that if $b\, \in \, \text{CMO}\left( {{\mathbb{R}}^{n}} \right)$ and $Ω$ satisfies certain minimal size condition, then the commutator generated by $b$ and ${T_{Ω }}$ is a compact operator on ${L^{p}}\left( {{\mathbb{R}}^{n}} \right)$ for appropriate index $p$ . The associated maximal operator is also considered.