Compact Commutators of Rough Singular Integral Operators. Issue 1 (1st March 2015)

• Record Type: Journal Article
• Authors: Chen, Jiecheng; Hu, Guoen
• Is Part Of: Canadian mathematical bulletin =. Volume 58:Issue 1(2015)
• Page Start: 19
• Page End: 29
• Publication Date: 2015-03-01
• Subjects: 42B20; commutator -- singular integral operator -- compact operator -- maximal operator; Mathematics -- Periodicals; Mathematics; Periodicals; 510.5
• DOI: 10.4153/CMB-2014-042-1
• Languages: English
• ISSNs: 0008-4395

Abstract

Abstract: Let $b\, \in \, \text{BMO}\left( {{\mathbb{R}}^{n}} \right)$ and ${{T}_{\Omega }}$ be the singular integral operator with kernel $\Omega \left( x \right)/{{\left| x \right|}^{n}}$, where $\Omega$ 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 ${{T}_{\Omega }}$ by integral operators with smooth kernels, it is proved that if $b\, \in \, \text{CMO}\left( {{\mathbb{R}}^{n}} \right)$ and $\Omega$ satisfies certain minimal size condition, then the commutator generated by $b$ and ${{T}_{\Omega }}$ is a compact operator on ${{L}^{p}}\left( {{\mathbb{R}}^{n}} \right)$ for appropriate index $p$ . The associated maximal operator is also considered.