Compact Commutators of Rough Singular Integral Operators

Chen, Jiecheng ; Hu, Guoen

Canadian mathematical bulletin, Tome 58 (2015) no. 1, pp. 19-29

Résumé

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.

Détail

DOI : 10.4153/CMB-2014-042-1

Mots-clés : 42B20, commutator, singular integral operator, compact operator, maximal operator