Compactness of the commutators of homogeneous singular integral operators

• DOI: 10.1007/s11425-015-5017-1
• Publication Date: 2015-04-16T19:04:13Z
• Authors: XiaoLi Guo; GuoEn Hu
• Keywords: Commutator; Unit sphere; Kernel (algebra); Operator (biology); Locally integrable function; Zero (linguistics)

Let T Ω be the singular integral operator with kernel $$\frac{{\Omega (x)}} {{\left| x \right|^n }}$$ , where Ω is homogeneous of degree zero, integrable and has mean value zero on the unit sphere S n−1. In this paper, by Fourier transform estimates, Littlewood-Paley theory and approximation, the authors prove that if Ω ∈ L(lnL)2(S n−1), then the commutator generated by T Ω and CMO(ℝ n ) function, and the corresponding discrete maximal operator, are compact on $$L^p \left( {\mathbb{R}^n ,\left| x \right|^{\gamma _p } } \right)$$ for p ∈ (1, ∞) and γ p ∈ (−1, p − 1).