Compact Commutators of Rough Singular Integral Operators

Abstract

AbstractLet b ∊ BMO(ℝn) and TΩ be the singular integral operator with kernel Ω(x)/|x|n, where Ω is homogeneous of degree zero, integrable, and has mean value zero on the unit sphere Sn-1. In this paper, using Fourier transform estimates and approximation to the operator TΩ by integral operators with smooth kernels, it is proved that if b ∊ CMO(ℝn) and satisfies certain minimal size condition, then the commutator generated by b and TΩ is a compact operator on Lp(ℝn) for appropriate index p. The associated maximal operator is also considered.