L 2(ℝ n ) Boundedness for the Commutators of Homogeneous Singular Integral Operators

Abstract

The commutators of singular integral operators with homogeneous kernel \( \frac{{\Omega {\left( x \right)}}} {{{\left| x \right|}^{n} }} \) are studied, where Ω is homogeneous of degree zero, and has mean value zero on the unit sphere. It is proved that Ω∈L(logL)k+1(Sn–1) is a sufficient condition such that the k-th order commutator is bounded on L2(ℝn).