$L^p({\Bbb R}^n)$ boundedness for the commutator of a homogeneous singular integral operator
Volume 154 / 2003
Studia Mathematica 154 (2003), 13-27
MSC: 42B20, 42B25.
DOI: 10.4064/sm154-1-2
Abstract
The commutator of a singular integral operator with homogeneous kernel ${\mit \Omega }(x)/|x|^n$ is studied, where ${\mit \Omega }$ is homogeneous of degree zero and has mean value zero on the unit sphere. It is proved that ${\mit \Omega }\in L(\mathop {\rm log}\nolimits L)^{k+1}(S^{n-1})$ is a sufficient condition for the $k$th order commutator to be bounded on $L^p({{\mathbb R}}^n)$ for all $1< p<\infty $. The corresponding maximal operator is also considered.
