$L^p({\Bbb R}^n)$ boundedness for the commutator of a homogeneous singular integral operator

Volume 154 / 2003

Guoen Hu Studia Mathematica 154 (2003), 13-27 MSC: 42B20, 42B25. DOI: 10.4064/sm154-1-2


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.


  • Guoen HuDepartment of Applied Mathematics
    University of Information Engineering
    P.O. Box 1001-747, Zhengzhou 450002
    People's Republic of China

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