Let gφbe the Littlewood-Paley operatorLet b be a locally integrable function on Rn, the commutator gφ,b generated by the function b and the operator gφ,b is defined byfor suitable functions f.Let TΩ,μ be the integral operators with variable kernelLet b be a locally integrable function on Rn, the commutator TΩ,b generated by the function b and the operator TΩ,b is defined byfor suitable functions f.In this thesis, the author studies the estimates for the commutators of Littlewood-Paley operator and the weighted estimates for the higher-order commutators of Littlewood-Paley operator, their corresponding endpoint estimates are also discussed. the author also studies the estimates for commutators of integral operators with variable kernel on Herz-Hardy space.This thesis consists four chapters.Chapter 1 presents the background and give out some necessary notations as well as definitions of spaces. In chapter 2, we consider the boundedness of the commutator gφ,b with the CBMO functions on Herz-type Hardy spaces .In chapter 3, we consider the boundedness of the commutator estimates for commutators of integral operators T with variable kernel on Herz-Hardy space. the variable kernel is proved that if the kernel satisfies the Dini-condition, then the commutators with variable kernel are bounded from H Kq,bα,p(Rn) into Kqα,p(Rn) and its endpoint estimates are also established.In chapter 4, we consider the boundedness of the higher-order commutator gmφ,b with the CBMO functions on Herz-type Hardy spaces.
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