CBMO Estimates For Two Classes Of The Commutators

Let g_φbe the Littlewood-Paley operatorLet b be a locally integrable function on Rⁿ, the commutator g_φ,b generated by the function b and the operator g_φ,b is defined byfor suitable functions f.Let T_Ω,μ be the integral operators with variable kernelLet b be a locally integrable function on Rⁿ, the commutator T_Ω,^b generated by the function b and the operator T_Ω,^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 g_φ,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 K_q,b^α,p(Rⁿ) into K_q^α,p(Rⁿ) and its endpoint estimates are also established.In chapter 4, we consider the boundedness of the higher-order commutator g_m^φ,b with the CBMO functions on Herz-type Hardy spaces.