The Boundedness Of The Commutator Of Generalized Fractional Integral Operator

The research of fractional integral operator and its related problems is one of the important topics in Harmonic analysis. This paper gives the boundedness of the commutator [b, T] on some spaces, where b(x) belongs to BMO, T is the generalized fractional integral operator.In 2002, Ding Yong, Lu Shanzhen gave the boundedness properties of higher order commutators generated by the homogeneous fractional integral operators on certain Hardy spaces. Motivated by this, in first chapter, we obtain the boundedness properties of higher order commutators of generalized fractional integral operators on the Hardy spaces and the Herz-type Hardy spaces.In second chapter, similar to the study of fractional integral operator, we give the weak LlogL estimate for higher commutators of generalized fractional integral operator.Tribel-Lizorkin space is one of important spaces. In 1995, M.Paluszynski gave that the commutator [b, I_α] of fractional integral operator is bounded from L^p to F_p^β,∞ , when 1 < p < +∞,0 < p < 1,b ∈∧_β, where ∧_β is homogeneous Lipschitz space and F_p^β,∞ is homogeneous Tribel-Lizorkin space. Motivated by this, in third chapter, we study the boundedness properties of the commutator of generalized fractional integral operator on Tribel-Lizorkin space.In [33], Lin Daoyong not only introduced many kinds of k-step Stein functions and k-step Littlewood-Paley functions, but also gave the relation of these functions and built the L^P estimates of k-step Littlewood-Paley functions. In the last chapter, we will discuss the boundedness properties of k-step Littlewood-Paley functions on Herz spaces.