Boundedness Of Commutators Of Generalized Fractional Integrals

In this paper, we study the boundedness of commutators of generalized fractional integrals.This paper is constituted with ffve chapters.First, we introduce the background and main results about commutators of generalized frac-tional integrals, we illustrate diffculties about establishing the boundedness of commutators ofgeneralized fractional integrals, ideas to solve them and main results.In chapter 1, we discuss that are bounded from Ln/α(Rn) into BMO(Rn).Respectively, the boundedness of commutator of fractional integral operator and the multilinearcommutator of fractional integral are essentially extended.In chapter 2, we establish the -boundedness and (Mpq,BMO)-boundednessof the multilinear commutator Lffbff which generated by a ffnite family of locally integral functions and the generalized fractional integralIn chapter 3, we prove that the Toeplitz-type operatorΘαb generated by the generalizedfractional integral, Calderón-Zygmund operator with standard kernel K and VMO function isbounded from Lp,. We also show that , the vanishing-Morreyspace, when it satisffes some conditions.In chapter 4, we establish the boundedness of the commutator [b,T] generated by a Lipschitzfunction and Calderón-Zygmund type operator T on the Lebesgue spacesand Hardy spaces.In chapter 5, we prove that multiplier operator is bounded on BMO(Rn), LMO(Rn) andCBMOp,λ(Rn) respectively, when satisfying some concellation conditions.