Boundedness Of Commutators Generated By Convolution Operators And Besov Functions

In this paper, we study the boundedness of commutators generated by Besovfunctions with some convolution operators ,including multiplier operator, singu-lar integral operator and fractional integral operator. This paper is organized asfollows.In chapter 1, we simply introduce the development in the study of the bound-edness of commutators of multiplier operators , singular integral operators andfractional integral operators. And then we introduce some correlative symbols andpropaedeutics.In chapter 2, we study the boundedness of multilinear commutator of mul-tiplier operators related to Lipschitz functions from Lp(R~n) to L2(R~n) . In thechapter, we prove that the operator [b,T] is bounded from Lp(R~n) to L2(R~n).In chapter 3, we discuss the boundedness of the commutators generated bysingular integral operators and fractional integral operators with Besov functions.In the first section, we prove that the operators [b,T] and [b,Iα] are boundedfrom Ld(R~n) to Lr(R~n). In the second section, we show that the commutator[b,T] is bounded from Ld(R~n) to F˙dβ?1/pn,∞(R~n), as well as Ibαfrom Ld(R~n) toF˙rβ?1/pn,∞(R~n).In chapter 4, we investigate the boundedness of commutators of multiplieroperators with Besov functions. In this chapter, we obtain the mapping propertiesof [b,T] from Ld(R~n) to Lr(R~n).