Boundedness Of Some Commutators On The Morrey-Herz Spaces

The core of this thesis,which consists of the five chapters,discusses mainly the bound-edness on the homogeneous Morrey-Herz spaces(denoted by MK_p,q^α,λ(Rⁿ))for some classes of higher order commutators which are generated by some common linear or sublinear op-erators.In Chapter 1,we define a class of new spaces named as homogeneous Morrey-Herz spaces MK_p,q^α,λ(Rⁿ)whose two special cases are homogeneous Herz spaces K_q^α,p(Rⁿ)and Morrey spaces M_q^λ(Rⁿ),and introduce some related concepts.And also,the Holder's, Minkowski's,Young-type inequalities and four interpolation theorems,which are hold on the Lebesgue spaces,are established on the homogeneous Morrey-Herz spaces MK_p,q^α,λ(Rⁿ) and the weak homogeneous Morrey-Herz spaces WMK_p,q^α,λ(Rⁿ).These results on the cor-responding homogeneous Herz space K_q^α,p(Rⁿ)and the weak homogeneous Herz spaces WK_q^α,p(Rⁿ)are also new.Since the Herz spaces are some kinds of generalization of the Lebesgue spaces and the weighted Lebesgue spaces with power weight,and the homo-geneous Morrey-Herz spaces are some kinds of generalization of the homogeneous Herz spaces and the Morrey spaces,so this work is of great meaningful.In Chapter 2,we obtain some boundedness results on MK_p,q^α,λ(Rⁿ)for a class of higher order commutators T_b,l^m and M_b,l^m generated by fractional integral operators T_l and maximal fractional integral operators M_l with function b(x)in BMO(Rⁿ)respectively.In Chapter 3,we investigate the boundedness results for some higher order commu-tators on MK_p,q^α,λ(Rⁿ).The commutators are generated by sublinear operator with rough kernels and BMO(Rⁿ)functions.Furthermore,on the fractional versions,the correspond-ing boundedness result on the homogeneous Morrey-Herz spaces is also obtained.In the meanwhile,we also establish the boundedness results of the higher order commutators generated by Hardy-Littlewood maximal operator with rough kernels and corresponding fractional maximal operator with rough kernels on MK_p,q^α,λ(Rⁿ).In Chapter 4,we obtain the boundedness on the homogeneous Morrey-Herz spaces for the fractional maximal multilinear commutators M_(?).In addition,we also discuss the boundedness results for a class of multilinear commutators generated by fractional integral operators and BMO(Rⁿ)functions on MK_p,q^α,λ(Rⁿ).In Chapter 5,we obtain a boundedness results of commutators of convolution opera-tors on MK_p,q^α,λ(Rⁿ)according to the corresponding boundedness of the sublinear operators on the homogeneous Morrey-Herz spaces.