In this paper,it is proved that the rough Hardy-Littlewood higher order maximal commutator Mmb,Ω is bounded on weighted Morrey-Herz space MK·α,λp,q(ω1;ω2).

The boundedness of some higher order commutators with variable kernels on the homogeneous Morrey-Herz spaces is obtained,where these commutators are generated by sublinear operators with variable kernels and BMO(Rn) functions.

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)(R~n))for some classes of higher order commutators which are generated by some common linear or sublinear op-erators.

In this paper,the author studies the boundedness of higher order commutators generated by the homogeneous fractional integral operator and BMO functions on weighed Herz-type Hardy space.

Some boundedness results are established in the setting of homogeneous Morrey- Herz spaces 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 operators M_l with function b(x)in BMO(R~n), respectively.

The boundedness of the higher order commutators generated by functions in Besov spaces and Calderón-Zygmund singular integral operators,and the boundedness of the corresponding higher order commutators of fractional integral operators are proved.

In this thesis, we studied some boundedness of Marcinkiewicz integrals on homogeneous kernel with high order commutatorsμ_(Ω,b)~m.