Some Results On Operator Weighted Boundedness

The theory of singular integral operators is not only a rather active area in harmonic analysis but in partial differential equations have also been remark-ably more applications. In this thesis, firstly, the basic knowledge and theories of the weighted Ap functions are introduced, and based on it, the weighted boundedness is considered further. Then, using the methods of real harmonic analysis, also by the decompositons of function spaces and the properties of weighted functions, the boundedness of some strong singular convolution oper-ator commutators in weighted Hadry space is discussed. And the boundedness of Marcinkiewicz integrals with variable kernel in weighted Morrey-Herz spaces is obtained. At the last, we show that Marcinkiewicz integrals are bounded in weighted weak Hadry spaces. These results supplement the theory of singular integral operators, and expand the application of weighted Ap functions.