函数空间的实变理论和算子的有界性一直是调和分析研究的核心内容之一. 交换子作为调和分析中一类重要的算子, 具有比它的生成算子更高的奇异性. 这要求人们引入新的适应于交换子有界性的函数空间. 本项目在申请人已建立的交换子的端点有界性基础上, 借助已有的欧氏空间上的Hardy空间与BMO空间的乘积空间的双线性分解的方法, 拟通过建立欧氏空间及一般的度量空间上的加权Hardy空间与BMO空间的乘积空间的双线性分解, 得到Calderón-Zygmund算子、Marcinkiewicz积分算子及分数次积分算子与BMO函数生成的交换子的双线性分解; 寻找加权Hardy空间的最大子空间, 使得交换子从该子空间到加权Lebesgue空间有界; 寻找BMO空间或Lipschitz空间的最大子空间, 使得前面提到的算子与此子空间中的函数生成的交换子从加权Hardy空间到加权Lebesgue空间有界.

Both the real-variable theory of function spaces and the boundedness of operators are always one of the core contents of harmonic analysis. As an important operator in Harmonic Analysis, the commutator is more singular than its generated operator. People need to introduce some new function space to study the boundedness of commutators. This project aims, on Euclidean space and metric measure space, via the bilinear decomposition of the product of functions in weighted Hardy space and BMO space, to establish the bilinear decomposition of commutators generated by Calderón-Zygmund operator、Marcinkiewicz integral operator and fractional integral operator and the function in BMO space; moreover, find the biggest subspace of weighted Hardy space such that the commutators are bounded from this subspace to weighted Lebesgue sapce; find the biggest subspace of BMO space or Lipschitz space, such that the commutators generated by operators and functions in this subspace are bounded from weighted Hardy space to weighted Lebesgue space.