算子的加权范数不等式一直是调和分析研究的核心内容之一，对它的研究不仅促进了函数空间和算子理论的发展，而且在探索偏微分方程和复变函数论中发挥了非常重要的作用。而Littlewood-Paley算子在研究乘子定理，发展Littlewood-Paley理论，解决Kato猜想中有着至关重要的应用。本项目拟通过加权Littlewood-Paley理论和加权原子分解理论，证明多参数Littlewood-Paley算子在加权Hardy空间的有界性，其中权函数为Muckenhoupt单权；利用随机二进网格、鞅差分解和Sawyer型测试条件，研究多参数Littlewood-Paley算子的双权T1定理；通过随机二进网格、鞅差分解、泛函能量、与仿增长函数相关的Sawyer型测试条件，探索单参数Littlewood-Paley算子、多参数Littlewood-Paley算子的双权Tb定理和双权局部Tb定理。

Weighted norm inequalities of operators are always one of core content in harmonic analysis, the study of these inequalities not only promote the development of theory of function spaces and operators, but also play a significant role in understanding partial differential equations and complex variable functions. And Littlewood-Paley operators have very important applications in studying multiplier theorems, developing Littlewood-Paley theory and resolving the Kato conjecture. The program will prove the boundedness of multi-parameter Littlewood-Paley operators on weighted Hardy spaces with Muckenhoupt one weights; applying random dyadic grids, martingale difference decomposition and Sawyer type testing conditions to study the two weight T1 theorems of multi-parameter Littlewood-Paley operators; by random dyadic grids, martingale difference decomposition, functional energy and Sawyer type testing conditions associated with para-accretive functions to verify the two weight Tb theorems and the two weight local Tb theorems of one-parameter Littlewood-Paley operators and multi-parameter Littlewood-Paley operators.