Littlewood-Paley平方函数及其相关算子的有界性研究

Littlewood-Paley平方函数及其相关算子的研究是调和分析的核心内容之一，它不仅在调和分析领域中的Fourier乘子理论、函数空间刻画、应用小波分析等研究中扮演着重要角色，而且在偏微分方程领域也有重要应用。本项目将主要采用稀疏控制理论、随机二进网等二进分析方法，研究几类重要Littlewood-Paley平方函数的有界性问题，主要包括如下三个方面的内容：一、研究几类经典平方函数的点态稀疏控制及其在一种拟Banach空间中的稀疏控制，建立这些平方函数的某些量化加权估计；二、研究几类双线性粗糙核平方函数的点态稀疏控制定理，并建立其量化加权估计；三、研究几类双参数平方函数在乘积Lebesgue空间、乘积Hardy空间等乘积空间中的有界性。本项目的完成将推进二进分析方法在算子有界性研究中的应用，进一步丰富和发展Littlewood-Paley算子理论。

The studies of Littlewood-Paley square functions and related operators have been one of the main research objects in harmonic analysis. They play a significant role not only in studying harmonic analysis, such as Fourier multiplier theory, characterization of function spaces, wavelet analysis, but also in studying partial differential equations. In this project, the random dyadic grids, sparse domination theory and other dyadic analysis methods will be mainly applied to study the following three problems about several important square functions: studying the pointwise sparse domination of several classic square functions, their sparse domination in Rearrangement Invariant quasi Banach spaces, and their quantitative weighted estimates; establishing the sparse domination and quantitative weighted estimates for several bilinear Littlewood-Paley square functions with rough kernel; studying the boundedness of several bi-parameter square functions in product Lebesgue space, product Hardy space and other product spaces. The completion of this project will not only promote the application and development of dyadic analysis methods, but also enrich and develop the theory of Littlewood-Paley operators.