算子有界性和函数空间实变理论一直是调和分析研究的核心内容. Riemann-Liouville (简记为R-L) 分数阶算子是调和分析中具有广泛应用的基本算子之一，而已有的函数空间理论并不适合于该算子的研究. 申请人与合作者在相关于各种算子, 尤其是对R-L分数阶算子的函数空间实变理论进行了系列研究. 本课题拟进一步发展适应于R-L分数阶算子的（弱）哈代空间及其对偶、Morrey空间、Besov空间和Triebel-Lizorkin空间及其变指标变形的实变理论, 其中包括各种极大函数特征、原子和分子分解特征及Littlewood-Paley特征等; 并用来研究R-L分数阶算子在这些空间中的有界性质以及紧性性质；建立对应的函数空间的Fréchet-Kolmogorov定理；研究基于R-L分数次微积分的分数阶方程的适定性问题，并由此进一步发展相关于高维R-L分数阶算子的函数空间的实变理论.

Both the boundedness of operators and the real-variable theory of function spaces are always one of the core contents of harmonic analysis. Fractional operator is one of the basic operators widely used in harmonic analysis, while the known function spaces are not suitable for Riemann-Liouville operators. The applicant and his collaborators have studied several operators, in particular, the real-variable theory of function spaces related to Riemann-Liouville fractional operators systematically. This project aims to establish the real-variable theory of (weak) Hardy, Morrey, Besov, Triebel-Lizorkin spaces and their variable exponents case, which fits the Riemann-Liouville fractional operators, including the characterization of maximal function, atomic and molecule, Littlewood-Paley theory. As applications, we will study the compactness and boundedness of Riemann-Liouville fractional operators on these new spaces; set up the Fréchet-Kolmogorov theorems on the corresponding function spaces; investigate the well-posedness of fractional equations based on Riemann-Liouville fractional calculus, and establish the new real-variable theory for function spaces in view of higher dimensional Riemann-Liouville fractional operators.