函数空间实变理论与算子有界性一直是调和分析研究的核心内容之一，而Lorentz空间与Musielak-Orlicz Hardy（简记为MOH）空间又是在研究算子有界性的端点和尖锐问题时起着关键作用的函数空间。申请人及其合作者已部分发展了(各向异性)欧氏空间上MOH空间及齐型空间上相关于满足球上强化非对角估计的算子的MOH空间实变理论。本课题拟在欧氏空间上，通过弱化增长函数的条件，进一步发展一套MOH空间实变理论；通过寻找增长函数的恰当形式，建立与之相适应的MOH型空间实变理论的统一框架，使之能够同时包含已有的MOH空间和变指标Hardy空间；发展一套变指标Lorentz-Hardy型空间实变理论；并在度量测度空间上发展一套完整的MOH型空间及适合于可相关于更多算子且更广的MOH型空间实变理论。作为应用，将上述所有这些函数空间实变理论应用于相关算子的有界性，特别是其端点和尖锐问题，的研究中。

Both the real-variable theory of function spaces and the boundedness of operators are always one of the core contents of harmonic analysis, while Lorentz spaces and Musielak-Orlicz Hardy spaces are function spaces which play a key role in the study for the endpoint or the sharp problems of boundedness of operators. The applicant and his collaborators have partly developed the real-variable theory of both Musielak-Orlicz Hardy spaces on (anisotropic) Euclidean spaces and Musielak-Orlicz Hardy spaces associated with operators satisfying the reinforced non-diagonal estimates on balls on spaces of homogeneous type. This project aims, on Euclidean spaces, via weakening the requirements of growth functions, to further develop a complete real-variable theory of Musielak-Orlicz Hardy spaces; via searching for appropriate forms of growth functions, to establish a corresponding unified framework of the real-variable theory of Musielak-Orlicz Hardy spaces such that it contains, at the same time, the known Musielak-Orlicz Hardy spaces and Hardy spaces with variable exponents; to develop a real-variable theory of Lorentz-Hardy type spaces with variable exponents and, on metric measure spaces, to develop a complete real-variable theory of both Musielak-Orlicz Hardy spaces and the ones associated with more operators and with more generality. As applications, this project aims to apply all the above real-variable theories of function spaces to the study for the boundedness, particularly the endpoint or the sharp problems, of the related operators.