函数空间理论及其应用一直是调和分析方向的核心课题之一. 作为研究算子端点(临界)情形有界性时Lebesgue空间的完美替代, Hardy空间和Campanato空间在调和分析及偏微分方程中起着至关重要的作用. 本博士学位论文致力于进一步发展和完善欧氏空间上Musielak–Orlicz–Lorentz Hardy空间 及Campanato型空间的实变理论及其应用. 本博士学位论文主要包括以下三个方面. 第一, 作为弱Musielak--Orlicz Hardy空间的自然推广, 引入了Musielak–Orlicz–Lorentz Hardy空间, 发展了其上一套完整的实变理论. 这部分工作的创新点在于克服了极大函数在Musielak–Orlicz空间相关空间有界性的缺失所带来的本质困难, 并巧妙地去除了Musielak–Orlicz函数为增长函数(即关于增长变量的凹性)这个已有工作的重要假设, 本质性推广了弱Musielak–Orlicz Hardy空间上的已有结果. 第二, 得到了Calderón–Zygmund算子及分数次积分算子在相关于等规格方体的John–Nirenberg–Campanato空间及其Hardy型前对偶空间上有界的充分必要条件. 其创新之处在于巧妙利用对偶定理提供了刻画Hardy型函数空间上线性算子有界性的一个新方法. 第三, 建立了球Campanato型函数空间上Calderón–Zygmund算子及Littlewood–Paley算子的有界性准则. 这部分工作的创新之处在于即使回到经典Campanato空间, 也扩展了g_{\lambda}^*算子有界性中$\lambda$的范围; 同时, 这些结论有着很强的一般性, 特别地, 应用到加权(变指标)Lebesgue空间和Orlicz空间共计8个具体函数空间时所获结果也是新的. 本文的工作极大地丰富了Campanato型空间及Hardy型空间上的实变理论并为调和分析、几何分析及偏微分方程等学科提供了更多的精细工作空间和理论工具.

      It is well known that the theory of function space and its application is one of the core topics in harmonic analysis. As perfect substitutions of Lebesgue spaces when studying the boundedness of operators in critical cases, both Campanato spaces and Hardy spaces play key roles in harmonic analysis and partial differential equations. This dissertation is devoted to further developing and improving the real-variable theory and its applications of Musielak–Orlicz–Lorentz Hardy spaces and Campanato-type spaces, which has important theoretical significance and application value. This dissertation mainly includes three parts. First, we introduce Musielak–Orlicz–Lorentz Hardy spaces which contain the known weak Musielak–Orlicz Hardy spaces as special cases. We then develop a complete real-variable theory of Musielak–Orlicz–Lorentz Hardy spaces. The major novelties of this part wort exist in that we overcome the obstacles caused by the deficiency of the boundedness of the Hardy– Littlewood maximal operator on associate spaces of Musielak–Orlicz spaces and then improve all the known corresponding results for weak Musielak–Orlicz Hardy spaces via moving the original assumption that the Musielak–Orlicz function is a growth function (namely concave for the growth variable). Second, we obtain the sufficient and necessary conditions of the boundedness of Calderón–Zygmund operators and fractional integrals on special John–Nirenberg–Campanato spaces via congruent cubes as well as on their preduals, some Hardy-type spaces. The major novelty of this part wort is that we provide a new method to characterize the boundedness of linear operators on Hardy-type spaces. Third, we study the mapping properties of Calderón–Zygmund operators and Littlewood–Paley operators on ball Campanato-type function spaces. The major novelty of this part wort is that we improve the existing results of Littlewood–Paley g_\lambda^*-function on Campanato spaces via widening the ranges of \lambda. In addition, these results are universal and when applied to a total of 8 specific function spaces such as weighted (variable) Lebesgue spaces and Orlicz spaces, all these results are new. These works greatly enrich the real-variable theory of Campanato-type and Hardy-type spaces and provide more work spaces and theoretical tools for some analysis disciplines such as harmonic analysis, geometrical analysis, and partial differential equations.