Hardy空间是分析学的重要研究内容之一,在调和分析、泛函分析、复分析等领域中有广泛的应用.抽象Hardy空间不仅保留了经典Hardy空间的主要性质,而且在这类空间上仍然可以建立实变理论、算子有界性理论等重要理论.因此,研究此类空间有很重要的意义.本项目拟在齐型空间背景下,对抽象Hardy空间进行深入的研究.首先,利用稠密性讨论的方法及测度的双倍性质,结合一种新的空间分环方法,给出变指标抽象Hardy空间的原子分解、分子分解及多参数抽象Hardy空间的原子分解；其次,借助原子分解、分子分解,建立这些空间上的算子有界性理论及变指标抽象Hardy空间上的对偶空间理论；最后,借助权函数性质及逆Hölder不等式建立这些空间上的加权理论.

Hardy space is one of the important contents in analysis. It is widely used in harmonic analysis, functional analysis and complex analysis. Abstract Hardy spaces retain the main properties of classical Hardy spaces. Moreover, we can establish some important theories in this kind of spaces, such as real variable theory and the theory of boundedness of operator. Therefore, it is very important to study this kind of spaces. This project intends to conduct in-depth research of abstract Hardy spaces under the background of homogeneous spaces. Firstly, using the method of density discussion, the double property of measure and a new spatial ring-dividing method, we obtain the atomic decomposition and molecular decomposition of variable exponent abstract Hardy spaces, and the atomic decomposition of multi-parameter abstract Hardy spaces. Secondly, via the atomic decomposition and the molecular decomposition, we establish the boundedness of operators of these spaces and the dual spaces of variable exponent abstract Hardy spaces. Finally, using the properties of weight and the reverse Hölder inequality, we establish the weighted theory of these spaces.