Heisenberg群上函数空间的实变理论和算子有界性是调和分析研究的重要内容之一, 同时实变理论为算子的有界性以及界的最佳性的研究提供了重要的研究工具。申请人及其合作者应用Heisenberg群函数空间实变理论在研究相应于sub-Laplace算子的最佳不等式方面已经获得了一系列的结果。本项目拟在之前工作的基础上进一步考虑并发展与Heisenberg型群相关的次Laplace算子、Grushin算子及其相应的分数次幂算子相关的函数空间的实变理论，并利用相关算子本身所蕴含的分析和几何结构和函数空间的实变理论进行最佳不等式的研究。作为应用，项目将最佳不等式应用于区域的几何刻画以及相关偏微分方程的正则性研究。

Both the real-variable theory of function spaces and the boundedness of operators on Heisenberg group are always one of the important contents in harmonic analysis, On the same time, the real-variable theory is an important tool in the study of the boundedness and sharpness of operators. The applicant and his collaborators have used the real-variable theory of function spaces on Heisenberg group to obtain a series of sharp inequality associated with sub-Laplacian operator. This project aims to develop a real-variable theory of function space associated with sub-Laplacian operator、Grushin operator and the corresponding fractional power operators on Heisenberg type group, moreover, we will study the sharp inequality by using the real-variable theory and the analysis and geometry structure contained in these operators. As applications, this project aims to apply all the above sharp inequalities to the study of the geometry characterization for domain and the regularity of some partial differential equations.