归因于Heisenberg群在数学方面的几何背景和在物理方面的量子力学背景，越来越多的研究希望调和分析在Heisenberg群上可以得到与欧氏空间类似的理论框架。与欧氏空间不同，Heisenberg群是非交换的，且其Fourier变换是一算子值函数，因此处理该群上的调和分析问题往往比较困难。. 本项目主要研究Heisenberg 群哈代空间上的两个调和分析问题，即该空间上的乘子定理以及哈代不等式。. 首先，我们结合特殊Hermite函数的性质以及Heisenberg群上带积分型余项的Taylor多项式，去估计函数Fourier变换在广义偏微分算子作用下的Hilbert-Schmidt范数，进而研究作用于哈代空间中函数的一般乘子算子的有界性。另外，本项目还将利用上述Hilbert-Schmidt 范数的估计，研究该空间上的哈代不等式。

Due to the geometric background in mathematics and the quantum mechanics background in physics, the Heisenberg group attracts more and more people’s interest to extend the theory of harmonic analysis, which has done on Euclidean space, to the Heisenberg group. Different from that in the Euclidean space, the Heisenberg group is of non-exchange, and its Fourier transform is an operator-valued function. Hence to solve these problems of harmonic analysis on this group will be more difficult.. This research contains two problems related to the harmonic analysis in Hardy space of the Heisenberg group, one is the multiplier theorem, the other is Hardy's inequality.. First of all, by the properties of the special Hermite function and the Taylor formula with integral remainder, the Hilbert-Schmidt norms of the Fourier transforms acted by the generalized partial differential operators are estimated. Then the boundedness of the general right-multiplier operator, which acting on functions in the Hardy space, is studied. Furthermore, by using these estimates，the Hardy's inequalities for functions in the Hardy space are established.