幂零李群上的调和分析是抽象调和分析的主要分支之一，在几何分析、非线性分析、表示论、量子力学、偏微分方程等许多领域发挥着重大的作用。本项目中，我们将在幂零李群上研究某些变指标的函数空间和多线性算子，并进一步探讨它们的应用。我们将在某些幂零李群上定义变指标的Hardy空间、Besov空间等，给出它们的刻画；研究海森堡群和四元数海森堡群或一般的海森堡型群（中心维数大于1）上的双线性谱乘子的有界性，多线性奇异积分算子的有界性，插值理论等，探讨这两类多线性算子之间的关系；研究海森堡型群上的仿积算子在所得函数空间上的有界性、核函数的性质等；并进一步把某些结果推行到一般的幂零李群上。探讨所得结果的应用。

Harmonic analysis on nilpotent Lie groups is one of the main branch of Abstract Harmonic Analysis.It plays an important role in Geometry Analysis,Nonlinear Analysis,Representation Theory,Quantum Mechanics,Partial Differential Equations etc..In this project,we will study some variable exponent function spaces, some multilinear operators, and their applications. We will define and characterize the variable Hardy spaces and Besov spaces on some nilpotent Lie groups; On the Heisenberg group, quaternion Heisenberg group or general Heisenberg type groups (the dimension of their center is bigger than 1), we will show the boundedness properties of bilinear spectral multipliers; study the boundedness of mutilinear singular operators,the interpolation theory, and the relationship between these two kinds of multilinear operators; we will study the boundedness of paraproducts and the properties of the kernel of the paraproducts.etc.; furthermore we will generalize some results to general nilpotent Lie groups, and present some applications.