Siegel型的幂零Lie群是Heisenberg群概念的推广，它是广义上半平面的特征边界，其四元数Heisenberg群也在其中. 本项目是运用Riesz位势、Riesz分数阶导数、超奇异积分、小脊变换、卷积投影以及非交换调和分析的理论研究四元数Heisenberg群上的Radon变换在各种意义下的逆算子表达式，并建立混合（p，q）型范数不等式. 进一步将这些问题推广到Siegel型的幂零Lie群上, 同时在不同函数空间中考虑Radon变换及广义Radon变换的连续性. 本课题是把近代欧氏空间上的Fourier分析与抽象的Siegel型幂零Lie群上的代数特征结合起来研究Radon变换的性质，进而讨论相关的发展型方程解的正则性估计，这对调和分析与偏微分方程相互交叉是大有益处的.

The concept of nilpotent Lie groups of Siegel type is an extension of the Heisenberg group, which can be realized as the distinguished boundary of generalized upper half-plane. The quaternion Heisenberg group is included as a special case. This project is to develop the explicit expression of inverse Radon transform in various sense on the quaternion Heisenberg group by using Riesz potential, Riesz fractional derivative, supersingular integral，ridgelet-like tranasform, convolution back-projection operator and the theory of noncommuitative harmonic analysis, and establish mixed norm inequalities of (p,q) type for the Radon transform. Furthermore, we also study this theory on the nilpotent Lie group of Siegel type. At the same time we consider the continuous proterties of the Radon transform and the generalized Radon transform in different function spaces. This project concerned with the properties of the Radon transform by the techniques that are intimately related to modern Fourier analys on Euclidean space and the algebraic features of the abstract nilpotent Lie group of Siegel type. We also discuss the regular estimates of the solution dispersive equation. It is important for us to combine harmonic analysis with partial differential equations.