Siegel型的幂零Lie群上的一些调和分析问题

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.

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.

在Siegel型的幂零Lie群上，结合Laplacian算子性质、Plancherel测度、热核的估计和齐次群上的Taylor展式的积分型余项给出了Radon变换的逆公式、有界性、Heisenberg–Pauli–Weyl不确定性不等式和特殊Hermite展开的Hardy不等式。应用Littlewood-Palay及各向异性Lusin面积函数理论和处理奇异积分算子的技巧，建立了各向异性的Musielak-Orliz Hardy 空间之积的原子特征、g函数特征、离散小波函数等实变特征。此外，我们还进行了欧氏空间的线、超平面间的Radon变换的研究，利用球上的Funk变换、Radon-John 的d维平面变换、Erdelyi-Kober分数次积分算子理论给出了精确的逆算子表达式。近年来调和分析理论被广泛应用于偏微分方程解的正则性问题的研究，利用Calderon-Zygmund理论，我们研究了Lipschitz区域上的强椭圆型方程组，得到在Dirichlet和Neumann边界为零的条件下强椭圆型方程组的预解算子的L^p估计，我们还研究了n维欧氏空间有界区域上低阶项属于 Kato-Stummel类的二阶强椭圆型方程组的Green矩阵，证明了其存在性和整体估计和L^p估计，并在此基础上得到该方程组的弱解的Schauder估计。在调幅空间上，我们研究了一类具有含时间势的的分数型Schr\"{o}dinger方程，利用波包变换和特征曲线法，得到与此类方程相关的传播子在调幅空间上的有界性估计。