幂零李群上次拉普拉斯算子的分析

Harmonic analysis on nilpotent Lie groups, which fokus on analysis of the sublaplacian in large part, is an important branch of harmonic analysis on Lie groups. We will concentrate on two classes of nilpotent Lie groups of step two, Siegel-type groups and H-type groups. They are generalizations of the Heisenberg group in different backgrounds. We will investigate some important problems in analysis of the sublaplacian, such as the restriction operator of spetral resolution of the sublaplacian and restriction theorems, the boundedness of Riesz means for spetral resolution of the sublaplacian, different versions, in especial the heat kernel version, of uncertainty principle, the joint spectrum of the sublaplacian and the differential operators with respet to central variables, characterization of joint eigenfunctions and irreducibility of eigenspace representations, generalized Cauchy-Szego projections and generalized Cauchy-Szego kernels, the Heisenberg Radon transform and the fractional Heisenberg Radon transform. These subjects root deeply in physics and geometry, and play an important role both in function theory of several complex variables and PDE. They are in the hart of harmonic analysis.

幂零李群上的调和分析是李群上的调和分析的主要分支之一。在相当大的程度上幂零李群上的调和分析围绕次拉普拉斯算子展开。本项目主要关注两类特殊的2步幂零李群：西格尔型群与H-型群。它们是海森堡群在不同情形下的推广。本项目研究关于次拉普拉斯算子的分析的若干重要问题，包括：次拉普拉斯算子谱分解的限制性算子与不同形式的限制性定理；次拉普拉斯算子谱分解的黎斯平均的有界性；不同形式的特别是使用热核的不确定性原理；次拉普拉斯算子与关于中心变量的微分算子的联合谱；联合特征函数的刻画与特征函数空间表示的不可约性；广义柯西-赛格投影与广义柯西-赛格核；海森堡拉东变换与分数次海森堡拉东变换。本项目研究的课题具有深刻的物理与几何背景，对多复变函数论与偏微分方程等学科分支有重要影响，属于调和分析的核心内容。

本项目研究海森堡型群上与次拉普拉斯算子相关的调和分析问题。研究工作在具有高维中心的海森堡型群上开展。主要研究内容和结果包括：1）次拉普拉斯算子和中心变量的偏拉普拉斯算子的联合泛函演算与Stein-Tomas型的限制性定理； 2）波方程的色散估计和Strichartz不等式；3）Hardy-Littlewood-Sobolev不等式的最佳常数与极值函数；4）Schrödinger极大函数的局部有界性估计；5）双线性Riezs平均的有界性。这些问题均属于现代调和分析的核心内容。