幂零流形上的热核分析与应用

In modern mathematics, nilmanifolds are important research areas, and they are extensively linked with other branches of mathematics. The objectives of this research project are to study the heat kernels of sub-Laplacians on nilmanifolds. This project consists of three parts. The first part is dedicated to Gaussian estimates and gradient estimates for the heat kernels of sublaplacians based on their explicit formulae. Inequlities of Poincaré type and Log–Sobolev type will be considered in the second part. In the last part, we discuss Talagrand type inequalities associated to the heat kernel measures on nilmanifolds. This research generalises the classical results on Riemannian manifolds to the sub-Riemannian setting, and leads our deep understanding to the structure of operators and equations on sub-Riemannian manifolds. The innovation of this research consists in (1) that the exact formulae of the heat kernels and related estimates are completely studied, (2) that Hamilton-Jacobi theory unifies each part of this research and (3) that the research methods and technical points cross many other branches of mathematics.

幂零流形是现代数学研究的重要领域, 并与其它数学分支建立了广泛的联系. 该项目的研究将围绕幂零流形上次Laplace算子的热核展开. 研究内容包括三部分: (1)次Laplace热核的Gauss估计和梯度估计; (2) Poincaré型不等式和Log–Sobolev型不等式; (3)热核测度的Talagrand型不等式. 该研究将使我们看到黎曼流形上经典的分析结果在幂零流形上的推广和刻画, 更深入地了解幂零流形上算子和方程的结构. 本项目的独到之处在于: (1) 次Laplace热核的精确解以及相关估计将完整地得到研究; (2)Hamilton-Jacobi理论统一了各项研究内容; (3)研究方法和技术手段与多个学科分支交叉渗透.

幂零流形上的次Laplace算子是现代数学研究的重要对象，它的研究与调和分析，偏微分方程以及多复分析有着紧密的联系。本项目借助于Hamilton-Jacobi变分方法，研究了三个自由度的二步幂零群上的次Laplace算子。我们得到了算子所对应Hamiton系统的测地线，极小作用函数以及热核的精确表达式。本项目的主要研究对象是二步自由群，它比Heisenberg型群具有更大的自由度，项目的重点和难点体现在自由群中任意两点间Carnot-Carathéodory度量的计算上。本项目还研究了次椭圆算子的周期解，它使我们更深入地理解算子的奇异测地线。研究对象具有物理和工程背景，包括带强非线性位势的Schroedinger算子和动能算子等。本项目的研究成果是进一步研究次椭圆算子的渐近估计，泛函不等式以及输运不等式等工作不可或缺的基础。研究成果以SCI论文的形式体现，截至目前，撰写并投稿4篇，其中1篇已被接受。