BANACH空间的非线性扰动保距映射和粗保距嵌入

The study of properties of classical metric-preserved embeddings, including interpolations of Banach spaces and local theory, has been an important research field since functional analysis was established as a branch of mathematics. The theory itself is one of the most essential and the deepest parts of functional analysis and related topics. The study of perturbed metric-preserved mappings has been active because of the broad background of perturbed metric-preserved mappings in both pure and applied mathematics. Generally speaking, where there is a difference, there is a perturbed metric-preserved mapping. Thus,they also have the broad background in experimental science. Another powerful "push"of the research of them has come from so called "coarse embedding" of "coarse geometry" established in the recent years, because coarsely metric-preserved mappings form a sdandard class of coarse embeddings. There are 6 goals for this project: (1) searching for strong stability characterizations of perturbed metric-preserved mappings; (2) proving the sharp formula of the weak stability for perturbed metric-preserved mappings; (3) studying strongly sharp stability formula of stably perturbed metric-preserved mappings; (4) investigating the strong stability characterizations of coarsely metric-preserved mappings; (5) looking for estimate formulae and sufficient conditions of strong stability of coarsely metric-preserved mappings and (6) existence of coarsely metric-preserved mappings from discrete metric spaces, especially, from bounded geometry space to super-reflexive spaces.

经典Banach空间保距嵌入理论以及他们的广义问题，它包括空间插值，局部理论等等，自泛函分析诞生就是一个重要的论题，构成了泛函分析中最本质最深刻的研究领域之一。近年来，人们发现以扰动保距嵌入及其稳定性研究为主体的论题不仅仅有着它深刻的理论价值，同时在基础数学和应用数学中有着广泛的应用背景。一般而言，哪里有误差哪里就有扰动保距映射。因此扰动保距在实验科学中随手可得；另外，近十年来粗几何的兴起，又赋予扰动保距的推广形式- - -粗保距映射研究的新的推动力, 因为"粗保距映射"是一类标准的"粗映射"。本项目主要目标：研究1）扰动保距的强稳定性特征；2）扰动保距映射的弱稳定性精确公式；3）一般条件下强稳定的扰动保距映射的精确估计；4）粗保距映射的稳定性特征条件和一般估计式；5）从度量空间到BANACH空间的粗保距映射稳定性条件；6）从离散度量空间，尤其是有界几何空间到超自反空间的"粗"映射的存在条件。

本项目研究紧紧围绕着“BANACH空间的非线性扰动保距映射和粗保距嵌入”展开， 主要研究成果发表在诸如JFA、 Studia Math 等12篇学术论文中，成果内容包括扰动保距映射（也称epsilon-等距）的稳定性和弱稳定性公式，超弱紧集及其在不动点理论、非紧性测度理论、赋范半群表示等方面的应用。最具代表性的成果为 1）给出了epsilon-等距的弱稳定性最优估计公式，它不仅仅统一了本领域80多年来的主要研究成果，同时也称为本领域研究的不可缺少的工具；2）给出了非紧性测度理论的泛函分析框架和表示，证明了“每个无穷维BANACH空间都存在着不等价的非紧性测度”，从而完全解决了自1978年以来的一个公开问题。另外，有14位博士生，16位硕士生参与了本课题的研究.