Harmonic Analysis and Green Functions on Metric spaces

度量空间上的调和分析和格林函数

• 批准号: 0100132
• 负责人: Nageswari Shanmugalingam
• 金额: $ 6.51万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-08-01 至 2003-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0100132&HistoricalAwards=false
• 关键词: Harmonic; Analysis; Green; Functions; Metric

In this project the principal investigator proposes to develop further the theory of analysis on abstract metric spaces by studying Green functions on such spaces. The principal investigator will initially focus on four problems. The first question seeks to find out whether in the framework of abstract metric spaces the corresponding harmonic functions satisfy the strong maximum principle. As a long range goal the principal investigator also hopes to determine whether the strong maximum principle holds for p-harmonic functions for general values of p. The second issue is to construct the Martin boundary for Gromov hyperbolic metric spaces that admit Green functions. It is also proposed to study some properties of Green functions, including the uniqueness property and the boundary Harnack principle. These two problems will be studied for two different constructions of Green functions, one - using the upper gradients, and the other - the Cheeger derivative. The fourth problem is to construct and study Brownian motion and the heat equation on abstract metric spaces.Potential applications of the research proposed in this project include connections between probability theory and analysis on abstract metric spaces. Such spaces arise in applications in physics and engineering, and hence the questions addressed in the project have a potential impact in physics and engineering as well.

在该项目中，主要研究者建议通过研究此类空间的绿色功能进一步发展对抽象度量空间的分析理论。首席研究人员最初将重点关注四个问题。第一个问题试图找出在抽象度量空间的框架中，相应的谐波函数是否满足了强大的最大原理。作为一个远距离目标，主要研究者还希望确定强大的最大原理在p的一般值中是否具有p谐波函数。第二个问题是为接纳绿色功能的Gromov双曲线度量空间构建马丁边界。 还建议研究绿色功能的某些特性，包括唯一性质和边界竖琴原理。将研究这两个问题，用于两个不同的绿色功能的构造，一个是使用上梯度，另一个是cheeger衍生物。第四个问题是构建和研究布朗运动和抽象度量空间上的热方程。该项目中提出的研究的潜在应用包括概率理论与抽象度量空间分析之间的联系。这些空间在物理和工程中的应用中出现，因此项目中提出的问题也可能影响物理和工程。