Harmonic Analysis and Green Functions on Metric spaces

在这个项目中的主要研究者提出进一步发展理论的分析抽象度量空间的研究绿色功能等空间。首席研究员将首先关注四个问题。第一个问题试图找出是否在抽象度量空间的框架下，相应的调和函数满足强极大值原理。作为一个长期的目标，主要研究人员还希望确定是否强最大值原理适用于p-调和函数的一般值p.第二个问题是构建马丁边界Gromov双曲度量空间，承认绿色功能。 研究了绿色函数的一些性质，包括唯一性和边界Harnack原理。这两个问题将研究两种不同的构造的绿色功能，一个-使用上梯度，和其他-Cheeger衍生物。第四个问题是构造和研究抽象度量空间上的布朗运动和热方程，本项目的研究潜在的应用包括抽象度量空间上概率论和分析之间的联系。这样的空间出现在物理和工程的应用中，因此该项目中解决的问题也对物理和工程产生了潜在的影响。

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.