Potential Theory on Metric Measure Spaces

首席调查员建议通过三个项目继续开展对公制空间的分析。第一个项目将研究一般度量度量空间上与Sobolev型函数空间相关的迹空间，该度量空间可能不具有黎曼结构；这里的想法是找出在给定域的边界上数据必须被确定得有多好，以便获得数据到域内部的相应扩展的可靠预测。第二个项目是在有界几何的度量度量空间中探索有界域的一致Martin边界。共形Martin边界一般不同于对应于拉普拉斯算子的经典Martin边界，它提供了与某些非线性偏微分方程势理论边界有关的更可靠的准则。第三个项目是在一般度量空间上构造一个分布导数结构，从而度量区域的边界，并确定这样的度量空间中的球面何时是可逆的。该项目中提出的研究的潜在应用包括随机过程与抽象度量空间中的分析之间的联系。抽象度量空间在物理学和工程学中有着广泛的应用，因此本课题所讨论的问题在物理学和工程学中也有可能产生影响。此外，这些项目将黎曼流形中散度型的次椭圆方程理论与Carnot-Carath空间的次黎曼几何中的退化椭圆方程理论统一起来，并且共形Martin边界的构造和研究即使在欧几里得环境下也是一个新的概念，将为研究拟共形映射的边界行为提供一个新的工具。

The Principal Investigator proposes to continue the program of developing analysis on metric spaces via three projects. The first project will study the trace spaces related to a Sobolev-type function space on general metric measurespaces that may not have a Riemannian structure; the idea here is to find out how well data have to be determined on the boundary of a given domain in order to obtain a reliable prediction of a corresponding extension of the data to theinterior of the domain. The second project is to explore the conformalMartin boundary for bounded domains in metric measure spaces of bounded geometry. The conformal Martin boundary is in general different from the classical Martin boundary corresponding to the Laplacian operator, and provides a more reliable gauge of the potential-theoretic boundary relevant to certain non-linear partial differential equations. The third project is to construct a distributional derivative structure on general metric spaces and thereby measure the boundary of domains and determine when spheres in such a metric space are rectifiable.Potential applications of the research proposed in this project include connections between stochastic processes and analysis in abstract metric spaces. Abstract metric spaces arise in applications inphysics and engineering, and hence the questions addressed in this project havepossible impact in physics and engineering as well. In addition, these projects unify the theory of subelliptic equations of divergence form in Riemannian manifolds with the theory of degenerate elliptic equations in the sub-Riemannian geometry of Carnot-Carath\'eodory spaces. Also, the construction and study of conformal Martin boundary is a new concept even in the Euclidean setting, and will provide a new tool in the study of boundary behavior of quasiconformal mappings.