Analysis on Alexandrov Spaces

在本研究中，我们建构了亚历山德罗夫空间分析的基础并加以发展。证明了紧化Alexandrov空间X的(1,2)-Sobolev空间W^<1,2>(X)嵌入L^2空间是紧化的。作为一个推论，我们得到了紧化Alexandrov空间上拉普拉斯谱的离散性。在这里，利用泛函分析将拉普拉斯函数定义为能量形式的无穷小发生器。对于Alexandrov空间上的DC函数，我们有另一个关于DC图的拉普拉斯函数的概念，叫做DC-拉普拉斯函数。研究了dc -拉普拉斯算子与泛函解析拉普拉斯算子之间的关系。此外，我们还证明了本征方程和热方程解在Alexaudrov空间上的连续性，以及热核的存在性。设A(n)是维数为n且曲率【大于等于】-1的紧Atexandrov空间的集合。我们证明了在A(n)上Kasue-Kumura引起的谱拓扑与Gromov-Hausdorff拓扑是一致的。我们引入了“里奇曲率有界以下的空间”的概念以及从这样一个空间映射到一个完整度量空间的能量。证明了庞加莱不等式及其能量测度的存在性。

In this study, we construct foundation of analysis on Alexandrov spaces and develop it.We prove that the embedding of the (1, 2)-Sobolev space W^<1,2>(X) of a compact Alexandrov space X into the L^2 space is compact. As a corollary, we obtain the discreteness of the spectrum of the Laplacian on a compact Alexandrov space. Here, the Laplacian is defined as an infinitesimal generator of the energy form, by using functional analysis. For a DC function on an Alexandrov space we have another concept of Laplacian depending on DC charts, called the DC-Laplacian. We investigate the relation between the DC-Laplacian and the functional analytic Laplacian. Moreover, we prove the continuity of the solutions of the eigen-equation and the heat equation on Alexaudrov spaces, and also the existence of the heat kernel. Let A(n) be the set of compact Atexandrov spaces of dimension n and curvature 【greater than or equal】 -1. We prove that on A(n) the spectral topology due to Kasue-Kumura coincides with the Gromov-Hausdorff topology. We introduce a concept of 'spaces of Ricci curvature bounded below' and energy of maps from such a space to a complete metric space. We prove the Poincare inequality and the existence of energy measure for it.