Analysis on Alexandrov Spaces

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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.