Sobolev spaces, Laplacian, and heat kernel on Alexandrov spaces

Abstract

We prove the compactness of the imbedding of the Sobolev space $W^{1,2}_0(\Omega)$ into $L^2(\Omega)$ for any relatively compact open subset $\Omega$ of an Alexandrov space. As a corollary, the generator induced from the Dirichlet (energy) form has discrete spectrum. The generator can be approximated by the Laplacian induced from the DC-structure on the Alexandrov space. We also prove the existence of the locally Holder continuous heat kernel.