We prove the compactness of the imbedding of the Sobolev space W1,20 (Ω) into L2 (Ω) for any relatively compact open subset Ω 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 Hölder continuous heat kernel.
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