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Heat kernel on a manifold with a local Harnack inequality. (English) Zbl 0845.58056

The author introduces a broad class of the so-called locally Harnack manifolds, which includes in particular the manifolds with \(C^k\)-bounded geometries, with (simply) bounded geometries and with weak bounded geometries. For those manifolds (i.e. not only for bounded geometries as previously) under a certain additional condition the author gives a direct proof of the fact that the heat kernel decays at least as fast as \(1/t^{1/2}\). Another sort of results for those manifolds deals with (classical and modified) isoperimetric properties. In particular a lower bound for the first Dirichlet eigenvalue is established, etc.

MSC:

58J65 Diffusion processes and stochastic analysis on manifolds
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
58J35 Heat and other parabolic equation methods for PDEs on manifolds
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