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Journal of the London Mathematical Society (1)

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OFF-DIAGONAL HEAT KERNEL LOWER BOUNDS WITHOUT POINCARÉ
THIERRY COULHON
Journal:

Journal of the London Mathematical Society / Volume 68 / Issue 3 / December 2003

Published online by Cambridge University Press:

17 November 2003, pp. 795-816

Print publication:

December 2003
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On a manifold with polynomial volume growth satisfying Gaussian upper bounds of the heat kernel, a simple characterization of the matching lower bounds is given in terms of a certain Sobolev inequality. The method also works in the case of so-called sub-Gaussian or sub-diffusive heat kernels estimates, which are typical of fractals. Together with previously known results, this yields a new characterization of the full upper and lower Gaussian or sub-Gaussian heat kernel estimates.