Abstract
In this survey we discuss heat kernel estimates of self-similar type on metric spaces with doubling measures. We characterize the tail functions from heat kernel estimates in both non-local and local cases. In the local case we also specify the domain of the energy form as a certain Besov space, and identify the walk dimension in terms of the critical Besov exponent. The techniques used include self-improvement of heat kernel upper bound and the maximum principle for weak solutions. All proofs are completely analytic.
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