On-diagonal lower bounds for heat kernels on noncompact manifolds and Markov chains, Duke Math. J., 89 (1997) This inequality can be easily proved by spectral theory ([C2]). It is the cornerstone of the semigroup version ([C2]) of a theorem by the second author that relates upper bounds for the heat kernel with Faber-Krahn type inequalities ([G2]). We will see that it gives a very easy approach to sup-lower bounds for the heat kernels, and more generally the kernels of symmetric Markov

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... s. In the case where (X, µ) is a Riemannian manifold M equiped with its natural measure, and T t is the heat semigroup, i.e. the heat kernel on a Riemannian manifold, the technique of [G2] also applies. Recall that if T t has a kernel p t (and this is the case as soon as T t 1→∞ < +∞), then In the sequel, sup x∈X p t (x, x) will simply mean +∞ if T t 1→∞ = +∞. Let D be a dense subset of D(A) \ {0} in L 2 that is also contained in L 1 . One easily deduces from (2.1):