In the setting of a manifold with doubling property satisfying a Gaussian upper estimate of the heat kernel, one gives a characterization of the lower Gaussian estimate in terms of certain Hölder inequalities.
Salahaddine Boutayeb. "Heat Kernel Lower Gaussian Estimates in the Doubling Setting Without Poincaré Inequality." Publ. Mat. 53 (2) 457 - 479, 2009.