Dimension-free Harnack inequality suffices for sharp upper Gaussian heat kernel estimates on infinitesimally Hilbertian metric measure spaces, with local logarithmic Sobolev inequality as an intermediate step, claimed new even in RCD(K,∞) spaces.
W ang, Equivalence of dimension-free Harnack inequality and curv ature condition, Integral Equations and Operator Theory, 48 (2004), pp
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From Harnack inequality to heat kernel estimates on metric measure spaces and applications
Dimension-free Harnack inequality suffices for sharp upper Gaussian heat kernel estimates on infinitesimally Hilbertian metric measure spaces, with local logarithmic Sobolev inequality as an intermediate step, claimed new even in RCD(K,∞) spaces.