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.
Grigor’yan , Heat kernel upper bounds on a complete non-compact manifold , Revista Matem´ atica Iberoamericana, 10 (1994), 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.