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Aim of this short note is to show that a dimension-free Harnack inequality onan infinitesimally Hilbertian metric measure space where the heat semigroupadmits an integral representation in terms of a kernel is suffcient to deduce asharp upper Gaussian estimate for such kernel. As intermediate step, we provethe local logarithmic Sobolev inequality (known to be equivalent to a lowerbound on the Ricci curvature tensor in smooth Riemannian manifolds). Bothresults are new also in the more regular framework of $RCD(K,\infty)$ spaces.

From Harnack inequality to heat kernel estimates on metric measure spaces and applications