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