  Metric measure spaces satisfying the reduced curvature-dimension condition
$CD^*(K,N)$ and where the heat flow is linear are called $RCD^*(K,N)$-spaces.
This class of non smooth spaces contains Gromov-Hausdorff limits of Riemannian
manifolds with Ricci curvature bounded below by $K$ and dimension bounded above
by $N$. We prove that in $RCD^*(K,N)$-spaces the following properties of the
heat flow hold true: a Li-Yau type inequality, a Bakry-Qian inequality, the
Harnack inequality.
