Li–Yau and Harnack type inequalities in RCD∗(K,N) metric measure spaces

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 KK and dimension bounded above by NN. 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.