Heat kernel on smooth metric measure spaces and applications

We derive a Harnack inequality for positive solutions of the $f$-heat equation and Gaussian upper and lower bounds for the $f$-heat kernel on complete smooth metric measure spaces $(M,g,e^{-f}dv)$ with Bakry-'Emery Ricci curvature bounded below. The lower bound is sharp. The main argument is the De Giorgi-Nash-Moser theory. As applications, we prove an $L_f^1$-Liouville theorem for $f$-subharmonic functions and an $L_f^1$-uniqueness theorem for $f$-heat equations when $f$ has at most linear growth. We also obtain eigenvalues estimates and $f$-Green’s function estimates for the $f$-Laplace operator.