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 bound estimates for the f-heat kernel on complete smooth metric measure spaces with Bakry-A parts per thousand mery Ricci curvature bounded below. Both upper and lower bound estimates are sharp when the Bakry-A parts per thousand mery Ricci curvature is nonnegative. The main argument is the De Giorgi-Nash-Moser theory. As applications, we prove an -Liouville theorem for f-subharmonic functions and an -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.