  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 $L^1_f$-Liouville
theorem for $f$-subharmonic functions and an $L^1_f$-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.
