  We study some function-theoretic properties on a complete smooth metric
measure space $(M,g,e^{-f}dv)$ with Bakry-\'{E}mery Ricci curvature bounded
from below. We derive a Moser's parabolic Harnack inequality for the $f$-heat
equation, which leads to upper and lower Gaussian bounds on the $f$-heat
kernel. We also prove $L^p$-Liouville theorems in terms of the lower bound of
Bakry-\'{E}mery Ricci curvature and the bound of function $f$, which generalize
the classical Ricci curvature case and the $N$-Bakry-\'{E}mery Ricci curvature
case.
