$L^p$-Liouville theorems on complete smooth metric measure spaces

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.