LpLp-Liouville theorems on complete smooth metric measure spaces

We study some function-theoretic properties on a complete smooth metric measure space (M,g,e−fdv) with Bakry–Émery 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 LpLp-Liouville theorems in terms of the lower bound of Bakry–Émery Ricci curvature and the bound of function f, which generalize the classical Ricci curvature case and the N-Bakry–Émery Ricci curvature case.