LP-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-Emery 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-Emery Ricci curvature and the bound of function f, which generalize the classical Ricci curvature case and the N-Bakry-Emery Ricci curvature case. (C) 2013 Elsevier Masson SAS. All rights reserved.