Lower bound estimates for the first eigenvalue of the weighted p-Laplacian on smooth metric measure spaces

New lower bounds of the first nonzero eigenvalue of the weighted p-Laplacian are established on compact smooth metric measure spaces with or without boundaries. Under the assumption of positive lower bound for the m-Bakry Emery Ricci curvature, the Escobar-Lichnerowicz-Reilly type estimates are proved; under the assumption of nonnegative infinity-Bakry Emery Ricci curvature and the m-Bakry-Emery Ricci curvature bounded from below by a non-positive constant, the Li Yau type lower bound estimates are given. The weighted p-Bochner formula and the weighted p-Reilly formula are derived as the key tools for the establishment of the above results. (C) 2015 Elsevier B.V. All rights reserved.