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--Émery Ricci curvature, the Escober--Lichnerowicz--Reilly type estimates are proved; under the assumption of nonnegative $\infty$-Bakry--Émery Ricci curvature and the $m$-Bakry--Émery 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.

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