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New lower bounds of the first nonzero eigenvalue of the weighted$p$-Laplacian are established on compact smooth metric measure spaces with orwithout boundaries. Under the assumption of positive lower bound for the$m$-Bakry--\'{E}mery Ricci curvature, the Escober--Lichnerowicz--Reilly typeestimates are proved; under the assumption of nonnegative$\infty$-Bakry--\'{E}mery Ricci curvature and the $m$-Bakry--\'{E}mery Riccicurvature bounded from below by a non-positive constant, the Li--Yau type lowerbound estimates are given. The weighted $p$-Bochner formula and the weighted$p$-Reilly formula are derived as the key tools for the establishment of theabove results.

Lower Bound Estimates for The First Eigenvalue of The Weighted $p$-Laplacian on Smooth Metric Measure Spaces