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Combined with our previous work \cite{LW19eigenvalue}, we prove sharp lowerbound estimates for the first nonzero eigenvalue of the weighted $p$-Laplacianwith $1< p< \infty$ on a compact Bakry-\'Emery manifold $(M^n,g,f)$, withoutboundary or with a convex boundary and Neumann boundary condition, satisfying$\text{Ric}+\nabla^2 f \geq \kappa \, g$ for some $\kappa \in \mathbb{R}$.

Sharp lower bound for the first eigenvalue of the Weighted $p$-Laplacian II