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

Combined with our previous work \cite{LW19eigenvalue}, we prove sharp lower bound estimates for the first nonzero eigenvalue of the weighted $p$-Laplacian with $1< p< \infty$ on a compact Bakry-\'Emery manifold $(M^n,g,f)$, without boundary or with a convex boundary and Neumann boundary condition, satisfying $\text{Ric}+\nabla^2 f \geq \kappa \, g$ for some $\kappa \in \mathbb{R}$.