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 $1<p<\mathrm{\infty }$ on a compact Bakry-'Emery manifold $(M^n,g,f)$, without boundary or with a convex boundary and Neumann boundary condition, satisfying $\text{Ric}+{\mathrm{\nabla }}^{2}f\ge \kappa g$ for some $\kappa \in \mathbb{R}$.