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In this paper, we study vanishing and splitting results on a complete smoothmetric measure space $(M^n,g,\mathrm{e}^{-f}\mathrm{d}v)$ with various negative$m$-Bakry-\'Emery-Ricci curvature lower bounds in terms of the first spectrum$\lambda_1(\Delta_f)$ of the weighted Laplacian $\Delta_f$, i.e.$\mathrm{Ric}_{m,n}\geq -a\lambda_1(\Delta_f)-b$ for $0<a\leq\dfrac{m}{m-1},b\geq0$. In particular, we consider three main cases for different $a$ and $b$with or without conditions on $\lambda_1(\Delta_f)$. These results areextensions of Dung and Vieira, and weighted generalizations of Li-Wang,Dung-Sung and Vieira.

$L^2_f$ harmonic 1-forms on smooth metric measure spaces with positive $\lambda_1(\Delta_f)$