Eigenfunctions of the weighted Laplacian and a vanishing theorem on gradient steady Ricci soliton

The aim of this note has two folds. First, we show a gradient estimate of the higher eigenfunctions of the weighted Laplacian on smooth metric measure spaces. In the second part, we consider a gradient steady Ricci soliton and prove that there exists a positive constant c(n)$c(n)$ depending only on the dimension n   of the soliton such that there is no nontrivial harmonic 1-form (hence harmonic function) which is in L^p$L^p$ on such a soliton for any 22<p<c(n).