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 ) 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 on such a soliton for any 2 < p < c ( n ) .