Upper bounds on the first eigenvalue for a diffusion operator via Bakry-\'{E}mery Ricci curvature II

Let $L=\mathrm{\Delta }-\mathrm{\nabla }\phi \cdot \mathrm{\nabla }$ be a symmetric diffusion operator with an invariant measure $d\mu =e^{-\phi }dx$ on a complete Riemannian manifold. In this paper we prove Li-Yau gradient estimates for weighted elliptic equations on the complete manifold with $\mathrm{\mid }\mathrm{\nabla }\phi \mathrm{\mid }\le \theta$ and $\mathrm{\infty }$-dimensional Bakry-'{E}mery Ricci curvature bounded below by some negative constant. Based on this, we give an upper bound on the first eigenvalue of the diffusion operator $L$ on this kind manifold, and thereby generalize a Cheng’s result on the Laplacian case (Math. Z., 143 (1975) 289-297).