Elliptic gradient estimates for diffusion operators on complete riemannian manifolds

In this note, we obtain the elliptic estimate for diffusion operator L = Δ + ∇ϕċ∇ on complete, noncompact Riemannian manifolds, under the curvature condition CD(K,m), which generalizes B. L. Kotschwar's work [5]. As an application, we get estimate on the heat kernel. The Bernstein-type gradient estimate for Schrödinger-type gradient is also derived.