Elliptic gradient estimates for the doubly nonlinear diffusion equation

In this paper we establish elliptic type gradient estimates for positive solutions to the doubly nonlinear diffusion equation on compact or complete noncompact Riemannian manifolds. As special cases, we can derive elliptic type gradient estimates for positive solutions to the p-Laplace parabolic equations and the fast diffusion equations. Harnack inequalities derived from these elliptic type gradient estimates can be used to compare solutions at the same time.