Generalized Li-Yau's inequalities on Finsler measure spaces

It is known that the Finsler heat flow is a nonlinear flow. This leads to the study of the linearized heat semigroup for the Finsler heat flow. In this paper, we first study its properties. By means of the linearized heat semigroup, we give two different kinds of generalized Li-Yau's inequalities for the positive solutions to the heat equation on $n$-dimensional complete Finsler measure spaces with Ric$_N\geq K$ for some $N\in [n, \infty)$ and $K\in \mathbb R$. These inequalities almost recover all known Li-Yau's type inequalities on complete Finsler and Riemannian manifolds with lower Ricci curvature bounds. In particular, we obtain some new Li-Yau's type inequalities on complete Finsler and Riemannian measure spaces both in negative and positive Ricci curvature. As applications, we obtain two generalized Harnack inequalities. Finally we give several equivalent characterizations of Ric$_\infty\geq K (K\in \mathbb R)$ by the linearized heat semigroup approach and their applications.