The Li-Yau Inequality and Heat Kernels on Metric Measure Spaces

Abstract

Let $(X,d,\mu )$ be a $RC{D}^{\ast }(K,N)$ space with $K\in mathbbR$ and $N\in [1,\infty )$. Suppose that $(X,d)$ is connected, complete and separable, and $\supp μ=X$. We prove that the Li-Yau inequality for the heat flow holds true on $(X,d,\mu )$ when $K\ge 0$. A Baudoin-Garofalo inequality and Harnack inequalities for the heat flows are established on $(X,d,\mu )$ for general $K\in \mathbb{R}$. Large time behaviors of heat kernels are also studied.

31 pages, J. Math. Pures Appl., to appear

References in corpus (2)

• Generalized Bochner formulas and Ricci lower bounds for sub-Riemannian manifolds of rank two
• On a new definition of Ricci curvature on Alexandrov spaces