Heat Kernel Bounds on Metric Measure Spaces and Some Applications

Abstract

Let $(X,d,\mu)$ be a $RCD^\ast(K, N)$ space with $K\in \mathbb{R}$ and $N\in [1,\infty]$. For $N\in [1,\infty)$, we derive the upper and lower bounds of the heat kernel on $(X,d,\mu)$ by applying the parabolic Harnack inequality and the comparison principle, and then sharp bounds for its gradient, which are also sharp in time. When $N=\infty$, we also establish a sharp upper bound of the heat kernel by using the dimension free Harnack inequality. For applications, we study the large time behavior of the heat kernel, the stability of solutions to the heat equation, and show the $L^p$ boundedness of (local) Riesz transforms.