Heat Kernel Bounds on Metric Measure Spaces and Some Applications

Let (X,d,mu) be a R C D (au)(K,N) space with and Na[1,a). 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. 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.