Heat kernel estimates for jump processes of mixed types on metric measure spaces

In this paper, we investigate symmetric jump-type processes on a class of metric measure spaces with jumping intensities comparable to radially symmetric functions on the spaces. The class of metric measure spaces includes the Alfors d-regular sets, which is a class of fractal sets that contains geometrically self-similar sets. A typical example of our jump-type processes is the symmetric jump process with jumping intensity e(-c0(x,y)|x-y|) integral(alpha 2)(alpha 1)c(alpha,x,y)/|x-y|(d+alpha) nu(d alpha) where nu is a probability measure on [alpha(1), alpha(2)] subset of (0, 2), c(alpha, x, y) is a jointly measurable function that is symmetric in (x, y) and is bounded between two positive constants, and c(0)(x, y) is a jointly measurable function that is symmetric in (x, y) and is bounded between gamma(1) and gamma(2), where either gamma(2) >= gamma(1) > 0 or gamma(1) = gamma(2) = 0. This example contains mixed symmetric stable processes on R-n as well as mixed relativistic symmetric stable processes on R-n. We establish parabolic Harnack principle and derive sharp two-sided heat kernel estimate for such jump-type processes.