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|}, ∫α1^α2} c(α, x, y)|x-y|^d+α}, ν (dα) where ν is a probability measure on [α1, α2]subset (0, 2), c(α, 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 γ₁ and γ₂, where either γ₂ gamma;₁ > 0 or γ₁ = γ₂ = 0. This example contains mixed symmetric stable processes on Rⁿ as well as mixed relativistic symmetric stable processes on Rⁿ. We establish parabolic Harnack principle and derive sharp two-sided heat kernel estimate for such jump-type processes.