Global heat kernel estimates for symmetric jump processes

In this paper, we study sharp heat kernel estimates for a large class of symmetric jump-type processes in Rᵈ for all $t>0$. A prototype of the processes under consideration are symmetric jump processes on Rᵈ with jumping intensity \[ 1/Φ (|x-y|) ∫ [α _1, α _2] {c(α , x,y)} {|x-y|d+α} ν (dα ) , \] where ν is a probability measure on [α ₁, α ₂] ⊂ (0, 2), Φ is an increasing function on [ 0, ∞ ) with c₁e^c₂rβ ≤ Φ (r) ≤ c₃ e^c₄rβ with β ∈ (0,∞ ), and c(α , x, y) is a jointly measurable function that is bounded between two positive constants and is symmetric in $(x, y)$. They include, in particular, mixed relativistic symmetric stable processes on Rᵈ with different masses. We also establish the parabolic Harnack principle.