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 ℝ^d for all t > 0. A prototype of the processes under consideration are symmetric jump processes on ℝ^d with jumping intensity where ν is a probability measure on [α1, α2] ⊂ (0, 2), Φ is an increasing function on [0,∞) with β Ie{cyrillic, ukrainian} (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 ℝ^d with different masses. We also establish the parabolic Harnack principle.