Heat kernel estimates and parabolic Harnack inequalities for symmetric Dirichlet forms

In this paper, we consider the following symmetric Dirichlet forms on a metric measure space (M,d,μ):E(f,g)=E(c)(f,g)+∫M×M(f(x)−f(y))(g(x)−g(y))J(dx,dy), where E(c) is a strongly local symmetric bilinear form and J(dx,dy) is a symmetric Radon measure on M×M. Under general volume doubling condition on (M,d,μ) and some mild assumptions on scaling functions, we establish stability results for upper bounds of heat kernel (resp. two-sided heat kernel estimates) in terms of the jumping kernels, the cut-off Sobolev inequalities, and the Faber-Krahn inequalities (resp. the Poincaré inequalities). We also obtain characterizations of parabolic Harnack inequalities. Our results apply to symmetric diffusions with jumps even when the underlying spaces have walk dimensions larger than 2.