Estimates of Dirichlet heat kernel for symmetric Markov processes

Abstract

We consider a large class of symmetric pure jump Markov processes dominated by isotropic unimodal Lévy processes with weak scaling conditions. First, we establish sharp two-sided heat kernel estimates for these processes in $C^{1,1}$ open sets. As corollaries of our main results, we obtain sharp two-sided Green function estimates and a scale invariant boundary Harnack inequality with explicit decay rates in $C^{1,1}$ open sets.

40 pages. The proof of Proposition 3.4 is corrected under the assumption that $D$ is $C^{1.1}$ open set. arXiv admin note: text overlap with arXiv:1402.4660