Dirichlet Heat kernel estimates for a large class of anisotropic Markov processes

Let Z=(Z^1, …, Z^d) be the d-dimensional Lévy process where Z^i's are independent 1-dimensional Lévy processes with identical jumping kernel ν^1(r) =r^-1ϕ(r)^-1. Here ϕ is an increasing function with weakly scaling condition of order α, α∈ (0, 2). We consider a symmetric function J(x,y) comparable to ν^1(|x^i - y^i|) if x^i y^i for some i and x^j = y^j for all j i 0 if x^i y^i for more than one index i. Corresponding to the jumping kernel J, there exists an anisotropic Markov process X, see . In this article, we establish sharp two-sided Dirichlet heat kernel estimates for X in C^1,1 open set, under certain regularity conditions. As an application of the main results, we derive the Green function estimates.