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

Let Z=(Z1,…,Zd) be the d-dimensional Lévy process where Zi’s are independent 1-dimensional Lévy processes with identical jumping kernel ν1(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 ν1(|xi−yi|)ifxi≠yifor someiandxj=yjfor allj≠i0ifxi≠yifor more than one indexi. Corresponding to the jumping kernel J, there exists an anisotropic Markov process X, see [32]. In this article, we establish sharp two-sided Dirichlet heat kernel estimates for X in C1,1 open set, under certain regularity conditions. As an application of the main results, we derive the Green function estimates.