Boundary Harnack Principle for Stable-Like Processes

Abstract

We establish the boundary Harnack principle for certain classes of symmetric stable-like processes in $\mathbf{R}^d$ on arbitrary open sets as well as censored stable-like processes on $\mathcal{C}^{1,1}$-domains. Using those results, we derive Dirichlet heat kernel estimates for killed stable-like processes and killed censored stable-like processes in $\kappa$-fat domains in terms of the surviving probabilities and the global transition density of the processes. For $\mathcal{C}^{1,1}$-domains, we derive explicit estimates of the Dirichlet heat kernel.