In this talk, we discuss the potential theory of Markov processes with jump kernels decaying at the boundary of the half space. The boundary part of kernel is comparable to the product of four terms with parameters appearing as exponents in these terms. We establish sharp two-sided estimates on the Green functions of these processes for all admissible values of parameters. Depending on the regions where parameters belong, the estimates on the Green functions are different. In fact, the estimates have three different forms depending on the regions the parameters belong to. As applications, we completely determine the region of the parameters where the boundary Harnack principle holds or not. This talk is based on joint works with Renming Song and Zoran Vondraček.