In this talk, I will present some results from ongoing projects on the potential theory of Markov processes with jump kernels decaying at the boundary. We study processes in an open set$D\subset \mathbb R^d$defined via Dirichlet forms with jump kernels of the form$J(x,y)=j(|x-y|)B(x,y)$and critical killing functions$\kappa(x)$. Here$j(|x-y|)$is the L\'evydensity of an isotropic stable process (or more generally, a subordinate Brownian motion) in$\R^d$. The main novelty is that the term$B(x,y)$tends to 0 when$x$or$y$approach the boundary of$D$. Under some general assumptions on$B(x, y)$, we construct the corresponding process and prove that non-negative harmonic functions of the process satisfy theHarnackinequality andCarleson'sestimate. We give several examples of boundary terms satisfying those assumptions. The examples depend on three parameters,$\beta_1, \beta_2, \beta_3$, roughly governing the decay of the boundary term near the boundary of$D$.

In the second part, we specialize to the case of the half-space $D=\mathbb R_+^d=\{x=(\wt{x},x_d):\, x_d>0\}$, the$\alpha$-stable kernel$j(|x-y|)=|x-y|^{-d-\alpha}$and the killing function$\kappa(x)=c x_d^{-\alpha}$,$\alpha\in (0,2)$, where$c$depends on$p$and$B$. Our main result in this part is a boundaryHarnackprinciple which says that, for any$p>(\alpha-1)_+$, there are values of the parameters$\beta_1, \beta_2, \beta_3$and$c$such that non-negative harmonic functions$f(x)$of the process must decay at the rate$x_d^p$if they vanish near a portion of the boundary. We further show that there are values of the parameters$\beta_1, \beta_2, \beta_3$and$c$for which the boundaryHarnackprinciple fails despite the fact thatCarleson'sestimate is valid.