  The goal of this paper is to prove the existence of heat kernels for two
types of purely discontinuous symmetric Markov processes in the upper
half-space of $\mathbb R^d$ with jump kernels degenerate at the boundary, and
to establish sharp two-sided estimates on the heat kernels. The jump kernels
are of the form $J(x,y)=\mathcal B(x,y)|x-y|^{-\alpha-d}$, $\alpha\in (0,2)$,
where the function $\mathcal B$ depends on four parameters and may vanish at
the boundary. Our results are the first sharp two-sided estimates for the heat
kernels of non-local operators with jump kernels degenerate at the boundary.
The first type of processes we consider are conservative Markov processes on
the closed half-space with jump kernel $J(x,y)$. Depending on the regions where
the parameters belong, the heat kernels estimates have three different forms,
two of them are qualitatively different from all previously known heat kernel
estimates. The second type of processes we consider are the processes above
killed either by a critical potential or upon hitting the boundary of the
half-space. We establish that their heat kernel estimates have the approximate
factorization property with survival probabilities decaying as a power of the
distance to the boundary, where the power depends on the constant in the
critical potential. Finally, by using the heat kernel estimates, we obtain
sharp two-sided Green function estimates that cover the main result of [31].
This alternative proof provides an explanation of the anomalous form of the
estimates.
