  In this paper we discuss non-local operators with killing potentials, which
may not be in the standard Kato class. We first discuss factorization of their
Dirichlet heat kernels in metric measure spaces. Then we establish explicit
estimates of the Dirichlet heat kernels under critical killings in $C^{1,1}$
open subsets of $\mathbb{R}^d$ or in $\mathbb{R}^d\setminus\{0\}$. The decay
rates of our explicit estimates come from the values of the multiplicative
constants in the killing potentials. Our method also provides an alternative
and unified proof of the main results of \cite{CKS10a, CKS10b, CKS-AOP}.
