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 [18], [19], [20].

中文翻译：

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具有致命性杀手的非本地操作员的Dirichlet热核的分解和估计

在本文中，我们讨论了具有杀伤力的非本地操作员，这可能不是标准的加藤级人员。我们首先讨论它们在度量度量空间中的Dirichlet热核的因式分解。然后我们建立了在临界杀灭下Dirichlet热核的显式估计。$C^{\mathrm{1个}，\mathrm{1个}}$ 的开放子集 ${\mathbb{[R}}^d$ 或在 ${\mathbb{[R}}^d\setminus \{0\}$。我们显式估计的衰减率来自于杀死电位的乘法常数的值。我们的方法还提供了[18]，[19]，[20]的主要结果的替代统一证明。