We apply the Davies method to give a quick proof for upper estimate of the heat kernel for the non-local Dirichlet form on the ultra-metric space. The key observation is that the heat kernel of the truncated Dirichlet form vanishes when two spatial points are separated by any ball of radius larger than the truncated range. This new phenomenon arises from the ultra-metric property of the space.

中文翻译：

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超度量空间上非局域 Dirichlet 形式的热核上界的 Davies 方法

我们应用戴维斯方法快速证明了超度量空间上非局部狄利克雷形式的热核的上估计。关键观察是当两个空间点被任何半径大于截断范围的球分隔时，截断狄利克雷形式的热核消失。这种新现象源于空间的超度量属性。