Abstract
We apply the Davies method to give a quick proof for the 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 a radius larger than the truncated range. This new phenomenon arises from the ultra-metric property of the space.
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Acknowledgements
The author thanks Jiaxin Hu and Alexander Grigor’yan for suggesting this topic, and Eryan Hu for discussions.
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The author was supported by National Natural Science Foundation of China (11871296).
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Gao, J. The Davies Method for Heat Kernel Upper Bounds of Non-Local Dirichlet Forms on Ultra-Metric Spaces. Acta Math Sci 40, 1240–1248 (2020). https://doi.org/10.1007/s10473-020-0506-x
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DOI: https://doi.org/10.1007/s10473-020-0506-x