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Anchored heat kernel upper bounds on graphs with unbounded geometry and anti-trees

Matthias KellerChristian Rose
Dec 2022
摘要
We derive Gaussian heat kernel bounds on graphs with respect to a fixedorigin for large times under the assumption of a Sobolev inequality and volumedoubling on large balls. The upper bound from our previous work [KR22] isaffected by a new correction term measuring the distance to the origin. Themain result is then applied to anti-trees with unbounded vertex degree,yielding Gaussian upper bounds for this class of graphs for the first time. Inorder to prove this, we show that isoperimetric estimates with respect tointrinsic metrics yield Sobolev inequalities. Finally, we prove that anti-treesare Ahlfors regular and that they satisfy an isoperimetric inequality of alarger dimension.
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