This paper proves two-sided estimates for the Dirichlet heat kernel on inner uniform domains in metric measure Dirichlet spaces satisfying the volume doubling condition, the Poincaré inequality, and a cutoff Sobolev inequality. More generally, we obtain local upper and lower bounds for the Dirichlet heat kernel on locally inner uniform domains under local geometric assumptions on the underlying space.

中文翻译：

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分形空间内均匀域中的狄利克雷热核

本文证明了满足体积倍增条件、庞加莱不等式和截止 Sobolev 不等式的度量度量狄利克雷空间中内部均匀域上狄利克雷热核的两侧估计。更一般地，我们在底层空间的局部几何假设下获得局部内部均匀域上狄利克雷热核的局部上限和下限。