We consider a general class of metric measure spaces equipped with a regular Dirichlet form and then provide a lower bound on the hitting time probabilities of the associated Hunt process. Using these estimates we establish (i) a generalization of the classical Lieb's inequality on metric measure spaces and (ii) uniqueness of nonnegative super-solutions on metric measure spaces. Finally, using heat-kernel estimates we generalize the local Faber-Krahn inequality recently obtained in [LS18].

中文翻译：

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Faber-Krahn 型不等式和度量空间上正解的唯一性

我们考虑配备有规则 Dirichlet 形式的一般度量空间类别，然后提供相关 Hunt 过程的命中时间概率的下限。使用这些估计，我们建立 (i) 度量度量空间上经典 Lieb 不等式的推广和 (ii) 度量度量空间上非负超解的唯一性。最后，使用热核估计，我们概括了最近在 [LS18] 中获得的局部 Faber-Krahn 不等式。