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Faber-Krahn type inequalities and uniqueness of positive solutions on metric measure spaces
Anup Biswas, Janna Lierl · 2018-10-27 · via math.PR updates on arXiv.org

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].

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