  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].
