In this talk, we study regular symmetric Dirichlet forms without killing on general metric measure spaces with doubling measures. For local Dirichlet forms, it is known that the on-diagonal upper heat kernel estimate (DUE) is equivalent to a weak localized Faber-Krahn inequality (FK). The Faber-Krahn inequality is also known to be equivalent to a local Nash inequality. When the index of the scale function exceeds 2, a cut-off Sobolev inequality is additionally required. For non-local Dirichlet forms, a similar result has been established under the assumption of a pointwise upper bound on the jump density.
We extend this analysis to the case where the jump kernel is not absolutely continuous. In this context, we show that the equivalence between DUE and FK fails. We then introduce a condition under which this equivalence holds, provided the cut-off Sobolev inequality is satisfied. Finally, we prove that this condition cannot be improved without further assumptions.