Stability of the Poincaré inequality under limits of Dirichlet forms

Determine whether the Poincaré inequality (PI) is preserved in the limit for Dirichlet forms: given a sequence of metric measure Dirichlet spaces that satisfy a uniform Poincaré inequality and whose associated Dirichlet forms Mosco converge (as in the Kuwae–Shioya framework under geometric assumptions), ascertain whether the limiting Dirichlet form also satisfies the same Poincaré inequality.

Background

Kuwae and Shioya showed that, under uniform Poincaré inequality and additional geometric conditions, the associated Dirichlet forms converge in the Mosco topology. What remains unresolved is whether the Poincaré inequality itself persists to the limit, i.e., whether the limit Dirichlet form also satisfies PI.

The present paper proves stability of heat kernel estimates under pointed Gromov–Hausdorff convergence and constructs a limit Dirichlet form satisfying comparable heat kernel bounds, suggesting broader stability phenomena. Nonetheless, the specific question of PI stability for the limit Dirichlet form is explicitly identified as unresolved.