Title: On-diagonal heat kernel estimates for regular Dirichlet forms on doubling spaces.
Abstract: In this talk, we will discuss on-diagonal heat kernel estimates (DUE) for regular Dirichlet forms without a killing part on locally compact separable metric measure spaces. We establish an equivalence between DUE and two functional inequalities — the Faber-Krahn inequality and the cutoff Sobolev inequality — assuming the usual tail estimates for the jump kernel without requiring the existence of a jump density. The first part of the talk will review classic and recent results for DUE in strongly local Dirichlet forms (e.g., energies associated with second-order uniformly elliptic operators) and forms with jump parts. The second part will introduce key ideas for the proof of the main result.