An approach to sub-Gaussian heat kernel estimates via analysis on metric spaces
Riku Anttila
TL;DR
This work addresses the longstanding problem of characterizing sub-Gaussian heat kernel estimates for strongly local regular Dirichlet forms on metric measure spaces with walk dimension $\beta>2$. It replaces the technical cutoff Sobolev inequality with a simpler cutoff energy condition (CE) and recasts the problem in a Poincaré-type framework analyzed via Hajłasz–Koskela methods, within a general $p$-energy structure. The authors prove that HK estimates are equivalent to doubling of $\mu$ together with a Poincaré inequality and CE, provide a quantitative two-measure Sobolev–Poincaré characterization, and establish regularity results including Hölder continuity of cutoff energies. The approach yields practical criteria, supports applications to reflected diffusion on complex domains, and extends to new examples such as Laakso-type fractals, offering insights toward the resistance conjecture and nonlinear energy settings.
Abstract
In this work, we establish a new characterization of sub-Gaussian heat kernel estimates for strongly local regular Dirichlet forms on metric measure spaces. Our formulation is based on the newly introduced cutoff energy condition, which offers a simpler and more transparent alternative for earlier technical energy inequalities, in particular the cutoff Sobolev inequality. The main idea of our approach is to reinterpret the cutoff Sobolev inequality as a Poincaré type inequality, and analyze it using Hajłasz--Koskela techniques from analysis on metric spaces. Applications of the new characterization are also discussed.