On-diagonal Heat Kernel Lower Bound for Strongly Local Symmetric Dirichlet Forms

TL;DR

This paper establishes on-diagonal heat kernel lower bounds for strongly local symmetric Dirichlet forms on general measure spaces, under minimal volume growth assumptions, extending previous results beyond volume-doubling spaces.

Contribution

It proves heat kernel lower bounds assuming only a Nash-type inequality and a mild volume growth condition, without requiring volume-doubling property.

Findings

Heat kernel lower bounds are obtained outside a properly exceptional set.

The results apply to spaces with volume growth bounded by a doubling function.

The approach extends known bounds to more general measure spaces.

Abstract

This paper studies strongly local symmetric Dirichlet forms on general measure spaces. The underlying space is equipped with the intrinsic metric induced by the Dirichlet form, with respect to which the metric measure space does not necessarily satisfy volume-doubling property. Assuming Nash-type inequality, it is proved in this paper that outside a properly exceptional set, given a pointwise on-diagonal heat kernel upper bound in terms of the volume function, the comparable heat kernel lower bound also holds. The only assumption made on the volume growth rate is that it can be bounded by a continuous function satisfying doubling property, in other words, is not exponential.