Stable characterization of diagonal heat kernel upper bounds for symmetric Dirichlet forms

TL;DR

This paper provides a stable characterization of on-diagonal heat kernel upper bounds for symmetric Dirichlet forms on metric measure spaces, using integral bounds and inequalities without requiring a density for the jump kernel.

Contribution

It introduces a set of optimal conditions involving integral bounds and inequalities that characterize heat kernel upper bounds without assuming jump kernel densities.

Findings

Conditions are stable under perturbations of the Dirichlet form.

The assumptions are shown to be essentially optimal.

The results apply to a broad class of metric measure spaces.

Abstract

We present a stable characterization of on-diagonal upper bounds for heat kernels associated with regular Dirichlet forms on metric measure spaces satisfying the volume doubling property. Our conditions include integral bounds on the jump kernel outside metric balls, a variant of the Faber-Krahn inequality, a cutoff Sobolev inequality, and an integral control of inverse square volumes of balls with respect to the jump kernel. Crucially, we do not assume that the jump kernel has a density, and we show that these assumptions are essentially optimal.