Scale-invariant boundary Harnack principle on inner uniform domains in fractal-type spaces

TL;DR

This paper establishes a scale-invariant boundary Harnack principle for inner uniform domains within metric measure Dirichlet spaces, including fractal spaces, under broad conditions like volume doubling and heat kernel bounds.

Contribution

It extends the boundary Harnack principle to fractal-type spaces without assumptions on the pseudo-metric, broadening its applicability.

Findings

Proves scale-invariant boundary Harnack principle in fractal spaces.

Applies to metric measure Dirichlet spaces with broad conditions.

No assumptions on the pseudo-metric induced by the Dirichlet form.

Abstract

We prove a scale-invariant boundary Harnack principle for inner uni- form domains in metric measure Dirichlet spaces. We assume that the Dirichlet form is symmetric, strongly local, regular, and that the volume doubling property and two-sided sub-Gaussian heat kernel bounds are satisfied. We make no assumptions on the pseudo-metric induced by the Dirichlet form, hence the underlying space can be a fractal space.