Uniform boundary Harnack principle for non-local operators on metric measure spaces

TL;DR

This paper establishes a uniform boundary Harnack principle for a broad class of non-local operators on metric measure spaces, including Euclidean spaces, under specific conditions related to jump measures and heat kernel bounds.

Contribution

It introduces a uniform boundary Harnack principle applicable to non-local operators on metric measure spaces, extending known results to Euclidean spaces with measurable coefficients.

Findings

Proves uniform BHP for non-local operators under jump measure and tail estimate conditions.

Extends BHP results to Euclidean spaces with divergence and non-divergence form operators.

Provides scale-invariant and uniform BHP for the first time in this context.

Abstract

We obtain a uniform boundary Harnack principle (BHP) on any open sets for a large class of non-local operators on metric measure spaces under a jump measure comparability and tail estimate condition, and an upper bound condition on the distribution function for the exit times from balls. These conditions are satisfied by any non-local operator $\mathcal{L}$ that admits a two-sided mixed stable-like heat kernel bounds when the underlying metric measure spaces have volume doubling and reverse volume doubling properties. The results of this paper are new even for non-local operators on Euclidean spaces. In particular, our results give not only the scale invariant but also uniform BHP for the first time for non-local operators on Euclidean spaces of both divergence form and non-divergence form with measurable coefficients.