Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Shiping Cao, Zhen-Qing Chen

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.

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

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 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.

Paper Structure

This paper contains 6 sections, 13 theorems, 164 equations.

Table of Contents

  1. Introduction
  2. Implications of heat kernel estimates ${\rm\bf HK}(\phi, t_0)$
  3. Uniform boundary Harnack principle
  4. A box method argument
  5. Boundary Harnack principle
  6. Examples

Key Result

Theorem 1.6

Suppose that $({\bf Jc})_{\bar{r}}$, ${\rm \bf (Jt)}_{\phi,\bar{r}}$ and ${\bf(EP)}_{\phi,r_0,\leq}$ hold for some $\bar{r},r_0\in(0,\infty]$. Then the uniform boundary Harnack principle holds for $\mathcal{L}$ with radius $R:= \min\{\bar{r}/2, r_0\}$ on any proper open set $D\subset \mathcal{X}$ in

Theorems & Definitions (44)

  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Remark 1.4
  • Definition 1.5
  • Theorem 1.6
  • Definition 1.7
  • Corollary 1.8
  • Definition 1.9
  • Corollary 1.10
  • ...and 34 more