Boundary Harnack principle and gradient estimates for fractional Laplacian perturbed by non-local operators

Abstract

Suppose $d\ge 2$ and $0<\beta <\alpha <2$. We consider the non-local operator ${\mathcal{L}}^b={\Delta }^{\alpha /2}+{\mathcal{S}}^b$, where $${\mathcal{S}}^{b}f(x):=\underset{\epsilon \to 0}{\lim }\mathcal{A}(d,-\beta ){\int }_{|z|>\epsilon }\left(f(x+z)-f(x)\right)\frac{b(x,z)}{|z{|}^{d+\beta }}dy.$$ Here $b(x,z)$ is a bounded measurable function on ${\mathbb{R}}^d\times {\mathbb{R}}^d$ that is symmetric in $z$, and $\mathcal{A}(d,-\beta )$ is a normalizing constant so that when $b(x,z)\equiv 1$, ${\mathcal{S}}^b$ becomes the fractional Laplacian ${\Delta }^{\beta /2}:=-(-\Delta {)}^{\beta /2}$. In other words, $${\mathcal{L}}^{b}f(x):=\underset{\epsilon \to 0}{\lim }\mathcal{A}(d,-\beta ){\int }_{|z|>\epsilon }\left(f(x+z)-f(x)\right)j^b(x,z)dz,$$ where $j^b(x,z):=\mathcal{A}(d,-\alpha )|z{|}^{-(d+\alpha )}+\mathcal{A}(d,-\beta )b(x,z)|z{|}^{-(d+\beta )}$. It is recently established in Chen and Wang [arXiv:1312.7594 [math.PR]] that, when $j^b(x,z)\ge 0$ on ${\mathbb{R}}^d\times {\mathbb{R}}^d$, there is a conservative Feller process $X^b$ having ${\mathcal{L}}^b$ as its infinitesimal generator. In this paper we establish, under certain conditions on $b$, a uniform boundary Harnack principle for harmonic functions of $X^b$ (or equivalently, of ${\mathcal{L}}^b$) in any $\kappa$-fat open set. We further establish uniform gradient estimates for non-negative harmonic functions of $X^b$ in open sets.

References in corpus (2)

• Perturbation by non-local operators
• Dirichlet heat kernel estimates for fractional Laplacian under non-local perturbation