Boundary Harnack Principle and Gradient Estimates for Fractional Laplacian Perturbed by Non-local Operators

Abstract

Suppose d ≥ 2 and 0 < β < α < 2. We consider the non-local operator $\mathcal {L}^{b}={\Delta }^{\alpha /2}+\mathcal {S}^{b}$, where

$$\mathcal{S}^{b}f(x):=\lim_{\varepsilon\to 0}\mathcal{A}(d,-\beta){\int}_{|z|>\varepsilon}\left( f(x+z)-f(x)\right) \frac{b(x,z)}{|z|^{d+\beta}}\,dy. $$

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)≡1, $\mathcal {S}^{b}$ becomes the fractional Laplacian Δ^β/2:=−(−Δ)^β/2. In other words,

$$\mathcal{L}^{b}f(x):=\lim_{\varepsilon\to 0}\mathcal{A}(d,-\beta){\int}_{|z|>\varepsilon}\left( f(x+z)-f(x)\right) j^{b}(x, z)\,dz, $$

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 [11] that, when j ^b(x, z)≥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 κ-fat open set. We further establish uniform gradient estimates for non-negative harmonic functions of X ^b in open sets.