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

Suppose $d\ge 2$ and $0<β<α<2$. We consider the non-local operator $\mathcal{L}^{b}=Δ^{α/2}+\mathcal{S}^{b}$, where $\mathcal{S}^{b}f(x):=\lim_{\varepsilon\to 0}\mathcal{A}(d,-β)\int_{|z|>\varepsilon}\left(f(x+z)-f(x)\right)\frac{b(x,z)}{|z|^{d+β}}\,dy.$ Here $b(x,z)$ is a bounded measurable function on $\mathbb{R}^{d}\times\mathbb{R}^{d}$ that is symmetric in $z$, and $\mathcal{A}(d,-β)$ is a normalizing constant so that when $b(x, z)\equiv 1$, $\mathcal{S}^{b}$ becomes the fractional Laplacian $Δ^{β/2}:=-(-Δ)^{β/2}$. In other words, $\mathcal{L}^{b}f(x):=\lim_{\varepsilon\to 0}\mathcal{A}(d,-β)\int_{|z|>\varepsilon}\left(f(x+z)-f(x)\right) j^b(x, z)\,dz,$ where $j^b(x, z):= \mathcal{A}(d,-α) |z|^{-(d+α)}+ \mathcal{A}(d,-β) b(x, z)|z|^{-(d+β)}$. It is recently established in Chen and Wang [arXiv:1312.7594 [math.PR]] that, when $j^b(x, z)\geq 0$ 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.

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