  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
$$\mathcal{S}^{b}f(x):=\lim_{\varepsilon\to
0}\mathcal{A}(d,-\beta)\int_{|z|>\varepsilon}\left(f(x+z)-f(x)\right)\frac{b(x,z)}{|z|^{d+\beta}}\,dy.$$
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 $b(x, z)\equiv
1$, $\mathcal{S}^{b}$ becomes the fractional Laplacian
$\Delta^{\beta/2}:=-(-\Delta)^{\beta/2}$. 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 [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 $\kappa$-fat open set. We further establish uniform gradient estimates
for non-negative harmonic functions of $X^{b}$ in open sets.
