  For $d\ge 2$ and $0<\beta<\alpha<2$, consider a family of non-local operators
$\mathcal{L}^{b}=\Delta^{\alpha/2}+\mathcal{S}^{b}$ on $\mathbb{R}^d$, where $$
\mathcal{S}^{b}f(x):=\lim_{\varepsilon\to 0}\mathcal{A}(d,-\beta)\int_{ \{z\in
\mathbb{R}^d: |z|>\varepsilon\}} (f(x+z)-f(x))\frac{b(x,z)}{|z|^{d+\beta}}\,dz,
$$ and $b(x,z)$ is a bounded measurable function on
$\mathbb{R}^{d}\times\mathbb{R}^{d}$ with $b(x,z)=b(x,-z)$ for every
$x,z\in\mathbb{R}^{d}$. Here ${\cal A}(d, -\beta)$ is a normalizing constant so
that $\mathcal{S}^b=-(-\Delta)^{\beta/2}$ when $b(x, z)\equiv 1$. It was
recently shown in Chen and Wang [arXiv:1312.7594 [math.PR]] that when $b(x, z)
\geq -\frac{\mathcal{A}(d, -\alpha)} {\mathcal{A}(d, -\beta)}\, |z|^{\beta
-\alpha}$, then $\mathcal{L}^b$ admits a unique fundamental solution $p^b(t, x,
y)$ which is strictly positive and continuous. The kernel $p^b(t, x, y)$
uniquely determines a conservative Feller process $X^b$, which has strong
Feller property. The Feller process $X^b$ is also the unique solution to the
martingale problem of $(\mathcal{L}^b, \mathcal{S}(\mathbb{R}^d))$, where
$\mathcal{S}(\mathbb{R}^d)$ denotes the space of tempered functions on
$\mathbb{R}^d$. In this paper, we are concerned with the subprocess $X^{b,D}$
of $X^{b}$ killed upon leaving a bounded $C^{1,1}$ open set $D\subset
\mathbb{R}^d$. We establish explicit sharp two-sided estimates for the
transition density function of $X^{b, D}$.
