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Dirichlet heat kernel estimates for fractional Laplacian under non-local perturbation

For $d\ge 2$ and $0<β<α<2$, consider a family of non-local operators $\mathcal{L}^{b}=Δ^{α/2}+\mathcal{S}^{b}$ on $\mathbb{R}^d$, where $$ \mathcal{S}^{b}f(x):=\lim_{\varepsilon\to 0}\mathcal{A}(d,-β)\int_{ \{z\in \mathbb{R}^d: |z|>\varepsilon\}} (f(x+z)-f(x))\frac{b(x,z)}{|z|^{d+β}}\,dz, $$ 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 ${\cal A}(d, -β)$ is a normalizing constant so that $\mathcal{S}^b=-(-Δ)^{β/2}$ when $b(x, z)\equiv 1$. It was recently shown in Chen and Wang [arXiv:1312.7594 [math.PR]] that when $b(x, z) \geq -\frac{\mathcal{A}(d, -α)} {\mathcal{A}(d, -β)}\, |z|^{β-α}$, 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 $X^{b, D}$.