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

For $d\ge 2$ and $0<\beta <\alpha <2$, consider a family of non-local operators ${\mathcal{L}}^b={\mathrm{\Delta }}^{\alpha \mathrm{/}2}+{\mathcal{S}}^b$ on ${\mathbb{R}}^d$, where $${\mathcal{S}}^{b}f(x):=\underset{\epsilon \to 0}{\lim }\mathcal{A}(d,-\beta ){\int }_{\{z\in {\mathbb{R}}^d:\mathrm{\mid }z\mathrm{\mid }>\epsilon \}}(f(x+z)-f(x))\frac{b(x,z)}{\mathrm{\mid }z{\mathrm{\mid }}^{d+\beta }}dz,$$ 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 $\mathcal{A}(d,-\beta )$ is a normalizing constant so that ${\mathcal{S}}^b=-(-\mathrm{\Delta }{)}^{\beta \mathrm{/}2}$ 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}$^b$admitsauniquefundamentalsolution$p^b(t, x, y)$whichisstrictlypositiveandcontinuous\mathrm{.}Thekernel$p^b(t, x, y)$uniquelydeterminesaconservativeFellerprocess$X^b$, 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 $X^{b,D}$.