Dirichlet Heat Kernel Estimates for ${\mathrm{\Delta }}^{\alpha \mathrm{/}2}+{\mathrm{\Delta }}^{\beta \mathrm{/}2}$

For $d\ge 1$ and $0<\beta <\alpha <2$, consider a family of pseudo differential operators ${\Delta^{\alpha} + a^\beta \Delta^{\beta/2}; a \in [0, 1]}$thatevolvescontinuouslyfrom$\Delta^{\alpha/2}$to$ \Delta^{\alpha/2}+ \Delta^{\beta/2}$. It gives arise to a family of L'evy processes {$X^a,a\in [0,1]\}$, where each $X^a$ is the sum of independent a symmetric $\alpha$-stable process and a symmetric $\beta$-stable process with weight $a$. For any $C^{1,1}$ open set $D$, we establish explicit sharp two-sided estimates (uniform in $a\in [0,1]$) for the transition density function of the subprocess $X^{a,D}$ of $X^a$ killed upon leaving the open set $D$. The infinitesimal generator of $X^{a,D}$ is the non-local operator ${\mathrm{\Delta }}^{\alpha }+a^{\beta }{\mathrm{\Delta }}^{\beta \mathrm{/}2}$ with zero exterior condition on $D^c$. As consequences of these sharp heat kernel estimates, we obtain uniform sharp Green function estimates for $X^{a,D}$ and uniform boundary Harnack principle for $X^a$ in $D$ with explicit decay rate.