Global Heat Kernel Estimates for $\mathrm{\Delta }+{\mathrm{\Delta }}^{\alpha \mathrm{/}2}$ in Half-space-like domains

Suppose that $d\ge 1$ and $\alpha \in (0,2)$. In this paper, by using probabilistic methods, we establish sharp two-sided pointwise estimates for the Dirichlet heat kernels of ${\Delta+ a^\alpha \Delta^{\alpha/2}; \ a\in (0, 1]}$onhalf-space-like$C^{1, 1}$domainsin${\mathbb R}^d$foralltime$t>0$. The large time estimates for half-space-like domains are very different from those for bounded domains. Our estimates are uniform in $a\in (0,1]$ in the sense that the constants in the estimates are independent of $a\in (0,1]$. Thus it yields the Dirichlet heat kernel estimates for Brownian motion in half-space-like domains by taking $a\to 0$. Integrating the heat kernel estimates in time $t$, we obtain uniform sharp two-sided estimates for the Green functions of $\{\mathrm{\Delta }+a^{\alpha }{\mathrm{\Delta }}^{\alpha \mathrm{/}2};a\in (0,1]\}$ in half-space-like $C^{1,1}$ domains in ${\mathbb{R}}^d$.