Heat Kernel Estimate for $\Delta+\Delta^{\alpha/2}$ in $C^{1,1}$ open sets

We consider a family of pseudo differential operators $\{\Delta+ a^\alpha \Delta^{\alpha/2}; a\in (0, 1]\}$ on $\bR^d$ for every $d\geq 1$ that evolves continuously from $\Delta$ to $\Delta + \Delta^{\alpha/2}$, where $\alpha \in (0, 2)$. It gives rise to a family of L\'evy processes $\{X^a, a\in (0, 1]\}$ in $\bR^d$, where $X^a$ is the sum of a Brownian motion and an independent symmetric $\alpha$-stable process with weight $a$. We establish sharp two-sided estimates for the heat kernel of $\Delta + a^{\alpha} \Delta^{\alpha/2}$ with zero exterior condition in a family of open subsets, including bounded $C^{1, 1}$ (possibly disconnected) open sets. This heat kernel is also the transition density of the sum of a Brownian motion and an independent symmetric $\alpha$-stable process with weight $a$ in such open sets. Our result is the first sharp two-sided estimates for the transition density of a Markov process with both diffusion and jump components in open sets. Moreover, our result is uniform in $a$ in the sense that the constants in the estimates are independent of $a\in (0, 1]$ so that it recovers the Dirichlet heat kernel estimates for Brownian motion by taking $a\to 0$. Integrating the heat kernel estimates in time $t$, we recover the two-sided sharp uniform Green function estimates of $X^a$ in bounded $C^{1,1}$ open sets in $\bR^d$, which were recently established in \cite{CKSV2} by using a completely different approach.