Heat Kernel Estimate for $\mathrm{\Delta }+{\mathrm{\Delta }}^{\alpha \mathrm{/}2}$ 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$\bR^d$forevery$d\geq 1$ that evolves continuously from $\mathrm{\Delta }$ to $\mathrm{\Delta }+{\mathrm{\Delta }}^{\alpha \mathrm{/}2}$, where $\alpha \in (0, 2)$\mathrm{.}ItgivesrisetoafamilyofL\acute{e}vyprocesses${X^a, a\in (0, 1]}$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 $\mathrm{\Delta }+a^{\alpha }{\mathrm{\Delta }}^{\alpha \mathrm{/}2}$ 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 ${\text{bR}}^d$, which were recently established in \cite{CKSV2} by using a completely different approach.