  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.
