We consider a family of pseudo differential operators {Δ + a α Δ α/2 ; a ∈ (0, 1]} on R d for every d 1 that evolves continuously from Δ to Δ + Δ α/2 , where α ∈ (0, 2). It gives rise to a family of Lévy processes {X a , a ∈ (0, 1]} in R d , where X a is the sum of a Brownian motion and an independent symmetric α-stable process with weight a. We establish sharp two-sided estimates for the heat kernel of Δ + a α Δ α/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 α-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 ∈ (0, 1] so that it recovers the Dirichlet heat kernel estimates for Brownian motion by taking a → 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