Heat kernel estimates for δ+δα/2 in C1, 1 open sets

We consider a family of pseudo differential operators {delta+a(alpha) delta(alpha/2); a is an element of(0, 1]} on R-d for every d >= 1 that evolves continuously from delta to delta+delta(alpha/2), where alpha is an element of(0, 2). It gives rise to a family of Levy processes {X-a, a is an element of(0, 1]} in R-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,C- 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 is an element of(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-1,C- 1 open sets in R-d, which were recently established in (Z.-Q. Chen, P. Kim, R. Song and Z. Vondracek, 'Sharp Green function estimates for delta+delta(alpha/2) in C-1,C- 1 open sets and their applications', Illinois J. Math., to appear) using a completely different approach.