Dirichlet heat kernel estimates for Δ^α/2 + Δ^β/2

For d ≥ 1 and 0 < β < α < 2, consider a family of pseudo differential operators {Δ^α + a^βΔ^β/2; a ∈ [0, 1]} on ℝ^d that evolves continuously from Δ^α/2 to Δ^α/2 + Δ^β/2. It gives arise to a family of Lévy processes {X^a, a ∈ [0, 1]} on ℝ^d, where each X^a is the independent sum of a symmetric α-stable process and a symmetric β-stable process with weight a. For any C^1,1 open set D ⊂ ℝ^d, we establish explicit sharp two-sided estimates, which are uniform in a ∈ (0, 1], for the transition density function of the subprocess X^a,D of X^a killed upon leaving the open set D. The infinitesimal generator of X^a,D is the nonlocal operator Δ^α + a^βΔ^β/2 with zero exterior condition on D^c. As consequences of these sharp heat kernel estimates, we obtain uniform sharp Green function estimates for X^a,D and uniform boundary Harnack principle for X^a in D with explicit decay rate.