Dirichlet Heat Kernel Estimates for Δ +δ

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