DIRICHLET HEAT KERNEL ESTIMATES FOR Δα/2 + Δβ/2

For d >= 1 and 0 < beta < alpha < 2, consider a family of pseudo differential operators {Delta(alpha) + a(beta) Delta(beta/2); a is an element of [0, 1]} on R-d that evolves continuously from Delta(alpha/2) to Delta(alpha/2) + Delta(beta/2). It gives arise to a family of Levy processes {X-a, a is an element of [0, 1]} on R-d, where each X-a is the independent sum of a symmetric alpha-stable process and a symmetric beta-stable process with weight a. For any C-1,C-1 open set D subset of R-d, we establish explicit sharp two-sided estimates, which are uniform in a is an element of (0,1], for the transition density function of the subprocess X-a,X-D of X-a killed upon leaving the open set D. The infinitesimal generator of X-a,X-D is the nonlocal operator Delta(alpha) + a(beta) Delta(beta/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,X-D and uniform boundary Harnack principle for X-a in D with explicit decay rate.