DIRICHLET HEAT KERNEL ESTIMATES FOR Δ~(α/2)+ Δ~(β/2)

摘要

For d ≥ 1 and 0 < β < α < 2, consider a family of pseudo differential operators {Δ~α + a~β Δ~(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 LeVy processes {X~a,a ∈ [0,1]} on R~d, where each X~α 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 C 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~(α,D) of X~α killed upon leaving the open set D. The infinitesimal generator of X~(α,D) is the nonlocal operator Δ~α + a~β Δ~(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~(α,D) and uniform boundary Harnack principle for X~α in D with explicit decay rate.