Sharp green function estimates for Δ+Δ^α/2 in C^1,1 open sets and their applications

We consider a family of pseudo differential operators {Δ+a^αΔ^α/2; a ∈ [0, 1]} on ℝ^d that evolves continuously from Δ to Δ + Δ^α/2, where d ≥ 1 and α ∈ (0, 2). It gives rise to a family of Lévy processes {X^a, a ∈ [0, 1]}, where X^a is the sum of a Brownian motion and an independent symmetric α-stable process with weight a. Using a recently obtained uniform boundary Harnack principle with explicit decay rate, we establish sharp bounds for the Green function of the process X^a killed upon exiting a bounded C^1,1 open set D ⊂ ℝ^d. Our estimates are uniform in a ∈ (0, 1] and taking a→0 recovers the Green function estimates for Brownian motion in D. As a consequence of the Green function estimates for X^a in D, we identify both the Martin boundary and the minimal Martin boundary of D with respect to X^a with its Euclidean boundary. Finally, sharp Green function estimates are derived for certain Lévy processes which can be obtained as perturbations of X^a.