Boundary Harnack principle for Δ+ Δ𝛼/2

For d≥ 1 and α ∈ (0, 2), consider the family of pseudo-differential operators \Δ + b Δ α /2; b∈ [0, 1]\ on Rᵈ that evolves continuously from Δ to Δ + Δ α /2. In this paper, we establish a uniform boundary Harnack principle (BHP) with explicit boundary decay rate for non-negative functions which are harmonic with respect to Δ +b Δ α /2 (or, equivalently, the sum of a Brownian motion and an independent symmetric α-stable process with constant multiple b1/α) in C1, 1 open sets. Here a “uniform” BHP means that the comparing constant in the BHP is independent of b∈ [0, 1]. Along the way, a uniform Carleson type estimate is established for non-negative functions which are harmonic with respect to Δ + b Δ α /2 in Lipschitz open sets. Our method employs a combination of probabilistic and analytic techniques.