Boundary Harnack principle for $Δ+ Δ^{𝛼/2}$

Abstract

For d ≥ 1 and α ∈ (0, 2), consider the family of pseudo-differential operators {Δ+bΔ 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+bΔ α/2 (or, equivalently, the sum of a Brownian motion and an independent symmetric α-stable process with constant multiple b 1/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 + bΔ α/2 in Lipschitz open sets. Our method employs a combination of probabilistic and analytic techniques. © 2012 American Mathematical Society.