A ug 2 00 9 Boundary Harnack principle for ∆ + ∆

For d ≥ 1 and α ∈ (0, 2), consider the family of pseudo differential operators {∆+b∆α/2; b ∈ [0, 1]} on R that evolves continuously from ∆ to ∆ + ∆. In this paper, we establish a uniform boundary Harnack principle (BHP) with explicit boundary decay rate for nonnegative functions which are harmonic with respect to ∆+b∆ (or equivalently, the sum of a Brownian motion and an independent symmetric α-stable process with constant multiple b) in C 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 nonnegative functions which are harmonic with respect to ∆ + b∆ in Lipschitz open sets. Our method employs a combination of probabilistic and analytic techniques. AMS 2000 Mathematics Subject Classification: Primary 31B25, 60J45; Secondary 47G20, 60J75, 31B05