Boundary Harnack Principle for Δ+δ
نویسنده
چکیده
For d ≥ 1 and α ∈ (0, 2), consider the family of pseudo differential operators {∆ + b∆α/2; b ∈ [0, 1]} on Rd that evolves continuously from ∆ to ∆ + ∆α/2. 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∆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 nonnegative functions which are harmonic with respect to ∆+ b∆α/2 in Lipschitz open sets. Our method employs a combination of probabilistic and analytic techniques.
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