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

Abstract

For $d\geq 1$ and $α\in (0, 2)$, consider the family of pseudo differential operators $\{Δ+ b Δ^{α/2}; b\in [0, 1]\}$ on $\R^d$ 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 nonnegative 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 $b^{1/α}$) in $C^{1, 1}$ open sets. Here a "uniform" BHP means that the comparing constant in the BHP is independent of $b\in [0, 1]$. Along the way, a uniform Carleson type estimate is established for nonnegative 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.