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

Abstract

For $d\geq 1$ and $\alpha \in (0, 2)$, consider the family of pseudo-differential operators $\{\Delta + b \Delta ^{\alpha /2}; b\in [0, 1]\}$ on $\mathbb {R}^d$ that evolves continuously from $\Delta$ to $\Delta + \Delta ^{\alpha /2}$. 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 $\Delta +b \Delta ^{\alpha /2}$ (or, equivalently, the sum of a Brownian motion and an independent symmetric $\alpha$-stable process with constant multiple $b^{1/\alpha }$) 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 non-negative functions which are harmonic with respect to $\Delta + b \Delta ^{\alpha /2}$ in Lipschitz open sets. Our method employs a combination of probabilistic and analytic techniques.