Boundary Harnack principle for $\Delta +{\Delta }^{\alpha /2}$

Abstract

For $d\ge 1$ 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 nonnegative 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 nonnegative 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.

36 pages, no figure